Outline of axiomatic relativistic quantum field theory

Outline of axiomatic relativistic quantum field theory R F STREATER Department of Mathematics, Bedford College (University of London), Regent's Park, London NW1

Abstract I n order that fields can be treated as quantum systems, quantum mechanics needs to be carefully formulated. After a brief introduction to the problem, $2 describes the Hilbert space approach of von Neumann and the C*-algebra approach of Segal. I n both cases symmetry groups and superselection rules can be described, and the latter allow a spontaneously broken symmetry. Section 3 describes the Wightman theory of fields, with the main results and recent generalizations. Section 4 describes the C"-algebra approach of Haag, which follows the same physical ideas but has become an independent theory capable of explaining superselection rules. Section 5 describes dispersion relations and the analytic properties in momentum space, with brief reference to the Froissart bound and other consequences of positivity. Finally, 96 describes the Euclidean approach leading to a history integral formulation. T h e review as a whole is a survey of the work over the period 1954-74. It is confined to relativistic theories, omitting applications to many-body theory and avoiding the actual solution of the field equations. That nonlinear field equations, at least in two- and three-dimensional space-time, have solutions has recently been proved by Glimm and Jaffe. This suggests that field theory will be solved and used to describe elementary particles within the next few years. This review was completed in September 1974.

Rep. Prog. Phys. 1975 38 771-846 52

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R F Streater

Contents

1. Introduction . 2. Quantum mechanics . 2.1. States and observables 2.2. Symmetries . 2.3. Superselection rules . 2.4. Cx-algebras and quantum mechanics 3. Wightmantheory . 3.1. Theaxioms . 3.2. The analytic programme . 3.3. The classical results . 3.4. Critique of the axioms 4. The C"-algebra approach . 4.1. TheHaagfield . 4.2. The freefields . 4.3. Development of the general theory 4.4. Spontaneous breakdown of symmetry 4.5. Weyl systems and the Clifford algebra 4.6. Algebraic theory of superselection rules 5. Analytic properties in momentum space 5.1. Dispersion relations . 5.2. Retarded functions . 5.3. T h e reduction formula . 5.4. Consequences of positivity . 6. Euclidean field theory . 6.1. The Schwinger functions . 6.2. The Nelson axioms . 6.3. The Osterwalder-Schrader axioms 6.4. Further developments References .

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1. Introduction Although the perturbation method of Feynman provides a workable theory of quantum electrodynamics and weak interactions, it is not capable of giving reliable results for strong interaction physics because, there, the expansion parameter is large, being about 15. Even when perturbation theory might be expected to work because of the small coupling constant, the renormalized perturbation series probably diverges. This has been proved by Jaffe (1965a) for a particular case. This means that the high-order corrections will be unreliable, especially at high energies where the lowest-order approximation is bad since it violates the principle known as unitarity. Apart from this practical aspect, the theory is not free from logical difficulties, since the necessary renormalization procedure involves the cancellation of infinite terms (Hepp 1966b). But the numerical success of quantum electrodynamics indicates that the theory is on the right track. Believing that the difficulties are of a mathematical rather than a physical nature, leading physicists like Wightman, Haag, Lehmann, Symanzik, Zimmermann and Bogoliubov were led to a re-examination of the mathematical formulation of quantum field theory, beginning around 1955. The ‘axioms’ are a list of properties a theory must have before it should be called a theory of quantized fields. Nonlinear quantum field equations exhibit ambiguities; the axioms, imposed on the solution, rule out unphysical interpretations of the formalism. If a solution satisfies the axioms, its general properties will be sensible from a physical point of view. Thus axiomatic quantum field theory is a framework encompassing a number of field theories, rather than a theory of any particular field subject to its particular field equations. Consequently, the general theory rarely, if ever, makes numerical predictions about ele-’ mentary particles; it leads to qualitative properties only. Quantum field theory is the result of combining the two great physical theoriesquantum mechanics and special relativity-with the principles known as causality and positivity of the energy. Causality-that no effect can precede its cause-takes a stronger form in a relativistic theory: the world line joining causally related events must be time-like; it is known that this principle cannot be satisfied in a Hamiltonian classical particle theory (Currie 1963) unless all interactions are absent. T h e successful classical theories that embody special relativity and causality, such as the Maxwell theory, are theories of jields. The field describes the situation locally, at each point, and propagates energy, and so information, from point to point at a speed not greater than the speed of light. I n a quantum field theory, causality may be expressed succinctly by requiring that operators A and B, that represent observations in space-like separated regions, should commute: A B =BA. This ensures that they are compatible observables in the sense of quantum mechanics. This leads us to the idea that observables should be localized; our task is to set up an axiomatic-that is, mathematical-scheme which embodies these ideas. We shall describe four axiomatic schemes-of Wightman, Haag, Segal and Nelson-and discuss the relations between them. The axioms of Bogoliubov and Shirkov (1959) and Lehmann et a1 (1955, 1957) (LSZ) are on a lower level of rigour in their original form, though a rigorous version can be derived from the Wightman axioms with a few extra assumptions (Hepp 1965, Epstein and Glaser 1971). T h e 52*

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axioms of Bogoliubov and Shirkov or LSZ can be used to remove the ambiguities in the renormalization procedure (Bogoliubov and Parasiuk 1957, Hepp 1965). They also lead to a convincing theory of dispersion relations, described in $5. Each axiom scheme has been proved to hold for a class of nonlinear fields in two-dimensional space-time (Glimm et a1 1973). Most of the results in Wightman theory obtained before 1963 can be found in the monographs by Streater and Wightman (1964) and Jost (1965). The algebraic theory is described by Araki (1969) and Emch (1972). Segal’s point of view is to be found in his 1963 book. I n this review, $2 introduces the notation and summarizes the algebraic approach to quantum mechanics. In $3 we survey the Wightman theory, and in $4 the algebraic theories of Haag and Segal. Section 5 covers the axiomatic treatment of dispersion relations, and $6 concerns the recent work on Euclidean field theory.

2. Quantum mechanics 2.1. States and observables A theory of quantized fields is in particular a quantum theory. This means (von Neumann 1955) that the states of the system can be represented by vectors 0,Y,. . . in a complex Hilbert space 2, with scalar product (@, Y ) , which we take to be linear in Y and antilinear in @, so that ( a @ ,P‘r> = .P(@, ‘r>

for any complex numbers a: and /3 (we denote the complex conjugate of a number 01 by E). T h e non-negative number (@, @)l/z is called the norm of @ and is denoted li@II. We interpret 1 (a, Y)lz as the probability of finding the state @ if the system is definitely in the state Y.Because of this, to represent a state, a vector @ must be normalized, ie ll@ll= 1, and the two vectors @ and exp (io)@ ( 0 real) represent the same state. T h e zero vector does not represent a state; it is certainly not the vacuum state! The set of vectors? {A@; h E c, A # 0) is called the ray through @, and the set of nonzero rays is called the projective space of Z, written&?. An observable A is taken to be a self-adjoint operator on P;it acts on Y , transforming it to A Y , which is also i n 2 . We interpret (Y, AY) to be the expectation value of the observable A in a sample of copies of the state Y. Since A is self-adjoint, (Y, A Y ) = ( A Y ,‘€7)

= (Y, AY)

so the expectation values are real, as required by their physical interpretation. Our notation and point of view differ slightly from those of Dirac (1932). The non-normalizable vectors introduced by Dirac do not lie in a Hilbert space, but in a suitably enlarged space (Roberts 1966a,b). We shall not use this extended space; it is not necessary for the description of operators with a continuous spectrum (which can be studied in Hilbert space); nor are non-normalizable vectors related to nonseparable Hilbert spaces. Dirac introduces two types of vector, the bra (( and the

t We use {x; y } to denote ‘the set of objects like x such that y is true’; to’; and C denotes the set of complex numbers.

E

means ‘belongs

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I).

ket This can be confusing, and we prefer to regard (@, Y) as the scalar product between two vectors, @ and Y ,in the same space X . We illustrate one of the pitfalls of operator theory with an example from elementary quantum mechanics. Suppose .%‘ is the set of wavefunctions Y (x) satisfying in which case we must identify two wavefunctions Y , @ such that

j; p ( X ) - q ~ ) p d3X=0 so that &‘=&(O, 1). The operator -iV cannot be applied to all vectors; VY has no a priori meaning unless Y is differentiable, and we then need VY E f l in order for the scalar product to make sense. As we can see from the identity

(a,

- iVY) = { - iV@,Y >- i [5(ij Y (x)]:

the operator -iV is Hermitian only if we impose suitable boundary conditions on @ and Y, such as periodicity in [0, 11, so that the boundary term vanishes. Returning to the general theory we see that it is important to specify the set of vectors D ( A ) on which an operator A is supposed to act. D ( A ) is called ‘the domain of definition of A’. We shall replace the vague term ‘Hermitian’ by the exact term ‘symmetric’. An operator A, with domain of definition D, is said to be symmetric if (AY, 0)= (Y, A@)for all Y and @ in D. We say an operator is bounded if there exists a number b such that llAY / / < bllY 11 for all Y in the domain of definition. T h e smallest b for which this inequality holds is called ‘the norm of the operator A’, written ~ ~ A ~ If an operator is bounded, it can be defined on the whole space .%‘ and we do not need to worry too much about its domain. A vector Y is said to be an eigenvector with eigenvalue a if AY = a y . This implies that Y E D ( A ) . We wish to interpret a as a possible exact value of the observable represented by A. If A is symmetric, the eigenvalues are real. A possibly complex number X is said to belong to the spectrum a(A) of A if A - AB has not got a bounded inverse, ie (A - XQ)-l is not bounded. This is true if X is an eigenvalue; so o(A) contains the eigenvalues of A. T h e fact is, there exist symmetric operators with complex numbers in their spectra; since we wish to interpret the spectrum physically as the set of possible values, symmetry is not a sufficient condition for an operator to represent an observable. Another weakness of symmetric as the appropriate definition is that if A is symmetric, exp (iA) need not be unitary, or even definable. To remedy these problems, we require a symmetric operator, to be an observable, to satisfy the following further condition, making it self-adjoint: if @ is such that {@, AY) = (@’,

Y)

for all Y G D ( A )and some a’, then @E D(A)and @’[email protected] tells us that D ( A ) contains as many vectors as it can, and that A is still symmetric on this large domain. If A is self-adjoint, the spectral theorem enables us to define any continuous function of A ; thus V ( t ) =exp (iAt) is unitary and V ( t ) V(t’)= V(t+t’)

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holds. I n this case we say that V ( t ) is a one-parameter group of unitary operators. Moreover, the spectrum of a self-adjoint operator is real (see Riesz and Sz-Nagy 1955). There has been some discussion (and some confusion) about the role of nonseparable Hilbert spaces in quantum field theory. Recall that an orthonormal basis { @ I } in a Hilbert space is a set of vectors such that

(at, @ j ) = a i j and such that any vector CD has an expansion CO

@=Eat ai where

Cila~12 0, is called the proper orthochronous group L t . I t does not contain time reversal or space reflections, and is believed to be an exact symmetry. Similarly B $ , denoting the proper orthochronous Poincar6 group, is an exact symmetry. If a theory is invariant under B\, then to each ray 0 E 22‘ and each (A, a ) E B\ is assigned the transformed state a’=T ( A ,a ) 0 , such that (i) (ii) (iii) (iv)

T(1,0) =identity T(&, az) T(&, a)= T ( A z h , Azal+ az) I(T(A, a ) a,T ( A ,a ) Y ) 1 2 = I ( 0 , Y ) [ 2 I ( Y ,T (A, a ) 0)l is a continuous function of a and A.

Property (iv) expresses the physical idea that a small change of coordinates in measuring 9should produce a small change in transition probabilities. A rule T that assigns to each group element (A, a ) a ray map T(A, a) satisfying (i)-(iv) is called a continuouJ projective representation on the space X . Two projective representations are called equivalent if there is an isometry Vfrom X I on to X Z(the spaces of the representations) which intertwines: VT(A,a)= T(A, a ) V for all (A, a) E B i . As far as their transformation laws under B\ are concerned, we cannot distinguish between equivalent representations, but it is not true that any

iAn anti-unitary operator Tis an invertible map satisfying (TY,T @ ) =(@, Y).

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two relativistic theories with equivalent representations of Pf, are physically equivalent. A fully interacting quantum field theory that admits a full description in terms of asymptotic particles has the same representation of 9: as the free quantum field describing the same particles. A field theory contains many observables not determined by the representation of the group (the field, for instance), and these serve to distinguish the theories. If T is a projective representation on X , and XocX is a subspace such that T (A, a ) X Oc Xo for all (A, a) E 9$, then we say X Ois invariant under T, and the action of T on X Ois called a subrepresentation of T. Clearly X itself is invariant under L+$,If X Ois invariant, and T i s isometric, then X $(the set of rays orthogonal to Xo) is also invariant. If YE‘ itself is the only subspace invariant under T, we say Tis irreducible, and it corresponds to an elementary system, in that the possible states are transformed into each other by changing coordinates. A stable atom, and even molecule, is an elementary system in this sense. By Wigner’s first theorem, for each (A, a) €9f, we can choose a linear operator T ( A , a) such that the ray through T ( A , a) Y is T(A, a) Y for allY, (Antilinear operators are needed for time reversal.) By the uniqueness of T up to a phase,

T(Ai, ai) T(A2,a2) = w(Ai, ai, A2, a2) T ( ( A i ,ai) (A2, ~ 2 ) ) where w E C and where I wI = 1; w is called a multiplier. T can be chosen such that w is continuous. Representations such that T can be chosen so that w = 1 are called true representations. For P$ these are the representations of integer spin. T h e Lorentz group Lf,, like its subgroup 0+(3), is doubly connected (it has continuous closed paths that cannot be shrunk to a point). To handle the non-true representations, we introduce the coveying group SL(2, C ) of L$. SL(2, C) is the set of 2x 2 complex matrices with determinant 1; the group law is matrix multiplication. SL(2, C) acts on R4 as follows. Let oo=(;

Y),

ol=(;

A),

Gz=(i

0

-1

01,

os=(o1 -1)0

and for each x E R4 let ,x be the 2 x 2 matrix X;”,=ox k p . Then ,x =,x*, and ,x determines uniquely. T o each A E SL (2, C ) define a transformation x + A(A)x = x’ by s’ =Ad*. Since det ,x = xPG; x, = det x,’ we see that A(A) is a Lorentz transformation. One can show A(A)E Lf+ and that A(AB)=A(A)A(B). T h e map A -+ A ( A ) is continuous, and as SL(2, C) is simply connected, we have that SL (2, C) is the covering group of L$ . If A (A) = A(B),then A = + B . The inhomogeneous SL(2, C), written ISL(2, C), is the semi-direct product SL(2, C) x R4, ie it is the set of pair (A,a), A E SL(2, C), a E W4, with multiplication law (-42, .2)

(A,a1)= ( A 2 - 4 1 , 4 4 2 )a1+ a2).

ISL(2, C ) is the covering group of 9$.. We may now state the theorem of Wigner: to each continuous projective representation T of &,. there exists a true continuous representation T of ISL(2, C) such that T(A(A), a) Y is the ray through T ( A , a) Y. It is not always possible to find a continuous true representation T ( A ,a) on z? itself (it is not possible for non-integral spin). By introducing SL(2, C) we avoid the (meaningless) use of two-valued continuous wavefunctions.

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T h e physically interesting representations of 9’$were classified by Wigner (1939). T h e list was completed by Bargmann (1947, 1954) and Gelfand and Naimark (1947); complete mathematical rigour was supplied by Mackey (1963). See Varadarajan (1970) for a general account. The inclusion of space and time reflections was done by Bargmann (1963) and Lever and Shaw (1974). We now describe Wigner’s results. First, the trivial representation, T ( A , a ) = 1 for all A, a E 9$. This is one-dimensional if irreducible, and the vector represents a vacuum state. More generally, given T ( A ,a), let us examine the subgroup of translations, R4, so A= 1. Then T(1, a ) defines a continuous four-parameter group; Stone’s theorem (Riesz and Sz-Nagy 1955) then asserts that the generators

a

P,=i - T ( l , up) aaa IO exist and are self-adjoint operators on 2 that mutually commute. We interpret Pfi= GPvPP,as the energy-momentum four-vector operator, which therefore exists in any translation invariant theory. P@is ambiguous up to an additive real c-number four-vector q, since T(1, a ) and exp (iq . a ) T(1, a ) both lead to the same ray map T for any q. For a representation to be physical it is necessary that, for some choice of q, the energy-momentum satisfy PO 2 0, (PO)22 P2 in the sense of expectation values

(0, PO 0)2 0,

(0[(P0)2- P2] 0)2 0.

T h e operator (PiLP,)112=mcommutes with T ( A ,a), and so is an invariant, called the mass, one of the Casimir operators of ISL(2, C) labelling the irreducible representation. T h e old term ‘rest mass’ for m is not used now, and pm = m/(1- v2)1/2,called the mass in ancient books, is not an invariant and is now called the momentum, since the speed of light is put equal to unity. I n the neighbourhood of A = 1, we can choose one A to correspond to each A, so T ( A ,0) can be parametrized by the six real parameters corresponding to rotations about three axes and Lorentz transformations along the three directions. The generators of these six one-parameter subgroups are the self-adjoint operators

Mi’’= - MY,, p , v = O , 1, 2, 3. Pauli and Liubanski define the four-spin

where E is the Levi-Civita tensor (completely antisymmetric, =0, for m > 0 is defined by WAWA=m2s(s+ 1)

_+

1). T h e spin

and is the second Casimir operator. It can have values 0, 4, 1, 14, . . .; if m > 0, the pair [m,s] determines the irreducible representation up to unitary equivalence. If m=O there are representations of ‘m-spin’ (Wigner 1963) as well as representations of helicity s=O, 5 $, -F. 1, . . . . Only the latter type appear as particles in nature (s= 4 being the neutrino, s = & 1 the photon); that the co-spin representations cannot appear as ‘quanta’ of a Wightman field was proved by Yngvason (1970). T h e representation T ( A ,a ) of a theory of elementary particles is reducible. T h e various stable particles appear as irreducible subrepresentations. T ( A ,a ) must also contain representations corresponding to scattering states of 2, 3, . , . particles. If we wish to describe a theory in which the smallest mass is positive, we may postulate

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the spectrum of m (a self-adjoint operator commuting with T ( A ,a)). I n this way we build experimental information into the theory without solving a dynamical theory (which would predict the spectrum). If U ( A )is a unitary representation of SL(2, C), we can obtain a unitary representation T ( A ,a) of ISL(2, C) by T ( A ,a)= U(A). Since then iBT(A, a)/BaP=O, Pp=O in this representation; its states have zero energy-momentum but nonzero angular momentum. Borchers (1962) has shown that these strange objects cannot occur in a Wightman theory. T h e irreducible representations of ISL(2, C) can be described in terms of relativistic wave equations (Bargmann and Wigner 1948), which transform according to representations 9 of SL(2, C). T h e finite-dimensional representations of SL (2, C) are described as follows. T h e vector space of H 3 1 M is spanned by complex-valued spinors tal:..ajl il. . .j,%,symmetric among the E and among the p; here ag = 1 or 2, i= 1, . . .,j 1 , and ,&= 1 or 2, i= 1, . . . ,j 2 . Under the element A, $ transforms to g[=Aala; * * Aallail Ajlj; * * * A&,jiIt u ; . uj,ji.. .jj,. (Recall that a bar denotes complex conjugate.) Under the action of (A, a ) E ISL(2, C), a spinor$eld transforms as f + [', where

['(x)

=

g[(A(A)-l(x- a)).

One then puts a scalar product on the space of $(x) so that this action is unitary, and imposes a wave equation or subsidiary condition to eliminate unwanted spins. I n this way, the action of ISL(2, C) on the set of solutions is unitary equivalent to one of the irreducible representations [m,s] of Wigner. For example, for spin 0, mass m, the real solutions to (0 + m2)+ = 0, transforming as +(x) + +(Ax + a), provide a realization of the representation [m,01; and the positive-energy solutions of Dirac's equation realize [m,81. Maxwell's equations define a space transforming as [0, 110 [0, - 11. Thus classical field theory is actually identical to the quantum mechanics of a single particle. T h e same representation [m, s] can be realized in many ways; they are not all physically equivalent if a meaning is attached to the localization defined by the point x E R4. Second quantization regards the classical field as an operator rather than a state, and replaces f a ( x )by an operatorjeld satisfying the same equation. This procedure enables us to describe states of n free particles, n=O, 1, 2, . . ., as vectors in one large Hilbert space, called Fock space. This is a necessary preliminary to a fully interacting theory since we need a formalism in which to describe states whose particle number changes with time, so that the particles appear and disappear. T h e free field does not in fact achieve this, since there, the particle number is conserved in time. How an interacting field theory achieves particle production, while retaining the asymptotic description (for large times) in terms of free particles of definite mass and spin, is the main success of the theory.

2.3. Superselection rules I n particle physics there are indications that there are many states, in the sense used so far-that is, rays in a Hilbert space-which are not physical states; they cannot even be approximated by physical states. It is believed, for instance, that any physically realizable state must be an eigenstate of charge, so that a superposition of states with different charge is not physically realizable. There is a similar rule for nucleon number and for the two types of lepton, the electron and muon. This rule leads to the

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absolute conservation of the quantum numbers Q (charge), B (baryon number), le (electron lepton number) and I, (muon lepton number). A conservation law occurring in this way is called a supenelection rule. T h e rule for charge is equivalent to the gauge invariance of the theory (Strocchi and Wightman 1975, to be published). I n the presence of a superselection rule, certain self-adjoint operators are not observable, eg the projection operator on to an unphysical state. Experimental proof of the nonmeasurability of an operator is not easy. But there is one superselection rule that can be proved theoretically from rotation invariance ; this is the univalence rule. Under a rotation of 2 x , vectors of integral spin are unchanged, while those of half-integral spin undergo a change of sign (which does not alter the ray). I n order for such rotations to leave physical rays invariant, there can be no mixing of integer and half-integer spins. It follows that (- 1)*, where F is the number of fermions, is conserved in time and defines a superselection rule. We note that Q, B , le, I,, and bounded functions of them, commute among themselves and with all observables. This led Wightman to formulate his hypothesis of commuting superselection rules. We give the equivalent form of Jauch and Misra (1961). Let d be the set of all bounded observables. Then the set d' of bounded operators commuting w i t h d should be Abelian (commutative). T h e general setting, then, is the following. The Hilbert space &' is a direct sum labelled by q= ( B , Q, Zp, Ze), the simultaneous eigenvalues of the of subspaces ZOs. This means that any vector @ E .xi' has an superselection operators: &'= expansion @ = I; Oq,where X 1/@",(12< CO, and Q4 is its component in Zq.The subspaces possess dense subspaces of physically realizable vectors and are called coherent subspaces (lately, superselection sectors). This direct sum reduces the set d of operators, ie A a O s . ~ifSA4is an observable. Haratian and Oksak (1969) have derived this set-up from the hypothesis that the set of physically realizable vectors span 2, and that the projection on to each physically realizable vector is observable. However, it turns out, in quantum field theory, that any local observable projection projects on to an infinite-dimensional space, and so one-dimensional projections need an infinitely extended apparatus for their measurement, and so are probably not observable. The algebraic theory of Haag (next section) offers an explanation of superselection rules which is more convincing; it is found that the law of commuting superselection rules may be violated, but only in the presence of parastatistics (Doplicher et a1 1969a, b, 1974, Druhl et a1 1970).

2.4. C"-algebras and quantum mechanics A more general approach to quantum mechanics than that afforded by Hilbert space theory was developed by Segal(1947). Consider the set of bounded observables, bounded symmetric operators on some Hilbert space. The set of all multinomial functions of these operators, with complex coefficients, forms a "-algebra; for convenience, we also add all norm limits of sequences of these operators (a sequence A , converges to A in norm if /lAn- A / ]-+ 0 as n -+ CO). We then obtain a set of operators d called the algebra of observables of the theory (though it contains non-Hermitian elements). d satisfies the axioms of a C*-algebra (see Lanford 1971, p140). Segal suggests that the primary object of quantum mechanics should be a C"-algebra, defined in some way, not necessarily as a set of operators on a particular Hilbert space. The observables of such a theory will be certain Hermitian elements of the algebra. I n practice, d will contain the identity 1.

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Given the algebra d of some system, a state of the system is then an expectation value p, assigned to each element A. It is to satisfy (i) p(A"A) 0 for all A E& (ii) p( AA)= +(A) for all h E C and p(A B ) = p(A) p(B) (iii) p ( l ) = l .

+

+

I n the Hilbert space version, where d is a set of operators on a Hilbert space 2 (so that it is a concrete C*-algebra), any vector Y E2 defines a state pr in this sense: P\r(A)=

e, AY).

Such a state is called a vector state o n d . A more general type of state is obtained as a statistical mixture. Let p be a density matrix on X, ie a symmetric operator with positive eigenvalues {A,) such that &A, = 1 including multiplicity, and the corresponding eigenvectors span &. Then, if A E&, Tr(pA) is finite and p(A)=Tr(pA) defines a state (here, T r denotes the trace of an operator, ie the sum of its diagonal matrix elements in any basis). If all A5 except one are zero, then a density matrix defines a vector state. Otherwise, p is an impure state. A state p is said to be impure if there are different states, pl and p2, such that p= A l p l + A2p2, Aj > 0, AI+ X 2 = 1 for some AI, A2. Otherwise, the state is said to be pure. States on a concrete C"-algebra given by a density matrix are called normal states. Non-normal states, even pure ones, arise in quantum field theory, and are responsible for superselection rules. If the algebra d of the theory is abstractly given, and is not a concrete algebra of operators on a Hilbert space, then the concepts of vector state and normal state have no absolute meaning, but must be defined relative to a representation. Let 2 be a Hilbert space, not necessarily separable. A representation T of d in 2 is a map v , assigning a bounded operator T(A)acting on X , to each element A of d , such that (i) n(A")= (n(A))" (ii) T ( AA + B ) = h ( A ) T ( B ) (iii) T ( A B )= T ( A )n(B).

+

A representation is said to be faithful if n(A)#O if AZO, and irreducible if .@ has no non-trivial subspace invariant under all v(A). If T is a representation of d in 2, then a vector Y of 2 defines a state py of I by pr(A) = (Y, v ( A )Y), We say p is a vector state relative to T . Similarly, a density matrix on 2 defines a state normal relative to T . T is irreducible if and only if ~(d)', the commutantt, consists of multiples of the identity (Lanford 1971, p157). A representation T is said to have cyclic vector Y if the set of vectors {T(A)Y?,A E&] spans 2. Two representations, 7 ~ 1and 772 in spaces 2 1 and 2 2 , are equivalent if there exists a unitary operator U from 2 1 to 2 2 , such that U q ( A ) U - l = T z ( A ) for all A E&. Otherwise they are said to be inequivalent. If T I and n-2 are inequivalent, then, in general, the vector states of v1 are not normal relative to v 2 (Kadison 1957). The Gelfand-Naimark-Segal (GNS) theorem states that any state p is a cyclic vector state for some representation, r Psay, in a Hilbert space 2,,with cyclic vector Yp, such that p ( A )= (Y r p ( A )Y ,,) ; v pis irreducible if and only if p is pure (Lanford 1971). Thus we recover the Hilbert space version of quantum mechanics, except that all states and representations are treated on the same footing.

,,,

+

If 9? is any set of operators on a Hilbert space, then 9?', its commutant, is the set of bounded operators A such that A X = X A for all X E5 .

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T h e theory of symmetries can be formulated in C*-language. If T i s a symmetry in the Hilbert space version, Wigner's first theorem says that there is a unitary (or anti-unitary) operator T implementing the symmetry. If A is an observable, so is TAT-1, the transformed observable. The transformation A + T(A)=TAT-1 is an automorphism, ie a one-to-one map of the algebra of observables satisfying

+

(i) T ( AA B )= k ( A )+ T ( B ) (ii) T (AB)= T ( A )T ( B ) (iii) T(A*) =T(A)*. (If T is anti-unitary, (i) reads 7(hA+B)=XT(A)+T(B).)

If a theory is invariant under a group G of symmetries, then there exists a group of automorphisms (79,g E G} of d which realizes the symmetry group : Tg, Tg,= Tg,g,. If T is an automorphism of an abstract Ca-algebra d , and n- is a representation of d , then T need not appear as an automorphism of the concrete algebra n-(d)(we denote the set of { r ( A ) ,A ~ dby}n-(d)).Indeed, if there are nonzero operators, A E.&, such that r(A)=0, then the set of such A , called the kernel of n-, ker n-, must be transformed into itself by 7,or else the action of T on r(d) will not make sense. This mechanism is probably not the mechanism for the breakdown of symmetries in relativistic theories, because the algebra d is often taken to be simple, and can be proved to be so under quite general conditions (Misra 1965, Borchers 1967). This means that the kernel of every representation (except n- = 0) consists only of the zero, and so every automorphism of d defines an automorphism of n-(d). There is another possibility, however. An automorphism T of a concrete C*-algebra d,acting in a?, is said to be implemented if there exists a unitary operator U on &' such that A,= U,AU,-1 (or an anti-unitary operator if T is an anti-automorphism). If d = I(%), the set of all bounded operators on a?, then every automorphism is implemented, but for other C*-algebras there usually exist non-implementable automorphisms. If d is an abstract C*-algebra, 7 an automorphism, and T a representation, then T might or might not be implemented in n-. If not, then a symmetry represented by T would not give rise to a unitary or anti-unitary operator on the representation space a?,,. Haag (1962) has suggested that an explanation of a breakdown of symmetry, such as occurs in nature, might be that there is an automorphism or automorphism group that is not implemented in the representations of interest. This is formulated for a relativistic theory by Streater (1965); the mechanism is known as spontaneous breakdown of symmetry (see 94.4). There is one important case when we can show that an automorphism T is implemented. Let p be a state of d , and define the transformed state p, by p7(A)=p(7(A)). We say a state p is invariant under T if p = p7. It is easy to show that if p is invariant under T, then T is implemented in the representation r pin the Hilbert space Z P ,and Y pis an invariant vector. A very similar theorem shows, in Wightman theory, that the theory has a representation of LY$ if the vacuum expectation values are invariant under LY\ (see 92). Indeed, if p is invariant under a whole group of automorphisms, then the group will be represented by unitary operators on 2,. Let us say that a theory is covariant (under a given group G containing time evolution) if the theory can be formulated in an algebraic way, and G is realized as automorphisms of the algebra d . Further, let us say that a representation r of d is covariant if the group of automorphisms is implemented in the representation n-. We also say that the theory is invariant under G in n-. Only in covariant representations does Wigner's theory of symmetry apply.

R F Streater

784

We can use the principles of PoincarC invariance and positive energy to help us select representations of physical interest. Consider an algebra s’ on which 9 k acts as a group of automorphisms. Let us say that a representation x is physical if: (i) &‘,, is separable; (ii) each (A,a) is implemented by U ( A ,a), say; (iii) we may choose U ( A ,a) such that it is continuous, and its energy momentum lies in the forward light cone. It is not necessary for a physical representation to have a vacuum state. We expect a realistic theory to have a series of physical representations, each corresponding to a coherent subspace (see $4.6). I n some models, P$ acts on s‘ continuously, ie

a)A-Aii -+ 0 as A + 1 and a + 0. (2.1) This implies that U ( A , a) can be chosen continuous in all representations x in which T is implemented. I n this case, Doplicher (1965) has given an algebraic condition on (d,T) which guarantees the existence of at least one physical representation x (except that might not be separable). Consider functions f on R4, such that its Fourier transform f satisfies f ( p )= 0 if p2 2 0, PO2 0. For every A E d and such f, construct the Bochner integral B = A ( f ) = S f ( x ) ~ ( l , x ) A d x ~ cd. The set of elements of the form CB, C E d ,B E a is invariant under left multiplication by d , and so forms a left ideal 4. Doplicher’s theorem states that there exists a physical representation if and only if 4#d. Dell’Antonio (1966) has attempted to replace the strong continuity condition (2.1) by a weaker condition; but since he used Doplicher’s ideal 4,which depends on (2.1) to give the integrals an algebraic meaning, the method is not convincing. Doplicher et al(1966) show that strong continuity of T(A, a) holds ifs’ is norm separable and possesses at least one physical representation. This includes the relativistic Fermi field, but not the boson field. Doplicher et al(1966) have remarked that if (2.1) does not hold, but& possesses a representation x in which continuous U ( A ,a) exist, then it is possible to modify the C*-algebrad so that (2.1) does hold. Let p be a Haar measure on 9, and f(A, a) be a continuous function of compact support. We form the concrete C*-algebra s’, generated by operators of the form

s U ( k +(A) Then d , has an automorphism d Sdepends on the choice of x .

u-v,.)f(A, a) d p ( 4 a).

group T(A, a) for which (2.1) holds. I n general,

Given a Cx-algebra s‘ and a group of automorphisms G, continuous in the sense of (2. l), Doplicher et al (1966) define a .covariance algebra, d x G, the semi-direct product of& and G. Every representation o f d x G gives rise to a covariant representation of d (one in which each T is implemented) and conversely. Kadison (1967) has shown that we cannot (in field theory) replace (2.1) by the norm condition IlT(1, a) - 111 -+ 0 as a + 0, where 11 /I is the operator norm on d as a Banach algebra. This condition is an abstract way of saying that the energy is bounded. But the spectrum of ~ ( 1a) , as an operator ons’ is not directly related to the physical energy-momentum spectrum (that of the implementing unitaries). The topic has been taken further by Borchers (1966, 1969) and Kraus (1970); see also Jadczyk (1969a, b).

Outline of axiomatic relativistic quantum jield theory

785

3. Wightman theory 3.1. The axioms The axioms (Wightman 1956, GArding and Wightman 1965) embody the idea of a field in quantum mechanics, consistent with causality and relativity. The dynamical variables are the field components +,(x, t ) , cy= 1,2, . . . This cannot itself be an operator, but only a bilinear form (Ascoli et a1 1970), ie (@, + , ( x ) Y ) is finite for suitable states @ and Y, Only suitable space-time averages, called 'smeared fields', such as +,(f)= f f(x)+,(x) d4x, can be operators. Wightman chooses the space Y for f; Y is the space of infinitely differentiable functions of rapid decrease, ie functions such that for all r, s

.

Ilf /Ir,s=S;P

2

1

( I + 1x1)" ax: *asy . . 8x2

1

rt< P 8'9 8

is finite. Such a norm is called a Schwartx norm (Schwartz 1950). The need to average smoothly may be inferred from the analysis of Bohr and Rosenfeld (1933, 1950), who argued that the electromagnetic field at a point cannot be measured; only its averages are observables. If is an observable field, we imagine that we can measure +,(f) by an apparatus located in any open set containing supp f (supp f is the closure of the set of points where f # O ) . From its heuristic definition as an integral, #,(f) must be linear in f. If the theory is to be relativistic, there should exist a continuous unitary representation of ISL (2, C) with a physical spectrum and unique vacuum (see $2.2), which transforms the field in the same way as the classical field. The most characteristic property is causality: two measurements, made inside space-like separated regions, must be compatible. For observable fields, this means they must commute. For unobservable fields, observable functions of them, such as currents (usually bilinear), should have this property. Wightman therefore requires that the fields should either commute, or anticommute, at space-like separation. I n order to formulate commutativity, Wightman needs to be able to form products of fields, and accordingly makes some hypothesis on the domains of definition (see $2.1) of the fields. The axioms are : (1) There is Hilbert space 2 carrying a continuous unitary representation U ( A , a) of ISL(2, C). The energy-momentum spectrum of U(1, a) lies in the forward light cone: PPPP20;PO>O. There is a unique vector Yo invariant under U(1, a). (2) T o eachf E Y , there is an operator +,( f)(where cy runs over a finite index set), with dense domain D,the same for all cy and f. YOlies in D,and D is invariant under each +,(f) and under every U ( A ,a). The Hermitian conjugate +$(f) of each field is included among the +/. (3) The field transforms under P$ according to

+,

U ( A ,4 +,(f)U-l(A,

c S,BW

4= B

4p(fA,a)

where fA, a(x)=f(A (A)-l(x- a)), and S is a finite-dimensional representation of SL(2, C). (4)(Causality) If the supports off and g are space-like separated, then

Mf) +/U 5 +a(d +m = [ + a ( f > , +a(g)l*= 0.

R F Streater

786 We choose the the rest.

+_

sign independently off and g,

(5) The vectors +,,(f1)

+ for some pairs of fields,

- for

. . . C$a,(fn)Yo span 2.

Axiom (5) is called cyclicity o f y o ; it states that we have included enough fields to get all the states, starting from the vacuum. We assume that, in axiom (4), the same sign, + or - , is chosen between all pairs from the same two irreducible representations of SL(2, C). Schneider (1969) has given weaker conditions under which this can be proved. Originally, Wightman added the postulate that the Wightmanfunctions, (Yo, +,,(fi) . . . +,,(fn) Yo), should be tempered distributions. Wyss (1972) has derived this from the other axioms. We often write

WO,

C$a,(fl)

* '

+a,(fn>

Yo)

=

S W O , d,, (4' ' ' da,(xn)Yo> f ( 4 * * ' ~ f ( ~d x n l .)

. . dxn

but the 'function'

W(x1, ' ' *,

Xn) = ( y o , +,,(Xl)

* *

C$a,(Xn)Yo)

is singular, and the integral has no precise meaning. The main theorem is due to Wightman (1956); it expresses the various axioms (Lorentz invariance, causality, mass spectrum, positivity of metric in Hilbert space, etc) as a list of properties of the Wightman functions. Two Wightman theories, with Hilbert spaces S t , fields +=.I, representations U ( A ,a){ and vacua 'Fog, i= 1,2, are unitary equivalent if and only if they have the same Wightman functions. TJnitary equivalence means that there is a unitary operator Vfrom 201 on to X 2 , such that VYol =Y?02, VU1(A,a ) V-l= U2(A, U ) and V+,l(f)V-1 = & 2 ( f ) for all 01 and f. The Wightman functions determine the theory in another sense too: given any set of tempered distributions having the properties listed below, then a Wightman quantum field theory can actually be constructed using them. T h e Hilbert space of states is explicitly constructed, and is separable (Borchers 1962, Ruelle 1962), and the action of the field operator +(f),and of @$.,is known on all states. Therefore one 'merely' needs to find a set of W functions to have a theory. This construction of the theory was given by Wightman (1956) and is described by Streater and Wightman (1964), where a simplification due to Baumann and Schmidt (1956) is included. A neat reformulation of the theory is due to Borchers (1962), well explained in the lectures of Lanford (1971). See also Maurin (1963). T o understand this theorem, note that the axioms imply that the smeared field operators +(f)(taking a scalar field for simplicity) can be multiplied and added, conjugated and multiplied by complex numbers. T h e algebra of field operators therefore imitates the following Borchers algebra: 9=C @ Y ( R * ) @ .Y(#?g)@ . . . (finite sums of test functions only). Multiplication is by tensor product, using f@g ~ Y ( R 4 ( 3 + k ) ) if f ~ Y ( R 4 1 ) and g E Y ( R ~ ~and ) , conjugation f+f" is defined by f*(xl, . . ., Xn)=f(xn, . . ., XI); if f E gP(W4), we can define an 'operator' on 9 byg +f @g for eachg €9.This operator will become the smeared field operator +(f). The groupB$, acts on D by transforming the function f -+fn, a. This will become the unitary operator U ( R ,a). The function 1, being invariant, becomes the vacuum; so far, 9 is the same for any scalar field, free or interacting. The dynamics is determined by the choice of scalar product on 9. A given set of Wightman functions W(x1, , . ., xn) determines this scalar product: if g=fi(Xl)@ . .@fn(xn) E Y ( R ~ " ) ,define

.

. ..

W(g)= J W(xl,. . ., x ~ ) ~ I ( x.~. ).fn(xn) d 4 ~ 1 d4xn.

Outline of axiomatic relativistic quantum field theory

787

Then W defines a normalized positive linear functional (a state) on g : W(l)= 1 and W ( g ” g ) 2 0, for all g E g.T h e scalar product is then defined from (f,g) = W(f*@g). If each Wightman function is Lorentz invariant, the action of 9’$+ becomes unitary on the space g. We may complete by adding its limit points in the topology defined by the scalar product, in the same way that the complete real number system is obtained from the rational numbers by adding Cauchy sequences. The resulting space, the completion of a, is the Hilbert space 2 of the quantum field. T h e set is a subset of Yf and can be taken as the domain D of the axioms. T h e field operator +(f)itself is the action g + f @ g . I n this way, Wightman theory tells us the answer to the puzzling question, what the Hilbert space and field operators actually are for an interacting field [for the free fields we know of the Fock space description as well (see 54.5),and this leads to a unitary equivalent theory to that given by the Wightman method]. It is a simple matter to find out the properties of the Wfunctions that are necessary and sufficient for the field theory so constructed to satisfy the axioms (1)-(5). The condition that the metric in Hilbert space is positive-definite leads to conditions like

which lead to inequalities involving W functions of various numbers of arguments (‘nonlinear’ conditions). If the field is a scalar field, the existence of U ( A , a) and the invariance of Yo mean that the W functions are Lorentz invariant distributions. These two are expressed in terms of the Borchers algebra by saying that W is positive and invariant. Translation invariance means that W(x1,. . ., Xn) is a function only of the difference four-vectors 52 =xi - X Z + I , say W(&, . . ., En-1). Let us now find the consequences of the spectrum condition. If f~ 9 is a test function, whose Fourier transformf(p) is zero outside a region A, then it can be seen that the operator +(f)changes the energy-momentum of a state only by vectors in A. For example, if A is not in the physical spectrum of states, then 4(f) Yo=O. I n this way one proves that the Fourier transform W(p1, . . .,pn-1) ofW(E1, . ., EPt-1) is zero unless pl, p2, . . .,pn-1 are all in the physical spectrum. It follows from this that

.

W(&’ . . ., &-I) = ( 2 7 p - n )

/%(Pi, . . .,pn-I)

(

exp -iEpj’fjp)

d4p1 .

. . d4pn-1 (3.1)

is a convergent Laplace transform also for certain complex (5 and that W(&, . . .,tlZ-1) is the boundary value of an analytic function of several complex variables. This is the key result of Wightman theory. T h e extension of the domain of analyticity of this function following from Lorentz invariance and causality is summarized in 53.2. Let us now consider the uniqueness of the vacuum. Bardakci and Sudarshan (1961) pointed out that if W1 and WZare two 9$.invariant positive functionals on the \ invariant positive functional for Borchers algebra, then XW1+ (1 A) Wz is also a 9 0 < X c 1. I n fact, this property is rather similar to the combination of two states to form a mixed state in statistical mechanics. Hepp et al (1961) pointed out that in general such linear combinations lead to theories which violate the axiom of uniqueness of the vacuum, even if W1 and WZbelong to theories with a unique vacuum. Hepp et a1 show that uniqueness of the vacuum is equivalent to a cluster property of the

-

R F Streater

788

W functions. This states that at large space-like distances, the W functions tend to factorize ( T o , +(fd ' ' - +(h> To) x ( T o , +(h+1> * * ' +(fnYo> as A --+ CO, where a is space-like vector and f h " ( x ) = f ( x + Aa). This holds for each n, j and fi, . , .,fn, and each a. It can be expressed as

P o , +(fd ' ' +(h) +(f.tL)' +(f3 Yo) *

+

I n probabilistic language, the correlation functions tend to zero at CO and the vacuum state is ergodic. One then sees that if 1" and "2 satisfy the cluster property, XW1+(1- h)W2 does not (unless W= Vi= WZ). Ruelle (1962) has shown that the uniqueness and cyclicity of the vacuum imply that the fields form an irreducible set of operators (any bounded operator commuting with them is a multiple of the identity). It is common practice to discuss the simple case of a neutral scalar field 4. Wightman (1956) postulated only that such a field be symmetric, rather than selfadjoint; as we saw in 52.1, observables need to be described by self-adjoint operators. Wightman raised the question whether a unique self-adjoint extension of +(f) is implied by the axioms. This was answered negatively by Borchers (1965b). Wightman has remarked, however, that a field has a unique extension from the original domain D to a domain containing vectors that can be denoted

{j+(XI)

. + ( x n ) Tof(x1,

a

*

xn) dx,

f E 9(R4")]

(Streater and Wightman 1964). There exist free fields of any spin satisfying these axioms, including the usual fields of spin 0, 8 and 1 (Streater and Wightman 1964, chap 2, Moussa 1973).

3.2. The analytic programme

If we replace gj, in equation (3 .l), by the complex four-vectors &=fj-iyj, the formula defines an analytic function of [ j , j = 1, . . ,,n - 1 for those <j such that the Since f l = O factor exp (-i Z j p f 5 j p ) is damping for all possible p j where @ # O . unless p i E V+,the closed forward light cone, we see that W (51, . . ., Cn-1) is analytic provided 73 E V+,the open cone. The set (&:qj E V+,j = 1, . . ., n- 1) is called the forward tube 7. The W distribution can be recovered as the boundary value of W as vi -+ 0 from the tube (§treater and Wightman 1964, chap 2). There is thus an analytic function for each of the n! permutations of the fields. The domains of analyticity are obtained by permuting the xj (not the &), and have no points in common, and contain no real points. The boundary values of these different functions coincide (up to a sign) when all the difference vectors X.I - xj are space-like, by causality. We should like to conclude now that (modulo C! 1) all functions analytically continue one another; that is, there should be one master analytic function W that equals each function in its own tube of analyticity. To do this, it is sufficient to show that the n ! functions coincide on a set of analyticity of 4(n- 1) real dimensions. Care is needed here, as we have not yet shown that the real space-like points { b : (3.1- xj)2 < 01 are points of analyticity. This is achieved by the 'edge of the wedge' theorem (Bogoliubov et a1 1958, Bremermann et al 1958): two functions, analytic in the forward and backward tubes respectively, and coinciding on a real open set 0, are analytic in a complex

Outline of axiomatic relativistic quantum Jield theory

789

neighbourhood of U, and analytically continue one another. A special case was proved by Dyson (1958a). A proof based on an idea of Glaser is given by Streater and Wightman (1964). Epstein (1960) generalizes the proof to the case when the boundary value is a distribution rather than a function, and can treat the case when the two tubes are not exactly opposite, but are angled. Borchers (1964), Araki (1963a) and Kolm and Nagel (1968) provide further generalizations. Another method of analytic continuation is provided by the Hall-WightmanBargmann (HBW) theorem (Hall and Wightman 1957), which makes use of Lorentz invariance. Suppose first that all the spins are zero; then W, and hence the analytic function W , is invariant under real and therefore complex Lorentz transformations, since the complex Lorentz group is an analytic group. T h e formula W(5)= W(A5) can be used to define W(5)at a point 1 not in the forward tube, provided A < is in the forward tube for some complex A. We can therefore extend the definition to the extended tube "I,the set of points obtained from Y by complex Lorentz transformations. T h e HWB theorem asserts that this continuation is one-valued: if we arrive at a point 5 two different ways, we obtain the same value for W (c). If the fields are not scalar, then the W functions are not invariant, but transform according to a finitedimensional representation of SL(2, C) ; the transformation law can be continued to the complex SL(2, C), giving the same result: the W has a one-valued analytic continuation to the extended tube (Streater and Wightman 1964, theorem 2.11). If n = 2, the extended tube is the plane, with the cut 5 2 2 0 removed in the one variable 52, where 5 =XI - x2 ; in particular a real space-like vector lies in the region of analyticity. T h e real points of the general extended tube were found by Jost (1957): (51, , . ., cn-l) E Y' if and only if (I;X&)2< 0 for all hi> 0. A strange converse to this was proved by Glaser (unpublished) and Streater (1962): a function analytic in Y and - Y , and at these Jost points, is analytic in Y' (even though it is not assumed to be a covariant function). Bros et a1 (1967) show more: if, in addition, the function is such that its boundary value is a tempered distribution, then it is necessarily a finite covariant, showing that Lorentz invariance is almost implied by causality and the spectrum condition (Streater 1972); the special case n = 1 was proved earlier by Bogoliubov and Vladimirov (1958); Oksak and Todorov (1970) extend the n= 1 case to non-distribution boundary values. Considering all permutations together, we see that the function W has a continuation into the union of the permuted extended tubes. Ruelle (Appendix to Jost 1965) has shown that the continuation is one-valued there, a fact shown independently by Tomozawa (1963), who also shows that the union of the permuted extended tubes is simply connected (all closed paths in it can be continuously shrunk to a point), T h e second part of the HWB theorem is that a Lorentz invariant function, analytic in Y, can be written as a function of the scalar products 5 6 . & inside the set obtained by letting and ( j vary over Y , I f n 2 6 and space-time = R4,then these scalar products are not functionally independent, and the theorem is slightly modified. The corresponding theorem for fields carrying spin has been obtained by Jost (1960). It is only inside the domain of analyticity that the functions are unambiguously given by functions of the invariants. The two-point function (Yo,$ ( X I ) $(x2)Y0) can be continued to a function of x = (XI - 4 2 , analytic unless z> 0. As we approach the real axis from above the cut, we obtain this Wightman function as its boundary; but as we approach the reals from the lower half-plane, we approach k (Yo, $ ( x 2 ) $(XI) YO}. T h e discontinuity across the cut z>O is the commutator or anticommutator (YO,[ $ ( X I ) , $(x2)]*Yo)= V , depending on the choice of sign in axiom (3). T h e

RF

790

Stveater

discontinuity vanishes if x < 0 by causality, in agreement with the fact that x < 0 is a point of analyticity and so of continuity. Consider now Cauchy’s integral formula

c XI-x where C is the contour shown in figure 1. If W(x)= 0 ( ~ - ( 1 + ~ as) )1x1 -+ CO, we can expand the large circle to CO, and in the limit, this gives no contribution to the integral. Letting E + 0, we obtain for the integrand the discontinuity across the cut: W(x+ie)- W(x-ie) -+ e(%). Thus we obtain the master analytic function in terms of its commutator

If W ( z )does not fall off as /xl-(l+E) as x --+ CO, this integral might not converge, and ‘subtractions’ are necessary (see $5.1).

Figure 1. The contour C.

For the free scalar field of mass m,

%?(A)= A(x, m2)=j €(PO) S(p2-&) where

E@’))

exp (-ips%) d4p,

A=X2

=pO/IpOI. T h e two-point function is

W ( z )= A+ (x, m2) = J 0 ($0) 6 (p2 - m2) exp ( - ip .x) d4p,

Z=X2

where B(pO)= 1 if p O > O , and O(pO)=O if pO 0 are the ‘wavefunction renormalization constants’ and pl( K 2 ) = 0 if K~ < mo2; p l is a positive tempered measure. This ‘already renormalized’ formula is due to Lehmann (1954). See GArding and Lions (1959) and Steinmann (1963) for a rigorous treatment. For the three-point function, W is a function of three complex variables (XI- ~ $ 2 , ( X 2 - X 3 ) 2 and ( X 3 - X 1 ) 2 . The domain, the extended tube, was determined by Hall (1957) and KallCn and Wightman (1958). T h e analogues of the A+ functions are the generalized singular functions

A+(a,x

2

, a,~b, c ) = j

8(p2-a) 8 ( q 2 - b ) S ( p . q - c ) e(p0) O(q0)

x exp[ - i ( p h

+ q&)1 d4p d4q

(Hall and Wightman 1955, KallCn and Wilhelmson 1959, Nieminen 1962). If a> 0, b > 0, A+ is the Fourier transform of a measure, zero outside the forward cone, and so is analytic in the forward tube. One can write the general function, invariant and analytic in the forward tube, as (Hall and Wightman 1955)

W@I, ~

JA+(a~

2 ~, 3 ) =

2 z3; , a,

6, c ) p(a, b, c) da db dc

where p = 0 unless a, b, c are physical values of p2, 4 2 and p .q. However, this representation does not embody the information coming from causality. T o do this, we must consider functions analytic in the union of the permuted extended tubes. Ka11Cn and Wightman show that this union is not a domain of holomorphy (a domain of holomorphy D is a region such that there exists a function, holomorphic in D, that cannot be continued past any point of the boundary). For a function of n complex variables, the analogue of Cauchy’s integral formula is the Bergman-Weil formula. It expresses the analytic function in terms of its values, not on the whole boundary, which has dimension 2n- 1, but on a subset called the distinguished boundary, usually of dimension n. T h e values on the distinguished boundary may be assigned arbitrarily; but the values on the rest are then determined. 6 has been found by Svensson T h e Bergman-Weil formula for the extended tube S (1962). He also showed that it is related by a Hankel transform to the Hall-Wightman representation. Ka11Cn and Wightman (1958) achieve the remarkable feat of finding the envelope of holomorphy ; that is, the smallest domain of holomorphy containing the union of the permuted extended tubes. For an account of the techniques needed, see also Wightman (1960a). Making use of this domain of holomorphy, Ka11Cn and Toll (1960) compute the distinguished boundary and write down the Bergman-Weil integral formula. Some simplifications to the Ka11Cn-Wightman calculation for the three-point function have been found (Jost 1965, Ruelle 1961, Streater 1960, Bros 1966). KallCn and Toll (1960), using a transform of their Bergman-Weil formula, obtain functions with nonzero mass thresholds, having singularities on the boundary of the domain of KallCn and Wightman. This shows that the x space analytic properties are not enlarged if the information about mass thresholds is included. T h e class of functions discusses by Ka11Cn and Toll is not the most general one having the given thresholds (unless these are zero). 53

R F Streater

792

The progress in the study of the four-point and higher functions has been very slight. Streater (1960) gave the envelope of holomorphy for the union of extended tubes taken two and four at a time; for the four-point function, these have been proved by Lavine (1961) in two-dimensional space-time, and by Bros et aZ(1964) and Bruning (1971) in four. Similar problems arise concerning the analytic properties of the retarded Green functions in momentum space ($5.2), where the mass thresholds influence the domain. One of the most interesting and powerful methods for studying the consequences of causality was discovered by Jost and Lehmann (1957) and generalized by Dyson (1958b). Consider a scalar field and the distribution

+

v.

F(x9 Y) = (a, [+(4,+(Y)l Causality implies that if x -y is a space-like four-vector, then F ( x , y) = 0. A distribution vanishing outside the double light cone is called a causal distribution. The typical causal distribution is A(x-y, K ) and sums of such for various K ; and as we saw just now, d.1 *(x-Y, K ) dK is the most general invariant causal distribution. We obtain a wider class of causal distributions by allowing the weight function p ( ~ to) depend on x and y ; it is a notvery-deep theorem that the integral

s

j p(x,y;

*(x-y,

dK provides a representation for any causal distribution, if we choose p ( x , y ; K ) suitably; the weight function p is not unique. The deep part of the J L D representation arises when we consider the Fourier transform K)

K)

QP, 4 ) = (274-4 J exp [i(P.+ qy)] F(x, y ) d4x d4y. The spectral properties of the energy-momentum entail that this function is zero outside a certain set in R8, depending on the energy-momentum content of the states CDand Y. The Fourier transform of the integral representation gives p ( p , q) = J” (P-zL, q + u ; which can be rewritten

K)

e ( u 0 ) 6(u2-

d 4 dK ~

K ~ )

E(p, q ) = j G ( ~ , p + q K, ) c ( u - q ) S [ ( U - ~ ) ~ - K ~d4u ] dK

(3.3)

for some weight function G. We can choose G to be zero outside the set of p + q , where P ( p , q) # 0 for some q; it is not difficult to see that this entails no loss in generality. One can also see that, for a given G, the corresponding F(p, q), as a function on RS, is zero if G(u,p+q, K ) is zero for all U,K on the two-pieced hyperboloid (q-u)2 =~ 2 T . h e deep part of the J L D theorem is that for any causal distribution F there exists a weight function G such that (3.3) holds and G(u, p + q, K ) = 0 unless both parts of the hyperboloid (q - u)2 = ~2 lie entirely within the support of P ( p , q) as a function of q with p + q given. I n this way one can find many of the combined consequences of the spectrum condition (in p space) and the causality condition (in x space). In particular, analytic properties of the retarded commutator

(274-4 j

@O-YO)

[+(4 4(Y)l y>exp [i(P.+PY)I

d4x d4r

Outline of axiomatic relativistic quantum field theory

793

can be read off from the zeros of (q-u)2- ~ 2 .Bros et al (1961) obtain the same domain by geometrical methods. See Wightman (1960a) for a proof of the J L D representation, and Woronowicz (1968) and SCnCor (1969) for generalizations. An interesting property of causal distributions is that their spectra cannot be arbitrary, but must be a union of hyperboloids. This fact is behind many of the results of 93.4 (Greenberg 1962a,b, Robinson 1962, Dell'Antonio 1961a,b, Araki 1963a, b, Borchers 1961, Kolm and Nagel 1968). T h e Wightman functions of a particular Wightman theory might well be analytic in larger regions than those found by general methods. T h e only singularities of the free fields are cuts for positive values of the invariants ( Q - X J ) ~ ; perturbation theory indicates that an interacting theory should have other singularities.

3.3. The classical results In a field theory, the dynamics, and also the interpretation of the theory at any time, is carried by the field. If T is a symmetry, then the unitary or anti-unitary operator T, given by Wigner's first theorem, must therefore transform the field locally, and in the correct manner. Thus, if +(x) is scalar, is parity; is time reversal. If

4 is pseudoscalar, Up+(xO, x) Up-l= -+(xO, - x )

is parity. It can be shown that U p and Uc, the charge conjugation, must be unitary, and UT anti-unitary. Note that if a UP, transforming the fields locally, exists, then parity is conserved. If we wish to impose P, c, T invariance separately on a field theory, then the components must fall into a representation of the full Lorentz group L. For fields of spin 4, $, . . . this means introducing a conjugate set of spinor components, so that there are twice as many components as there are spin degrees of freedom. However, it is possible to define 0, the anti-unitary PCT operator, even when there are no 'doubled' components. For a general spinor we choose

See Streater and Wightman (1964) for a justification of this choice for spins 0, Q and 1. T h e phase seems arbitrary; but it is this transformation, with this phase, that is proved to be a symmetry of any field theory by the PCT theorem (Luders 1954, Pauli 1955). T h e most general proof (Jost 1957, Streater and Wightman 1964) uses the fact that the Wightman functions are analytic at Jost points; it therefore follows from a weaker hypothesis than causality, namely, weak local commutativity (WLC) : e o , 41(.1)

* * *

+n(xn) Yo>= iF

( T o , 4n(xn)

.-

*

+1(x1> T o )

where F is the number of fermion fields, holds if x1 . . , xn is a Jost point (Jost 1957, Dyson 1958a). Jost (1963) has given examples of field theories satisfying WLC and all the axioms except causality. So WLC is not a very ambitious way to express causality.

794

R F Streater

To prove PCT, the assumption that the field transforms according to a Jinitedimensional representation of SL(2, C)is essential; Oksak and Todorov (1968, 1969) have given an example of a local infinite-component free field for which the PCT transformation on the particle states is not a local transformation of the field. The spin statistics theorem says that fields transforming as k under the Lorentz group cannot commute at space-like separation if j + K is not integral, and cannot anticommute ifj+K is integral. Since the theory postulates that fields either commute or anticommute, we obtain the usual quantization rules, and, in particular, the Pauli exclusion principle for electrons and protons. Historically, the theorem was proved by Fierz (1939) and Pauli (1940) for free fields, and by Luders and Zumino (1958) and Burgoyne (1958) for general theories, interacting or not. The proof again uses analyticity and covariance, combined with the positivity condition for the two-point function. A deeper justification for the choice of either commutator or anticommutator comes from Haag’s theory of superselection rules ($4.6). Again, the theorem only holds for finite-dimensional representations of L ; many authors have shown that infinite-component fields could violate the spin statistics theorem, eg Feldman and Matthews (1967a) give a clear treatment. Streater (1967a) extended the Wightman axioms to include infinite-component fields, and constructed free fields satisfying the axioms but having the wrong connection between spin and statistics. Dell’Antonio (1961b) showed that the Hermitian conjugate of a field must have the same relation, commutation or anticommutation, with a given field, as the field itself. The final point, made by Luders and Zumino (1958), concerns the commutation relations between different fields. The ‘usual rule’ is for two fields, both of nonintegral spin, to anticommute, and for a field of integral spin to commute with any other field at space-like separation. Luders asked if it was possible to have other theories in which, for example, different spin-$ fields commute at space-like separation, or two different spin-0 fields anticommute. It turns out that this is quite possible, provided that the theory is furnished with suitable conservation laws. These take the form of even-odd selection rules : the Wightman functions vanish unless they contain even numbers of certain fields. This means that there can be no transitions between states with an even number of these particles, and states with an odd number. T h e total Hilbert space splits as a direct sum of these even-odd subspaces. One would expect these to be superselection rules, but whether they are or not depends on the choice of measurement theory of the model. Luders showed that in any theory related to which does not postulate the usual rules, one can introduce new fields +a by a Klein transformation, so that the satisfy the usual commutation relations. A Klein transformation has the effect of changing the sign of some of the fields 4, on some of these subspaces. It is not unitary; but it should be possible to prove that it can be chosen so as not to change any observable of the theory. This point remains unsettled. See Doplicher et al (1969a,b, 1971), however. The rigorous version is due to Araki (1961a). See Streater and Wightman (1964) for the details. We now turn to the theory of Borchers classes of fields. I n classical mechanics one can use any one of a number of different sets of canonical coordinates. T h e allowed coordinates are related by canonical transformations; that is, changes in coordinates preserving the Poisson bracket. If ($1, . . .,phi, 41, . . ., q ~ is) one set of I f coordinates, and (pi, . . .,ph7,ql, . . ., qk) another, then the primed coordinates are functions of the unprimed, and {pl, si}=6 y holds. I n quantum field theory, the analcgue of a canonical transformation is a change from one set of canonical

+:

+&

Outline of axiomatic relativistic quantum field theory fields, +, relation

T,, to

another, [+a(f,

&

T:,

795

such that the equal-time canonical commutation

9, q ( g , 41 = - ihZ%, f f (4g ( 4 d3X

is preserved. Here,

J

+ a ( f , 9 = + a ( X , t > f(4 d3X and 2 is a renormalization constant. I n the Wightman formulation, one does not assume canonical commutation relations between the field and its conjugate, at least, in four-dimensional spacetime. I t is probable that the field is too singular, and that it is a distribution in the time variable, and so it cannot be evaluated at t=0. This shows up in perturbation theory in the divergence of the renormalization constant 2. I t is hoped that this is taken care of in the Wightman formulation by smearing in time as well as space. Instead of the canonical commutation relations, one postulates merely commutativity at space-like separation, leaving open the nature of the singularity at equal space-time points. If [+1(x),+z(y)]*=O for x-y space-like, we say that $1 and 4 2 are relatively local. I n this definition we assume that 4 1 and 4 2 act on the same Hilbert space &', with a common dense domain D (of the axioms), are covariant under the same representation U ( A ,a ) of ISL(2, C), and satisfy all the axioms except (4). If (4,) is a complete set of local fields, so that they satisfy axioms (4)and ( 5 ) , and $1, $2 are local relative to (4J, then in some sense $1 and $2 are local functions of the (4,): #6(X)=Fkh(x),+2(x),' .), i= 1,2, ' * * * I n this formulation it is not rigorous, and it is hard to make mathematical sense of this equation. Borchers (1960), by reformulating it, has proved the natural result of and $ 4 ~are ) relatively local. Taking $1 = $2, his result shows this equation: $I(.) that each is local. Thus all fields local relative to a given set (4,) are themselves local and relatively local. This class of fields is called the Borchers class, K(+,) of the theory defined by {+a}. We shall see that any field from K{+,) with the same quantum numbers, and creating the same one-particle state when applied to the vacuum, gives the same scattering matrix, and could be used to calculate it. Thus one can prove that any set of fields local relative to each other (carrying all the quantum numbers) is as good as any other. As an example of Borchers classes, consider the 7-n system. Here one usually introduces the fields T+,TO, $p, t,hn and their conjugates. We assume that these fields are local and relatively local. We note that the fields ~ f#p, and their Hermitian conjugates T-, $: can create all the quantum numbers when acting on the vacuum. They therefore satisfy axiom ( 5 ) , and by Ruelle's (1962) result form an irreducible set; we could use these fields to define the theory. Alternatively, the fields $n and r,hP could be used. Note that this does not prove that TO is a bound state of p, p or ii, n (though it might be) in the conventional meaning of the term. For, in Wightman theory, one smears the fields in space and time to get a complete set of operators. The canonical fields $n and $p at a sharp time still might not be complete, even if they exist. I n conventional field theory, T is a bound state of p, 5 only if the field T is not needed for a complete set of operators at a given time. Proof of Borchers theorem again uses the analytic properties of the Wightman fields (Streater and Wightman 1964, theorem 4.21). Note that a Borchers class is defined by a complete set of local fields, not by an individual field.

+,

R F Streater

796

A rather similar but less precise result was obtained in perturbation theory (Kamefuchi et aZl961). I n a space-time of dimension > 3, Epstein (1963) (and independently Schroer) has obtained the complete characterization of the Borchers class of the free boson field of mass m : any field in K(+)is a Wick-Wightman polynomial; that is, a finite sum of Wick ordered powers in the field and its space-time derivatives. I n fourdimensional space-time the Borchers class is restricted to polynomials because the singularities of infinite sums are locally worse than distributions. I n two dimensions the singularities are not as severe; Jaffe (1965b) defined ‘entire functions’, of the free field in this case, and showed that they form the Borchers class. This led him (Jaffe 1966b) to consider generalizing the Wightman axioms in four-dimensional spacetime to more singular ‘functions’ than distributions (§3.4), so as to incorporate more general local functions into the Borchers class. Haag’s theorem (Haag 1955, Hall and Wightman 1957) is one of the most widely misquoted results of the subject (Bjorken and Drell 1965). Suppose that a field theory is not very singular, so that the fields do not need smearing in time, and that polynomials in the operators

+

+ u ( f , t )= f +a(% t)f(“) d3% f E 9 ( R W 3 > make sense when applied to the vacuum, Suppose that at some time t the field f ,f ~ form 9 an irreducible set (eg we must include the conjugate operators ~ $ ~ (t), fields rr,=SL/S$, among the to get irreducibility). Suppose, as is usually done in the interaction picture, that at t=O the canonical field variables +,(f,0), r,(g, 0) are related by a unitary operator (the Dyson U matrix) to the canonical variables of a free field of mass mu. Haag’s theorem then asserts that + u ( ~ ,t ) are free fields of mass mu. Thus the interaction picture exists only if there is no interaction! This is not a disaster for the theory; it means that non-free representations of the canonical fields are required, and there are many of these. For a proof of Haag’s theorem, see Streater and Wightman (1964, theorem 4.16). The shortest proof uses the theorem of Schroer (1958, unpublished) and Jost (1961), part of which was found also by Federbush and Johnson (1960): if a relativistic quantized field has the same two-point function as the free field, then it is the free field. Csikor and P6csik (1967) generalize this: if any one 2n-point function is free, the field is free. It has been proved for massless fields by Pohlmeyer (1969). A more general question was raised by Hall and Wightman (1957); they proved that if two sets of irreducible fields, not necessarily free, are related at t = 0 by a unitary operator, then the first four Wightman functions of the two theories are equal. It is an open question whether then all Wightman functions are equal. Haag’s theorem can be traced to the requirement of translation invariance and the fact that local interactions ‘polarize the vacuum’; that is, do not annihilate it. An algebraic proof has been given by Emch (1972). Guenin (1966) and Segal (1967) suggested that Haag’s theorem can be avoided if we do not insist that time translations be implemented by unitary operators, but regard them as automorphisms. This has been justified by Segal(l967) and Glimm and Jaffe (1972) for the two-dimensional relativistic fields. A consequence of the generalized Haag’s theorem, noted by Fabri and Picasso (1966, Fabri et a l 1967), is that a broken symmetry cannot be implemented by a unitary operator, even a time-dependent one. For, if a multiplet of fields is transformed locally by a unitary operator V at t = 0 (the usual assumption about an internal

Outline of axiomatic relativistic quantum field theory

797

symmetry), and also the canonical conjugates, then the first four Wightman functions are the same for the fields and the transformed fields. This means that all masses, vertex functions and elastic scattering amplitudes are invariant under the transformation, contrary to the breaking of the symmetry. It follows that such a Y does not exist, and that every broken symmetry is spontaneously broken as well; that is, is not implemented even by a time-dependent unitary operator. T h e last major result is scattering theory. It is clear from observation that the interaction between particles is negligible, at large times before or after the collision, and that the particles behave as free particles. I n some old-fashioned formulations of quantum field theory, the disengagement of the particles from each other’s influence was helped along by the arbitrary insertion of a time-dependent factor in the interaction, which adiabatically switched it off at large times. Here, adiabatic means that the switching on and off is done smoothly (in quantum mechanics, a rapid switching on can cause effects, such as the reflection of the wave, which are unnecessary complications). For example, one might choose exp ( - aItl)HI as the interaction, where a > 0, t is the time, and H I the desired interaction, letting cy. ---t 0 at the end of the calculation. This adiabatic switching is completely abandoned in modern quantum field theory. Its place is taken by the asymptotic theorem which automatically ensures the interpretation of the theory in terms of asymptotic particles. I n potential scattering, in a theory with a short-range potential V , it is possible to prove that the particles behave as free particles at large times, in the sense that exp (iHt) exp (- iHot) Y converges (strongly) as t --+ & CO. For an account, see Prugovecki (1971). Long-range forces like V = l/r give trouble and need special treatment (Dollard 1964). I n axiomatic quantum field theory, causality, together with the condition that the lowest mass in the theory is positive, excludes the possibility of long-range field correlations. Haag (1958) realized that this was enough to establish an asymptotic condition, provided one can ‘solve the one-body problem’; that is, construct a creation operator for a single particle. This will always be possible if we make the more detailed assumption about the spectrum, namely that the possible values of PfiP,Lare 0 (from Pp=O, the vacuum), m2 (an eigenvalue of finite multiplicity) and 2 M2 (the beginning of the continuum), where M > m . I n this case, the particle of mass m is not orthogonal to because of axiom (5). all vectors of the form BYo, where B = + l ( f l ) . . , &(fn), fj €9, We can extract a one-particle creation operator from B by picking a part of it that creates mass m: let f ( x ) €9be such that f ( p ) is zero except in a neighbourhood of p2= m2, so that f ( p )= 0 if p = 0 or $ 2 2 M2. Then J U(x)BU-l(x)f(x)d4x creates a one-particle state when applied to the vacuum. Ruelle (1962) showed that by averaging this over the Lorentz group one can construct creation operators for a dense set of one-particle states. I n field theory we cannot write exp (iHt) exp ( - iHot) as in potential theory, because of Haag’s theorem, which says that the free and interacting time evolutions cannot both be implemented in the same representation of the field variables. Haag (1958) avoids this problem as follows. Letf(x, t ) be a classical solution of the KleinGordon equation of the form f(p)=F(p)QO) S((pO)Z-pZNow define the time-dependent operator

m2),

FEY(R3).

798

R F Streater

whereB(x)= U ( x )BU-l(x), U(x)=exp (-ip.x). T h e time dependence of Bf(t)is due to the time dependence of B(x0, x), which follows the full Hamiltonian of the theory, combined with the time evolution of f ( x 0 , x), which follows the Klein-Gordon equation backwards in time, and so imitates the operator exp (iHot). It can be shown (Ruelle 1962) that B f ( t ) Y ois a one-particle state, independent of t , but the product of several creators, as in Y ( t ) = B f i ( t ). . . Bfn(t)Yo,does depend on t ; the limits lim Y ( t ) = Y * t+*m

exist, and transform under P$ as states of n particles with wavefunctions the same as those of n one-particle states Bf,(t)Yo,j= 1, . . ., n. We interpret these as the ingoing and outgoing states. It is worth emphasizing that the asymptotic states are normalizable vectors in &', the Hilbert space of the Wightman theory, and transform covariantly under U ( A ,a) of the Wightman theory. T h e asymptotic free fields +*,that are constructed as operators creating and destroying these asymptotic particles, also act on 8'.Haag's (other) theorem says that +*(x, 0), T * ( x , 0) are not related to the interacting field variables at t=O by a unitary operator; this does not mean that the Wightman field and its asymptotic fields act on different Hilbert spaces (though their domains of definition could be different). It does mean that the wave operators do not exist. T h e ingoing states Y- span a subspace of 2, called 8'-, and the outgoing states Most states used in experimental physics are made out of ingoing span a subspace 8'+. stable particles, and they make transitions to outgoing stable particles. T h e scattering S is a unitary operator S maps &'+ to 8'-and is defined by SY+=Y-. If 8'+=%-, operator on 8'*, so the equation &'+=%- is called unitarity. A stronger postulate than unitarity is asymptotic completeness: &'= &+ = &'-. It means that every vector in 8 has an interpretation as a superposition of ingoing particle states; the same vector has a (different) interpretation in terms of outgoing states. There is good experimental evidence for asymptotic completeness for theories not involving massless particles (for which the Haag-Ruelle asymptotic theorem does not hold). It can be proved that the S matrix obtained in a Wightman theory is invariant under LP$ and PCT, and that PCT transforms X + on to 8-in the expected way. It can also be shown (Ruelle 1962, Wightman 1962) that the asymptotic state Y* is actually independent of the choice of creation operator B from among all fields in the same Borchers class, provided it creates the same particle. Thus axiomatic field theory treats all fields on the same footing, and does not distinguish between elementary and composite particles. This distinction can only be made within the context of a specific dynamical theory. That the S matrix is invariant under choice of 'interpolating field' in the Borchers class was proved by Borchers (1960) in the LSZ formalism, where an asymptotic condition is postulated rather than proved. T h e first ingredient in the proof of the asymptotic condition is an estimate for the size of the solutions of the Klein-Gordon equation, namely:

+

+*

,&fI

sup

I

t ) < ct-3'2

X

which Haag proved in the stationary phase approximation, and which can be understood in terms of the spread of the wave packet. Ruelle (1962) gave a geometrical proof, without, however, giving all the details. Araki generalized this method to a wider class of solutions. A complete proof was written out by Jost (1966). T o prove

Outline of axiomatic relativistic quantum field theory

799

the Lorentz invariance of S, it is necessary to obtain the same estimate for the Lorentz transformed solution f A ( x , t ) =f (Ax), A E L . This can be achieved by Jost’s method, and has been written out by Streater (1967b) T h e second ingredient in the proof of the asymptotic condition is an estimate on the correlations of the field at large space-like separation. T h e correlations are measured by the truncated Wightman functions introduced by Haag (1958), which in perturbation theory correspond to the sum of only connected diagrams, and were inspired by the ‘linked cluster expansion’ of statistical mechanics. T h e truncated functions are the Wightman functions with the contribution from the vacuum intermediate state removed in a way symmetrical under permutations. Thus, for example, if (Yo, B(x)Yo)= 0, then the truncated four-point function is (for a boson creation operator B )

(YO, B(xl)B(x2)B(X3)B(x4) YO)- (YO, B(xl)B(x2)YO) ( T O , B(x3)B(x4)YO) - ( T O , B(xl)B(x3)T O ) (‘YO, B(x2)B(x4)YO)- (YO, B(xl)B(x4)yO) x (Yo, B(x2)4%) To). For Fermi fields there is a change of sign for every odd permutation. T h e general definition of the truncated function W~(x1,. . ., xn) in terms of the Wightman function W(x1, . . ., xn) is given inductively by

W(x1, *

xn)= wT(x1,

-

-2

x n ) + z P ( I ) WT(xZ1, .

xij).

I

WT(.

.

xZjk).

Here the sum is over all partitions

I = (il,. . ., ijJ (ijl+l. . . ij2) . . . (ijk-l+l. . . ijk) of (1, , . ,,n) into K parts, for TZ> K > 1. I n each part the numbers are written in increasing order. For bosons, P ( I )= 1; for fermions, P ( I )= 1 if the permutation

(i, . . 21, *

-9

ql,

.,]I, *-

-1

* * - 9 z.fk

. . ., n

is even, and P ( I ) = - 1 if it is odd. T h e WT are Lorentz covariant, and their Fourier transforms are zero outside the physical spectrum, and indeed the point Pfl=0 is missing. They satisfy causality in the sense that they analytically continue one another in the union of the permuted extended tubes. T h e estimate needed to prove the asymptotic condition is the ‘almost local’ property WT(x1-k at,

. . ., xn+ an) if there is a mass gap, and a polynomial decrease if massless particles occur. This result forms the basis of the rather neat proof of the Haag-Ruelle theorem given by Hepp (1966a) which is to be recommended as a readable account of relativistic and nonrelativistic scattering theory. Details of the proof of Lorentz invariance have

R F Streater

800

been worked out by Streater (1967b). Truman (1975, to be published) has extended the theory to non-causal fields, whose commutator vanishes sufficiently rapidly at space-like separation. A detailed study with generalizations has been made by Herbst (1972). T h e rate at which the states Y (t) converge to their asymptotic limits Y* is rather slow, t-(8-1)/2, where s is the dimension of space. That it converges at all can be traced to the spread of the wave packet. If s = 1, the theory fails. Hepp (1966a) remedies this by considering wave packets with no common velocities, so that the particles do not remain close together. We call two solutions of the Klein-Gordon equation, f(x, t), g(x, t), non-ovdapping if the supports of f(p, w ) and g(p, w ) are non-overlapping in velocity space; ie if (p, up)is such thatf(p, w ) # O , and (q, U,) is such that z(q, wq)#O, then p/wp#q/wq. Hepp shows that if €31, . . ., B , create non-overlapping wave packets, then the convergence to the asymptotic states is faster than any polynomial (Buchholz 1974). For these states, Hepp proves a form of the LSZ asymptotic condition, allowing a proof of dispersion relations from the Wightman axioms (see 55.1). Crichton and Wichmann (1963) give a series of cluster properties that a good S matrix ought to satisfy. They argue that the outcome of a scattering process between a group of particles (of nonzero mass) should be nearly independent of the presence of other particles far away, or of the previous presence of other particles in the region, in the distant past. For example, if fl, , . ,,fm, gl, . . .,gm are wavefunctions in 9’(R3), andE(p)=exp (ip.a)f(p), then lim

+(fl,

a4m

- ’ .,fk,W+I,

. - .,fklgl, should hold, where If1, . . .,fj>*

*

’

.,fiJgr,’ * .,gj,g;+1,

-

*

.,g:>-

-

gj>- +denotes the in (out) state with these wavefunctions. These were proved by Hepp (1964a), who also proved a time-like clustering property (Hepp 1964b). Araki and Haag (1967) give a scheme for relating cross sections directly with expectation values of observables ; the particle states are interpreted as states with specified localization centres moving out with given velocities. Hepp (1965) has derived the existence of poles in the S matrix, corresponding to the ‘one-particle exchange’ Feynman diagram. To some extent this justifies one of the most useful techniques of S matrix theory, the ‘one-particle exchange’ model. =+(fi,

a

‘9

+

3.4. Critique of the axiom T h e Wightman axioms define a rather tight mathematical scheme, despite the very general nature of the physical principles they attempt to embody. T h e axioms are mutually consistent, since free fields, Wick polynomials and generalized free fields (Cook 1953, Greenberg 1961) satisfy them. Generalized free fields (also called quasi-free fields) are defined by specifying the one-point function (a constant) and a two-point function satisfying positivity (jf(x)f(y)W(x-y) dx dy> 0 for all f ~ ( 9 4 ) ) , Lorentz invariance, and analyticity in the cut (x-y)2 plane. T h e n-point function is defined by the requirement that its connected part, the truncated n-point function, vanishes for n 2 3 ; the field is then obtained uniquely from the reconstruction theorem. For example, if we choose the Kalltn-Lehmann weight function p ( ~ 2 )to be 6 ( K~ - m12)+ 6 ( ~ 2 -m22), then the corresponding generalized free field is the sum of two free fields of mass ml amd mz; the Haag-Ruelle scattering theory can be carried

Outline of axiomatic relativistic quantum field theory

801

through, and the resulting S matrix is the unit operator, as for any generalized free field representing particles. While this example is therefore physically trivial, it does show that it is possible for two particles to be completely described by a single field. If the Ka11Cn-Lehmann weight function is continuous, there are no particles associated with the corresponding generalized free field; the interpretation in terms of unstable particles is not adequate, since in nature unstable particles always decay eventually into particles. Thus in a Wightman theory there is no fixed relationship between the number of fields and the number of particles. Licht (1965) has set up an asymptotic condition for generalized free fields, and has determined its Borchers class: it is the set of ‘Wick ordered’ polynomials defined by Greenberg (1961). Apart from the above simple examples, self-interacting boson fields in twoand three-dimensional space-time also satisfy the Wightman axioms (Glimm et al 1973). Since there are generalized free fields that are not asymptotically complete, the latter axiom is independent of the Wightman axioms. One easily constructs ) generalized free fields that do not make sense at sharp time: one chooses p ( ~ 2 such ) = 00. Thus, in general, a Wightman theory may be inconsistent with that j p ( ~ 2 dK2 the canonical formalism. Haag and Schroer (1962), to rule out possible unphysical features, introduce primitive causality (the time slab axiom) : the fields should form an irreducible set when smeared with test functions nonzero only between two given space-like non-intersecting surfaces. This axiom reflects the idea that the motion of the field in time should be governed by a differential equation of hyperbolic type, and so the field at a later time should be a function of the field and its time derivatives at t = 0 . Haag and Schroer (1962) show that if p ( ~ 2 contains ) a continuous part it is possible for a generalized free field to violate primitive causality. Details have been worked out by Mollenhoff (1969). Jost (1965) has shown that a field is a generalized free field if and only if its commutator [$(x), $ ( y ) ] is a c-number. Licht and Toll (1961) obtain the same result by assuming [+(x), $ ( y ) ] depends only on x - y . Dell’Antonio (1961a) considers fields zero for all momenta p such that p2 > M2, and Greenberg (196213) and Robinson (1962) fields that vanish in a space-like open set in momentum space. They show that all such fields are quasi-free. Epstein and Wightman (1960) study the equation (3.4) (O+m2)$(x)=X: 40(x)3: where 40is a free field of mass m,and seek a solution + ( x ) , assuming that the Wightman theory defined by 4 has a stable particle of mass m with unit multiplicity. This equation can be regarded as the first approximation to a self-interacting equation in which $ is replaced by $0 on the right-hand side. Epstein and Wightman prove there is no such solution to (3.4). T h e reason is easy to see if we use the theory of Borchers classes. If ( o + m 2 ) + = j ( x ) = h : c$o(x)~: and the current j is complete, then it defines a Borchers class containing $0, and so gives no scattering. But also, if j is not complete, it gives no scattering (Araki et al 1961). But a direct calculation of the scattering amplitude using the LSZ reduction formula shows it is not zero-a contradiction. Araki et al (1961) generalize this to ‘almost local’ currents; they also show that if there is non-trivial scattering, then the solution to (0 + m2)$ = j , with j given, is unique if it exists, unlike the classical theory, when the solution is unique only up to the addition of free fields.

R F Streater

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Greenberg (1962a) proves two folk lemmas: if the interacting field, or outgoing field, can be written as a finite expansion in the ingoing field, then there is no scattering. Greenberg and Licht (1963) show that there is no scattering if the truncated functions vanish beyond some order. Robinson (1965) improves this by showing that, in this case, the field is quasi-free. I n an attempt to construct quantum fields by entirely new methods, Greenberg (1961) introduced 'Lie fields', ie a Wightman field 4J such that

4J(Y)l= J F (x, Y , z>4J ( z )d4z for some F. He later (Greenberg 1968) showed they lead to no scattering. Other simplifying assumptions also lead to contradictions. Wightman (1964) has shown that the energy-momentum spectrum is additive (if PI and P2 are in the spectrum, so is PI+ P2). This means that energy cannot be a bounded operator, a result proved more abstractly by Kadison (1967). For a neat proof, see Borchers (1968). One can show that the energy-momentum spectrum cannot be simple (it must have multiplicity), and that angular momentum is not bounded. Since these facts are clear for the asymptotic states, the proof is easy for any theory for which we can solve the one-body problem, and hence, by the Haag-Ruelle theory, the scattering problem. One might try to construct models in which local commutativity is replaced by something weaker. But Pohlmeyer (1968) shows that if the commutator, instead of being zero, falls off as a Gaussian in space-like directions, or even (Borchers and Pohlmeyer 1968) faster than exponential, then the field is local anyway. An exponential fall-off is probably a real generalization, however. If [+(x), 4J(y)]= 0 on a space-like open set of x -y, then the field is local (Wightman 1960a,b). This theorem is called the global nature of local commutativity. See Streater and Wightman (1964) for a simple proof. This discussion shows that the Wightman axioms form a tight and stable scheme: extra simple assumptions beyond the Wightman axioms often lead to trivial theories, whereas apparent generalizations lead us back to a Wightman theory. Many of these results can be traced to a simple and remarkable theorem (Reeh and Schlieder 1961, Streater and Wightman 1964, p138): let O c W 4 be an open set, and let B(0) denote the class of test functions in Y(W4) zero outside U . If 4J is a Wightman field, let 9(0)denote the algebra of polynomials in smeared with functions in 9 ( 0 ) . T h e n P ( 0 ) YOspans &. This theorem can be formulated and proved in the algebraic approach and has been generalized by Borchers (1968), where a converse is found. This theorem appears to contradict common sense, since the state PYo, PEB(U) is localized in U , and states localized elsewhere 'ought' to be orthogonal. But, in fact it can be proved that there is no exact quantum position operator in a relativistic theory with positive energy (Newton and Wigner 1949, Wightman 1962); because of analyticity, the state PYo has a small but nonzero tail, enough to approximate any vector in 2. Knight (1961) has given a definition of localization of states that is appropriate for quantum field theory: Y is localized in 0 if [$(XI,

+

(y",

+(fd

* *

+(fdy">= P o , +(fd '

*

W n ) Yo>

for allfi . . .fn with support space-like with respect to U ; if the state Y is prepared from the vacuum in region 0,and is then measured by observers space-like to 8 , the

Outline of axiomatic relativistic quantum field theory

803

state looks like the vacuum. Localized states of this form have been studied by Licht (1963), who notes that they do not form a linear set. Further ideas on the localization of states have been developed by Haag and by Schlieder (1965), based on the idea that the number of states in a finite region, with finite maximum energy, should be finite. Schlieder (1965) proves a rigorous form of this (compact replacing finite), provided no infinite-component fields enter (see also Haag and Swieca 1965). Infinite-component fields were treated systematically by Gelfand and Yaglom \ with some space-like momenta, (1948). These fields described representations of 9 thus violating the spectrum condition. Second quantized co-component free fields were written down by Nambu (1968) and by Feldman and Matthews (1967a,b). These models either have infinitely degenerate mass states, or violate causality: Grodsky and Streater (1968) show that any co-component field, irreducible under L$, satisfying spectrum condition and causality, must have co-degeneracy of mass, whether thefield is free or interacting. This shows that a local interaction cannot split the masses of the Nambu free fields. This result has been generalized (Oksak and Todorov 1970) to reducible fields. For free fields, the proof is more elementary (Abarbanel and Frischman 1968). A more detailed property of a field than the time slab property, and one which better reflects the hyperbolic propagation property, is the idea of ‘causal completion’. Let 0 be a region of space-time, and 0’ its ‘causal complement’, ie the set of points space-like to every point of 0. Then U” will be called the causal completion of 0. Properties of the wave equation hint that any field operator localized in U” should be a function of the field operators in 0. The first result of this kind was by Borchers (1961), who has shown that a Wightman field 4 smeared with all functions nonzero in U , a time-like tube, is an irreducible set of operators. A time-like tube is a set of points (x, t ) with \.x-al< r for some a and r ; clearly O ” = R4 in this case. This is an opposite kind of result from the time slab property, which as we saw is not valid for some generalized free fields, and so cannot be proved from the Wightman axioms. Araki (1963a) and Kolm and Nagel (1968) have generalized Borchers’ result, showing completeness in more general regions. For fairly general regions 0,Araki (1964b) has shown that free fields in 0” are functions of the free fields in 0; this is called the diamondproperty. Because even a neutral Wightman field in general is only symmetric (not selfadjoint), the concept of ‘functions of the field’ is ambiguous, and the question is better formulated in the algebraic approach, as was done by Haag and Schroer (1962). For free fields, various self-adjointness questions have been solved. Borchers and Zimmermann show that YOis an analytic vector for the free field, ie

converges for some A # 0. This implies that $(f ) has a unique self-adjoint extension, and that, iff and g have space-like separated supports, then the spectral resolutions of +(f) and $ ( g ) commute. Gachok (1965) has generalized this; one gets the same result if Yo is a quasi-analytic vector, ie

converges for a X # 0. Wightman theories satisfying Gachok’s condition can be reformulated in terms of local C”-algebras (54.1).

R F Streater T h e assumption that the field is a ‘tempered distribution’, ie has test functions in Y (R)4, is a technical point, corresponding to considering only renormalizable theories.

Guttinger (1958, 1966) has remarked that in the perturbation theory of a nonrenormalizable theory the various contributions are not tempered, since they grow worse than polynomially in momentum space. He suggested that for such fields test functions should be in 9 in momentum space, to reflect the bad high-energy behaviour. T h e trouble with this idea is that the test functions, being analytic in space-time, cannot be chosen zero outside regions, and causality becomes hard to formulate. Constantinescu (1971) has solved this for certain classes of analytic test functions, using the concept of support of an analytic functional. Bardakci and Schroer (1966) have studied non-renormalizable theories in the ladder approximation, and come across non-distribution Wightman functions. Bardakci and Schrwr (1966) remark that causality can still be formulated for certain classes of theories that are not distributions, and that non-temperedness cannot lead to exponential growth of the scattering matrix elements provided the theory exhibits the analytic properties of a Wightman theory. Wightman (1956) originally postulated that the expectation values are distributions, not necessarily tempered. He showed that the positivity axiom implied that the two-point function is automatically tempered. He hoped to show the same for all functions, but this problem remains open. T h e first published proof of analyticity of non-tempered W functions is by Borchers (1964). He also shows that timesmearing a field Jrj(x, t ) h(t) dt, h E 9 ( R ) converts it from a distribution to an infinitely smooth operator function of the remaining variables x. This is because smearing with h(t), which acts as an energy cut-off, is also a momentum cut-off because IpI < IEI; and operator fields with a sharp enough momentum cut-off are c m functions of x. T h e most cogent generalization is due to Jaffe (1966a)b). We seek a class S of test functions ‘smoother’ than those in 9, so that more singular generalized functions than distributions can be applied, but such that S n 9 is dense in S, so that there are enough localized test functions to formulate causality. Jaffe found that there was no best space (in the sense of smallest), but that the Gelfand-Shilov (1964) spaces S a , 01 > 1 are satisfactory. T h e choice of 01 determines the local singularity structure of the Wightman functions. The behaviour at 00 can again be postulated to be tempered or not. Jaffe establishes most of the results of the usual theory, and defines entire functions of the free field. These functions enlarge the Borchers class beyond the polynomials, but the complete Borchers class within the Jaffe classes has not yet been determined; such fields are called strictly local. Oksak and Todorov (1970) consider the possibility that an co-component field might belong to the Jaffe class, but are able to generalize the Grodsky-Streater theorem to this case; all such fields have infinite mass degeneracy. One might try to generalize the axioms by omitting the uniqueness of the vacuum. Borchers (1962) and Reeh and Schlieder (1962) noticed that in this case the field is reducible. They (and Maurin 1963) show that in this case it is possible to factorize out the operators commuting with the field, to obtain a theory satisfying all the axioms, including a unique vacuum. Thus possible multiplicity of the vacuum is not a serious obstacle to setting up a theory. Modificatons to the axioms are to be expected in quantum electrodynamics; whereas the electromagnetic field F,, ought to obey the Wightman axioms, the unobservable fields A , and zju(x) (the vector potential and electron fields) do not (Ggrding and Wlightman 1965). Perturbation theory suggests that in a Hilbert space

Outline of axiomatic relativistic quantum jield theory

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approach (positive-definite metric) # does not anticommute at space-like separation. Even for the free A , field, one has a negative result (Ferrari et al 1974). Assume: (i) There exists a representation U ( a ,A ) of ISL(2, C). (ii) The potentials A , are tempered distributions and are covariant under translations U(1, a ) A,(x) U-1(1, a)=A,(x+n). (iii) The fields Fpv(x)= 8,A 8"A,are covariant under U ( A ,0). (iv) U ( A ,U ) Yo=Yo for a normalizable Yo. (v) (Yo,A,(x) A " ( y )Y O= ) W,, is analytic in the extended tube. (vi) Maxwell's free space equations hold. Then W , L, = 0. The Gupta-Bleuler formalism (GBrding and Wightman 1965) provides one way out; it is probably not the final axiomatic scheme for the electron field. y-

4. The C*-algebra approach 4.1. The HaagJield The algebraic developments in quantum mechanics ($2.4) have been used by Haag and others to set up an axiomatic scheme for quantum field theory similar to, but not equivalent to, the Wightman axioms. The essential ingredient which makes a quantum theory into a quantum field theory is the notion of localization. Consider, for example, a neutral scalar Wightman field 4. The operator +(f) is not bounded, and it would be convenient to consider in its place bounded functions of it; let us suppose this is possible. Since +(f ) represents something with a direct interpretation (the field strength weighted with f), the various bounded functions F of it are interpreted as the corresponding functions of this. But, for most purposes, including scattering, a detailed assignment of an interpretation for each Hermitian operator is not necessary; the most important piece of information is that an operator F(+(f)) is localized in the region where f # 0, the support off. I f f and g have space-like separated supports, so that +(f) and +(g) commute, we would expect that bounded functions like exp (i+(f)) and exp (i$(g)) would commute. While this is also a reasonable property physically (it asserts the simultaneous measurability of two bounded measurements made at space-like separation), it does not quite follow from causality expressed in terms of the Wightman fields. Because of this and other technical difficulties, Haag (1959) set up an independent set of axioms involving bounded observables only. These axioms are still being developed, but always embody the following ideas. T o each open bounded set 0 in space-time is associated a *-algebra d ( 0 ) of bounded operators of which the selfadjoint elements represent observables that can be measured in 0. I n a field theory, d ( 0 ) will be generated by the bounded (operator) functions of the smeared fields $(f) for all f zero outside 0; there is no loss in generality in assuming each local algebra d ( 0 ) to be a C"-algebra ($2.4). Haag and Schroer (1962) assume each d ( 0 ) to be a von Neumann algebra, denoted W (0) : let 3 be a set of bounded operators on a Hilbert space containing the identity, and denote E' its commutant, the set of bounded operators commuting with every element of S. S is called a von Neumann algebra if 3 = E". More generally, 3"is called the von Neumann algebra generated by E. Various physical principles can be expressed in terms of the local algebras.

R F Streater

806

PoincarC invariance leads to the existence of an automorphism .(A, a) of d,the C”-algebra generated by all d (0). These automorphisms should realize the group 9 $ , that is, ~(ni, ai) 7(A2, a2)=7(Ai, A2, ai+Aiaz) should hold ; and moreover T

(A, a) d ( U ) =d(AU + a)

is required by the interpretation of & ( U ) as observables in the space-time region ( 0 ( A O + a ) being the PoincarC transformed rCgion). A theory having such a set of automorphisms will be called Poincare‘ covariant. A stronger requirement, coming from Wigner’s formulation of symmetry (92.2), is that there should exist a continuous unitary representation U ( A ,a) of ISL(2, C) in the Hilbert space of the theory, which implements the automorphisms

U ( A ,U ) B U - I ( A , u)=T(A(A), a) B for all B E&. If this is so, we say the theory is invariant under P\. Similarly, we can formulate covariance and invariance under other groups. More detailed properties of U ( A ,a) can be postulated, just as in Wightman theory, such as spectrum condition, uniqueness of the vacuum, and existence of stable particles of specified mass and spin. T h e vacuum is not cyclic for the observable algebra if there are superselection rules. For one can show from the spectrum condition (Araki 1964a, proposition 2, corollary 1) that if the vacuum is cyclic, then d is irreducible, the analogue of Ruelle’s theorem in Wightman theory. The superselection operators, on the other hand, commute with all observables (92.3), so their presence rules out irreducibility. Observables are neutral, and do not carry superselection numbers like charge, etc; that is, they do not change the charge of the states on which they act. So we relax the cyclicity axiom for d . Causality is expressed by requiring that if 81 and 02 are space-like separated, then d(01)c d ( U 2 ) ’ . Haag and Schroer (1962) suggested the diamond property, d ( 0 ) = &(Of’), where 0” is the causal completion of 0 (93.4). This implies primitive causality: the observables associated with any time slab generate the whole algebra. The examples of generalized free fields violating this (Haag and Schroer 1962, Mollenhoff 1969) also give examples of algebraic theories violating it. If the theory d originates from a field theory, we would expect & ( U 1 U&) to be the algebra generated by d(01) and d ( S 2 ) . This is called the Feld property. Kibble has argued that this might not hold for a theory with a gauge field; if 0 = U@(is not simply connected, one can imagine observables in 0 that cannot be deduced from information gathered in each 0, separately. Indeed, the failure of the field property was found in a simple gauge theory for regions with holes in two-dimensional spacetime (Streater and Wilde 1970). Because of the physical interpretation of d ( 0 ) as being generated by the observables in 0, we require that U l c U z implies d ( U 1 ) c d ( o 1 2 ) . This property is called isotony. Another property which might be true in realistic models is duality. Duality expresses the idea that what can be measured in 0 is everything compatible with what can be measured outside; in symbols, d ( f l ’ > ’ = d ( O ) . Here, 0’ is the causal complement (93). If this is so, a theorem of von Neumann implies that d ( 0 ) is weakly closed, since it is a commutant of a set of operators. If the algebras d ( 0 ) are not weakly closed, we can replace them, in discussions, by their weak closures,

Outline of axiomatic relativistic quantum field theory

807

the W*-algebras they generate. T h e weak closure of d ( 0 ) is denoted by "(O), and duality may now be expressed by the equation "(0') = "(0)'. Then

~ ? ( ~ ) = g ( u ) n w ( o '"(0) ) = nw(0)' is the centre of "(0). Things in 3(0) can be measured both inside and outside U, and are thus associated with the boundary. Araki has suggested a connection with superselection rules. For example, we can measure the total charge Q in a region 0 c R3, either by measuring the charge density in 0 and integrating, or by measuring the electric field outside and using Gauss's theorem. We can express the absence of such observables by stating: 9 ( 0 ) consists of scalars. I n this case, g ( 0 ) is called afactor. Haag and Kastler (1964) distinguished between d,the smallest C*-algebra containing the local algebras & ( U ) , and d",which contains global observables like the total charge, which require an infinite volume of space to measure.

4.2. Thefreefields Araki (1963b, 1964b) has constructed "(0) explicitly for the free scalar field of mass m. H e finds that " ( U " ) is a factor if 0 is a connected open region with smooth boundaries. Osterwalder (1973) has given a simple proof of duality (and hence the diamond property), first proved by Araki (1964b), who showed also that the factors "(0) are of type I11 for certain regions. A simple proof of this was given by Dell'Antonio (1968) ; S t ~ r m e(1966) r proved it from quite general considerations. A physical interpretation of the surprising result, type 111, has been offered by Licht (1965) (see $4.3). That the local algebras 9 ( 0 ) exist, are local and relativistic, and causal, can be proved for free fields by the method of Borchers and Zimmermann (1964). For a Wick polynomial, not of even degree, Langerholc and Schroer (1965) show that the corresponding local algebras are the same as those generated by the free field; thus the whole Borchers class of fields is associated with just one local ring system. T h e free relativistic Dirac field is not an observable field, and so cannot be used to illustrate local observables; rather it is an example of a jield algebra containing charge raising and lowering operators. T h e local field algebra S ( 0 ) is generated by all operators ($*(f),f E 9(0)}. F,the C*-algebra generated by all the .F"(), is known to be simple (has no non-trivial two-sided ideals). It contains the even CAR algebra, generated by even powers of $ and #*. (Note that the anticommutation rules imply that $(f) is bounded.) This is also a simple algebra (Doplicher and Powers 1968). Although the even sub-algebras commute at space-like separation, it is not observable, since it is not gauge invariant, ie invariant under $ exp (ia)$, $*+ exp ( - ia) i,h. Gauge invariant operators contain the same number of $ and 9" fields. T h e gauge invariant sub-algebra is not simple (Araki, in Appendix to Haag and Kastler 1964). --f

4.3. Development of thegeneral theory We now discuss the independence and interrelation among the suggested axioms. Using the spectrum condition, Borchers (1966) has shown that energy and momentum are global observables, in the sense that bounded functions of them belong to W = d " , the von Neumann algebra generated by the local observables; so energymomentum is a limit of local observables. This is obvious if d is irreducible, since 54

808

R F Streater

then si’” is the set of all operators. If d is reducible, the choice of U (1, a ) implement’ with si’) then ing space-time translations is ambiguous, since if V E ~(commutes VU(1, a)AU-l(l, a)V-l= U(1, a)AU-l(l, a ) for all A €si’,so VU could be chosen instead of U. Borchers’ result then says that it is possible to choose V such that U(1, a ) is continuous in a, and is weakly inner, ie lies in a”’. He also proves that 2”= W n W ’ , the centre of 3,is pointwise invariant. This was first proved by Araki (1964a, proposition l), who analysed the general situation when the vacuum is not unique. He showed that in this case the algebra 9 is reduced by a direct sum of Hilbert spaces &’= @a#O,. Each S a is invariant under U ( a ) and 9,and contains a unique vacuum. Thus there is no real loss in generality in assuming a unique vacuum (except that it has not yet been proved that each ZO, is invariant under the U ( A , 1) of ISL(2, C ) if the theory is Lorentz covariant as well). Using causality as well as the spectrum condition, Araki (1964a) shows that W a , the algebra 9 acting on Za, is a factor of type I in von Neumann’s classification. This result is the algebraic analogue of the reduction theory of Borchers (1962) and Reeh and Schlieder (1962) for a Wightman theory with several vacua. It is possible to prove that the C*-algebra of the canonical commutation or anticommutation relations is simple (Slawny 1971, 1972). Misra (1965) has proved that any Haag field is simple, basing his proof on the usual assumptions and an extra hypothesis of a technical nature. Borchers (1967) has removed this extra assumption; the theorem is the following. Suppose the set (W( U ) > of von Neumann algebras satisfies isotony, translation invariance, the spectrum condition, with weakly inner unitary group of space-time 9(8 + x)}). translations, causality and weak additivity (ie for any open 0,9= { uZEW( Then if 9 has a trivial centre, it is simple. If W has a non-trivial centre, we can apply the reduction theory of Araki, and apply the Misra-Borchers theorem to each component in the reduction. Wightman (1964) makes the assumptions of isotony, PoincarC invariance, causality and the field property, and postulates that 9 is reduced into irreducible components by the direct sum over superselection sectors (coherent subspaces). He then obtains a form of the Reeh-Schlieder theorem, and shows that the local algebras cannot be Abelian. He also shows that, if W is irreducible, the intersection of the W ( O ) , as U run over all open sets containing a point x E R4,consists of multiples of the identity. W ( 0 ) would contain an operator B located at x; and B(x) If this were not true, nos% would not only be a Wightman field at each point, but also a bounded one. Thus Wightman’s result that this is impossible is the modern version of a result of Lehmann (1954), which says that an interacting quantum field must be at least as singular as a free field. Kadison (1963) proves that the local algebras are not of finite type [under the same assumptions as Wightman (1964)l. , Schlieder (1965) proves the following lemma: if A E W ( U 1 ) and B E ~ ( O Z ) where 01 and 82 are space-like separated, then AB cannot vanish unless A or B is zero. This allowed Roos (1970) to demonstrate a version of causality related to statistical independence, suggested by Haag and Kastler (1964). Let 81 and 0 2 be space-like separated, and si’(01) and d (8 2 ) be C”-algebras of observables associated with them. Suppose w l is a state on &(@I),and w2 a state on d ( O 2 ) . Can these be independently assigned? In other words, does there always exist a state on & such

Outline of axiomatic relativistic quantum jield theory

809

that w reduces to 01 onsal(01) and to w2 on d ( 0 2 ) ? The answer to this is yes (Roos 1970). Moreover, o can be chosen to be w l @ w 2 acting on the algebrad(01)@.d(Oz), which is homomorphic to d ( 0 1 ~ 0 2 ) . [See the Corrigendum by Roos (1970), however.] Another property of interest is related to duality, and is called extended locality: if 01 and 0 2 are space-like separated, then d(01) n d ( 0 2 ) should contain only multiples of the identity. Physically, this means that anything that can be measured in 01 cannot be measured in 0 2 (except the identity), and conversely (Schoch 1968). Landau (1969) derives extended locality from translation invariance, isotony, causality and weak additivity (at least when 01 and 02 are double cones based on a time-like line), assuming that W is a factor, and that the following ‘asymptotic condition’ holds : as a + CO in space-like directions, for every Y with compact energy-momentum. Licht (1963) has considered the notion of strictly localized states in the algebraic approach. Following the ideas of Knight (1961) (see §3), he says that a state Y is strictly localized outside the region 0 if (Y, AY) = (YO,AYo) for all A E “(0). He shows that if W is a partial isometry in W ’ ( 0 ) with W”W=1 (so that WW” is a is strictly localized outside U. Such an isometry has the projection), then Y = WO property ACD2) ifAEW(0). (WCD~,AWCDZ)=(CD~,

This leads to the definition: a bounded operator W is said to be strictly localized outside 01 if this equation holds for all 01, @Z E # and all A E W ( 0 ) . Licht then shows that W is strictly localized outside 0 if and only if W”W= 1 and W E W’(0). Moreover, for every state Y, strictly localized outside 0,there exists a unique operator WE W’(0), strictly localized outside 0, such that Y = WO. W is not the only operator creating Y from the vacuum; it is the only one in W’(0). Strictly localized states do not form a linear set. W has an interesting physical interpretation. Given P Ea’(@)), there exists a partial isometry WP E Wf(O1), such that Wp*Wp=l andWpWp*=P. In quantum mechanics, a projection P represents a question q ; if Pa=@, then q is true in the state CD, while if PCD=O, q is false in the state CD. We interpret the corresponding Wp as an apparatus in 0’which alters any state CD to a state WpCD such that q is true:

PWpQ = WpW$ WpCD = WpQ. Moreover, the change, from CD to WpCD, does not affect any measurement in 0. Borchers et a2 (1963) examine whether the postulate concerning the existence of a normalizable vacuum is reasonable, or whether it should dissolve into the continuum in the presence of zero-mass states. Perturbation theory suggests not, though it indicates that the one-particle state might well disappear as a sharp eigenvalue of P”PP.The latter phenomenon has been studied by Schroer (1963) in exactly soluble models; a charged particle receives a cloud of ‘infra-particles’ which smear its mass, Borchers et aZ(l963) remark that the existence of a vacuum is an independent axiom; the theory obtained from a state of a single charged particle by applying Lorentz transformations and the free electromagnetic quantized field is relativistic, local, and has a physical energy-momentum spectrum, but has no vacuum state. They also give examples in which the energy-momentum is in the physical cone, but the Lorentz automorphisms are not implemented by unitary operators; that is, Lorentz symmetry is spontaneously broken (see 54.4). For the free field, they show that there exists a

R F Streater

810

functional Eo on the algebra d such that for any Y, and space-like a, lim (Y,

U(ha) A U-1 (Aa)Y) =Eo(A) for all A €d.

a-+m

Eo is interpreted as a vacuum state. By the Gelfand-Naimark-Segal theorem (see $2.4), there exists a Hilbert space s o , containing a vector Yo, and a representation nof d,such that E@)= (YO,T ( A ) Y o ) .We say that Yo is obtained from Y by the ‘method of large translations’: U(Aa) translates the material in the state Y ‘behind the moon’, leaving a vector which looks like the vacuum as far as local measurements are concerned. I n the limit h --+ 00, we obtain Yo, possibly in a new Hilbert space. This idea was further developed by Borchers (1965a, 1967), who met technical diffi00 in a culties in his attempt to prove that (Y, U (ha) A U-1 (ha)Y) converges as A general field theory. If this could be proved, then the axiom of the existence of a vacuum would be redundant. Araki (1964a) shows .--)

U ( h a ) A U - l ( h a ) - ( Y , U ( h a ) A U - l ( h a ) Y ) --+ 0 weakly as h ---f 00, and relates this ‘weak cluster’ property to the fact that 9 is a factor. Borchers (1965a) proves that if A is local, then A (x) YOand A”(x) YOhave the same energy-momentum spectrum and multiplicity. This is related to the PCT theorem. Araki (1969) has developed a scattering theory in the algebraic framework. If we postulate that there is an isolated mass m in the spectrum of finite multiplicity, and the vacuum is cyclic, then we can construct a ‘creation operator’ B = SA (x) f ( x ) dx U(1, x) A U - l ( l , x), and f ( p ) = O except from any local operator A, where A(%)= near the mass shellp2= mz. Just as in Wightman theory (33.3) one defines the creation operators B$(t),for any solution fa of the Klein-Gordon equation, by t-)

Suppose Bz(t)YOis the one-particle state with wavefunction go,(p). One then shows that the n-particle states

Igl@

. . . @ g,)*=

lim BT(t) . . . B:(t)Yo

t-e

m

converge strongly as t _+ CO. These states turn out to be independent of the choice of A,f and fa individually, depending only ong,(p). Thus it is possible to define the S matrix without a specific interpretation of each operator A E & ; this is in the spirit of the theory of Borchers classes (93.3). It is not possible to prove the spin statistics theorem in a general Haag field (Streater 1967a) because of the possibility of infinite-component fields. These are ruled out by an extra physical condition proposed by Haag and Swieca (1965); it is an open question whether their condition is sufficient to prove the spin statistics theorem in general. A related condition, finite degeneracy of mass eigenvalues, does lead to the theorem for the asymptotic particles (Epstein 1967b), and also the PCT theorem. T h e one apparent drawback of Haag’s theory is that observables themselves cannot create particles carrying superselection quantum numbers, and that any axiomatic scheme involving unobservable operators is ad hoc. Haag and Kastler (1964) point out that the vacuum sector, containing only states of zero charge, does in fact contain all information about the states carrying superselection quantum numbers too. This is demonstrated by an argument using large space translations. Consider the sequence --f

Outline of axiomatic relativistic quantum field theory

811

of states in the vacuum sector XO,representing a particle in a fixed open set 0, together with an antiparticle further and further away. T o an observer in 0, the end of the sequence looks more and more like the one-particle state in 8. Similarly, states of charge 2, 3, . , . can be constructed (approximately) in X O ,and their scattering cross sections measured (Araki and Haag 1967). This is, in fact, how charged states are made in practice. Haag and Kastler argue that, although the different coherent subspaces X q carry mutually inequivalent representations rrg of the C*algebra d,all representations are physically equivalent in the following sense. Any actual experiment measures the expectation values f1, . . .,fn, say, of a finite number of observables A I , . . ., A , with a finite error < E, and therefore establishes that the state could be any w such that

Iw(At)-fil

<E,

If w is a vector state in a representation X Osuch that

.

i = l , 2 , . .,n.

T ~it,can

be proved that there exists a vector

Y in the vacuum space

lw(Aa)-(Y, rro(Ai)Y)I

+,(XI ~ X

P (-iQA)=#a(A)

(XI.

This use of Noether’s theorem is not correct in quantum field theory; in general, j f i ( x , t ) will need ‘time smearing’; and then the integral over the whole of R3 will not make sense. So we define a localized charge Qa,O=J

.(t) dt Jj0(

for each local A and suitably chosen test functions a and 8. They also need to assume that A + A T and time evolution are continuous in norm. With these hypotheses they show that if A ---t A T is spontaneously broken-that is, not implemented by unitary operators U ( T )such that U(T)yO=TO-then m=O, ie there can be no mass gap in the spectrum. This is Goldstone’s theorem. T h e actual presence of particles of zero mass was soon proved, by Ezawa and Swieca (1967). If there is a spontaneous breakdown of symmetry, then the vacuum is ‘degenerate’ in a particular sense. If T is an automorphism commuting with time evolution ut, and p is a state (on the C*-algebra d)invariant under crt but not under T , then the state pT defined by p,(A) = ~ ( T ( Ais) )a vacuum state. This means that pT is invariant under crt:p,(crt(A)) = p,(A). Hence we get a distinct vacuum for each broken symmetry transformation. These vacua will belong to difJevent irreducible representation spaces o f d , and (in general) each such space will containa uniquevacuum. This phenomenon is illustrated in the free boson field of zero mass (Streater 1965). I n this model, the Lagrangian density is 8,48p+, which is invariant under the one-parameter group of transformations +(x) -+ +(x) + 7 , E ~R. This group commutes with the Poincart group, which in the relativistic Fock representation of the free field is implemented by unitaries U ( A ,a), with a unique vacuum vector. T h e vacuum Yo, however, is not invariant under the automorphism T,, defined by these transformations, and new vacua, Y,,, can be defined in new representation spaces. All these new representations are B$.covariant, and satisfy the spectrum condition (Streater 1965). Segal(l962) has, however, proved the uniqueness of the positive-energy representation under continuity assumptions which do not apply to this model. One proves that, although the automorphism T,, is not implementable, its action on a single local algebra d ( 0 ) is implementable, and we say that the two representations of the whole algebra d are locally equivalent. Borchers’ (1965a) general theorem that there is only one representation in the ‘local equivalence class’ with positive energy-momentum is not true without some further conditions that rule out zero mass, as this example shows. There have been several models (Boulware and Gilbert 1962, Higgs 1964) in which an apparent symmetry of the Lagrangian is broken by the vacuum state ot the solution, while no Goldstone bosons appear as physical particles. These models avoid the Goldstone theorem because they involve a ‘gauge field’ which, like the electromagnetic potential in the Coulomb gauge, does not commute with the field at space-like separation, I n a covariant gauge (with indefinite metric), causality is restored; the Goldstone boson duly appears, in the unphysical mode. For a review, see Kibble (1967). T h e photon certainly is not a Goldstone boson, contrary to the early hopes of some physicists. I n fact, Maison and Reeh (1971) prove that Goldstone particles must have zero helicity. Coleman (1966) proved, not quite rigorously, that the invariance of the vacuum

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is the invariance of the world. A rigorous version of this was formulated and proved by Landau and Wichmann (1970). Suppose the local algebras are W*-algebras, with translation invariance, satisfying the spectrum condition with a unique vacuum w (up to a phase), and suppose causality holds. Let V be a unitary operator leaving Yo invariant, such that VW(0)V-1=9(0). Then V commutes with U(a), and is therefore a symmetry. A similar theorem also holds in Wightman field theory. Kraus and Landau (1972) show that, if there is a complete set of asymptotic states, then these form a natural domain in which the local charges are essentially self-adjoint; and the exponentials of the charges form a representation of the symmetry group. See also Orzalesi (1970).

4.5. Weyl systems and the Clajford algebra I n the most important known examples of Haag fields (the free fields and the selfinteracting boson field in two-dimensional space-time), the C*-algebra is of a very special nature: it is generated by the canonical field variables, and thus can be chosen to be the same whatever the choice of Hamiltonian. I t is not known whether this Cx-algebra is suitable for an interacting theory in four-dimensional space-time. I n a canonical field theory, the field and its canonical conjugate have a meaning as operators when smeared in the space variable only. This is probably not true in a theory with an infinite wavefunction renormalization. We now define and study the canonical Cx-algebra. If a system has a finite number of degrees of freedom, we may label the variables (41, . . ., qn) and their canonical conjugates ( p l , . . .,pn). The Heisenberg relations [ p i , qj] = - ih 661 may be written n

[C%Pi,

CPjqjI= -ift

C O1lPi. i=l

.

Thus to each U = (011, . . ., a n ) E Rn and v = (PI, . ., fin) E Rn are associated the operators d = 201fpz and 6 = X&j, such that [d, 61 = - ih (U,v ) , where (U,v ) = 2;gaipi.

T o each pair product

(U,v )

we can associate a complex vector z = U + iv E Cm, with scalar R

(z’, x) = 3;

xj.

j=l

To each x E C n associate the operator 9 = ZZ - 6. Using the Heisenberg relations, these d satisfy

[$,,$’I

= ih ((U, v’) - (U‘, v ) ) = 2ih Im

(z’, x).

(4.1)

The advantage of using the complex space C n is that it treats p and p on a more symmetric footing; this is necessary in a relativistic theory, since Lorentz transformations mix the p and the q (Segal 1963). We see further that only the imaginary part of the complex scalar product enters (4.1). Now, I m (x’,x) defines a symplectic form on C”,ie it is linear in each variable, and Im ( z , x’)= -1m (x’,x). I t is also nondegenerate, ie I m (x,x‘) = 0 for all z E C n implies z’ = 0. A symplectic form corresponds to the Poisson bracket of a classical phase space. Since the operatorsp and q cannot both be bounded, the CCR (4.1) is rather ambiguous, because of domain problems. Writing W(x)= exp (is), and B(x’, x) being a symplectic form on @a, eg B(x’, x ) = 2 I m (x’,x), Segal writes the Heisenberg

Outline of axiomatic relativistic quantum jield theory

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relations in W e y lform :

W(x)W(x')=exp (iB(z, x')/2) W(x+z'). (4 * 2) This would follow immediately from (4.1) if we could expand the exponentials, which is not always valid. Thus a rigorous treatment starts with C" and a given nondegenerate symplectic form B ( z ,x') on C",and postulates (4.2), which states that W(x) defines a multiplier unitary representation of the additive group C". The Schrodinger representation of the Weyl-Segal relations (4.2) is the representation on L2 (Wn) given by W ( z ) Y ( x ) = Y ( x + u ) if x = u ~ T w n W(x)Y(x)=exp (iv.x)Y(x) if x = i v ~ i R n . This representation is continuous, and the generators of W(Au) and W(iAv) are p=iR a/&, q=x. The Stone-von Neumann uniqueness theorem asserts that any continuous unitary representation of the Weyl relations is unitary equivalent to the Schrodinger representation, or to a direct sum of Schrodinger representations, if it is reducible. Slawny (1972) has proved that any representation of (4. Z), whether continuous or not, leads to a set of operators W(x)that generate the same C*-algebra, called the C*-algebra of the CCR over C". Conversely, any representation of this algebra leads to a representation of the CCR (not necessarily continuous.) Continuous representations, in the sense that for fixed x, W (Ax) is a continuous operator function of A, are of interest, since only then do the generators x and - ;ha/ ax exist. For a field theory, or any system with infinitely many degrees of freedom, we have an infinite-dimensional real vector space & (in place of C.), furnished with a corresponds to the classical phase space nondegenerate symplectic form B (XI,272). of the system; in a relativistic scalar field, Jtd is the space of one-particle wavefunctions parametrized, for example, by the Cauchy data +(x, O)=f, d(x, 0) =g, B(+l,q52) being the Wronskian j (4142 - (b42) d3x. Unlike the case of finitely many degrees of freedom, there exist more than one continuous representation of (4.2), when z runs over &. Such representations were encountered by Friedrichs (1953), and called myriotic fields. The set Jtd determines the test functions that can be used to smear the field; if 4(x), ~ ( xare ) canonical fields, then a typical element of A is a pair (f,g)corresponding to the operator

s

x = J 4 4 g (4d3X - 4 ( x ) f ( x >d3x = 7r ( g )- 4 ( f ). If Jtd is the Hilbert space L2(W3), we interpret the 'modes' pi, qj as ~ ( f g ) , +(fj), where ( f i ) is a real basis in Lz(R3). Girding and Wightman (1954a) choose for Jtd the set of finite linear combinations of a particular set of basis vectors in a Hilbert space, and classify all representations. Haag (1955) showed that there is a continuous infinity of inequivalent representations, and gave physical situations in which they occur. Segal (1958) arrives at a classification of the representations when & is a Hilbert space. This is more convenient in a relativistic theory than the Girding-Wightman choice, since any Lorentz transformation which acts on & mixes infinitely many modes together. Other systematic treatments are those of Araki (1960b), Lew (1960) and Fukutome (1960). Segal (1958), and later Klauder et a1 (1966), gave a useful criterion for the equivalence of representations of the CCR for which the Hilbert space is an infinite tensor product space

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. . ., each factor carrying the operators corresponding to one mode. See Wightman (1968) for an account. T h e Fock representation of the CCR, made rigorous by Cook (1953), is appropriate for the free-field dynamics. ~8 is the Hilbert space H which carries a unitary representation U ( R ,a) of @+. T h e Hilbert space Z in which the field operators act is the symmetric Fock space over H, namely Z = C @ H 0( H @ H ) s @. , Here, C, the one-dimensional Hilbert space, contains the Fock vacuum Yo ; H is the one-particle space itself; (H@H)sis the symmetrized two-particle space appropriate to identical particles, obeying Bose statistics. T h e remaining terms in the direct sum are the symmetrized three-particle, four-particle, , . spaces. T o describe the field we take H = L 2 ( W 3 ) , and define creation operators for each f € L Z ( W 3 ) , real, by H1@H2@

..

.

. . ., x,)=(n!)-1/2 C f ( x j )0"-1 (XI, . . ., gj, . . ., x), j where a general vector is a sequence (00, 01,. . .) of symmetric functions of 0, 1, 2, . . . (a*(f)@)"( X I ,

vectors, and gj means we omit this argument. T h e operators + ~ ( f and ) then defined in terms of ax (f)by

~ ( fare)

h ( f ) 2-1/2 (a* (f)4- a ( f > ) rp(f)=i2-1/2(ax(f)-a(f)). This representation has the Foch property (+p(f) + irrp(f)) YO= 0. Conversely, any irreducible representation possessing a vector with the Fock property is unitary equivalent to this one (Glimm and Jaffe 1972). Cook (1953) showed that these operators +p(f) and n ~ ( fpossess ) well-defined exponentials. The relativistic Fock representations (there are several) are not Fock strictly speaking, but are obtained from it by a change of localization. For example, for spin 0, H=Lz(R3, d3p), and the relativistic quantized field is $(f,o ) = + ~ ( D - l f ) ,n ( g , 0) = ?rF(Dg), where D = (p2+ m2)1/4. This has a meaning provided D-lf and Dg E Lz. Segal (1958, 1963) has introduced two other equivalent formulations, the real wave and the complex wave formulations. A most useful formulation for solving nonlinear field equations is the stochastic field approach of Nelson (1973a, b) ($6). From the free fields +p, T F it is possible to construct unitary operators W satisfying the Weyl relations. From a two-component scalar field Wilde (1971) has constructed two local theories, each describing two spinless particles of the same mass, which are inequivalent and not relatively local. T h e Fock representation possesses a number operator N , which generates phase transformations [exp (iNa) @I. ( X I , . . ., x), = exp (ina)@m ( X I , . . ., x,) for (00,01, . . .) E X . Dell'Antonio et a1 (1966) show that the Fock representation is the only one in which N = Ca"(fd a(fd (4.3) i

converges strongly for every choice of basis (ff}of 2'. Chaiken (1968) defines the number operator as the generator of gauge transformations, rather than by the usual formula (4.3). He finds that these automorphisms are continuously implemented by unitary operators in some non-Fock as well as in the Fock representation; only in the latter, however, is the generator positive. This theory involves measures on infinitedimensional vector spaces. For an account see Gelfand and Vilenkin (1964). For

Outline of axiomatic relativistic quantum field theory

817

later work, see Kristensen et a1 (1964, 1965, 1967), Chaiken (1968), Hegerfeldt and Klauder (1970), and Guichardet (1966). Segal (1956a) introduced the Weyl algebra to describe boson systems in a way independent of the representation. Given ( 4 ,B), let V be a finite-dimensional subspace of &, and let dv be the ?+‘*-algebra generated by an irreducible representation of the CCR over V. This is unique up to unitary equivalence, so that different choices of representation at this stage leads to isomorphic C*-algebras. But, in order to proceed to the next step, the inductive limit, a definite choice of each C*-algebra d~ must be made for each V. Once this choice has been made, we have for each V a map WV:V -+ d y given by z Wv(z),satisfying (4.2). If V Ic V2, we may use the maps Wv,and Wv, to define an isomorphism Tvlv2 between d v l and a sub-algebra of d v z ;namely Wv,Wv,-l maps WV,(VI) to dv,,and has a unique extension to the W*-algebra generated by W v , ( V I ) ,namely, dv,.This isomorphism TV1V2 allows us to identify d v l as a sub-algebra of dv,,and Wv, as a restriction of WV,. The C*algebra obtained as the completion of Udv is denoted d . The Weyl algebra consists of the C*-algebra d together with the map W : & +d,obtained by piecing together W ) ,is independent, up the various Wv. Segal proves that the final result, the pair (d, to isomorphism, of the particular choices of dv at each stage. This algebra d has the property that any canonical transformation on L.@ (a real linear transformation T preserving B : B(Tz, Tz’)=B(z, z’)) induces an automorphism of the Weyl algebra. This also holds for the smaller algebra advocated by Slawny (1972); but it is not true that every canonical transformation induces an automorphism of the algebras of the free field involving local von Neumann algebras, since here, weak closures over infinitely many degrees of freedom are involved, and, in general, an automorphism of a C*-algebra is not extendable to an automorphism of its weak closure. If .d is the Weyl algebra, r a representation, and T a canonical transformation, then We say that r and we get a new representation rs by the definition r,(A)= ..(.(A)). nr are symplectically related. The automorphism T is unitarily implemented in the representation r if and only if r and rr are equivalent. Shale (1962) proves that an automorphism T , induced by a canonical transformation T, is implemented in the Fock representation r p if and only if 1+ T or 1- T is of Hilbert-Schmidt class (as a real operator on d ) ;a canonical transformation T is implemented in all representations symplectically related to Fock if and only if 1+ T o r 1- T i s of trace class. A useful role (Araki 1960b) is played by the generating functional of a cyclic representation (the analogue of the characteristic function of a measure). Let W ( f ) be a representation of the Weyl relations in which W (Af) is continuous in A, and let Yo be a cyclic vector. Define E ( f )= ( Y o , W ( f )YO);it satisfies --f

(iiij

E(X;C)

is continuous in A.

For the Fock representation, choosing YOto be the vacuum, we obtain

E (f)= exp (- B llf1I2). Conversely, given a functional E satisfying (i), (ii) and (iii), there exists a Weyl system W ( f )(ie a continuous representation of the Weyl algebra) with cyclic vector YO, such that E(f)=(Y?o, W(f)Yo). Moreover, E defines (W,Yo) uniquely up to unitary equivalence. T h e special class of representations obtained where E ( f )= exp S d3x F ( f ( x ) )for

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some continuous F was investigated by Araki (1960~)and by Gelfand and Vilenkin (1964). T h e problem is a special case of finding representations of current algebra (Streater 1969, Araki 1970, Newman 1972). We now describe the relativistic free boson field in these terms [see Kastler (1965) for a more general treatment]. JZ is a set of real solutions to (0 + m2) 4 (x, t )= 0. T h e symplectic form B is the Wronskian of two solutions :

B (4, #) = . L = t (4$ - $$I d3X, $(x) -+ $(Ax+a). B is the This form is invariant under the natural action of 9$: , Poisson bracket of the classical system with canonical variables (4(x, 0), ~ ( xO)=$). T o specify d,we introduce the real scalar product

and a complex structure J (analogous to multiplication by i) : J ( 4 , $) = (04, - 0-14)' where D = ( - A + m2)1/4. This defines an action on d,since the pair (4, 4) determines a unique solution of the wave equation. Then J 2 = - 1; we choose those 4 for which (4, 4) and Y is not specified. T h e main criticism of it as an axiom is, however, that it, or a rigorous form of it, should be proved and not postulated. This was achieved by Hepp (1966a); see Hepp (1966a) for a superb review. For a spin-0 field 4, Hepp introduces the ‘creation operator’

Similar objects are defined for any spin. T h e smearing in four dimensions ensures that this has a meaning in any Wightman theory. T h e factor exp [it(pO- U,)] picks out (by the method of stationary phase) the energy P O = wp=(p2+m2)1/2 as t + f 00, and so we may expect this energy-momentum relation to be exact in the limit. Indeed, Araki and Haag prove that, strongly, Y(t)=B(f1, t)

.

*

a

B(f,, t)Yo

---f

I fi@ . . . @ f n ) * = Y * ,

t

---f

i.Co

if the ft are ‘non-overlapping’ (93.3). If theft overlap, the convergence is weak:

(a>*, Y ( t ) )-+ 1. This arises as follows. If an invariant amplitude A has no kinematical singularities, and is multiplied by a polynomial covariant in the momenta to give an S matrix element, then A must decrease at infinity to avoid conflict with the Froissart bound. If sA(s) also decreases at CO, we obtain a relation S I m A(s‘)ds‘=sA(s)lo=O.

Outline of axiomatic relativistic quantum field theory

83 1

A superconvergence relation expresses a kinematical zero at CO. Bell (1969) has given a clear derivation of the Froissart bound for particles of any spin. The use of positivity can lead to absolute numerical bounds on the 7-7 scattering amplitude (Raszillier 1972, Common and Yndurain 1970). Jin and Martin (1964) use positivity to obtain lower bounds to the scattering amplitude T ( s , t ) for fixed s as t+ CO, The Pomaranchuk theorem (Martin 1973) states that if the total cross sections for particle-particle, U, and for particle-antiparticle, C, converge to limits as s + CO, and if Re T(s, t ) / I m T log E + 0 as s + CO, then U, = Ca. For a more detailed review of positivity, see Martin and Cheung (1969) and Martin (1969,1973).

6 . Euclidean field theory 6.1. The Schwinger functions If (et, x1,x2, x3) is a real Lorentz vector, and we write ict=x4, as was done before the war, then (XI, x2, x3, x4) is a complex vector which transforms under the complex rotation group; or rather, under the subgroup of it that leaves xl, x2, x3 real, and x4 imaginary. This subgroup is isomorphic to L\, and it would seem that little has been gained from this trick. If, however, t is imaginary, then ict is real, and complex Lorentz transformations of (et, x1, x2, x3), that leave xO=ct imaginary and x real, form a group isomorphic to SO(4). This is the idea behind Euclidean field theory. If 4 is a Wightman field, then the Wightman functions (Yo, XI) ~ ( x z ) .. . 4 (x,) Yo)have analytic continuations into ‘extended tubes’. These domains include the Euclidean points, at which the variables ,$‘ = -:x are purely imaginary. The analytic function evaluated at Euclidean points is called the Schwinger function of order n, and is denoted S,. The Hall-Wightman-Bargmann theorem assures us that the function is invariant under complex Lorentz transformations, including the subgroup, isomorphic to S0(4), leaving the set of Euclidean points invariant. Thus each S, is invariant under the Euclidean group E4. If the field + ( f , O ) = J $(x, 0) f ( x ) d3x, f E 9 ( 0 7 3 ) at t=O has a meaning as an operator with a suitable domain, then we may relate Sn to this field and the Hamiltonian H of the theory. We have

J Sn(x1, 71; x2, 7 2 ; . - .; X n , T n ) f i ( X l ) . =(TO,

4 ( f L 0) exp [-(72-71)

provided that

7,

H ] 4 ( f 2 , 0)

-

.fn(xn) d3Xl . exp [-(Tn-Tm-l)

. d3xn H]4(fn,o)~O>

> ~ % - 1 >. , . > 71 holds. This condition ensures that

is bounded, since H 2 0. Indeed, exp ( - t H )= V ( t ) forms a contraction semigroup, ie ( a ) ll Wll Q 1 (b) V(O)=l (c) V ( t 1 ) V( t 2)= V(tl+ tz), t, tl, t2 2 0 ( d ) B ( t )is strongly continuous.

R F Streater

832

Symanzik (1966, 1969) remarked that, in perturbation theory, the Schwinger functions for a theory such as the 4 4 4 interaction satisfy a positivity condition of the form

1 +(jfll(xl) Sl(x1) d4Xl + jf21(xl)f22(x2) SZ(x1, x2) d4Xl d4X24+ j f n l ( x ~ ) . . .fnn?t(xn) &(XI,

+ n,m>l

jfnl(X1)

. . ., xn) d 4 q . . . d4x,+Hermitian

. .fnn(xn)fml(Y1) . . .fmm(Ym) Sm+n(X1,

.)

.

a

conjugate) xn,yl,

.

.ym)

x d4x1 . . . d4ym> 0

for all test functions f t j E Y ( R 4 ) . This condition occurs in probability theory as the positivity condition obeyed by the moments of a random field. T o explain this, we must introduce ideas and results from probability theory. p). Here, Y is a specified T h e basic notion is the idea of aprobability space ( 0 , 9, set of subsets of the space R ; the elements of Y are called the measurable sets of R. I t is postulated that Y satisfies the axioms of a r~ field (Reed 1973). that is, to each set A E Y is T h e symbol p stands for a probability measure on 9; assigned its measure, p ( A )> 0, subject to the condition that p is countably additive, and that p(R)= 1. Given a probability space (R, 9, p), we may interpret the elements A E Y as events, and p ( A ) as the probability that the event A will occur. A random variable is then any measurable function F from (Cl, 9)to R. (We say F is measurable relative to Y if, for each measurable subset S of R, F-l(S) €9.) If F1, . . .,Fn are random variables, such that F1F2. , . Fn is absolutely integrable over R, then Jn F1 , . . Fn dp, written s(F1. . . Fn), are called moments of F1, . . ., F,. I n particular, 8’(F ) is called the expectation of F, also called the mean of F. Random variables can be regarded as self-adjoint operators on the Hilbert space L2(R, p ) ; if F is a random variable, we may define a self-adjoint operator P by

I n this form, F is already diagonalized. If we have several random variables on R, they give rise to commuting self-adjoint operators which are simultaneously diagonalized on L2(R, p). If we have enough random variables Fj, then they span L2(Cl, p). I n this case the function 1 acts as a cyclic vector, YO, and

. . Fn) = (Yo, Pi . . . Pm YO)^. The ‘truncated expectation values’ of the operators PI,Pz, . , ., defined define the cumulants of the random variables FI,F 2 , . . . . s(F1.

as on p799,

The converse of the map F -+ is a form of the spectral theorem; namely, given an arbitrary family of commuting bounded self-adjoint operators {Aj),there exists a p) and random variables Fz, such that the map Ft -+ At measure space (R, 9, defines a W* isomorphism between the von Neumann algebras generated by and At. For unbounded At there are difficult domain problems, and simultaneous diagonalization is not always possible (Nelson 1959). Segal (1970) has shown that the Wick powers :#(f,O),: of a free relativistic boson field .in two dimensions, smeared with f E a ( R ) , of positive mass, are random variables on a space R, often called Q space. He has also shown that these random variables lie in all LP(0, p), 1< p < CO. But for three or four dimensions, Segal (1970) has shown that :+(f,O),: is not a random variable, no matter what vacuum is used in the definition of : :.

Outline of axiomatic relativistic quantum jield theory

833

A random process is a collection of random variables {at,t E [a, b ] } on a probability p). Usually, t is interpreted as the time. A randomjield is a map @ space (0, 9, from Rn to random variables @(x). We shall need something even more general, a random distribution, defined as follows. We are given a probability space (a, 9, p), and, to each f E 9 ( R n ) , a random variable @(f) on Q, such that f --f @( f)is linear and continuous in measure (Simon 1973). Using other test function spaces, such as 9(W),other types of random distribution can be defined. Of great interest is the Sobolev space 2 - 1 , consisting of those functionsf of finite Sobolev norm: llf\l-1< CO, where the inner product is

+

(f,g)-1= J g ( x ) ( - A mZ)-lfdnx. A random distribution is said to possess moments of all orders if &(@(f~).. .O(fm)) < 00 for all choices of test function f1, . . .,fm and all m. If the moments exist, they satisfy the positivity coming from the statement &(la01 +Q,( f l ) + @ ( ( f Z l )

@(fzz)+

* *

.12)>.0.

This leads to a condition on the moments B(@((xl). . . @(xm))that is identical to the condition on the Schwinger functions found in perturbation theory by Symanzik. This strongly suggests that the Schwinger functions are the moments of some random distribution, at least for Wightman theories with a Lagrangian (the Glimm-Jaffe models are Lagrangian, and so too are free fields in any number of space-time dimensions). If we are given a set of distributions E(f1@ . . . @fm), m=O, 1, 2, . . . satisfying the positivity conditions, are they the moments of some random distribution? T h e answer is nearly, in the same way that in a theory of a neutral field the Wightman functions determine a symmetric field operator, but not necessarily a self-adjoint one. But, if the Schwinger functions satisfy a condition, known as quasi-analyticity, Gachok (1965) has shown the uniqueness of the corresponding random field. I n particular, if all the truncated functions beyond the second vanish, then Gachok‘s conditions hold, and there is a unique random distribution Q, possessing the given first and second moments; in this case, each random variable @(f) is Gaussian. This result is the Abelian analogue of the result of Borchers and Zimmermann (1964). To understand Nelson’s axioms, we must define the concept of Markov process. Let Q be a space, and B a collection of functions, Q + R. It can be proved that there exists a unique smallest U field of subsets of Q, Y B ,say, such that each function in B is measurable relative to 913. We call 9~the ofield generated by B. It is intuitively clear that knowledge about a random system changes the probability of events, and consequently the expectation of random variables. This is expressed p ) be a probability space, and suppose 9 1c 9. mathematically as follows. Let (Q, 9, Theorem. For each F E L ~ ( Q9, , p ) , there is a unique function G, measurable p ) we have relative to 9’1, such that for all U EL“ (Q, 9, ~ ( F u ) ~= ( G u ) . [See Simon (1973) for an outline proof.] T h e function G depends on F and on 9 1 , and we write ( F I 9’1) for G. It is the conditional expectation of F , given the outcomes of the events in 9’1. If now t --+ @ ( t ) is a random process over R, let 9’t be the U field generated by @ t , and 9,- the U field generated by {@t$:- CO < t’ Q t } . We say that @ is a Markov process if

,w(s)l

9 ( - m , t ] ) = W

(s>l

9’d

a34

R F Streater

for all s > t and all t E R. Thus, in words, a process is a Markov process if the conditional expectation of the future, @(s), s > t , given the present and past, depends only on the present, at, and not on the past. For a random distribution f ---f @(f),f E g ( R n ) we may introduce an analogous property. Let O c R n denote an open set, and 80 its boundary; let Y ( 0 )denote the a-algebra generated by {@(f), fGB(0)); if K c R n , we define Y ( K ) to be 9 ' ( K ) = n o 2 K 9 ( 0 ) We . say that @ is a Markov field if, for every open 0 and

gEq

~ 4 o- was),

(@ (g)I Y ( 0 a@)>= (@ ( g )I 9'(80)) holds; in words, the expectations of the fields outside 8, given their values on 0 and 80, depend only on their values on 80. This idea is due to Nelson (1973a, b). If the map f+ @(f) is continuous when 9 ( R n ) is furnished with the Sobolev norm l[flI-1= [JJ(X) ( - A + mZ)-lf(x) dnxI1'2 then it can be extended to a process over Z - 1 . I n that case the test function space 2 - 1 actually contains some distributions localized on sets of dimension n - 1, eg f(x) @ S(xn- t ) is localized on the plane xn= t. I n that case, the Markov property can be given a direct meaning in terms of random variables localized on such boundaries. T h e first result of Euclidean field theory is that for the free scalar field of mass m 2 0 (if n > 2) or m.> 0 (if n = 2), the Schwinger functions are the moments of a real random distribution Q, over &-1(Rn); @ satisfies the Markov property, and is covariant under Euclidean group En. This means that when we regard @(f) as a self-adjoint operator on L2(Q p), there is a unitary representation U ( R ,U ) of En on L2(Q,p), such that U(& a) @ (f)U - W , 4 = Q,(fR,a> wherefR,@(x)=f(R-l(x - a)).

6.2. The Nelson axioms Nelson took the essential properties of the free field as axioms also for interacting fields. Of course, we must omit the property of being Gaussian! We give the axioms in the form used by Simon (1973). (1) There is a probability space (a, 9, p) and a random distribution @ over Z-I(P),such that 1 is cyclic for the algebra of bounded measurable functions of @(f),f E Z - 1 in L2(R, p), and f --+ @(f) is continuous in measure. (2) T h e field is covariant under a continuous representation U of the full Euclidean group En (including reflections). (3) T h e field is Markov. I n order to state the next two axioms, we need a result which follows from (l), (2) and (3). Let Eo be the projection in L2(R) on to the subspace S spanned by all bounded measurable functions of all @ (f),where f is zero outside the plane xn= 0. Then Pt=EoU(l, a ) Eo\@,where n--l

a=(O

*

. .o, 0, t )

is a strongly continuous contraction semigroup, with Pt = P-6. This means that Pt=exp ( - ItlH) for some positive self-adjoint H on b. For a proof, see Nelson

Outline of axiomatic relativistic quantum field theory

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(1973a). Let f E SP(Rn-1). Then f ( x 1 , . . ., xn-1) @ is in .%“-1(Rn), and @o(f) may be defined as the random variable @ (f @ 6). Let Q (f)denote the form domain of @ o ( f ) ; that is, Q(f)={uEL2(R): fi@,o(f)uELl}. We can now state Nelson’s fourth axiom. (4) There exists a Schwartz norm 11 /I and an integer 1, such that for allf E S”(Rn-1): (a) Q ( f )13 D(Hl/2),the domain of Ht/2

*

(b) @o(f 1< ( H + Q)zlfl. ( 5 ) Ut acts ergodically on Q ; that is, the only sets in S” invariant under Ut for all t are of measure 0 or 1. These axioms hold for the free field (Nelson 1973b); moreover, the moments of any theory satisfying (1)-(5) exist and are the Schwinger functions of a unique Wightman theory (Nelson 1973a). These axioms are designed for theories in which the fields at sharp time make sense. I n perturbation theory, this occurs when the wavefunction renormalization constant 2-1 is finite and so Z f O . If Z=O it is probable that the more general Euclidean field, over B(Rn), should be used. See Nelson (1973a) for the axioms in this case; he proves that these axioms also imply the existence of a Wightman theory. Axiom (3) is the Euclidean analogue of local commutativity, and also leads to the semigroup property for Pt. Of course, axiom (2) leads to Lorentz covariance, once we have analytically continued to the Minkowski region. Axioms (1) and (4) lead to the distribution nature of the Wightman functions. Axiom (S), which can be separated off, leads to the uniqueness of the vacuum. T h e spectrum condition has no direct Euclidean statement; the energy is positive by construction, once we have the semigroup Pt. T h e full spectrum condition then follows from Lorentz invariance. T h e free scalar field satisfies the Nelson axioms, with covariance function g(@(f)@(g))=J f(x) ( - A+m2)-1g(x) dnx. If we have a Nelson field, we may obtain another by altering the measure. Nelson defines a multiplicative functional to be a function F on R, such that, for every finite partition I , of Rn into disjoint measurable sets, there are functions F,, functions of the fields in I, such that F=II,F,, and F,>O, & ( F ) = l . If p is the measure of a Markov field, then U = F p satisfies all the Nelson axioms except covariance. This happens in the P(4)z theory, where

for some volume cut-off V. Interacting fields are obtained by taking the limit Y + CO, in complete analogy with the thermodynamic limit of classical statistical mechanics (Symanzik 1969, Nelson 1973a,b, Guerra et aZl975, Newman 1973). One might ask what extra conditions beyond the Wightman axioms a field theory needs to satisfy in order for its Schwinger functions to define a Nelson theory. T h e answer is not known, but if there is a mass gap in the spectrum, Simon (1973) has given a set of properties which suffices. Apart from the Z# 0 condition, which ensures that fields at sharp time exist, Simon requires that exp (- t H ) should be a positivitypreserving semigroup. I n the theory of Markov chains, this condition ensures that an underlying process exists. Simon’s axioms are, for n = 2: (1) The GArding-Wightman axioms hold for q5(x, t ) .

R F Streater

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(2) For each f E ~ ' ( R )there is a self-adjoint operator @o(f), an integer I and a Schwartz norm /I 11, such that Nelson's (4)holds. Also, (U,%(f) v ) , U E &(f) is defined uniquely by the U,v E a(HL/2),and O(h)=J_", exp(iHt)@o(ht)exp(-iHt) dt, hgfa(R2). (3) The W*-algebra A! generated by @ o ( f ) is Abelian, with the vacuum YOas cyclic vector. (4) For any two positive elements F, G of A!, ( F Y o , exp ( - t H ) GYo) 2 0 for all t 2 0. ( 5 ) H has a mass gap and ZZO. Simon (1973) proves that these are equivalent to a slight modification of Nelson's axioms.

6.3. The Osterwalder-Schrader axioms Osterwalder and Schrader (1973a, c) asked: what are the necessary and sufficient conditions on the Schwinger functions for them to be the analytic continuations of the Wightman functions of a field theory? This leads to a more general theory than the Nelson theory. Simon (1973) has argued that, at least in one-dimensional spacetime, it is actually possible to find an OS theory not satisfying the Nelson axioms. T h e reason is that there is no reason, in a general Wightman theory not defined by a Lagrangian, why the Schwinger functions should satisfy the positivity condition of moments of random fields. Instead, they satisfy a more general positivity condition, discovered by Osterwalder and Schrader (1973a). A slight technical error in lemma 8.8 of Osterwalder and Schrader (1973a) is corrected in (1973c, 1975). T h e main properties of the Schwinger functions for real scalar fields, with their connections with the properties of the Wightman functions, are given by the following table : Euclidean region

i

A distribution property Euclidean covariance A positivity property

Minkowski region Distribution property Covariance Positivity Spectrum

{ ] + symmetry+-+ { ) + locality { )+cluster +-+{ )+cluster To be more precise, let us define

and to each f E Y(Wy),

and introduce a set of norms on Y ( R y ) by

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Also, introduce further subspaces of 9'(R4n), namely Y(R4