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J'¢(x)· ( (j¢(x) st. + ata (j
hence, the associated EulerLagrange field equations (3.1) are not deterministic because a free constant of integration must appear in a solution ¢(x; t) for prescribed initial data [¢(x; 0), ¢(x; 0)]. Let us consider the Hamilton formulation correspondent of the action principle (3.14). In place of the action functional (3.10) associated with the Lagrange formulation, we must employ the canonical field variable action functional
S = S[¢(x; t), n(x; t)] == ("
(J
(3.26) obtained by using (3.16) to eliminate the Lagrangian from (3.10) and defined on the domain of real ntuple functions ¢(x; t) of continuity class C 1 with respect to t that reduce to the prescribed boundary values ¢(x; t') = ¢'(x) and ¢(x; t") = ¢"(x), fixed ntuple functions of x, and n(x; t) of continuity class CO with respect to t (unconstrained at t = t' and t = t"). The 2n firstorder Hamilton canonical field equations (3.18) are equivalent to the canonical field variable action principle: (jSj(j¢(x; t)
= 0,
(jSj(jn(x; t)
= 0,
(3.27)
where the functional derivatives are taken according to the general definition in Appendix A with:F = {a(x; t) : a(x, t) E C 1 with respect to t for t' ~ t ~ t", er(x; t') = c(x: t") = O} in the case of the ¢(x; t) differentiation and with :F = {a(x; t) : a(x; t) E CO with respect to t for t' ~ t ~ t"} in the case of the n(x; t) differentiation. With these definitions of functional differentiation, the canonical field variable action principle (3.27) states that S has an extremum for a physical pair of ntuple functions ¢(x; t), n(x; t). Because ¢(x; t) and n(x; t) play asymmetric roles in (3.26) and (3.27), the canonical field variable action principle does not provide a logical starting point for a discussion of canonical transformations.
60
3
Classical Field Theory
3. Neologized Poisson Formulation
A useful feature of the Hamilton canonical field equations (3.18) is that they allow the time rate of change of any observable J = J[¢, n], defined here as an infinitely differentiable real ltuple functional in the canonical field variables, to be expressed directly in terms of ¢ and n as di/dt
== J = [f, H]
(3.28)
with the fieldtheoretic Poisson bracket of two observables introduced as [f, g] ==
I(
6f 6g 6g 6f ) 6¢(x) . 6n(x)  6¢(x) . 6n(x) dx.
(3.29)
For example, consider the observable f =
II (t¢(x) .
a(x, y) . ¢(y)
+ tn(x) . y(x,
I
+ (~(z)·
+ ¢(x) . P(x, y)
. n(y)
y) . n(y)) dx dy
¢(z)
+ 1](Z)'
n(z)) dz
+ 0),
(3.30)
where «(x, y), P(x, y), and y(x, y) denote real n 2dyad distributions (generalized functions) in x, y with a(x, y) and y(x, y) symmetrical under a simultaneous interchange of their (continuous) arguments and (suppressed) indices, ~(z) and 1](z) denote real ntuple distributions in z, and 0) denotes a real ltuple constant; the Poisson bracket of the observable (3.30) and the Hamiltonian observable l1 (3.19) is the observable [f, H]
=
II
(¢(x) . «(x, y) . n(y)
+ n(x) . P(x,
y) . n(y)
+ [¢(x) . P(x, y) + n(x) . y(x, y)J
. [V
2 /, ( ) _ 'I' Y
OU(¢(y))]) d d o¢(y) X Y
+ I (~(Z) . n(z) + 1](z) .
[V 2¢ (Z)  O~~~;)]) dz.
11 For the Hamiltonian (3.19) to be an infinitely differentiable functional in the canonical field variables, u must be an infinitely differentiable function of 4>.
3. Neologized Poisson Formulation
61
In particular, the observable constants of the motion are quantities which have a zero Poisson bracket with the Hamiltonian. The set of all observables is closed under ordinary addition and Poisson bracket combination, in the sense that both the sum and the Poisson bracket of any two observables are observables. Furthermore, the Poisson bracket combination law (3.29), associating a new observable functional of ¢ and 71: with an ordered pair of observable functionals of ¢ and 71:, has all the properties required of a combination law for a Lie algebra, properties (1.28)~(1.30). Hence, the set of all observable functionals subject to ordinary addition and Poisson bracket combination (3.29) constitutes a Lie algebra, and any subset of observable functionals that is closed with respect to ordinary addition and Poisson bracket combination consitutes a Lie subalgebra of the Lie algebra of all observable functionals. The subset of observables having the generic form (3.30) constitutes an infinitedimensional Lie subalgebra because an infinite number of real constants are required to specify «(x, y), P(x, y), and so forth. In subsequent discussion, we refer to the set of all observable functionals having the generic form (3.30) as the infinitedimensional Lie algebra Ills' the fieldtheoretic analog of the (2n 2 + 3n + I)dimensional Lie algebra of classical mechanics denoted by d s • The main structure of Ills is evident in the subset with ~(z) = I1(Z) == 0, W = 0; this subset of observable functionals, which we denote by Ill.', is closed with respect to ordinary addition and Poisson bracket combination, and hence constitutes an infinitedimensional Lie subalgebra of Ills; the subset with «(x, y) = P(x, y) = )I(x, y) == 0 also constitutes an infinitedimensional Lie subalgebra of Ills' which we denote by Ill.". Because the Poisson bracket of an observable in Ill.' and an observable in Ill." is an observable in Ill.", Ills is the semidirect sum of Ill.' and Ill.",
To the Lie algebra .91 of a subset of observable functionals is associated a conjugate Lie algebra .91* consisting of all observable functionals which have a zero Poisson bracket with all elements of d. As in classical mechanics, the observable constants of the motion constitute the Lie algebra d H *, conjugate to the Idimensional Lie algebra consisting of elements proportional to the Hamiltonian, .91H == {(real constant parameter) x H}. However, in contrast to classical mechanics where (2n  1)
62
3
Classical Field Theory
functionally independent constants of the motion are generally associated with the Hamiltonian as a complete set of solutions to the firstorder linear homogeneous partial differential equations [Ii' H] = 0, in classical field theory there is no established theorem which gives the number of functionally independent constants of the motion, independent of the detailed form of the Hamiltonian. For a generic linear field theory, the firstorder linear homogeneous functional differential equations [Ii' H] = 0 admit an infinite number of functionally independent observable functions J, as constants of the motion, exemplified by1 2 (i= 0, ±l, ±2, ...) in the case of the simple linear field theory with H =
J(tn. n + t JlVi ¢.Vi ¢) dx.
(3.31) (3.32)
Recent work 1 3 supports the conjecture that a generic nonlinear field theory also admits an infinite number of functionally independent observables as constants of the motion.l" Associated with the infinitedimensional Lie algebra .s1H * [which contains infinitely differentiable functions of (functionally independent) observable constants of the motion], we hav the Lie algebra or;{ H* consisting of a maximal set, closed under Poisson bracket combination, of linearly independent constants of the motion (exclusive of H and the trivial generic observables that are constants of the motion for any H, as exemplified by S ¢ . Vi n dx for i = 1,2,3). For a generic linear field theory, d H * is an infinitedimensional Lie algebra, as exemplified for the simple linear theory with the Hamiltonian (3.32) by the set (3.31) (the element i = 0 being excluded), for which [Ii ,fj] = 0. It is likely that d H * is infinitedimensional for a generic nonlinear field theory. Major importance would be attached to a general proof of the infinite dimensionality of d H* and to a general algorithm for constructing its elernents.!" '2 Negative powers of the negative Laplacian are defined by inserting Dfunctions and using (_\72)' D(x) = (27T)3J(k ·k)'e'k.x dk=(47T)'(X,2+ X22+ X32)'/2. 13 P. D. Lax, Comm. Pure Appl. Math. 21,467 (1968). 14 A general algorithm for the construction of such observable functionals would serve to elucidate the nature of singularityfree global solutions to nonlinear classical field equations; what is needed is a general integration theory for firstorder linear homogeneous functional differential equations of the form [[;, H] = 0 with H prescribed.
3. Neologized Poisson Formulation
63
A transformation of all observables J
= J[r/J, nJ ~ I, = Jr[r/J, nJ
9
= g[r/J, nJ ~ e, = g.['/1, nJ
(3.33)
such that fo =1,
(3.34)
go =g,
is said to be a canonical transformation generated by the observable w = w[r/J, nJ if the transformed observables satisfy the firstorder total differential equations
dfjd: = [fr' wJ dgJdr = [gr' wJ
(3.35)
A transformed observable is given explicitly in terms of the original observable by the Maclaurin series derived from (3.34) and (3.35),
I, = J
r2
+ r[f, wJ + 2! [[f, wJ, wJ + ...
6w 6w ( f (6n(x) . 6r/J(x)  6r/J(x) . 6n(x») dx f. 6
== ,exp r
6.
)
(3.36)
As readily verified by direct computation, we have (3.37) for r ~ 0, so canonical transformations in classical field theory preserve the Lie algebraic structure for a subset of observable functionals that is closed with respect to addition and Poisson bracket combination: Lie algebras are mapped into isomorphic Lie algebras by. canonical transformations. By putting h = j and 9 = H in (3.37), it follows that the time rate of change of any canonically transformed observable is prescribed by a dynamical equation of the form (3.28), (3.38) By specializing (3.37) to the cases for which h = c, a quantity independent of r/J and zr, we find that [f,gJ = c
=>
[fpgrJ = c.
(3.39)
64
3 Classical Field Theory
In particular, for the components of the field amplitude and momentum density ntuples, we have the basic Poisson bracket relations [;(.~),
ly)] = [n;(x), n/y)] = 0,
[ i(X), nly)] = 8ij f>(x  y),
(3.40)
so that (3.39) yields
[tJx), t/y)J = [nt/x), ntly)J = 0,
(3.41)
[ri(x), ntj(y)] = 8ij8(x  y).
(3.42)
If two observablesj"and g are regarded as functionals of the canonically transformed components of the field amplitude and momentum density ntuples, their Poisson bracket computed with respect to t and n t equals their Poisson bracket computed with respect to and n, 8f 8g 8g 8 f( 8rCx) • 8nrCx)  8rCx) . 8nrCx) x 8f 8g 8g 8f ) = f (8(x) . 8n(x)  8(x) . 8n(x) dx, f)d
(3.43)
the proof being analogous to the proof for (1.45) in classical mechanics. Hence, the Poisson bracket in an equation can be computed with respect to 0 n t for any value of 1" ~ O. By doing this in (3.38) and specializing to the cases for which j', equals each of the 2n components of t and nt' we see that canonical transformations also preserve the form of the Hamilton canonical field equations, (3.44)
Under canonical transformations generated by an mdimensional Lie algebra d composed of all real linear combinations of m linearly independent observable functionals W l, ..• , W m ' a generic observable f = fe, n] is transformed to
!a = (exp IX • G)f, in which the generators for canonical transformations
(3.45)
3. Neoloqized Poisson Formulation G i
=
65
j'(bn(x) bW b . b¢(x) i
bW i b) b¢(x) 'bn(x) dx
(3.46)
are m firstorder functional differential operators that satisfy commutation relations of the form GiGj 
GjG i
=
m
I
k~l
Cijl,G k·
(3.47)
All properties derived for the transformed observables (3.36) are featured by the transformed observables (3.45). As an illustration of canonical transformations generated by a finitedimensional Lie algebra, consider the 3dimensional subalgebra of Ills' d =
+!Xz Wz
{!X1W 1
Wz == W3
t
+!X3
J
W3:
W1
(¢(x)· ¢(x)
== t
J
(n(x)' n(x)  ¢(x)· ¢(x)) dx,
+ n(x) . n(x)) dx,
J
== 1 ¢(x) . n(x) dX}'
(3.48)
F or this d the generators (3.46) are
J = 1f G = 1f
G1
=1
(¢(x) . b/(jn(x) + n(x) . (j/(j¢(x)) dx
Gz
(¢(x) . (j/(jn(x)  n(x) . (j/b¢(x)) dx
3
(n(x) . (j/(jn(x)  ¢(x) . (j/(j¢(x)) dx
(3.49)
and satisfy (3.47) with C1Z3 = CZ 31 = C31Z = 1; thus, (3.45) with (3.49) gives the effect of SL(2, R) canonical transformations on the space of observable functionals. It is evident that the dynamical evolution of all observables, as prescribed by (3.28), can be viewed as a canonical transformation parametrized by 1: = t withft = fr[¢(x; 0), n(x; 0)] ==f[¢(x; r), n(x; t)J and generated by the Hamiltonian with W = H[¢(x; 0), n(x; O)J. This special canonical transformation takes ¢(x; 0) ~ ¢,(x; 0) == ¢(x; t) and n(x; 0) ~ n,(x; 0) == n(x; r), with the Poisson bracket in (3.35) computed
3
66
Classical Field Theory
most conveniently with respect to the "original" canonical field variables (¢(x; 0), n(x; 0)). The dynamical evolution of all observables is expressed in the canonical transformation form (3.36) as f
=
f[¢(x; t), n(x; t)]
= (ex p t
x f[¢(x), n(x)J!(X);(X; 0)
I (b~~)
.
b¢~X)  b~%) . bn~X)) dX) (3.50)
n(x);n(x; 0)
Since the Lie algebra d H * is infinitedimensional for a generic linear field theory and conjectured to be infinitedimensional for a generic nonlinear field theory, the notion of canonical transformations generated by an mdimensional Lie algebra must be extended to the case of an infinitedimensional Lie algebra in order to apply to an d H * in classical field theory. We have GiH = for the generators associated with d H *, and thus the Hamiltonian is invariant for canonical transformations generated by d H* for o: such that the sum in (3.45), o: . G == If=l rxi G i , converges. The Lie algebra d H * is referred to as the symmetry algebra for the classical field theory because the dynamics of observables is unchanged by canonical transformations generated by d H * with H" = H, and the associated Lie group t§H*' with its linear representation on the space of observable functionals t§H* = {(exp rx . G)}, is called the
°
symmetry group.
An especially simple kind of dynamics arises if H is an element of the Lie algebra Ills' that is, if H has the form of the righthand side of (3.30), as it does generally for any linear field theory with the EulerLagrange field equations (3.1) linear and homogeneous in ¢ = ¢(x, t). [For example, the Hamiltonian (3.32) is the element of Ills with rx(x, y) =  V2 b(x  y)l, f3(x, y) == 0, y(x, y) = b(x  y)l, ~(z) == 0, I'/(z) == 0, w = O.J Because the components of ¢(x) and n(x) are elements of Ills" (with «(x, y) = f3(x, y) = y(x, y) == 0, W = 0, and one component of either ~(z) or I'/(z) equal to b(x  z), all other components of ~(z) and I'/(z) == 0) and their Poisson brackets with elements of \lIs are elements of Ills", it follows from (3.45) that elements of Ills generate canonical transformations for which the components of ¢ix) are inhomogeneous linear expressions in the components of ¢(x) and n(x) , and likewise for the components of n,,(x). In particular, the dynamical evolution generated by an H in Ills is characterized by the components of ¢(x; t) being inhomogeneous linear expressions in the components of ¢(x; 0) and
67
3. Neoloqized Poisson Formulation
n(x; 0), and likewise for the components of n(x; t). Moreover, if H is an element of 2Is ' all observables in 2Is at t = 0 remain in 2Is for t > 0, and any observable that is not initially in 2Is does not enter the algebra. Dynamical containment in 2Is is featured by all elements in the Lie algebra if H is in 2Is '
I[ 4>(x; t), n(x; t)] =.j,[4>(x; 0), n(x; 0)]
=
II C!4>(x; 0) . «(x, y; t) . 4>(y; 0) + 4>(x; 0) . f3(x, y; t) . n(y; 0) +!n(x; 0) . y(x, y; t) . n(y; 0)) dx dy + I (~(z; t) . 4>(z; 0) + 1](Z; t) . n(z; 0)) dz + w(t),
(3.51)
because of the Lie algebra addition and Poisson bracket closure properties.
Chapter 4
quantum Field Theory
1. Neologized Feynman Formulation
Again taken as a primitive physical notion, the state of a continuous dynamical system with an infinite number of degrees of freedom is realized at any instant of time by prescribing the complex wave functional 'P = 'P[ 4>] that depends on the field amplitude real ntuple 4> = 4>(x) = (4)l(X), ... , 4>n(x)). The associated positivedefinite quantity 1'P[4>]1 2 is interpreted as the relative probability density for a physical measurement fixing the state at the field amplitude ntuple 4> = 4>(x). This role assigned to a complex wave functional is the basic postulate in quantum field theory. The system dynamics is described by assigning the wave functional a timedependent form 'P = 'P[ 4>; t] (C 1 with respect to t). A physically admissible evolution in time takes the specific linear form of a dynamical principle of superposition
'P[4>"; t"] = IK[4>", if/; t"  t']'P[4>'; t']!0(c/>,)
(4.1)
for all t" ~ t' and initial wave functionals 'P = 'P[4>'; t']; in (4.1), the integration is over all 4>' = 4>'(x), the 4>' space infinitesimal volume element is represented symbolically as !0(4)') = N IT all
68
n
IT d4>/(x) x i= 1
(4.2)
69
1. Neologized Feynman Formulation
with the components of x = (Xl' Xl' X 3) taking on all real values in the infinite product, and the normalization constant N is independent of ¢'. Viewed physically in terms of a sufficiently accurate finitedimensional approximation to ¢'space, the infinitesimal volume element (4.2) is a displacementinvariant Haar measure, like D(fJ) in the propagation kernel representation (2.13), but with the integration variable ¢' in f0(¢') parametrized by the three real coordinates in x. l To guarantee the general consistency of (4.1) for all 'P[¢'; t'], the propagation kernel K[¢", ¢'; t"  t'] must satisfy a semigroup composition law K[¢", ¢'; t"  t'] = IK[¢", ¢; t"  t]K[¢, ¢'; t  t']f0(¢)
for t'
~
t
~
(4.3)
t", as well as the initial value condition
lim K[¢", ¢'; t] ' ....0
= b[¢"  ¢'] == N l
n i=lnb(¢i(X)), n
all x
(4.4)
where the ofunctional on the righthand side of (4.4) is such that b[¢"  ¢'] = 0
I b[¢]f0(¢) = 1.
for
¢" # ¢',
(4.5)
Repeated application of (4.3) shows that the propagation kernel for finite values of (t".can be expressed as an iteration of the propagation kernel for infinitesimal values of (t"  t'),
n
K[¢", ¢'; t"  t']
= I (JilK[¢(X; t' + MAt), ¢(x; t' + (M  1) At); At])
n
Nl X
M=l
f0(¢(x; t'
+ MAt)),
(4.6)
1 Precise criteria for existence and methods of construction of displacementinvariant measure for integration over fields have been given by K. O. Friedrichs and H. Shapiro, "Seminar on Integration of Functionals." Courant Institute of Mathematical Sciences, New York Univ., 1957; K. O. Friedrichs and H. 'N. Shapiro, Proc. Nat. Acad. Sci. 43, 336 (1957); N. N. Sudakow, Dokl. Akad. Nauk SSSR 127, 524 (1959); B. S. Mitjagin, Uspekhi Math. Nauk 26, 191 (1961). The general formulations of displacementinvariant measure on infinitedimensional topological function spaces have been reviewed by N. Dunford and J. Schwartz, "Linear Operators, I," p. 402 if. Wiley (Interscience), New York, 1958; E. J. McShane, Bull. Amer. Math. Soc. 69, 597 (1963); J. M. Gelfand and N. J. Wilenkin, "Verallgerneinerte Funktionen, IV," Hochschubiicher Math. Bd. 50, (1964).
4
70
Quantum Field Theory
where the notation is analogous to that employed in writing (2.6) with the time variable in ¢(x; t) serving as a virtually continuous parameter for the intermediate integrations between t' and t" as.M == [(t"  t')/N] + 0, the fixed ntuple field amplitude arguments in the propagation kernel being prescribed by ¢(x; t') == ¢'(x)
¢(x; t") == ¢"(x).
and
(4.7)
In analogy to Feynman's prescription (2.7), the infinitesimal propagation kernel in (4.6) is postulated to have the form K[¢(x; t
+ At), ¢(x; t); M] = AI exp(iL[¢, 4>] At/fl)
(4.8)
in which
¢ = t[¢(x; t + At)
4> = (At)I[¢(X; t + At) ¢(x; t)],
+ ¢(x; t)],
(4.9)
and the normalization constant A integral representation' K[¢", ¢'; t"  t']
= A(At).
=
The Feynman functional
f (exp is/h)'XJ(¢) '6
(4.10)
follows in the limit N ~ 00, At + 0 of (4.6) with (4.8), where the action functional S = S[¢(x; t)J defined by (3.10) appears in the form of a Riemann sum, and 'XJ(¢) in (4.10) denotes an infinitesimal volume element for the integration over the class of real ntuple functions ((j = {¢ = ¢(x; t) for t' ~ t ~ t": ¢(x; 1') = ¢'(x), ¢(x; t") = ¢"(x)}. In symbolic notation, we have 'XJ(¢)
=%
n
Il allx Il i=l II d¢i(X; t)
t'
(4.11)
with the generic normalization factor
% = %(t"  t') == lim([A(At)r
(4.12)
independent of ¢(x; t). By introducing
$ = $(x; t) == ¢(x; t) 
[«t  1')¢"(x)
+ (t"
 t)¢'(x»/(t"  t')],
(4.13) 2
R. P. Feynman, Phys. Rev. 80,440 (1950); 84, 108 (1951).
1. Neoloqized Feynman Formulation
71
as have 1)(
J(exp '§
iSjh)1)(¢)
(4.14)
with the integration over the space of real ntuple functions C.!J
= {¢ = ¢(x;
t)
for
t' ~ t ~ til: ¢(x; t')
= 0 = ¢(x; til)}
Displacementinvariant for any fixed a in C.!J, 1)(¢ +a) = 1)(¢),
(4.15)
the infinitesimal volume element in (4.14) can be viewed physically as a leftinvariant and rightinvariant Haar measure for the integration over a large but finitedimensional Abelian topological subgroup of C.!J, like D(fj) in (2.13).1 Hence, the functional integral representation of the propagation kernel (4.14) is defined uniquely to within a normalization factor independent of
=
II G
(4.16)
72
4
Quantum Field Theory
modulo an additive real constant and a transformation of the form (3.7) with X a general quadratic functional of ¢(x); in (4.16), p(x, y), (I(x, y), and ,(x, y) denote real n 2dyad distributions (generalized functions) in x, y with p(x, y) and ,(x, y) symmetrical, (I(x, y) antisymmetrical, under a simultaneous interchange of their (continuous) arguments and (suppressed) indices, and ~(z) denotes a real ntuple distribution in z; it should be noted that a Lagrangian of the form (4.16) generally produces nonlocal, inhomogeneous, linear EulerLagrange field equations (3.1). If S = S[¢(x; t)J is a general quadratic, then we have S[¢(x; t)J
=
S[¢c(x; t)J
+ S[¢(x; OJ
(4.17)
where ¢(x; t)
== ¢(x; t)  ¢c(x; t)
with
¢c(x; t)
the classical" history" satisfying the action principle (3.14), f>S/f>¢(x;
(4.18)
t)14>=4>c = 0,
or equivalently, with the Lagrangian given by (4.16),
f(p(x, y) . ¢c(y; t) + (I(x, y) . cPcCy; t)  ,(x, y). ¢c(y; t) dy
+ ~(x) =
0,
(4.19)
subject to the boundary conditions ¢c(x; t') = ¢'(x) and ¢c(x; t") = "(x). With ¢(x; t) an element of rc, it follows that the variable ntuple function ¢(x; t) is an element of rs, that is, ¢(x; t') = ¢(x; i") = O. Hence, by setting the fixed G in (4.15) equal to G(x; t) = [«t  t')¢"(x)
+ (t"
 t)¢'(x»/(t"  1')J  ¢c(x; t),
(4.20)
we find that the displacementinvariant measure 'l)(¢) = 'l)(¢). Therefore, (4.14) produces the result K[¢", ¢'; t"  t']
= Y(exp
is[c(x;
OJ/h)
(4.21)
with the factor
ff§ (exp is[¢(x; t)]/h)'l)(¢) independent of ¢', ¢" and absorbed into (4.12) to give the normalization factor vV in (4.21). As a prime example, the Lagrangian
1. Neologized Feynman Formulation L
=
73
f (! ¢. ¢  t(V¢)' (V¢) 1 itl m/¢/) dx,
(4.22)
which specializes (4.16) to a local linear theory of uncoupled field amplitude components, leads to a propagation kernel of the form (4.21) with S[¢c(x; t)]
= fitt(!¢/,(x)( _V 2 + m/)l/2(tan[( _V 2 + m/)l/2(t"  t,)])l¢/,(X)
+ !¢;'(x)(  ¢/,(x)( 
+ m/)1/2(tan[(  V 2 + m/)1/2(t"  t,)])l¢;,(X) V 2 + m/)l/2(sin[(  V 2 + m/)l/2(t"  t')])l¢/(x))dx. '1 2
(4.23)
The normalization factor jj7 = jj7(t"  t') associated with (4.22) and (4.23) cannot be expressed as a finite function of (t"  1'), as is generally the case for such a normalization factor in the Feynman formulation of quantum field theory. However, since the state functional at any instant of time is physically significant modulo a complex constant independent of ¢, this feature of the formulation does not impair practical calculations. The standard procedure is to require normalization of the wave functional at the time of interest, J1 'P[¢; t] 1 2£&(¢) = 1, concomitant with the definition of expectation values (defined by (4.54) below) and other quantities amenable to experimental measurement at time t. As in quantum mechanics with the Feynman functional integral representation for the propagation kernel (2.13), we can extract useful dynamical information from (4.14) for quantum fields by asymptotic techniques and other suitable analytical procedures if S = S[¢(x; t)] is not a general quadratic expression that can be evaluated explicitly. For example, if the action functional takes the form S = So + Sint, where So = So[¢(x; t)] is simple [that is, a general quadratic functional in ¢(x; t)] and Sint = Sint[¢(X; t)] is relatively small compared to So for typical ¢(x; t), then the propagation kernel can be evaluated approximately by working out the leading terms in the expansion of (4.14), K[¢", ¢'; t"  t']
= k~o(k!)l(ilh)k ff9(Sint)k(eX p
iSolh)'j)(¢). (4.24)
4
74
Quantum Field Theory
The latter perturbation expansion, directly related to the conventional iteration solution for the wave functional in the Schrodinger formulation [as shown by formula (4.52)], provides an immediate conceptual basis for a large fraction of the calculations performed in quantum field theory. In particular, Feynman's important work on quantum electrodynamics" stemmed from considerations based on formula (4.24).4 Applications of the expansion (4.24) lead one to study Feynman operators in field theory, functional integrals of the form
f A(exp is/h),!)(¢) == (1)''; t" IA[1'(x; t)JI1"; t')s,
(4.25)
where S = S[1'(x; t)J and A= A[1'(x; t)J are generic realvalued functionals of 1'(x; t). The "functional integration by parts" lemma derived in Appendix 0 applies to such functional integrals in the theory with displacementinvariant Haar measure, and it allows one to express exact equations involving Feynman operators without having to do any explicit functional integration. For example, we obtain the operator
EulerLagrange field equations
f
JS
~
J1'(x; t) (exp is/h),!)(1')
==
(1'' ; r Ib1';:; t)' 1"; t l
=0
for all
x
and
t'
< t < t",
(4.26)
and the Feynman operator equations
f [b1'i~S.)t 1'/Y; s)  ih bij b(x  b(t  s)1(exp iSjh),!)(¢) == (1'' ; r I[b1'i~~; t) l' j(Y; s)  ih bij b(x  y) b( t  s)111"; t') s
y)
X,
=0
for all
x, Y
and
t' < t,
s < t",
(4.27)
3 R. P. Feynman, "Quantum Electrodynamics." Benjamin, New York, 1961, and papers reprinted therein. 4 Interesting recent work based on the Feynman perturbation expansion (4.24) has been published by E. S. Fradkin, Nucl. Phys. 49, 624 (1963); Acta Phys. Hung. 19, 175 (1965); C. S. Lam, Nuono Cimento 50A, 504 (1967).
1. Neologized Feynman Formulation
75
As in quantum mechanics, exact equations such as (4.26) and (4.27) facilitate the determination of Feynman operators and the computation of terms in (4.24) without explicit functional integration. Thus, for example, if the action is the time integral of (4.22), Eqs. (4.27) yield
(  :t22
+ V2

m/)(¢"; t" l¢i(X; t)¢iy; s)1 ¢'; t'h
= iii
(jij (j(x  y) (j(t  s)(¢"; t"
111
¢'; t')s,
(4.28)
in which the propagation kernel (4.21) with (4.23) appears as the Feynman operator (¢"; t" III ¢'; t')s on the righthand side; the causal Green's function of perturbation theory is obtained by integrating the vacuum expectation value of (4.28) subject to appropriate boundary conditions," obviating calculation of the causal Green's function by explicit functional integration. 6 It is also noteworthy that the connection between Feynman operators and the more conventional differential operators of quantum field theory [expressed by Eq. (4.53)] is readily established as the analog of the relationship derived in Appendix F for the operators in quantum mechanics; hence, Feynman operator equations, such as (4.26), (4.27), and (4.28), imply corresponding Heisenberg operator equations for the field variables. The canonical variable version of the "sumoverhistories" functional integral representation for the propagation kernel follows in quantum field theory as K[¢", ¢'; t"  t']
=
f
(exp iSjli)'1)(¢, n),
(4.29)
cex~
where the canonical field variable action functional
S = S[¢(x; t), n(x; t)] defined by (3.26) appears in the form of a Riemann sum, and '1)(¢, n) in (4.29) denotes an infinitesimal volume element for integration over the class of real ntuple functions f(i
= {¢ = ¢(x; t) for
t'
~
t
~
t": ¢(x; t')
= ¢'(x), ¢(x; t") = ¢"(x)}
and the space of real ntuple functions :F 5 6
= {n = n(x;
t)
for
C. S. Lam, Nue!. Phys. 87, 549 (1967).
t'
~
t
~
r"},
P. T. Matthews and A. Salam, Nuovo Cimento 2, 120 (1955).
76
4
Quantum Field Theory
the field momentum density ntuple being unconstrained at t = t' and t = t", In symbolic notation, we have
~(¢, n) = Cn,,, Dx iU d¢i(x; t)) CJ1"NIl i[ldrei(X; t»)
(4.30)
with a generic normalization constant N appearing in the infinite product of momentum density infinitesimal volume elements; thus, the measure in (4.29) is displacementinvariant for any fixed (J E rg = {¢ = ¢(x; t) for t':;:; t s; t" : ¢(x; t') = 0 = ¢(x; t"n and WE fF, ~(¢
+
(J,
t:
+ w) =
~(¢,
re).
(4.31)
It follows from (4.31) that the Feynman functional integral representation (4.29) yields the representation (4.14) for field theories described by a Hamiltonian of the form
H=
J(ire'
t:
+a
. re) dx
+
V
(4.32)
where a = a(¢, V¢, ...) is a real ntuple function of ¢ and spatial derivatives of ¢, and V = V[¢] is a real ltuple functional of ¢ (that is, bV/bn(x) == 0). More generally, for a Hamiltonian which is not of the form (4.32), the dynamical laws (4.14) and (4.29) are inequivalent. Thus, for the Hamiltonian
H
=
f(rel V2¢ 2 
re2 V2¢ l ) dx
(4.33)
straightforward evaluation of (4.29) yields the propagation kernel
== (r¢")2 == (r¢")l
K[¢", ¢'; t"  t'] = b[r¢"  ¢'], (cos[(1"  t') V2] )¢ l "  (sin[(1"  t') V2] )¢ / (4.34) 2 2 (sin[(t"  t') V ] )¢ l " + (cos[(t"  t') V ] )¢ z",
while (4.14) is illdefined with the Lagrangian obtained by putting (4.33) into (3.20) identically zero." Both Feynman functional integral 7 If, however, we disregard the 7T" iT 2 canonical field equations derived from (4.33), the ~" ~2 equations follow from the (essentially unique but canonically unrelated) Lagrangian
L
=
f
t (,p'~2  ~,,p2 
(\7,p,)2  (\7,p2)2) dx
for which (4.14) yields a propagation kernel of the form (4.21), welldefined but distinctly unequal to (4.34).
2. Neologized Schriidinqer Formulation
77
representations exist and are unequal for certain canonically related Lagrangians and Hamiltonians. It is necessary to appeal to the observable physics in order to determine whether (4.14) or (4.29) is the appropriate dynamical law for such quantum field theories. A formal quasiclassical approximation for the propagation kernel (4.14) is derived in the same fashion as (2.39) for the propagation kernel (2.13) in quantum mechanics. If the action evaluated at the classical "history" is large in absolute magnitude compared to h, we obtain K[¢", ¢'; til  t '] ~.AI [det(b 2 SJ r
1 2(exp /
is[¢c(x; t)]/h),
(4.35)
where .AI = .AI (til  t') is a normalization factor (generally not expressible as a finite function in field theory), and det(b 2 SJ denotes the infinite product of all eigenvalues AI' of the symmetric matrix kernel det(b
2
SJ == Il AI' 00
1'=1
must exist modulo formal (not necessarily finite) normalization depending on (til  t') for the quasiclassical approximation (4.35) to have applicability in quantum field theory. 2. Neologized Schrodlnger Formulation
An alternative formulation of the dynamical law for a quantum field theory is obtained if (4.8) is used to derive a linear functional differential equation for the wave functional, rather than a functional integral representation for the propagation kernel. By differentiating (4.1) with respect to t" and letting t' + t" in the integral, we find that 8\f'[¢"; t"]/8t"
=
f A[¢", ¢']\f'[¢'; t"]E0(¢'),
(4.36)
in which A[¢", ¢']
== lim
,it~O
8K[/"" /"'. I'lt] 'f' , 'V ,
8(l'lt)
= lim{[Oh)lH  AI dA JK[¢", ¢'; I'lt]}. M~O
d(l'lt)
(4.37)
4
78
Quantum Field Theory
In (4.37), we have employed (4.8) and (4.9) with ¢"(x) in place of ¢(x; t + At) and ¢'(x) in place ¢(x; r), and thus the Hamiltonian (3.16) appears in terms of ¢ == H¢" + ¢') and ¢ =.=(At)I(¢"  ¢'). To resolve the indeterminacy in (4.37) as At > 0 (stemming from the Hamiltonian's dependency on ¢), we note that <5
_ _ (<5[¢"  ¢']) = lim
<5¢"(x)
<5K[rI."
.i»:
'I' , 'I' ,
At]
<5¢"(x)
M>O
= lim {i(At) oL[¢, ¢] K[¢", ¢'; At]} "'1>0
h
o¢"(x)
= hi (lim n(x») <5[¢"  ¢'], M>O
(4.38)
where n(x) is the field momentum density ntuple (3.15), expressed in terms of ¢ = H¢" + ¢') and ¢ = (At)I(¢"  ¢'). It follows from (4.38) that the field momentum density ntuple multiplied by o[¢"  ¢'] is well defined as At > 0 and given by lim n(x») <5[¢"  ¢'] ( M>O
=
ih[<5jo¢"(x)] (<5[¢"  ¢']),
(4.39)
but powers of n;(x) =.= <5Lj<5¢i(X) in the Hamiltonian in (4.37) cannot be replaced simply by corresponding powers of ih <5j<5¢;"(x), as evidenced by the terms (in 1 0 2Ljo¢;(x) 8¢i(X) + i(n At) 1 8 2 Lj8¢i(X)2) in K[¢", ¢'; Atr 1 8 2K[¢", ¢'; At]j8¢;"(x)2. It is these extra pure imaginary terms that counter the normalization term AI dAjd(At) in (4.37) as At > 0, so that subject to a suitable renormalization transformation (2.15), (4.37) yields A[¢", ¢'] = (in) 1 H[¢"(x),  in 8j8¢"(x)] 8[¢"  ¢']
(4.40)
for a certain symmetrical ordering of ¢, t: product terms in H[¢, n]. Thus, the functional differential operator H[¢(x),  in <5j8¢(x)] is Hermitian on a space of sufficiently smooth complexvalued functionals of ¢(x), functionals which are absolutesquareintegrable over all ¢(x) with the displacementinvariant measure (4.2) and a suitable rule for normalization. By putting (4.40) into (4.36), we obtain the Schrodinger equation for the wave functional ih (iN'[¢; t]jot) = H[¢(x), ih 8j8¢(x)]'P[¢; t].
(4.41)
2. Neologized Schriidinqer Formulation
79
Note that the semigroup composition law (4.3) and the initial value condition (4.4) are sufficient to give the propagation kernel in terms of an unspecified quantum Hamiltonian H (namely, the generator associated with the oneparameter dynamical semigroup) as K[¢", ¢'; t  t'] = (exp i(t  t')H/Ii) <5[¢"  ¢'],
(4.42)
the formal dynamical equation for the wave functional as lJI[¢; t] = (exp i(t  t')H/Ii)lJI[¢; t'],
(4.43)
and the abstract Schriidinqer equation as ili(olJl[¢; t]/ot) = HlJI[¢; t],
(4.44)
where H is a functional differential operator in ¢(x) and <5/<5¢(x); the origin and structure of the quantum Hamiltonian H
= H[¢(x), iii <5/<5¢(x)],
(4.45)
the classical Hamiltonian suitably ordered to be Hermitian and with n(x) replaced by  iii <5/<5¢(x), is elucidated by the postulate for the infinitesimal propagation kernel (4.8). The iteration solution for a wave functional satisfying Eq. (4.41) with H = H o + Hint> where H o = H o[¢, n] is simple (that is, quadratic in the canonical field variables) and Hint = H int[¢] is relatively small compared to H 0 for typical values of the canonical field variables, is derived in manifest analogy to the iteration solution for the wave function in quantum mechanics described in Appendix H. With
u, == Ho[¢(x),
iii <5j<5¢(x)],
(4.46)
the dynamical evolution of the interaction wave functional 'l'int[¢; t]
is given by
== (exp iH o t/li)lJI[¢; t]
'l'int[¢; t"]
= U(t", t')lJI int[¢; t'J,
U(t", t') = T(ex p  i (Hin,($(X; t))dt/fz),
(4.47) (4.48) (4.49)
in which the chronological ordering symbol T arranges noncommuting factors to the left with increasing values of t, and the operator field amplitude ntuple is introduced as <j>(x; t) == (exp iHo t/Ii)¢(x)(exp  iHo t/Ii).
(4.50)
4
80
Quantum Field Theory
From (4.48) we obtain the iteration solution in its usual perturbationtheoretic form by expanding the Uimairix (4.49), V(t", t')
= kto(k!)\i/h)kT( 
{'Hinl(<j)(X; t)dtr
(4,51)
The operator V( 00,  (0), which maps 'P inl[ ¢;  00J into 'P inl[¢; + 00J according to (4.48), is referred to as the Smatrix in quantum field theory. Finally, by inverting the definition (4.47) and recalling the propagation kernel equation (4.1), we obtain the formula K[¢", ¢'; t"  t'J
= L (k !)l(i/h)\exp 00
k=O
 iHo"t"/Ii)
x T(  {'Hint(<j)"(X; t» dtf(ex p iHo"t'/h)
x c5[¢"  ¢'J,
(4.52)
where H o" and <j)"(x; t) are defined by (4.46) and (4.50) with ¢(x) replaced by ¢"(x); formula (4.52) is equivalent to (4.24) because of the analog of (F.8) for Feynman operators in field theory,
f A(exp iSo/h)~($) == i«, t" jA[¢(x; t)JI ¢'; t')so ~
= (exp
iHo"t"/h)T(A[«j>"(x; t)J)(exp iHo"t'/h)
x c5[¢"  ¢'].
(4.53)
With the state of the system represented by a wave functional 'P[¢; tJ, the expectation value of an observablef[¢(x), n(x)J is postulated as
(4.54)
for a suitable (Hermitian and experimentally appropriate) ordering of ¢, n product terms, if such terms with noncommuting factors appear in f[¢(x), n(x)]. As in previous functional integrals, ~(¢) in (4.54) denotes the displacementinvariant measure (4.2) for integration over all fields ¢ = ¢(x); the wave functional in (4.54) is assumed to be normalized to unity, S I'P[¢; tW~(¢) = 1. Again it is Eq. (4.39), derived from the Feynman formulation, that elucidates the origin and structure of the functional differential operator associated with an observable in (4.54).
2. Neologized Schrodinqer Formulation
81
Stationary states in quantum field theory are described by wave functiona1s of the form 'P[¢; t]
= (exp iEtjh)V[¢]
(4.55)
that satisfy (4.41), and hence V [¢] is an eigenfunctional solution to the Schrodinger stationary state equation HV[¢]
= H[¢(x), ih <5j<5¢(x)]V[¢] = EV[¢]
(4.56)
associated with the constant energy eigenvalue E. If the quantum Hamiltonian in (4.56) admits a complete orthonormal set of complexvalued eigenfunctionals {VI'[ ¢]} associated with the discrete set of energy eigenvalues {EI'}, 8 we have HVI'[¢]
= EI'VI'[¢]'
I
JV/[¢]VvE¢][email protected](¢) = <51'",
VI'[¢"]V/[¢']
I'
= <5[¢"  ¢T
(4.57)
Then it follows that the propagation kernel in (4.1) has the formal representation K[¢", ¢'; t"  t ']
= I (exp iEit"  t')jh)VI'[¢"]V/[¢']' I'
(4.58)
which satisfies the initial value condition (4.4) because of the completeness relation in (4.57) and is the formal solution to the propagation kernel equation implied by (4.41) and (4.1), (ih(ojiJt")  H[¢"(x), ih <5j<5¢"(x)])K[¢", ¢'; t"  t']
= 0,
(t" > 1').
(4.59) With only one eigenfunctional associated with the vacuum state energy Eo min; {EI'}' (4.58) yields the formulas
=
Eo = VO[¢"JVO*[¢']
 lim (hjs) In K[¢", ¢';  is], s« cc
= lim {(exp Eosjh)K[¢", ¢';  is]}. s~
(4.60)
(4.61)
00
8 Ordinarily, the spectrum of the quantum Hamiltonian in field theory is continuous if x ranges over the unbounded threedimensional Euclidean space R 3 • A discrete spectrum of energy eigenvalues is usually induced if x ranges over a large region "r c R 3 with periodic boundary conditions on the field amplitude ntuple; the summations over JL in Eqs, (4.57) and (4.58) are then interpreted in the symbolic sense (that is, representing integrations appropriately weighted with the number density of energy eigenvalues) in the limit 1/ + R 3 •
4
82
Quantum Field Theory
Similar relations for the E" and U,,[4>] of other stationary states are obtained by successively removing terms from (4.58) and rewriting (4.60) and (4.61) with the subtracted propagation kernels; thus, energy eigenvalues and eigenfunctionals can be extracted by simple limiting procedures from a closedform expression for the propagation kernel, as provided by the Feynman functional integral representation (4.14). For example, from the propagation kernel (4.21) with (4.23), we obtain Eo =  lim «hjs) In JV(  is»
U o[4>"]Uo*[4>']
=)] (ex p
(2h)1 f(4);''(x)(V 2 + m/)1/24>;"(x)
+ 4>/(x)( 
= (Phase
.n
cons t) ,=1
(4.62)
V2
+ m/)1/24>/(X»)
dx
(4.63)
(ex p (2h)1 f4>i(X)(V 2 + m/)1 /24>;(X)dX). (4.64)
The Hamiltonian related canonically to (4.22) by (3.16) is
(4.65)
and direct computation shows that (4.64) is, indeed, the vacuum state solution to Eq. (4.56), h2 s s f ( "2 f>4>(x) . f>4>(x)
l I n
+ 2 (V4>(x»' (V4>(x» + 2 i~l m/4>i(xf
x Uo[4>] = Eo Uo[4>], with the vacuum state energy,
)
dx
(4.66)
83
2. Neologized Schriidinqer Formulation
an infinite constant." Not a measurable quantity in itself, the vacuum state energy provides the reference value for definition of the observable energy (E~  Eo) associated with the Hamiltonian eigenfunctional U~[¢]. In the case of Hamiltonians that feature a term quadratic in the momentum density ntuple, as exemplified by (4.65), the stationary state eigenvalue equation HUJl[¢J = EJlUIl[¢J is reduced to an eigenvalue equation for (E~  Eo) by putting UIl[¢J == fl~[¢]Uo[¢]. As an example, for the Hamiltonian (4.65) we have 2flJl (  2h 2 b¢;(X)2 b + h¢Jx)( J
~
ifI
V
2
2 1/2
+ m,)
bflJl) b¢i(X) dx
= (E~
 Eo)flJl,
(4.67)
with the solutions for fl~ = flJl[¢J readily obtainable as polynomial functionals in the field amplitude ntuple,
n, = where
~(x)
f ~(x)
. ¢(x) dx
satisfies h(  V 2 + m/)1/2~i(X)
= (E I  EO)~i(x),
1 h2 fl 2 = 2 ¢(x) . (x, y) . ¢(y) dx dy   (E 2  E o) I 2
J
(4.68)
f.2: (ii(X, x) dx, ,= n
1
where (ij(X, y) = C;(y, x),
[(  V/ +m i 2)1/2 + ( V/ +m/)1/2J(ij(X' y) = (E 2 
Eo) (ij(x, y), ....
(4.69) Specialized to the n = 3 case with all m i equal to zero, these stationary state solutions appear with slight modification in the quantum theory of electromagnetic radiation, outlined in Appendix K. Closely related stationary state solutions appear for certain quantum field theories of a general nonlinear character, as shown in Appendix L. The Schrodinger formulation also supplies a dynamical description for quantum fields without a classical analog, systems that cannot be 9 The fact that the quantity {f U*[.p]HU[.pjfi(.p)/f IU[.pW:z'(.p)} is stationary with respect to variations of U[.p] about solutions to Eq. (4.56) precludes existence of finiteenergy eigenfunctionals for a large class of local field theories. For the method of proof, see G. Rosen, J. Math. Phvs. 9, 804 (1968).
4
84
Quantum Field Theory
treated by the Feynman formulation with a propagation kernel given by (4.14). Generally, the state of such a quantum field is described by an abstract Hilbert space vector'P = 'P(t) that satisfies the Schrodinger equation in a'Pjot
= H'P,
(4.70)
where the quantum Hamiltonian H must be prescribed without appeal to an equation of the form (4.45). For example, the field theory for neutrinos features the quantum Hamiltonian H
=
f ljJt(q)DljJ(q) dq
(4.71)
with ljJ(x) a twocomponent field operator that satisfies the anticommutation relations
+ ljJ/y)ljJ;(x) = 0 ljJ;(x)ljJ/(y) + ljJ/(y)ljJi(X) = oij o(x ljJ/(x)ljJ/(y) + ljJ/(y)ljJ/(x) = 0, ljJ;(x)ljJiY)
y)
(4.72)
in which ljJ /(x) is the Hermitian adjoint of the operator ljJ i(X) on the Hilbert space of 'P, i, j = 1, 2, and the matrixdifferential operator D in (4.71) is the oneneutrino quantum Hamiltonian expressed by the righthand side of (2.71). The field theory for electrons also features a quantum Hamiltonian of the form (4.71), but ljJ(x) a fourcomponent field operator that satisfies anticommutation relations (4.72), i,j = 1,2,3,4, the matrixdifferential operator D in (4.71) being the oneelectron quantum Hamiltonian expressed by the righthand side of (2.72). Although it follows from (4.70) that the generic dynamical evolution of the state vector 'P(t)
= (exp i(t  t')Hjn)'P(t')
(4.73)
is consistent with (4.43), a dynamical law of the form (4.1) and the concomitant notion of a propagation kernel do not apply to a quantum field without a classical analog. Hence, a Feynmanformulation is not available for such a quantum field.' 0 Notwithstanding the unavailability of a Feynman formulation, an abstract version of the 10 A heuristic "sumoverhistories" representation for the dynamics of the electron quantum field (patently distinct from a Feynman "sumoverhistories") was given by W. Tobocman, Nuooo Omenta 3, 1213 (1956).
3. Neologized Dirac Formulation
85
iteration solution (4.48), (4.51) obtains for a state vector 'P satisfying (4.70) with H = H, + Hint and provides the basis for perturbationtheoretic calculations involving such a quantum field. 3. Neologized Dirac Formulation By evoking the dynamical equation (4.43), the generic expectation value equation (4.54) can be recast in the form
f'P[¢; O]*f'P[¢; O]EC(¢)
(4.74)
with H defined by (4.45) and the Hermitian observable
f== (exp itH/h)j[¢(x), ih <5jb¢(x)](exp itH/h) =j[, rc], (4.75)
= (x; t) == (exp itH/h)¢(x)(exp itH/h)
(4.76)
1t
= 1t(x; t) == (exp itHjh)( ih b/J¢(x))(exp itHjh),
(4.77)
embodying the quantum field dynamics, and the wave functional in (4.74) fixed at the initial instant of time t = O. Equations (4.74) and (4.75) are mathematical statements for the Heisenberg picture of quantum field theory with the time rate of change of any expectation value (4.74)
<J> = f 'P[¢ ; 0] *1'P[¢; O]EC( ¢)
(4.78)
obtained by differentiating (4.75),
dildt == f' = [f, H],
(4.79)
where the Dirac bracket of two observable operators is defined as in quantum mechanics, [f, g]
== (ih)l(fg gf).
(4.80)
Thus, for example, the components of (4.76) and (4.77) have the Dirac brackets [i(X; r),
s.,
with the identity operator understood to appear on the righthand side of the latter equation. In quantum field theory the observable constants of the motion are quantities which have a zero Dirac bracket with the quantum Hamiltonian (4.45).
86
4
Quantum Field Theory
The set of all Hermitian observables is closed under operator addition and Dirac bracket combination, in the sense that both the sum and the Dirac bracket of any two observables are observables. Hence, the set of all Hermitian observables subject to operator addition and Dirac bracket combination constitutes a Lie algebra. Moreover, any subset of Hermitian observables that is closed with respect to operator addition and Dirac bracket combination constitutes a Lie subalgebra of the Lie algebra of all Hermitian observables. Equations (3.28) and (4.79) for the time rate of change of observables in field theory are identical except for the meaning of the bracket combination law, according to Poisson in classical field theory with (3.29) and according to Dirac in quantum field theory with (4.80). A Lie algebraic structure for the set of all observables is provided in either case by the bracket combination law, but the entire Lie algebraic structure for classical fields with (3.29) cannot be preserved for quantum fields with (4.80) by the Dirac correspondence
[I, 9 J = h
=
[f, gJ
= h,
(4.82)
even though extra special properties of the Poisson bracket for fields hold good with proper ordering under the Dirac correspondence. As for dynamical systems with a finite number of degrees of freedom, only a certain elite subset of the fieldtheoretic observables follows the Dirac correspondence (4.82), in the sense that for any triplet of observables f, g, h in the subset with [I, gJ = h the triplet of associated Hermitian observables f, g, h is such that [f, gJ = h. Because the Dirac correspondence carries Eqs. (3.40) into Eqs. (4.81) for components of the canonical field variable ntuples, the elements of the infinitedimensional Lie algebra 'lIs [observables of the form (3.30») follow the Dirac correspondence with the unique Hermitian observable f
II (<j>(x; t)· «(x, y)' <j>(y; t) + <j>(x; t)· f3(x, y) 're(y; t)
=±
+ re(x; t) • P'(x,
f
y) • <j>(y; t)
+ re(x; t) • y(x, y) • re(y; t»
+ (~(z)' (z; t) + 11(Z)' re(z; t»
dz
+ co
dx dy
(4.83)
associated with the generic element (3.30), where P;j(x, y) == Pj;(Y, x) in (4.83). The subset of Hermitian observables having the generic form (4.83) is closed with respect to Dirac bracket combination, and hence
3. Neologized Dirac Formulation
87
constitutes an infinitedimensional Lie algebra 'lIs, the quantum correspondent of Ills. Since the Lie algebraic structure of \!Is with the Poisson bracket combination law is identical to the Lie algebraic structure of 'lIs with the Dirac bracket combination law, the Lie algebras Ills and 'lIs are isomorphic. However, in analogy to the situation that prevails in quantum mechanics, observables with higher order ¢, 11: product terms cannot be assigned an Hermitian ordering which validates the Dirac correspondence (4.82) for them and other observables. To the Lie algebra .91 of a subset of fieldtheoretic Hermitian observables that is closed with respect to operator addition and Dirac bracket combination, there is associated an infinitedimensional conjugate Lie algebra .91* consisting of all observables which have a zero Dirac bracket with all elements of d. As in quantum mechanics, the observable constants of the motion constitute the Lie algebra d H * == {f: [f, H] = O} conjugate to the onedimensional Lie algebra consisting of elements proportional to the quantum Hamiltonian, d H == {(real constant parameter) x H}. Associated with the infinitedimensional Lie algebra d H*, we have the Lie algebra d H * consisting of a maximal set, closed under Dirac bracket combination, of linearly independent constants of the motion (exclusive of H and the trivial generic observables that are constants of the motion for any H). If the action S = S[¢(x; t)] is a general quadratic expression in ¢(x; t), that is, if the Lagrangian is of the form (4.16), the operator EulerLagrange field equations (4.26) are equivalent to inhomogeneous linear equations in the operator field amplitude ntuple (4.76), namely
I(p(x, y). ct>(y; t) + 8(x, y). (y; 1)  ((x, y). (y; t» dy + ~(x) = 0) (4.84) and for such a generic linear field theory d H * is an infinitedimensional Lie algebra; it is likely that d H * is also infinitedimensional for a generic nonlinear field theory. In general, the classical and quantum Lie algebras d ,,* and d H * are not isomorphic for there occur elements of d H * with ¢, 11: product terms of higher order than bilinear in the canonical variables. However, .cd H * and d H * are generally isomorphic for a linear field theory, as exemplified by the field theory associated with the simple Hamiltonian (3.32) for which d H * consists of all real
4
88
Quantum Field Theory
linear combinations of the Hermitian observable constants of the motion (i
= ± I, ± 2, ... ). (4.85)
It is an open question whether it is possible to prescribe Hermitian orderings which take each element of d H * into an element of d H * in such a manner that d H * and d H * are isomorphic in the case of a generic nonlinear field theory. A transformation of all observables f = f[q" 1t] ~ r, = ft[q" 1t]
9 = g[q" 1t] ~ s. = gt[q" 1t] such that
fo
= f,
(4.86)
(4.87)
go =g,
is said to be a canonical transformation generated by the Hermitian observable W = w[¢(x), ih (j/(j¢(x)], considered here to be independent of t, if the transformed observables satisfy the firstorder total differential equations (4.88)
A transformed observable is given explicitly in terms of the original observable by the Maclaurin series derived from (4.87) and (4.88), I, = f =
+ ref, w] + (r 2/2!)[[f, w],
(exp iTw/h)f(exp  irw/h).
w] + ... (4.89)
As for canonical transformations in quantum mechanics, we have the preservation of Dirac bracket relations by canonical transformations in quantum field theory, (4.90)
Thus, Lie algebras of Hermitian observables are mapped into isomorphic Lie algebras by canonical transformations. Again as in quantum mechanics, the time rate of change of any canonically transformed observable is prescribed by a dynamical equation of the form (4.79), (4.9J)
3. Neologized Dirac Formulation
89
as seen by putting h =t and g = H in (4.90). By specializing (4.90) to the cases for which h = c, a quantity independent of
= c = Eft' gr] = c.
(4.92)
In particular, for the components of the field amplitude and momentum density ntuples, the Dirac bracket Eqs. (4.81) yield [
(4.93)
[
(4.94)
Specializing to the cases for which f r equals each of the 2n components of
= [
Tcr(x; t) = [1tlx; t), Hrl
(4.95)
In quantum field theory one defines the transformed wave functional at the initial instant of time as
'¥r[¢; OJ == (exp irwjh)'¥[¢; 0],
(4.96)
so that expectationvalues (4.74) are invariant with respect to canonical transformations, (4.97) Hence, canonical transformations in quantum field theory have no effect on observable predictions. For canonical transformations generated by an mdimensional Lie algebra d composed of all real linear combinations of m linearly independent Hermitian observables WI>"" W m with each Wi = H'i[¢(X), ih bjb¢(x)] independent of time, we have [Wi' Wj]
= 
m
L:
k~l
Ci j k W k
(4.98)
with the structure constants Cijk = Cjik satisfying the quadratic Lie identities (1.52); the associated generators for canonical transformations Gi
== (ijh)w i = (ijh)wi[¢(x), ih bjb¢(x)]
(4.99)
4
90
Quantum Field Theory
are m functional differential operators that satisfy the commutation relations m
GiGj GjG i = LCijkGk. k=l
(4.100)
As for systems with a finite number of degrees of freedom, the classical generators (3.46) differ strikingly from the quantum generators (4.99) for canonical transformations in field theory, the latter generators exhibiting a general nonlinear dependence on bjb¢(x) with n(x) and bjbn(x) absent. A linear representation of the mparameter Lie group associated with the Lie algebra .91 is provided by linear differential operators of the form (exp a' G) ==
L (N!)l(a' G)N, 00
N=O
parametrized by a real mtuple a = (aj, ... ,am)' Under canonical transformations generated by .91, a generic Hermitian observable (4.75) is transformed to f~
= (exp
a . G)f(exp  a' G)
(4.101)
with all properties derived for the transformed observables (4.89) featured by the transformed observables (4.10 1), and the transformed wave functional at the initial instant of time defined as I.f'~[¢;
0] == (exp a' G)lJl[¢; 0].
(4.102)
As an illustration of quantum canonical transformations generated by a finitedimensional Lie algebra, consider the threedimensional subalgebra of ms ,
W
1
Wz
==  1 4
If(ZIi i5¢(x) b . b _ b¢(x)
== 
w3 ==
f (zIi b¢(x) s . b + ¢(x) . ¢(x)) dx b¢(x)· ,
4
¢(x) . ¢(x) ) dx,
iii f( () b )} 4" ¢(x)' b¢(x) + b¢(x) . ¢(x) dx,
(4.103)
3. Neologized Dirac Formulation
91
which is the quantum correspondent of (3.48). For this sf the Dirac bracket closure relations (4.98) are and the generators (4.99),
iJ(h (5 . (5 + h (5¢(x) (5¢(x)
Gj =  
4
iJ(h (5 ' (5 (5¢(x) (5¢(x)
G 2 = 4 G3
=

!
¢(x)· ¢(x)) dx
h [ ¢(x)· ¢(x)) dx
(4.105)
~ J(¢(x) . (5¢~X) + (5¢~X) . ¢(X») dx,
satisfy the commutation relations (4.100) with the independent nonzero structure constants C 123
=
C2 3 1
=
C 3 12
=
I.
Because the Lie algebra (4.103) is a subalgebra of '2ls , it is isomorphic to the corresponding subalgebra of 2(." namely (3.48), and thus d defined by (4.103) is isomorphic to the Lie algebra for SL(2, R). The generators (4.105) give a representation of SL(2, R) canonical transformations on the space of wave functionals according to (4.102). The dynamical evolution of all observables, prescribed by (4.79), can be viewed as a canonical transformation parametrized by time and generated by the quantum Hamiltonian, as seen by putting r = t, w = H = H[¢(x), ih bj(5¢(x)], and f, ==};[¢(x), ih (5j(5¢(x)] == f[ <j>(x; t), It(x; t)] in (4.88). Under this special canonical transformation, we have <j>(x; 0) = ¢(x) ~
92
4
Quantum Field Theory
of Hermitian observables is unchanged by these canonical transformations, the Lie algebra d H * is referred to as the symmetry algebra for the quantum field theory, and the associated Lie group ~H*' with its linear representation on the space of wave functionals ~H* = {(exp IX· G)}, is called the symmetry group. Finally, we note the especially simple kind of dynamics that arises if H is an element of the Lie algebra Ws discussed above, that is, if H has the form of the righthand side of (3.30) and H the righthand side of (4.83), as in the case of a generic linear field theory with operator EulerLagrange field equations of the form (4.84). Because of the isomorphism of Ws to ms ' the quantum dynamics associated with an H in Ws is identical to the classical dynamics associated with the corresponding H in ms for corresponding observables in Ws and ms ' In particular, the components of cj)(x; t) are inhomogeneous linear expressions in the components of cj)(x; 0) = 4>(x) and 1t(x; 0) = iii bj(j4>(x) with the same timedependent coefficients as in the classical theory, and likewise for the components of 1t(x; t). Dynamical containment in W s is featured by all elements in the Lie algebra if H is in Ws , f[cj)(x; t), 1t(x; t)]
== ft[cj)(x; 0), 1t(x; 0)]
=t
If (cj)(x; 0)· IX(X,
+ 1t(x; 0) . P'(x,
+ cj)(x; 0)· P(x, y;
t)· 1t(y; 0)
y; t) • cj)(y; 0)
+ 1t(x; 0)· y(x, y; + 1](z; t)
y; t) • cj)(y; 0)
t)· 1t(y; 0» dx dy
. 1t(z; 0» dz
+ w(t),
J
+ (~(z;
t). cj)(z; 0) (4.106)
where P:/x, y; t) == Pji(y, x; t) and with IX(X, y; t), P(x, y; t), y(x, y; t), ~(z; t), 1](z; t), and w(t) the same as in (3.51) because of (4.79), the Lie algebra addition and Dirac bracket closure properties, and the isomorphism of Ws to ms ' A generic linear field theory has solvable dynamics on the quantum as well as on the classical level because 'lIs is isomorphic to ms '
PROSPECTS Notwithstanding the elegance and simplicity of its present form, the conceptual framework of quantum field theory is likely to undergo dramatic changes in future years if quantum field theory is not superseded entirely by a new and different dynamical theory for elementary particles. In quantum electrodynamics the technical procedure called "renormalization" is required to derive finite predictions for experimentally observable quantities from the mathematical formulations. The status of quantum electrodynamics with renormalization has been described by HeitIer and by Jauch and Rohrlich: The ambiguities can always be settled by applying a certain amount of .' wishful mathematics," namely, by using additional conditions for the evaluation of ambiguous integrals. Unless such conditions are used, the results of the theory may contradict its very foundation .... Clearly such a mathematical situation is unacceptable. On the other hand, these difficulties do not prevent us from giving a theoretical answer to every legitimate question concerning observable effects. These answers are, wherever they can be tested, always in excellent agreement with the facts, and no serious discrepancy exceeding the limits of accuracy of the calculation has so far been discovered. 1 With respect to the applications, i.e., the actual description of physical quantities, we have here one of the bestestablished physical theories. Whenever the theory is subjected to an experimental test, we find the theoretical prediction in complete agreement with the experimental result. The accuracy 1 W. Hertler, "The Quantum Theory of Radiation," p. 354. Oxford Univ. Press (Clarendon), London and New York, 1954.
93
94
Prospects
in the agreement is limited only by the experimental error and the endurance and ingenuity of the computer. With respect to the fundamental concepts, on the contrary, we are not so fortunate. The theory is incomplete insofar as we are forced to introduce the charge of the electron and the masses of electron and photon as phenomenological quantities. The value of these quantities cannot be deduced from the theory but must be accepted as empirically given. This point of view enabled us to carry through the program of renormalization which was essential for the removal of the divergences in the iteration solution." Thus, the renormalization procedure in quantum electrodynamics can at best be accepted as tentative and heuristic. The alternative negative attitude, outright rejection of renormalized quantum electrodynamics, is in fact advocated by Dirac: It seems to be quite impossible to put this theory on a mathematically sound basis. At one time physical theory was all built on mathematics that was inherently sound. I do not say that physicists always use sound mathematics; they often use unsound steps in their calculations. But previously when they did so it was simply because of, one might say, laziness. They wanted to get results as quickly as possible without doing unnecessary work. It was always possible for the pure mathematician to come along and make the theory sound by bringing in further steps, and perhaps by introducing quite a lot of cumbersome notation and other things that are desirable from a mathematical point of view in order to get everything expressed rigorously but do not contribute to the physical ideas. The earlier mathematics could always be made sound in that way, but in the renormalization theory we have a theory that has defied all the attempts of the mathematician to make it sound. I am inclined to suspect that the renormalization theory is something that will not survive in the future, and that the remarkable agreement between its results and experiment should be looked on as a fluke."
If the renormalization procedure in quantum electrodynamics is in essence illegitimate, the entire conceptual framework of quantum field theory must be suspect. With regard to this possibility, we recall the opinion of extreme dissent by Einstein: 2 J. M. Jauch and F. Rohrlich, "The Theory of Photons and Electrons," pp. 415416. AddisonWesley, Cambridge, Massachusetts, 1955. Reprinted by permission. 3 P. A. M. Dirac, Sci. Amer. 208, 50 (May 1963).
Prospects
95
Is it conceivable that a classical field theory permits one to understand the atomistic and quantum structure of reality? Almost everybody will answer this question with" no." But I believe that at the present time nobody knows anything reliable about it. This is so because we cannot judge in what manner and how strongly the exclusion of singularities reduces the manifold of solutions. We do not possess any method at all to derive systematically solutions that are free of singularities. Approximation methods are of no avail since one never knows whether or not there exists to a particular appoximate solution an exact solution free of singularities. For this reason we cannot at present compare the content of a nonlinear classical field theory with experience. Only a significant progress in the mathematical methods can help here. At the present time the opinion prevails that a field theory must first, by "quantization," be transformed into a statistical theory of field probabilities according to more or less established rules. I see in this method only an attempt to describe relationships of an essentially nonlinear character by linear methods." 4 A. Einstein, "The Meaning of Relativity," p. 165. Princeton Univ. Press, Princeton, New Jersey, 1966. Reprinted by permission.
Appendix A
Functional Differentiation
Let the ordered array of Cartesian coordinates x = (XI' ... , x m ) represent a point in the mdimensional Euclidean space R m , and let f(x) = UI(X), .,. In(x)) denote a real ntuple function of x. Afunctional F = F[f(x)], a rule which assigns a number F to a function f(x), is continuous at f(x) if I lim F[f(x)
,0
+ sa(x)] = F[f(x)]
(A.I)
for all real ntuple functions of x, c(x) = (al(x), ... , an(x)), in a prescribed space 9'" = {a(x)}. Quite generally, an appropriate function space 9'" for a welldefined definition of continuity (A. I) is such that the f(x) domain of F[f(x)] and F[f(x) + sa(x)] coincide for all c(x) in 9'"; for example, 9'" might contain all functions of continuity class eN (perhaps also of" compact support," a(x) equal to zero outside a certain compact region associated with each function) that vanish at prescribed boundary values of x. While the domain of a functional is in general a class of functions, the sum of two functions in the domain not I For the original versions of functional continuity and functional differentiation, see V. Volterra... Theory of Functional and of Integral and IntegroDifferential Equations." Dover, New York, 1959.
96
Functional Differentiation
97
necessarily being in the domain, :F is patently a space of functions, closed under addition.f For example, the functional F[f(x)] = ( .. , (f(X)' f(x) dx ==
l l
fo '" fo ttl (};(X»2) (iiI dXj)
(A.2) is continuous for all continuous I(x) with :F == {CT(X) : CT(X) E CO}. A functional F = F[f(x)], continuous at I(x), is differentiable at f(x) if 1 ((djde)F[f(x)
+ eCT(x)])E~O ==
f
Rm
CT(X)' [bF/bf(x)] dx
== tmCtlCTi(X)[bFjbfi(X)]) (}j/x j)
(A.3)
exists as a linear functional in CT(X), the latter ntuple function again being any element in the prescribed :F. From the implicit definition (A.3) it follows immediately that functional differentiation is a linear operation,
that obeys the differentiation product rule, [b/bf(x)] (F lF 2 ) == [bFl/bf(x)] F 2
+ Fl [bF 2/bf(x)].
(A. 5)
In fact, for a differentiable function of a functional, 1> = 1>(F) 1>(F[f(x)]), we have b1>/bf(x) = (d1>/dF) [bF/bf(x)].
=
(A.6)
Illustrations of functional derivatives calculated according to (A.3) are provided by the following formulas: _b_
J f(x') . f(x') dx' = 2f;(x)
f5};(x) n
n
compact in
0
Rm ,
for for
x
x
E
n
if n,
(A.7)
:F == {CT(X): CT(X)E CO},
2 A comprehensive survey of special spaces .'F for distributions [specialized linear functionals with F[f(x) + ea(x)] == F[f(x)l + sF[a(x)] for all a(x) in 3"J appears in L. Horrnander, .. Linear Partial Differential Operators," pp. 3362. Academic Press, New York, 1964.
Appendix A
98
l5/j(a)/I5j/x) = l5ij l5(x  a) == l5ij
_15_ l5j/x)
m
TI b(x k=!
f j=!f 88xf (xj? af(X? dx' = 2 f ~2j/X) 8xj j=! oXj 8x j
k 
ak ) ,
for
x
E
11
g;
== {a(x): zr(x) E C l , c(x) = 0
for
x EOn},
(A.8)
n, (A.9)
_15_ (exp  c f f(x')· f(x') dX') 15!;(x) Rm
= 2Ch(X)(exp g;
C
tJ(x') ·f(x') dX}
== {a(x): fRma(X) . a(x) d x <
(A. to)
(jJ}.
Note that it is unnecessary to give a meaning to the socalled variational symbol 15 when 15 stands alone although the variational form of (1.3), obtained by expanding F[f(x) + ea(x)] as a Maclaurin series in s and setting w(x) == I5f(x),
bF[f(x)] == F[f(x)
=
f
Rm
+ ~r(x)]
 F[f(x)]
bf(x)· [()F/bf(x)] dx
+ O(bf(x) . of(x)),
is quite useful as a computational device. A functional F an extremum at f(x) if
bF/l5f(x) = 0
(A.ll)
= F[f(x)] has (A.12)
when the lefthand side is evaluated atf(x). Higherorder functional differentiation is defined by iteration of (A.3). Thus, the secondorder functional derivatives of F = F[f(x)J are given implicitly by
Functional Differentiation
99
It follows from (A.13) that secondorder functional differentiation is symmetrical, linear, and obeys the iterated differentiation product rule,
()2Fjbf(x) bf(y)
== b 2Fjbf(y) bf(x),
(A.14)
(A.I6) The variational form of (A.13), obtained by making use of (A. I I), facilitates actual computation of secondorder functional derivatives,
(
sr )
b b};(x)
=
t~ J! b};(x) blj(y) bfiY) dy + O(bf(y)' bf(y». n
b
2F
(A.17)
The most important secondorder functional differential operator is the functional Laplacian
(n b b rR~ '. dx ()f(x) bf(x) t~ lim I s:
•
yex i=!
_
()2
)
b};(x)(jJ:(y)
(A.IS)
dx,
which plays a key role in quantum field theory. In analogy to the Taylor expansion of an analytic function of several real variables, by definition, an analytic functional F = F[f(x)] admits a Volterra expansion! about a prescribed function f = f(x), F[f(x)]
=

F[f(x)]
1
+ 2!
I
+
f
()F "ij'()
R~ U
R~x R~
X
I __. (f(x)  f(x» ff
2F
,b (l(x)  f(x»' bf( ) bf( ,)
. (f(y)  fey»~ d x dy
X
+ ....
.}
dx
I
_
f=f
(A.19)
Other basic notions from the theory of a function of several real variables with partial differentiation carryover to the theory of functionals with functional differentiation. Thus, for example, a homogeneous functional of order IY., F = F[f(x)] == raF[Af(x)] for all real
100
Appendix A
A > 0, satisfies the linear functional differential equation first noted by Volterra.i r'
f
Rm
(A.20)
f(x)' [<5Ff<5f(x)] dx = rf.F,
and, conversely, the general integral to (A.20) is given by a homogeneous functional of order «. The manifest analogy with the theory of a function of several real variables can also be exploited to establish other general integrals to linear functional differential equations; for example, we have <5Ff<5f(x)
+ f(x)F = 0
admitting the general integral F = (const) exp 
t
for all
f
Rm
x
E
R;
f(x)' f(x) dx.
(A.21)
(A.22)
3 Equation (A.20) is obtained most readily by applying the general parameterdifferentiation formula
(d/dA) F[f(x; A)] =
that follows from (A.3) and
r [SF/Sf(x; A)] . [Of(x; A)/OA] dx JRm
F[f(x; A+ e)] = F[f(x; A) + e Of(x; /..)loA] + 0(e 2 )
(A.23)
Appendix B
Linear Representations of a Lie Group
Consider a set <§ = {T(a)} of linear operators' T(a), labeled by m real essential parameters, the components of the mtuple a = (a l , ... , am)' <§ is a linear representation of an mparameter Lie group ie: 1. T(a) is an analytic function of a about a identity operator; thus, T(a)
=
0 with T(O)
=
1, the
= 1 + a . G plus terms quadratic and of higher order in a (B.1)
where the generators appear as (B.2) The m linear operators (B.2) are linearly independent. 2. Each T(a) has a unique inverse T(a)l == T(ii) T(ii)T(a)
= T(a)T(ii) = 1.
E <§ such
that (B.3)
1 See, for example, N. 1. Akhiezer and 1. M. Glazman, "Theory of Linear Operators in Hilbert Space," pp. 3039. Ungar, New York, 1961. In physical applications, T(et) is either a squarematrix that acts on a finitedimensional vector space or a differential operator that acts on an infinitedimensional function space. • 2 For the sake of practical transparency, our conditions on the group elements T(et) are somewhat more specific than necessary.
101
102
Appendix B
3. For any pair of elements T(a) and T(f3) product T(a)T(f3) = T(y)
In
'fl, their ordered
(B.4)
is contained in 'fl with the mtuple of parameters y = Yea, fi) = a
+ 13
plus
(8.5)
(terms bilinear, quadratic, and of higher order in a and 13) as a consequence of (B.I). Because of (B.4), we have
so by setting rx = f3 = (B.2), we obtain
°and making use of
2 a T (Y)) G·G= (  . J DYj OYj. y=O L
+
(8.5) and the definition
Im ( a2Yk ) Gko k= 1 oaj of3j a=p=O
(8.7)
From (B. 7) it follows that m
c,c,  c,c, = I
k=1
CijkGko
(B.8)
where the structure constants appear as Cjjk ==
(a:i2~pj  O:;~kp)FP=O == Cjjko
(8.9)
It is readily seen that the structure constants satisfy the quadratic Lie
identities,
m
I
"=1
(Cjjl,Chkl
+ CjkhChil + CkihChj/) == 0,
(8.10)
by adding the i,j, k cyclic permutations of the commutator of (B.8) with G, and evoking the linear independence of the C'5. Lie's main theorem is that an array of constants Cijk ==  Cjik' [i, i. k = 1, . ° • , m], satisfying (B.lO) permits the Eqs. (B.8) to be solved for m linear operators G i related by (B.2) to a T(a) satisfying Eqs. (B. 1), (B.3), and (B.4); furthermore, by evoking a trivial relabeling of the elements of q; with a parameter transformation of the form a  t a + (terms quadratic and of higher order in a), elements of q; can be expressed canonically as
Linear Representations of a Lie Group T(a)
= (exp
a . C)
==
103
L (N !)l(a . ct. 00
N=O
(B.ll)
The set of all linear combinations of the generators, Lcd = {a' G} with elements parametrized by a, is closed with respect to ordinary operator addition and operator commutation, and hence constitutes a Lie algebra with the Lie product prescribed as the commutator of a pair of elements. A solution to Eqs. (B.8) for the generators, such that no G, is the zero operator, yields a representation of the Lie algebra." If the structure constants are such that C ij k does not vanish for all values of j and k with i any fixed value, we have the socalled adjoint representation with the generators mdimensional matrices composed of the elements (GJjk =  Cijk, as readily seen by writing (B.8) with matrix indices and recalling (B.10). The associated adjoint representation gives the generators as the differential operators G, =  L},k=l CjjkX j %xk on the space of infinitely differentiable functions of x = (x, ... , x m ) . Lie's main theorem asserts that a representation of the Lie algebra provides a linear representation of the Lie group '§ for an appropriate set of a in R m , with elements in '§ given canonically by (B. 11). If the appropriate set of a in R m is a compact point set, the Lie group is said to be compact; otherwise, the Lie group is noncompact. To obtain all elements for certain Lie groups, it is necessary to augment the set {(exp a' G)} in an obvious way (see Example 4 below). Linear representations of Lie groups are illustrated by the following examples: 1. Cijk = 0, the mdimensional Abelian Lie group. According to (B.8), all m generators commute with one another. Of particular interest is the differential operator realization C i = 0/ ox i v for which the set of differential operator group elements (B.11), T(a) = expect . %x), 3
(B.12)
All Lie algebra representations are isomorphic to the Lie algebra of real
mtuples with the addition
and the Lie product
a
+ (3 == (rXl + (31 ..... rXm + (3m) [a.
(31. =
m
L
sJ> 1
C'jk rX , (3 j
The integrability property (1.30) is satisfied for this Lie product because of (B.IO). Notice that for m = 3 and e'jk = e'jk. this Lie product is the 3tuple crossproduct of Cartesian vector analysis.
Appendix B
104
translates the mtuple argument of a C'" functionf(x) by a, T(a)f(x) = f(x + a). In order for all elements in the group to be included, each of the m parameters ai must assume all finite real values, and hence the mdimensional Abelian Lie group is noncompact. The special case m = 1 is closely associated with solutions of firstorder (ordinary or partial) linear differential equations. 2. m = 2 with Cl2 1 = and CI22 = I, the proper affine Lie group on real numbers. It is easy to verify that the quadratic Lie identities (B.IO) are satisfied. The generator equations (B.8) produce the single relation GI G2  G2 GI = G 2 , which admits the twodimensional matrix representation
°
(R13)
By putting this twodimensional representation into (B.11), we obtain T(a) = exp
(~
(R14)
The group parameters a l and a2 assume all finite real values, so this Lie group is noncompact. Note that the representation (B.14) acts on
(7) to produce proper affine transfor
the space of vectors of the form
mations of the real number x+ [ea1x + az(e at  l)/a l ]. 3. m = 3 with C i j k = 8 i j k, the LeviCivita symbol, the Lie groups SU(2) and SO(3). That such an array of structure constants satisfies the quadratic Lie identities (RIO) is verified immediately by recalling the relation I,l=l 8ijh8hkl = ~ik ~jl  ~il ~jk' With C i j k = 8 i j k, generator equations (R8) become GI G2

G2 GI = G3
G 2G 3

G 3G 2
= GI
G 3 GI

GI G 3
= G2 •
(B.l5)
We have the twodimensional matrix representation G2
1(01 1)o '
=2
G3
1
=2
(i
0
(B.16)
Linear Representations of a Lie Group
105
as well as the threedimensional adjoint representation GI
=
(~ ~  ~), o
1
(~ ~
o. =
0
1
G,~ (! ~ ~)
0
(B.17)
By putting the twodimensional representation (B.l6) into (B.1l), we obtain I ( T(a) = exp 2
 ia

.
la l
+3 az
0) _i(sin tlal) ( I
lal
a3
al + iaz
(B.I8)
lal == (a/ + a/ + a/)l/Z, a parametric representation of all 2 x 2 unitary matrices of determinant one, the Lie group SU(2).4 All elements in the group are included if a is restricted to a sphere about the origin of radius 2n, lal ::::; 2n, with all points on the spherical surface lal = 2n being identified with the group element
0). (1 o 1 '
thus, SU(2) is compact and simply connected. By putting the threedimensional adjoint representation (B.l7) into (B.ll), we obtain
a 3 a z) (RI9) 0 aI' ( a a 0 z l a parametric representation of all 3 x 3 real orthogonal matrices of determinant one, the Lie group SO(3).4 All elements in the group are T(a)=exp
0 a3
4 The nomenclature symbol S stands for .. special" and means .. determinant equal to one." The determinant of an exponentiated traceless matrix [such as (B.l8), (B.l9), or (B.26)] equals one because we have
det(exp M) = limN_oo deteCt + N 1M )N) = limN_oo (det(t + N ' M » N = limN_oo (1 + N 1 (tr M) + O(N  2»N =exp(trM)
Appendix B
106
included if o: is restricted to a sphere about the origin of radius n:, I'l.l == (:x/ + :x/ + 'l./)1/2.( tt, with each pair of opposite points on the spherical surface l:xl = tt being identified with a single group element (which squares to the identity because T(:x)T( 'l.) = 1); hence, SO(3) is compact and doubly connected. Since linear representations of both SU(2) and SO(3) follow from the structure constants Cijk = 8 i jk' the Lie algebras for SU(2) and SO(3) are isomorphic, which implies that the Lie groups SU(2) and SO(3) are isomorphic in the neighborhood of the identity. On the other hand, the appropriate parameter set domains and topological character (manifest by the identification of points with group elements on the critical spherical surfaces) are different for the complete sets of group elements, and so the isomorphism between the Lie groups SU(2) and SO(3) is local but not global. We also note that the differential operator solution to (B.15),
(B.20)
the associated adjoint representation, yields the set of differential operator group elements T(:x)
=
exp (
.I_
I,
J, k  1
8ijk'l.iXP3/0Xk»)
(B.21)
eYe
that rotates the 3tuple argument of a Ituple function f(x), T(ex)f(x) = f(x'), where x' is related to x by the inverse of the associated SO(3) matrix of the form (B.19),
(B.22) 4. m = 3 with Cijk = 8 ij k T k ,Tk == (_I)kt, the Lie group SL(2, R). To show that this array of structure constants satisfies the quadratic Lie identities (B.I 0), we first establish the formula 3
I
h=l
CijhChkl
= (l)k(b ik b j l

b il b j k),
(B.23)
Linear Representations of a Lie Group
107
from which (RIO) follows by cyclic permutation of i, j, k and addition. The generator equations (B.8), G1 G z 
c, G1 = G3
Gz G 3

G3 Gz
=
G1
G 3G1

G1 G 3
=
G z ,
(B.24)
admit the twodimensional matrix representation
I)o '
G3
="21(10
(8.25)
By putting the latter set of generators into (Bi l l), we obtain
T(a)=ex p 1( (;(3 :x1(2) 2 :Xl + :Xz a3
=(COSh~(;(/_(;(/+(;(/)I/Z)(~ ~) Z + (Sinh Ha/  a/ + a 3 )1/2) ( a3 (:x1 2  a/ + a/)1/2 , :Xl + a2 for = (cos
~ (:x/ + a/  a/)1/2)(~
~)
+ (Sin !( a/ + a/  a/)l/Z) (;(3 (a 1 z+IX/a3 2)I/Z
(a/  a/ + :x/) ~ 0,
a 1+a Z for
a1 
:xz) IX 3 (IX 1 Z  az 2 + IX3Z) ~ 0,
(8.26) a parametric representation of 2 x 2 real matrices of determinant one, a part of the Lie group SL(2, R).5 All elements of SL(2, R) are included in the augmented set {texp « . G), (exp a . G)} with the components of IX assuming all finite real values for which (:x/  IX/ + IX/)~ _7[z with opposite points on the hyperboloid surfaces IX 2 =
±(a/
+
a/ +
7[Z)I/Z
5 The trace of matrices of the form (B.26) is greater or equal to  2; to get the matrices with trace less than  2, we must also consider the derived set {(exp ex' G)}.
Appendix B
108
identified with a traceless matrix that occurs in both subsets {(exp (X • G)} and { (exp (X' G)}; thus, SL(2, R) is noncompact and has the connectivity of a periodic slab. By making the formal parameter replacement ((Xl' (X2' (X3)+ (i(Xl' (X2' i(X3), one obtains (B.18) from (B.26), and so SU(2) is the (unique) compact complexextension of SL(2, R). An immediate general classification of mparameter Lie groups follows from Cartan's metric, defined in terms of the structure constants as the symmetric m x m array gij
==
m
k,
It> 1Cik1Cj 1k•
(B.27)
We have (gij) = 0 for the mdimensional Abelian Lie group (Example 1 above), (gij) = diagf l , 0...1 for the proper affine group on real numbers (Example 2 above), (gij) = diag'" 2, 2, 2...1 for the Lie groups SU(2) and SO(3) (Example 3 above), and (gij) = diag r2, 2, 2...1for the Lie group SL(2, R) (Example 4 above). Cartan's metric (B.27) is negativedefinite for compact Lie groups and indefinite for noncompact Lie groups. By making use of (B.1O), we find that the quantity aijk
==
m
I
h=l
Cijhghk
=
m
I
r,s,t==l
(CrisCSjtCtkr 
CrjsCsitCtkr)
(B.28) is, in general, a totally antisymmetric array of real constants, changing its sign under transposition of any two indices. An mparameter Lie group is semisimple if the m x m array (B.27) constitutes a nonsingular matrix, det(g i) =I O.
(B.29)
Thus, the mdimensional Abelian Lie group and the proper affine Lie group on real numbers are not semisimple, while the Lie groups SU(2), SO(3), and SL(2, R) are semisimple. The symmetric matrix inverse of Caftan's metric (B.27), customarily denoted gij, exists for a semisimple Lie group and satisfies m '" f...,g ikgkj=b j.i
(B.30)
k~l
In terms of s" and the generators for a linear representation of a semisimple Lie group, the quadratic Casimir invariant is defined as C ==
m
L
gijGiG j •
i,j= 1
(B.31)
Linear Representations of a Lie Group
109
Because it commutes with all the generators, CGkGkC=
m
I
i, j, Il= 1
gii(CjkIlGiGh+CikhGhG)
m
I
(gii Cikll i,j, h> 1 m
I
(gii o"
t, j, h, 1= 1
+ gillCik)Gj Gh + gihgjl)aikl Gj Gh (8.32)
as a consequence of the antisymmetry of (B.28), the quadratic Casimir invariant plays a central role in the representation theory for semisimple Lie groups." 6 For general discussions of Lie group representation theory, see L. S. Pontrjagin, "Topological Groups." Princeton Univ. Press, Princeton, New Jersey, 1946; R. E. Behrends, et al., Rev. Modern Phys. 34, I (1962) and works cited therein.
Appendix CHaar Measure
Functions of the linear representations of Lie groups discussed in Appendix B can be" averaged over the group" by defining appropriate integrals with respect to the mtuple parameter o: There is a general method for the computation of an infinitesimal volume element in
E
J
rs
(C.2)
for all T([3) E
110
Haar Measure
III
group, the leftinvariant integral (CI) requires an infinitesimal volume element such that (C3) for all fixed T(fJ) E q}, while the rightinvariant integral (C2) requires an infinitesimal volume element such that (C4)
for all fixed T(fJ) E q}. An infinitesimal volume element with the property (C3) or (CA) is a leftinvariant or rightinvariant Haar measure? on q}. To obtain general formulas for the leftinvariant Haar measure associated with a linear representation of an mparameter Lie group, first differentiate (8A) with respect to fJ i v set fJ = 0, and make use of Eqs. (8.2) and (8.S); the m operator equations that result are
T(rx)G i =
m
I
j=l
Pij(rx)[oT(rx)/orx j ]
(CS)
with the m x m array Pij(rx) == oyjofJilp=o. Next, multiply (CS) by T(ii) from the left and make use of the relation derived from (8.3) and (8A), T(y)T(rx) = T(fJ); the resulting equations are
T(fJ)G i =
m
i.
I pi rx )(OfJk/oa)[oT(fJ)/OfJkJ k=!
(C6)
where o: and yare viewed as independent mtuples with fJ = fJ(rx, y) prescribed by the group multiplication. Finally, replace a by fJ in (CS) and compare these equations with (C6) to obtain m
I
j=!
Pij(rx)(OfJk/Orx) = Pik(fJ)
(C7)
for all y independent of «. In view of (C7), the leftinvariant Haar measure is defined to within normalization by
DL(T(a» == [det(pi/rx)r!
m
I1 drxk, k=!
(C8)
2 The classic abstract discussion of Haar measure is that of P. R. Halmos .. Measure Theory," pp. 250265. Van Nostrand, Princeton, New Jersey, 1950.
Appendix C
112
for it follows from (CS) and (C?) that DL(T(Y)T(a))
= DL(T(fJ))
=[det(pi/!mr
m
1
TI dfJk
k=l
= [det(pi/a))T 1 [det(ofJdoa)r 1
m
TI
k=l
dfJk
(C9)
for all y independent of a. A similar calculation produces the formula for the rightinvariant Haar measure, DR(T(a))
=[det(wij(a))]l k=l TI dak, m
(CIO)
to within normalization and with the m x m array wij(a) given implicitly by Gi T(a)
=
m
I
j=l
(Cll)
wi/a)[oT(a)/oaj].
The leftinvariant and rightinvariant Haar measures (CS) and (ClO) are related by the equation DL(T(a)) = DR(T(a)) where T(a) = T(a)l since we have m
wi/a)
= I
k=l
(CI2)
pik(a)oa)oak'
because of (CS) and (CIl). If '§ is compact, the leftinvariant and rightinvariant Haar measures (CS) and (C 10) are identical. 2 The Haar measure formulas in the preceding paragraph do not depend on the parametric labeling of the elements of '§. Subject to a parameter transformation a + a' = a + (terms quadratic and of higher order in a) which relabels the group elements with T(a') T(a) , both Haar measures (CS) and (CIO) transform as scalar invariants, D'(T(rx')) = D(T(rx)), as a consequence of the Pij(a) and wij(a) parameter transformation character implied by (CS) and (CII). However, the actual computation of Pij(rx) and wi/a) is simplified for '!J with m ;;,: 3 by employing the parametric labeling for which the elements of '!J are given by the canonical expression (B.I I). Then the formula for the
=
Haar Measure
113
partial derivatives of an exponential operator function.' applies and produces" 1
(oj8ct i)(exp a : G) = ~ (exp(l  s)ct· G)Gi(exp sa : G) ds
==
t
1
(exp s« . G)G;(exp(1  s)« . G) ds, (C.13)
where s is a dummy ltuple integration variable. Hence, by putting (B.ll) into (C.S) and (C.ll), we have the more explicit equations for Pij(ct) and Wj/ct), m
j!;/i/ct) = G, =
fo(exp 1
sa . G)G/exp sa . G) ds
I 1 Wi/ct) J0 (exp s« . G)G;(exp m
1
j;
sa . G) ds.
(C.14)
Haar measures for the linear representations of Lie groups mentioned in Appendix B are obtained to within normalization as follows: 1. For the mdimensional Abelian Lie group with all G i commuting and the elements of <:9' expressed canonically as (B.12), we have
8T(ct)j8ctj = GiT(ct) = TCct)G i.
(C.IS)
Hence, both m x m arrays Pij(rx) in (C.S) and wij(ct) in (C.ll) equal b ij, and so the leftinvariant and rightinvariant Haar measures (C8) and (CIO) are given by
D(T(ct» =
m
TI
k;l
drxk'
(CI6)
2. For the proper affine Lie group on real numbers with the representation (B.14), we find 3 For example, R. M. Wilcox, J. Math. Phys. 8, 962 (1967) and works cited therein. 4 A direct proof of (C.13) follows from the equation
where A is a real ltuple parameter; since the squarebracketed quantity in the latter equation vanishes for A = 0, the equation implies that the squarebracketed quantity vanishes for all Aand, in particular, for A= I.
Appendix C
114
by working out terms in (CS) or the left member of (CI4). Hence, the leftinvariant Haar measure (CS) is (CI7)
On the other hand, by working out terms in (CII) or the right member (CI4), we find
Hence, the rightinvariant Haar measure (CIO) is (CIS)
Note that the Haar measures (CI7) and (CIS) ate different for this qj. 3. For the Lie groups SU(2) and SO(3) with the representations (B.IS) and (B.19), the calculations are facilitated by making use of the fact that IX1, IX2' IX 3 enter with rotational symmetry." Since the weight factors [det(p uC IX))]  1 and [det( (I) ij( IX))] ) in (CS) and (CI 0) can depend only on IIXI == (IX/ + IX/ + IX/) 1/2, we set IX = (0, 0, IIXI) in (CI4) and 5 It is not difficult to demonstrate the invariance of (CS) with respect to proper rotations of the 3tuple parameter IX = (lXI, IXl' IX3) in the SU(2) representation (B. IS). To show that Dd T(IX"» = DdT(IX) for IX: = Ri, IXj with RijR;k ~ 6j ko det(Rij) = + I, we first associate the proper rotation matrix (Rij) (an element of SOD)) with a 2 x 2 unitary matrix of determinant one T«(3) (a fixed element of SU(2)) by solving the equations
where the generators are given by (8.16) as the Pauli matrices times the numerical factor (/12); the latter equations prescribe n(3) in terms of (RiJ uniquely to within a (± I) numerical factor and imply that n(3)T(IX') ~~ n(3)(exp RijIXjG;)
Next, we replace right to obtain
IX
by
IX'
=
T(IX)T«(3).
in (CS) and move n(3) through the equation from left to R;J(IX)Gj
from which it follows that
~
Pij(IX')[oT(IX)/olX/l
Haar Measure
115
solve for the weighting factors associated with the SU(2) representation (B.18), [det(puCa))]l
= [det(wij(a))]l = (2(sin !Ial)/ lal)2.
(C.19)
 As a consequence of the local isomorphism of SU(2) and SO(3), the identical weighting factor is obtained by solving (C.14) with the SO(3) representations (B.19) and (B.2l). Thus the leftinvariant and rightinvariant Haar measure, the same for the linear representations of SU(2) and SO(3), is given by D(T(a»
= (2(sin !lal)/laI)2 da, da 2 da 3 •
(C.20)
Observe that the Haar measure (C.20) vanishes on the critical spherical surface lal = 211: for SU(2), but is finite on the critical spherical surface lal = 11: for SO(3). 4. For the Lie group SL(2, R) with the representation (B.26), the calculations can be related directly to the SU(2) calculations in the preceding example by making the analytic continuation which takes (3.18) into (B.26), namely: a l + ia l , a2 + a2 , a 3 + ia 3 • By applying this analytic continuation to (C.20) , we find that the invariant Haar measures are equal and given by D(T(a»
= D(±(exp a . G»
for
(a/  a/
+ a/) ~ O.
(C.21)
and hence that det(pij(IX')) = det(pij(IX}}. Thus, we have 3
DL(T(IX'}}
== [det(pu(IX'}}] 1 II dai' = k==l
DL(T(IX}}.
It is interesting to note that explicit computation with (C.S) can be avoided if we use the fact that DL(T(IX}} == DR(T(ex», because of 8U(2) being compact 2 ; we
have DL(T(IX')) = D L(T(j3}T(IX'}} = DdT (IX}T(j3» DL(T(IX)}.
== D R(T(rx)T(j3»
= DR(T(IX})
==
Appendix D
Functional Integration by Parts Lemma
Here we establish the lemma 1: If the functional derivative 6F/6j(x) and the functional integral S% F D(f) both exist for a certain functional F = F[f(x)] and function space :¥ with the measure displacementinvariant (D(f + 0") = D(f) for any fixed (J E iF) then
J~[<5F/6f(x)]D(f) = O.
(D.1)
The notation is that of Appendix A. To prove the lemma, we observe that tnf(x)
+ O"(x)]D(f) =
fy;F[f(X)
+ (J(x)]D(f+
(J) = tnf(X)]D(f)
(D.2) with the first equality in (D.2) a consequence of the measure being displacementinvariant, and the second equality in (0.2) the result of replacing the dummy integration variable, (f + (J) +f Since the last member of (D.2) is independent of (J, (0.1) follows by taking the functional derivative of (D.2) with respect to c(x) and setting (J = O. 1
116
G. Rosen, Phys. Rev. Lett. 16, 704 (1966), and works cited therein.
Functional Integration by Parts Lemma
117
To illustrate the utility of the lemma, let us use it to derive the Wiener correlation function"
bet', t"; t) ==
I f(1')f(t") A(f) :F
=
t
min(t', t")
0
for
t', t"
~
~
t,
(D.3)
where:? = U=/(r) for 0 ~ r ~ t : fer) E c°,f(O) Ituple variable; the Wiener measure in (0.3),
A(f) == (ex p 
O} with r a real
=
f~(df(r)jdr)2 dr)D(f)
(0.4)
with DU + rr) == D(f) displacementinvariant for any fixed a E.'IF is normalized to produce
I
:F
We have
A(f) == 1.
tf(t') A(f) = 0
for
(0.5)
0 ~ t' ~ t,
(0.6)
H
because the integrand in (0.6), F = F[f(r)] = .f(t')exp (d/(r)jdr)2 dt ; is an odd functional offer). By applying the functional integration by parts lemma to this F, it follows that S:F l5Fjl5/U")D(f) = 0, and we obtain
f)l5(t'  t")
+ 2f(t')[d 2f(t")fdt"2]) A(f) =
0
for
0
t', t"
~
t.
(0.7)
In view of (0.5), (0.7) becomes
tf(t')(d2f(t")fdt,,2) A(f) =
~
t l5(t'
 t")
for
0
~
t', t"
t,
(0.8)
which implies that
f(t')(df(s)fds) A( f) = t f: F ' . 0
~
for for
0 0
~ s, < ~
t
~
t'
s
~t
~
t,
(0.9)
because the lefthand side of (0.9) vanishes if t' = O. The expression (0.3) for the Wiener correlation function follows by integrating (0.9) with respect to s from 0 to t" 2 For example, see I. M. Gelfand and A. M. YagJom, J. Math. Phys. 1, 48 (1960).
Appendix E
Relativistic SumoverHistories for the OneDimensional Dirac Equation
Heuristic interest is attached to the onedimensional Dirac equation! (i(%t)
+ irx(%x) 
f3)IjJ(x; t) = 0,
(E.!)
where ljJ(x; t) is a twocomponent wave function, x is a real Ituple coordinate, and the 2 x 2 Hermitian matrices rx and f3 satisfy the conditions
rx2=f32=(~
~),
rxf3+f3rx=(~
~).
(E.2)
For a prescribed initial state ljJ(x; 0) == ljJo(x), the solution to (E.l) ljJ(x; t) =
foo
K(x  x', t)ljJo(x') dx'
00
(E.3)
involves the propagation kernel K(x, t) =
if3)[JO((t2  X2)1/2)IT(t Ix!)]
(~ rx!. 
ot
ox
IT(u) ==
1
1
o
U>o u=O U <0.
(EA)
1 Physical units are chosen such that the speed of light, the mass of the electron, and h equal one.
118
Relativistic SumoverHistories
119
That (EA) satisfies the equations derived from (E.l) and (E.3), (i(ojot)
+ irx(ojox) 
P)K(x, t)
=0
lim K(x, t) = (D(OX)
(E.5) (E.6)
tO
is verified immediately by evoking the zeroorder Bessel function representation 2 J ((t2 _ o
x
2)1 /2)(1(t _
I I) =! foo [sinCe + 1)1/2t][cos kx] dk (E.7) x n (k 2 + 1)1/ 2 00
for t> O. On the other hand, by substituting the Maclaurin series representation for J o into (EA), we find that
if3)
~ n+l( (t+rxx) 2 2n K(x,t)=n~o(l) 22n+l(n+1)!n!+22n(n!)2 (t x) (E.8) for t> Ix\. Feynmarr' has noted a remarkable" sumoverhistories" representation for this propagation kernel, namely: K(x, t)
nU'(
J1,
.)=
r, s, e  s
I
.0
= lim :ff(xje, tie; e),
~ (N++(r'S;R)
L.
R=O
N _+r,s; ( R)
N + _(r, s; R»)(ie)R, N __ (r,s;R)
(E.9) (E.10)
in which N~'1(r, s; R) denotes the number of "histories" composed of s unit steps from the origin to the (positive or negative) integer r [t(s + r) unit steps forward with Ar = + 1 and !(s  r) unit steps backward with Ar = lJ that have R "reversals" (that is, changes in the sign of successive Ar) and where '1 is the sign of the initial step and ~ is the sign of the final step. Thus, at any instant of time the electron 2 I. S. Gradshteyn and I. M. Ryzhik, "Table of Integrals, Series, and Products," p. 472. Academic Press, New York, 1965. 3 R. P. Feynman and A. R. Hibbs, "Quantum Mechanics and Path Integrals," p, 35. McGrawHili, New York, 1965.
Appendix E
120
is considered to move forward or backward at the speed of light, with the spin state associated with the instantaneous velocity." From the definition of the N'e, it follows that"
+ 1; R) = N + _ (r, s + 1; R) = N _ +(r, s + I; R) = N _ _ (r, s + 1; R) = N + + (r, S
+ N _ + (r  1, s; R  I) N + _ (r  I, s; R) + N __ (r  1, s; R  1) N + +(r + 1, s; R  I) + N _ +(r + 1, s; R) N + _(r + 1, s; R  1) + N _ _ (r + I, s; R) N + + (r  I, s; R)
(EJ 1)
for all s ;:, I, and hence the definition (E. 10) implies that
ff(r,s+ l;e)=
(~
~)ff(r 
1, s; e) +
(~
~) ,.;f"(r + 1, s; e), (E.12)
which, with r = xle and s = tie, produces the relation
(e
f f~,
~., e) = e
(10
ie)"ff ((X  e) , !.,e) O s e.
0)" ) ff ((X + e) , !.,e. 1 e
(E. 13)
e
4 Compare with P. A. M. Dirac, "The Principles of Quantum Mechanics," pp. 260262. Oxford Univ. Press, New York and London, 1947. 5 By symmetry, we also have
N++(r, s; R) =N __ (r, s; R)
and
N+ _(r, s; R) =N_+(r, s; R) =N+ _(r, s; R) =N_+(r, s; R)
Combinatorial analysis shows that N + +(r, s; R) =
gcs + r 
R)  1, !RHHs  r  R), !R  l}
and N + _(r, s; R) =
for R > 0, where
{~(s
+ r  R)  !, !R 
, °+
{m n} "" (m
n) !/m! n!
mHs  r
r:
R)  L !R 
!}
for m, n nonnegative integers, otherwise.
An elegant derivation of these formulas for N + + and N + Bock (unpublished),
_
has been made by Charles
Relativistic Sumover Histories
121
By writing Taylor expansions for the terms in (E.13), we have X
a (X, t) X t) ; c + c ::;.ff ; c + O(e 2)X (X , t ; c) (, ot e c c c c s
(E.14) and so
~X(~,~;c) = (~
 (b
~)xG,~;c) _~) :x x(~, ~; c) + O(c)xG, ~; c). (E.15)
The latter relation shows that (E.9) satisfies (E.5) with and
p= ( 10
1)o .
(E.16)
Moreover, from the definition of the N's and (E.9) it follows that
lim K(x, t) = lim $'(~' 1; c) t+O
£+0
= (D(X) o
E
0)
b(x) ,
(E.17)
122
Appendix E
and thus (E.9) satisfies (E. 6). Hence, the relativistic sumoverhistories (E.9) gives the propagation kernel for the onedimensional Dirac equation. Unfortunately, a generalization of (E.9) for three spatial dimensions has not been found.
Appendix F
Feynrnan Operators
Consider a generic functional A = A[q(t)] that depends on q(t) for t' ~ t ~ ttl and is analytic' about q(t) == O. Suppose further that the Feynman operator
rA(exp iSjh)D(q) == (q"; ttl IA[q(t)]J q'; t')s
(F.l)
.~
exists for A. The Volterra expansion (A.19) for A[q(t)] with q(t) == 0 takes the generic form 00
A[q(t)] = kf:
o
t ... t t"
t"
(}(k)(t 1 , ••• , tk)q(t 1 )
· · ·
q(t k) dt 1
'"
dt k,
(F.2)
where (suppressed) indices on the distributionvalued (}(k),S and generalized coordinate ntuple q's are understood to be contracted. Since the (}(k),S are independent of ij, it follows from (F.l) that
(q"; t"IA[q(t)]1 q'; t')s
=I 00
k=O
1
f ···f (}(k)(tl,.·.,tk)(q";t"!q(tl)···q(tk)lq';t')sdtl···dtk· r:
r"
t'
t'
(F.3)
From a practical point of view, there is no loss in generality in assuming that
A = A[q(t)] is analytic about q(t) ~ O. If not originally so, A can always be modified to be analytic about q(t) ~ 0 without inducing a physically significant change in the
associated Feynman operator.
123
Appendix F
124
Recalling the propagation kernel representations
K(q", s': t"  t') = (q"; t" III q'; t')s
= (exp
iH"(t"  t')lh) !J(q"  q'),
(FA)
where H" == H(q",  ih 818q") is the Hermitian quantum Hamiltonian operator associated with the action functional S, we have
(q"; t"lq(t 1)
q(tk)lq'; t')
== J'§q(tl)
q(tk)(exp iSlh)D(q)
XK(q(k) , q(kl)·t. 'lk ... dq(k) =
s
t,
lkl
=
f··· f K(q", q(k); t" 
tiJq(k)
)·q(k1)···q(l)K(q(1) ' q'·t. t')dq(l) '11
f··· J (exp  iH"(t" 
tiJlh) !J(q"  q(k»)q(k)
x (exp  iH(k)(tik  tik_ Jlh) !J(q(k) _ q(kl))q(k1) ... q(l) x (exp  iH(l)(t il  nih) !J(q(1)  q') dq<1) .. 'dq(k) = (exp iH"t"lh)q"(tik)q"(tik_J'" q"(tiJ
x (exp iH"t'lh) c5(q"  q')
q"(t) == (exp iH"tlh)q"(exp iH"tlh).
(F.5)
(F.6)
By substituting the final member of (F.5) into (F.3) and introducing the chronological ordering symbol T, which arranges noncommuting factors to the left with increasing values of t,
we obtain the formula that relates Feynman operators to the more conventional differential operators of quantum mechanics',
(q"; t" IA[q(t)]lq'; t')s =
(exp  iH"t"lh)T(A[q"(t)])(exp iH"t'lh) !J(q"  q').
(F.8)
If timederivatives of q(t) appear in the functional A[q(t)], the timedifferentiations transfer to the 8(k),S in (F.2), and it is clear that they must be put to the left of the T symbol; thus, for example,
Feynman Operators T(ii(ti)q(t Z))
125
== (d Z/dtiZ)T(q(ti)q(t Z))
= (d z /dt/)(a(t i  tZ)q(ti)q(t Z)
+ a(tz  ti)q(tz)q(td)
= a(ti  tz)ii(ti)q(tz) + a(t z  ti)q(tz)ii(ti)
+ b(ti
 t Z)(4(ti)q(tZ)  q(t Z)q(t1))
(F.9)
in which for for
1
a(u) ==
t
o
for
u>0 u= 0 u < O.
Equation (F.8) can be used to translate Feynman operator equations into a more conventional form that involves the familiar differential operators of Schrodinger and Heisenberg. For example, the operator EulerLagrange equations (q"; t"lbS/bq(t)1 q'; t')s = 0
(F.lO)
ii(t)+oV(q)/Oq!
(F.ll)
produce q=q(t)
=0
for the action functional S=
('(t4'4 V(q)) dt.
(F.12)
The Feynman operator equations (q"; t" I[bS/bq;(t 1)]q/tZ)!q'; t')s = iii bij b(t 1  tz)(q"; t" 111 q'; t')s (F.l3)
produce T(iii(ti)qj(t Z) + OV(q)/Oq;\
q=q(t,)
qj(t z)) =  iii bij b(t1  t z), (F.14)
126
Appendix F
where the identity operator is understood to appear on the righthand side; by making use of (F09), the operator equations (F.I3) become
ott, 
t2)(ii;(tt)
+ a(t 2 + o(tt
+ av(q)/oqil
tt)qj(t2)(ii;Ctt)
q=q(tll
)qj(t2)
+ av(q)/Oqi/
 t 2)(qi(tt)qj(t2) 
qi t2)qi(tt»
q=q(tt}
)
=  iii Oij o(tt  t 2)
(F.15)
and reduce to the Heisenberg commutation relations (F.16) because of the operator EulerLagrange equations (F.1I). Finally, we note that in the case of the action functional (F .12) with V(q) =
n
t L w/q/, i=t
(F.17)
the Feynman operator equations (F.B) take a form which is translated by (F08) into the operator equations (d 2/dt/ + W?)T(Qi(tt)qit2» = iii oij o(tt  t 2)0 (F.18) Equations (F.18) enable one to determine the causal Green'sfunction of perturbation theory, the ground state expectation value ofT(Qi(tt)qit2»' 2 2 R. P. Feynrnan and A. R. Hibbs, "Quantum Mechanics and Path Integrals," p. 180. McGrawHill, New York, 1965.
Appendix G Quantum and Classical Statistical Mechanics
Consider a dynamical system described by a Hamiltonian of the form H = U + V, where the kinetic U = U(p) is a function of a generalized linear momentum ntuple and the potential energy V = V(q) is a function of a generalized linear coordinate ntuple. For the quantum statistical description of the system, we have a (microcanonical) ensemble of normalized wave functions l/JJl = l/JJl(q; t) that satisfy the Schrodinger equation ih ol/Jjot = [U( ih ojoq) + V(q)]l/JJl
(G.l)
with J1 an enumerator index. Expectation values of observables are defined by!
f
<J>~ == L (OJl l/J/(q; t)f(q,  ih ojoq)l/Jiq; t) dq Jl
(G.2)
with the canonical variables in f(q, p) ordered to give an Hermitian operator." The (OJl's in (G.2) are probability weights for the states in 1 In order to have an enumerable set of states for H which admit running waves, socalled box normalization is understood here. 2 Our formalism does not require a specification of the Hermitian ordering rule (or rules); instead, the ordering of quantum observables is essentially free to be determined appropriately by experiment.
127
Appendix G
128
the ensemble, positive constants that sum to unity, (G.3) Let us introduce the hypercharacteristic function'
f
(GA)
with the properties"
0; t)
= l.
(Go5)
Expectation values are obtained directly from the hypercharacteristic function according to the formula
= [I( 
i %~
+ thl1, 
i 0/011  th~)$~]~=o=~
0
(Go6)
To prove (G.6), we first note the operator relations
+ thl1) exp[i~ . (q + thl1)] = exp[i~ . (q + thl1)]( i o/OIJ  th~) exp[i~ (q + thrO] = exp[i~ . (q + thl1)]( 
(  i %~
(
i %~
+ q + hl1)
(G.?)
0
i 0/011),
from which it follows that for a generic f(q, p) with the arguments ordered in a definite way,2 we have the operator relation f(  i O/o~
+ th'1,
 i 0/011 
+ th'1)] i %~ + q + hn, 
th~)exp[i~ . (q
= exp[i~ . (q + thr,)]f( 
i 0/0'1).
(G.8)
3 This function is the double Fourier transform of the quasiprobability density defined by E. Wigner, Phys. Rev. 40, 749 (1932). For a pure state (that is, one value of f.l. entering the summations in (G.2) and (G.4) with the W/l equal to unity) wave packet, the quasiprobability density
is significant in magnitude through a q, p phase space volume about equal to (21Th)" and displays a mean motion along the classical trajectory in phase space. 4 The hypercharacteristic function also possesses nonholonomic properties, as exemplified by the bounding condition I~I:;;;; 1; other nonholonomic properties follow from the results ofG. A. Baker, Jr.,Phys. Rev. 109, 2198 (1958); T. Takabayasi, Progr. Theor. Phys. 11, 341 (1954).
129
Quantum and Classical Statistical Mechanics
Next, we let both sides of (G.8) act on I/!iq + hn ; t), multiply from the left by OJItI/!It*(q; t), integrate over q and sum over J1 to get
=
L OJIt It
J
I/!/(q;
t)exp[i~' (q + th/1)]
x [f( i ojo~ + q
+
hn, i ojO/1)I/!It(q
+
hn; t)] dq,
(G.9)
where use is made of the definition (G.4) with a displacement of the integration variable q ~ (q + th/1). The square bracket term in (G.9) equals
[f(  i ojoe + q + h/1' ih ojoq)l/!lt(q + hn: t)]
= U(q + hn,
ih ojoq)l/!iq
+ hn ; t)],
(G.lO)
independent of how the arguments are ordered in f( q, p), and so from (G.9) and (G.2) we obtain formula (G.6). In view of the Schrodinger equation (G.I), the dynamical evolution of the hypercharacteristic function (and thus of expectation values (G.6)) is expressed by ocfJ~jot
= (ih[U(  i 0;0/1 + the)  U( i 0;0/1  the)
+
V(ia;o~th/1) V(io;o~+th/1)]cfJ~.
(G.II)
Essentially timedependent problems in quantum statistical mechanics can be solved by integrating (G. 11) subject to a prescribed initial form cfJie, /1; 0), but equilibrium stationary states are not determined completely by the static solutions to (G.11), requiring an additional postulate for the theoretical determination of statistical equilibrium. With no condition imposed on the value of p, the probability density for values of q is
P~(q; t) == L OJIt II/!It(q; tW It
= (2n)n
Jei~' qcfJ~(~, 0; t) d~.
(G.12)
The formalism for classical statistical mechanics appears as the classical limit, h ~ 0, of die preceding formalism for quantum statistical mechanics. Expectation values in classical statistical mechanics are identified with the classical limit of the expectation values in quantum statistical
130
Appendix G
mechanics given by (G.6), <J>o == lim <J>h = [f(  i aja~,  i a/017)<])o]~=o=~, h.... O
(G.l3)
in which the characteristic Junction of classical statistical mechanics, <])0
=
<])o(~,
17; t) == lim <])h(~' 17; r),
(G.14)
h ....O
satisfies the limiting form of (G.ll), 5 o<])%t =
i[~
. grad U( i 0/017) 17 . grad V(  i
%~)]<])o,
(G.l5)
where ntuple gradients of the energy terms are denoted by grad U (p) == oU/op and grad V(q) == oV/oq. The probability density for values of q in classical statistical mechanics, with no condition imposed on the value ofp, is identified with the classical limit of (G.l2), PO(q; r) == lim Ph(q; t) = (2n)n h.... O
f ei~·q<])o(~, 0; t) d~.
(G.16)
We introduce the Fourier transform of <])0' W = W(q, p; t) == (2n)2n
f ei~.qi~·P<])o(~, 17; t) d~ d17,
(G.17)
a real normalized function because of the properties of the hypercharacteristic function (G.5), W(q,p; t)* = W(q,p; t), f W(q,p; t) dq dp = 1. (G.18) Interpreted as the phase space probability density for values of q and p over the (microcanonical) ensemble, it can be shown that the quantity (G.l7) is nonnegative as a consequence of definitions (GA) and (G.l4).4 In terms of the probability density (G.l7), Eqs. (G.l3), (G.l5), and (G.l6) reduce to the familiar form for classical statistical mechanics: <J>O
=
(G.l9)
fJ(q,p)W(q,p; t)dqdp,
oW/ot = (grad U(p»' oW/oq
+ (grad
V(q»· oW/op
= [H,
W],
(G.20) PO(q; t)= f W(q,p; t) dp.
(G.2l)
5 Note that for U(p) quadratic in p and V(q) quadratic in q (that is, for Han element of the Lie algebra d,), Ii cancels out in the differential operator on the righthand side of (0.11), and the dynamical equations for
Appendix H
The Iteration Solution of Perturbation Theory
Consider a dynamical system described by a Hamiltonian of the form H = H o + Hint> where H o = Ho(q, p) is simple (that is, quadratic in the canonical variables) and Hint = Hinlq) is relatively small compared to H o for typical values of the canonical variables. The Schrodinger quantum dynamical description of the system is based on the equation ih 8lj;/8t = (H,
+ Hint)lj;
(H.l)
satisfied by the wave function lj; = lj;(q; t), where
H, == Ho(q,  ih 8/8q).
(H.2)
It follows from (H.I) that the interaction wave function lj;int = lj;inlq; t) == (exp iHot/h)lj;
(H.3)
satisfies the equation with so that
(HA)
q(t) == (exp iHot/h)q(exp  iHot/h) Hint(q(t»
= (exp iHot/h)Hinlq)(exp
 iHot/h).
(H.5) 131
132
Appendix H
Subject to a prescribed form for the interaction wave function at t = t', the formal solution to (HA) at a subsequent t = til is given by
l/Jint(q; til) = T(ex p  i (Hint(q(t))dt/h)l/Jiniq; 1'),
(H.6)
in which the chronological ordering symbol T arranges noncommuting factors to the left with increasing values of t,
T(Hint(q(t l))··· Hint(q(tk))) == Hint(q(tik))'" Hint(q(t i)),
(H.7)
We verify that (H.6) satisfies (HA) by noting the relation
T( exp  i ('+EHint(q(t)) dt/h)
Jt~"Hiniq(t)) dt/h),
(H.8)
Hint(q(t"))) T( exp  i (Hint(q(t)) dt/h) = O.
(H.9)
= (1  ieHint(q(t"))/h + O(e2))T( exp 
i
which implies that
(ih
a~" 
The iteration solution' of perturbation theory is obtained by expanding the exponential in (H.6),
l/Jint(q; t") = k~o(k!)l(i/h)kT(  ('Hint(q(t)) dtfl/Jiniq; 1'). (H.10) Moreover, by inverting the definition (H.3) and recalling the propagation kernel equation
l/J(q"; t") =
JK(q", q'; t"  t')l/J(q'; t') dq',
(H.U)
we obtain the formula
K(q", q'; t"  t')
= L (k!)l(i/h)k(exp iHo"t"/h) 00
k=O
x T(  ('Hint(q"(t)) dt)\ex p iHo"t'/h) x (j(q"  q') 1
F. J. Dyson, Phys. Rev. 75, 486 (1949).
(H.12)
The Iteration Solution of Perturbation Theory
133
where H o" == Ho(q", ih ojoq")
(H.l3)
q"(t) == (exp iHo"tlh)q"(exp  iHo"tjh).
According to the formula (F.8), the terms in (H.12) are Feynman operators computed with respect to the main part of the action functional S = So + Sint associated with H = H o + Hint, So = So[q(t)]
=
r t"
(H.14)
Lo(q(t), 4(t» dt,
where Lo(q, 4) = p . 4  Ho(q, p) with p eliminated in favor of q and q by evoking the ntuple Hamilton canonical equation 4 =oHjop = oHojop. In view of(H.12) and (F.8), we have? K(q", q'; t"  t')
=
L (k!)  l(ijht(q"; t" I(SinJkl q"; t')so 00
k=O
= (q"; t",I(exp iSintjh)1 q'; t')so = (q"; t" 111 q'; t')s'
(H.15)
where Sint == S  So = Sint[q(t)]
= 
r
J Hin,(q(t» t'
dt.
(H.16)
Conversely, the standard form of the interation solution (H.IO) foIIows deductively from (H.15). 2 Direct applications of(H.l5) are worked out by R. P. Feynrnan and A. R. Hibbs, "Quantum Mechanics and Path Integrals," pp. 120161. McGrawHill, New York, 1965.
Appendix I
Existence of Spatially Localized SingularityFree Periodic Solutions in Classical Field Theories
A necessary condition for the existence of spatially localized singularityfree periodic solutions to EulerLagrange classical field equations is readily obtainable by applying dilatation (scale) considerations to the variables in the action principle. In particular, the dilatation considerations can be applied to the action principles for Einsteintype unified theories,' in which the existence of spatially localized singularityfree periodic solutions is crucial to the physical interpretation. We illustrate the dilatation considerations that lead to the necessary condition for existence of spatially localized singularityfree periodic solutions by treating the generic local theory based on the Lagrangian density
!£ = t4>· 4>  t L 'V 3
i; 1
i
¢ ' 'Vi ¢  U(¢),
(1.1)
where ¢ = ¢(x; t) is a field amplitude real ntuple and u(¢) is a selfinteraction energy density real ltuple. The action principle C5SjC5¢(x; t)
=0
S
= S[¢(x; t)] ==
f J!£ dx dt T
o
(1.2)
1 A. Einstein, .. The Meaning of Relativity," pp. 139166. Princeton Univ. Press, Princeton, New Jersey, 1955.
134
SingularityFree Periodic Solutions
135
for a solution periodic in time, ¢(x; t global condition
+ T) =
(dS[¢(..1.x; t)J/dA).l=l
¢(x; t), implies the
=0
(1.3)
for a solution localized in space, where A is a real positive parameter associated with the specific variation of the field amplitude ¢(x; t) ~ ¢(..1.x; t) == ¢(x; t) + (j¢(x; t). Provided that the x integrations converge, we have
f
¢(AX; t)· ¢(AX; t) dx
= A 3
f
¢(x; t)· ¢(x; t) dx
and similar relations for the other terms in thus, (1.3) works out to give
(f
(i¢·
Sfi' dx
¢ + !itlVi¢· Vi ¢ + 3U(¢))
(1.4)
with ¢ = ¢(..1.x; t);
dx dt
=0
(1.5)
with ¢ = ¢(x; t). An alternative way to derive (1.5) is to contract the field equations prescribed by (1.1) and (1.2), (1.6) with the real ntuple functions (2:}= 1 Xi Vi ¢) and integrate the resulting equation over all x and over the range 0 to T for t. On the other hand, if we work out (1.7) or if we contract (1.6) with ¢ and integrate, we obtain
(1.8) again provided that the x integrations converge and T is the period of the solution. Finally, by subtracting twothirds of (1.5) from (1.8), we find the global condition
( f (~i~ V
i¢·
Vi ¢
+ ¢. (ou(¢)jo¢)
 2U(¢)) dx dt
= 0, (1.9)
which implies that !¢ . (ou(¢)/o¢) < u(¢)
(1.10)
Appendix [
136
for some values of ¢ is a necessary condition for existence of a localized singularityfree periodic solution to Eqs. (I.6). Hence, such solutions are precluded in theories with positivedefinite selfinteraction energy densities of simple algebraic form, as exemplified by
(1.1 1) with m", g real positive constants. Spatially localized singularityfree periodic solutions have been found, however, for local field theories with Lagrangian densities of the form (1.1) that feature more complicated positivedefinite selfinteraction energy densities (satisfying the condition (1.I0)V 2
G. Rosen, J. Math. Phys. 9, 966 (1968), and works cited therein.
Appendix J
Rigorous Solutions in Essentially Nonlinear Classical Field Theories
Exact solutions to the field equations of an essentially nonlinear theory follow from the reduction of the system of partial differential equations in the spacetime independent variables to a system of ordinary differential equations in one independent variable, there being a welldeveloped general integration theory for systems of nonlinear ordinary differential equations.' Ignoring trivial reductions in which the field is prescribed to depend only on a single spacetime coordinate, one must apply a grouptheoretic method'" 3 (based on the invariance character of the specific field equations) in order to accomplish the reduction to a system of ordinary differential equations. We illustrate the grouptheoretic method of reduction by considering classical field equations of the form (J.l) 1 For example, A. R. Forsyth, "Theory of Differential Equations," Vols. II and III. Dover, New York, 1959. 2 A. J. A. Morgan, Quart. J. Math., Oxford Ser. 2, 250 (1952). 3 G. Birkhoff, "Hydrodynamics," Chaps 4 and 5. Princeton Univ. Press, Princeton, New Jersey, 1960.
137
Appendix J
138
where the components of the real ntuple field amplitude ¢ are scalar functions of the spacetime coordinates t == Xo == Xo, x == (Xl' X2' X3) == (x', x 2, x 3), and u(¢) is a selfinteraction energy density real ltuple. Manifest Lorentzinvariance of the n field equations (1.1) implies/ the existence of solutions of the form
¢ = «(r),
(J.2)
in which the Lorentzinvariant scalar argument of ()( is r
== tx~x~ == t(  t 2 + x/ + x/ + x/).
(J.3)
Indeed, by putting (J.2) into (1.1), we obtain the system of ordinary differential equations for ()( = ()((r) , (1.4) Less obvious reductions of Eqs. (1.1) to systems of ordinary differential equations depend on the specific algebraic character of the selfinteraction energy density. For u(¢) a homogeneous function of the field amplitude ntuple, for all real
A. > 0,
(1.5)
where y is the degree of homogeneity, the grouptheoretic method of reduction is based on the invariance of Eqs. (J.l) under spacetime dilatation transformations (J.6) From the invariance under dilatation transformations (J.6), it follows that if ¢(x") is a solution to (J.l), then so is A.2/(Y2)¢().X~) for all real ). > O. Moreover, a general theorem" guarantees existence of selfsimilar solutions to (J.l) such that for all real
). > 0
(1.7)
if the selfinteraction energy density has the homogeneity prescribed by (J.5). Furthermore, it is possible to reduce Eqs. (1.1) to systems of ordinary differential equations for specific selfsimilar solutions." For example, if we seek a selfsimilar solution that depends on r(>O) defined by (1.3) and (k~x~) with the k's constants, it must take the form
¢ = r l /(2 
Y){J(s),
[y
=f. 2],
(J.8)
Rigorous Solutions in Classical Field Theories
139
where the argument of 13 is the dilatationinvariant scalar s == (k llx ll )/r 1 / 2 • (1.9) Indeed, by putting (J.8) into (1.1), terms in powers of r cancel out, and we obtain the system of ordinary differential equations for 13 = f3(s), 2f3
Il 1 2 d 3 df3 (kllk 4 S) ds 2 4 S ds
(3  y)
_ ou(f3)
+ (2_y)2f3Jj3'
(1.10)
If alternative selfsimilar forms are taken in place of the ansatz (J.8), the systems of ordinary differential equations that result admit determination of other rigorous solutions to Eqs. (1.1) with (1.5).
Appendix K
Quantum Theory of Electromagnetic Radiation
Let us consider the electromagnetic field associated with a prescribed static solenoidal current density j = j(x) = Ul(X), j2(X), jix)), oj(x)jot == 0 == V . j(x), and a vanishing electric charge density. Such a field is a linear superposition of free radiation and magnetostatic parts. Letting ¢ = ¢(x) = (¢l(X), ¢ix), ¢3(X)) denote the electromagnetic vector potential and n = n(x) = (n1(x), nix), 1t 3(x)) the canonically conjugate electric vector field, the Hamiltonian is given by H
J
= H[¢, n] = (1n . n + t(cur1¢) . (curl o) + ¢ . j) dx.
(K.l)
In order to complete the dynamical description for the electromagnetic radiation, the Hamilton canonical field equations derived from (K.l), ¢(x)
= 8Hj8n(x) = n(x)
1i:(x) = 8Hj8¢(x)
=
curl(cur1¢(x))  j(x),
(K.2) (K.3)
must be supplemented by the physical condition on the electric vector field for a vanishing charge density, V· n(x) = 140
o.
(KA)
Quantum Theory of Electromagnetic Radiation
141
Equations (K.2) and (K.3) combine to yield an inhomogeneous linear wave equation for the vector potential, ¢(x)
+ curl(curl cf>(x)) =  j(x).
(K.5)
By introducing the transverse part of the vector potential
in which the transverse bfunction has the Fourier representation b:.J(X) = (2n)3
J{bij 
[k;kjf(k' k)]}(exp ik· x) dk,
(K.7)
we find that Eq. (K.5) splits up to give ¢tr(x)  V 2 cf> tr(x ) = j(x)
(K.8)
V . ¢(x) = O.
(K.9)
It should be noted that the consistency of (K.4) for all time is guaranteed
by (K.2) and the latter dynamical equation (K.9). To effect quantization of the electromagnetic radiation field theory, we satisfy the canonical variable commutation relations [cPi(X; t), 1t j(Y; t)] == (ih)l(cP;(X; t)1t/y; t)  1t/Y; t)cP;(x; t)) =bijb(XY)
(K.lO)
at the instant of time t = 0 by taking the vector potential to be diagonal for all x, cP(x; 0) == cf>(x), and representing the electric vector field by means of a functional differential operator, 1t;(x; 0) =  ih bfbcf>i(X).
(K.I I)
Then from (K.I), we obtain the quantum Hamiltonian H
= H[cP(x; O),1t(x; 0)] =
h2 s s 1 J( "2 bcf>(x) . bcf>(x) + 2 cf>tr(x) . (  V
2)cf>tr(x)
+ cf>tr(x) . j(x)
)
dx,
(K.12) where the transverse part of the vector potential (K.6) is brought in through the relations curl cf>(x) == curl cf>tr(x) and V . j(x) == O. In view
Appendix K
142
of (K.4) and (K.ll), physically admissible wave functionals 'P[¢; t] must satisfy the condition
v . 15'P[¢; t]/15¢(x) =
O.
(K.13)
Hence, physically admissible wave functionals depend only on the transverse part of the vector potential,
'P[¢; t] == 'P[¢!'; t],
(K.14)
because (K.l3) implies 15'P[¢ + VX; t] = _ V' 15'P[¢ + VX; t] == 0 15X(x) 15¢(x) for arbitrary real Ituple functions X = X(x), and shows that
'P[¢
+ Vx;t]
is independent of X. Stationary states of the field are represented by wave functionals 'P[¢; t] = (exp iElLt/h)UIL[ ¢ ] that satisfy the Schrodinger equation. Thus, we have (K.l5) where (K14) requires UIL[ ¢] to depend only on the transverse part of the vector potential, (K.16) Subject to (K.16), the ground state eigenfunctional solution to (K.15) with the functional differential operator quantum Hamiltonian (K.12) is given to within normalization by
and the associated ground state energy eigenvalue is
(K.17)
It is only necessary to work out the second functional derivatives of
(K.17) and to recall the definition 15 2Uo/15¢i(x)2 = lim 15 2Uo/15¢i(x) 15¢;(y) y_x
Quantum Theory of Electromagnetic Radiation
143
in order to verify that (K.17), (K.l8) satisfy (K.15) with (K.12). The first term in the ground state energy (K.l8) is the zeropoint energy of the electromagnetic radiation vacuum, an unobservable infinite constant, while the second term in (K. I8) is the classical Ampere selfinteraction energy of the current density,' expressed in the familiar form  I [j(x) . j(y)/8n Ix yl]dx dy
by using the relation (V 2)1£5(xy)=(4nlxyl)I. To obtain the excited state eigenfunctional solutions to (K.15) with E" > Eo, it is convenient to introduce the quantity 0" = O,,[] == U,,[¢]/Uo[ ] == O,,[¢tr] which satisfies the simpler functional differential equation I
±(
i= 1
h
2
2
8 ° "2 + h[¢lr(x)(  V2)1/2 + Hx)( _ V2)  1/2] £50,,) dx
2 b¢i(X)
b¢i(X)
= (E"  Eo)O"
(K.19)
as a consequence of (K.15) and the form of (K.l2). Solutions to Eq. (K.l9) are readily obtained as polynomial functionals in ¢tr. For example, the onephoton state solution takes the form (K.20) where ljJ(x) is the onephoton wave function, a solenoidal solution to the relativistic wave equation (K.21) which admits bounded solutions for all finite real positive values of (E I  Eo); the twophoton state solution is O 2 = t II [¢tr(x) X
+ j(x)( 
dx dy  fh
2(E
2 
r
V x2 I] 'ljJ(x, y) . [¢tr(y) E O) 
3
1
Ii~/ii(X, x)
+ ( V/)1 j(y)]
dx
(K.22)
with the twophoton wave function ljJ(x, y) a solenoidal (that is, symmetrical (that is, ljJij(x, y) = IjJjJy, x) solution to the relativistic wave equation
Lr =1 iJljJi/x, y)/iJx i = 0 = LI =1 iJljJi/x, y)/iJy) n[(  V/)1/2
+ ( V/)1/2]ljJij(X, y) = (E 2 
E o)ljJiix, y)
(K.23)
1 For a recent and novel derivation of this electromagnetic radiation energy term, see J. Schwinger, Phys. Rev. 173, 1264 (1968).
144
Appendix K
which admits bounded solutions for all finite real positive values of  Eo); solutions to (K.19) describing three or more free photons are given by higher order polynomial functionals in ¢lr. This socalled coordinatediagonal representation for the quantum theory of electromagnetic radiation, with the quantum Hamiltonian prescribed as the functional differential operator (K.l2), can also be employed in theories of chargedparticle interaction. In particular, the coordinatediagonal representation has been applied recently to certain basic problems in quantum electrodynamics.f
(E2
2 G. Rosen,Phys. Rev. 172, 1632(1968); Nuovo Cimento Yl, 870.(968);J. Franklin Inst, 287, 261 (1969).
Appendix L
Stationary States in Quantum Field Theories
Consider the class of local field theories based on Lagrangian densities of the form !£ =
tcP . cP 
f(V4»' (V4»  u ==
n
t I [(cPY i=l
(V4»2]  u,
(L.I)
where ¢ denotes a real ntuple field amplitude and the selfinteraction energy density u = u(¢) is a real ltuple function of ¢. From the canonically related Hamiltonian for such a theory, H
J
= H[¢, n] = (tn'
tt
+ !(V¢) . (V¢) + u) dx,
(L.2)
we obtain the quantum Hamiltonian
H:IE H[¢(x),  ifu5j8¢(x)]
=
h2 8  . 8 + 1 ¢(x) . (  V J( 2 8¢(x) 8¢(x) 2
2)¢(x)
+ u(¢(x)) )
dx
'
(L.3)
and for the ¢dependent part of a stationary state wave functional 1fI[¢; t] = (exp iEl'tjh)UI'[¢]' we find the eigenfunctional equation (L.4) 145
Appendix L
146
with Ep. the constant energy eigenvalue. Physically admissible solutions to Eq. (LA) are such that /Up.[¢JI 2 , the relative probability density for locating the state at the field amplitude ntuple ¢ = ¢(x), vanishes for unbounded field magnitudes,
IUp.[¢JI 2 =
lim
(q, . q,)+ 00
o.
(L.5)
The J1 = 0 vacuum state solution to (LA) is associated with the energy Eo == minp.{Ep.}; once Uo[¢J is obtained, the general stationary state eigenfunctional problem reduces to solving the equation
h
2
J.i~l n
(

1 b
20p.
2 b¢;(X)2 
b(in Uo[¢J) so, ) b¢i(X) b¢;(x) dx = (Ep.  Eo)Op. (L.6)
for Op. = Op.[¢J == Up.[¢J/Uo[¢J because the lefthand side of (L.6) works out to give
h2
Ltl ( t
U
o [b2Up./b¢ i(X)2J + !U0 2Up.[b 2U 1
O/b¢;(X)2J)
= U o2(UoHUp. 
Up.HUo)
dx
(L.7)
with H prescribed by (L.3). Approximate physical solutions to Eq. (LA) with the Hamiltonian operator (L.3) have been obtained for certain generic classes of entire analytic u(¢) by applying the RayleighRitz procedure for functionalities.' In the following, we discuss a class of nonlinear field theories for which exact physical solutions to Eq. (LA) are obtainable in closedform. These theories feature a selfinteraction energy density that converges rapidly for large field magnitudes to the form representative of a linear field theory of uncoupled field amplitude components, a u = u(¢) expressible as n
U
=! L m/¢/ + r5If i= 1
(L.8)
where r5If = r51f(¢) is a continuous singularityfree real Ituple function such that 1 G. Rosen, Phys. Rev. Let. 16, 704 (1966); Phys. Rev. 173, 1680 (1968), and works cited therein.
147
Stationary States in Quantum Field Theories
lim ('00
lim [¢2i!l1(¢)] = 0 ''>00 [(¢. ¢)[In ¢ . ¢]i!lI(¢)] = 0
if n = 1, if n = 2,
(L.9)
lim [(¢. ¢)i!lI(¢)] = 0 if n ~ 3. ('00 For a continuous i!lI(¢) that tends to zero for large field magnitudes with the asymptotic behavior prescribed by (L.9), we have existence of the function  Joo I¢  ¢'I i!lI(¢') d¢' 00 j(¢)
== 
~ J[ln (¢ 2n
r(tn  1) "/::2n n 2
J
for
n= 1
¢,). (¢  ¢')]i!lI(¢') d¢'
for
i!lI(¢') d¢'
( [(¢ _ ¢'). (¢ _ ¢')r/2)
1
for
n
=2
n
~
3
(L.IO) that satisfies the ndimensional Poisson equation n
L
i= 1
82j(¢)/8¢/
=
2i!l1(¢).
(L.ll)
The asymptotic behavior of j( ¢) for large field magnitudes follows from (L.9) and (L.IO) asj(¢) ex I¢I if n = l,j(¢) ex [In (¢ . ¢)] if n = 2, and j(¢) ex (¢ . ¢)1(n/2) if n ~ 3. In terms of the function (L.IO), the J1. = 0 vacuum state solution to (LA) with (L.8) in (L.3) is U o[ ¢ ] = ex p( 
2~ J;~ ¢;(x)( 
2+ m/)1/2¢;(x) dx  :2 Jj(¢(x)) dX)'
V
(L.12) where the limit B ~ 0 is understood to be taken as the final step in a computation involving Uo[ ¢ ] , in conjunction with a limit representation of the (lfunction limt>o (l(t)(x) = (l(x) such that? (l(tiO) =
B
1•
(L.13)
2 An immediate way to secure this relation is to introduce the wavenumber cutoff limit representation
S(,i x) =
Jlk l"K(exp ik . x) dkj(27T)3
Appendix L
148
To verify that (L.12) is an exact solution to (LA), one simply computes ilU o[¢] il¢;(x) 2U il O[¢] (j¢i(X)2
=
{_!(_ V +
{I
=  h [(
+
2
Ii
1 Ii 2
[( 
V
2
V2
m/)1/2¢i(X) _ ; 01(¢)1 }U o[¢]' Ii O¢i
+ mi)
1/2
s (j(x)]x=o  li 2
s () 021(¢)/ <E)
0 o¢/
+ m/)1 /2¢i(X)]2 + o(e)}u o[ ¢ ] ,
(L.15)
and makes use of(L.ll) and (L.13) to obtain in the limit s + 0 2 1i (j2Uo[~] + ! [( _ V2 + m/)1 /2¢i(X)]2U o[¢]) ;=1 2 (j¢i(X) 2
J{±(
+ o7t(¢(X))U o[¢]} dx =
Eo U o[¢],
(L.16)
where the (unobservable) vacuum state energy appears as the infinite constant Eo = (lij2)
±[(
i= 1
V2
+ m/)1 /2(j(x)]x=0
Jdx.
(L.17)
By substituting (L.12) into Eq. (L.6), we find
JI
n (
i= 1

2
(j20 11_ + [ Ii¢,(x)( _v2 2 2 il¢;(x)
1i
01 1 + m/)1 /2 + e .:.
] _ ilO_ 11 ) dx O¢i
(L.18) an equation which facilitates determination of excited state eigenfunctional solutions to (LA), UIl[¢] = UO[¢]OIl[¢] with Ell> Eo. Solutions to (L.18) are continuous in e about e = 0, and therefore the limit e + 0 obtains validity in the equation itself. Hence, the field theory is effectively linear with twoparticle and multi particle state solutions exhibiting no interaction between the quanta.
Index
A Action functional, 7, 11,55, 59 principle, 8, 11,56, 59 Algebra, see Lie algebra Anticommutation relations, 84
c Canonical equations, see Hamilton canonical equations Canonical transformations, 12, 1620, 4650, 6366, 8891 Canonical variables, 9, 11, 30, 41, 59 Characteristic function, 130 Chronological ordering, 4041, 7980, 132133 Class, 7,96 Constants of motion, 13, 42, 62, 85, 87 Contact transformations, see Canonical tranformations D
Dilatation transformations, 56, 134136, 138139
Dirac, 94 bracket, 42, 85 correspondence, 4345, 8687 electron, 4041, 84, 118122 Dynamical law, 5, 52 Dynamical principle of superposition, 22, 68 Dynamics, 5, 22, 52 E
Einstein, 95 field theories, 134 Electrodynamics, quantum, 9394, 144 Electromagnetic radiation, quantum theory of, 140144 Energy canonical, 6, 54, 57 density, selfinteraction, 53 eigenstates of, see Stationary states eigenvalues, 3839, 8182 multiplets, degenerate, 39 observable, 83 149
Index
150 Energy (conz.) potential, 5 vacuum, 81 Enveloping algebra, 13 EulerLagrange equations, 5, 30 for fields, 52, 74 Newtonian form of, 5 Expectation values, 38, 80, 127 F
Feynman operators, 2930, 7475, 123126 Feynman postulate, 24, 30, 70 Field amplitude, 52 canonical variables for, 57 linear, 53 local, 53 momentum density, 57 nonlinear, 53, 56 semilinear, 54 solutions for, 134139 velocity, 52 Functional, 96 analytic, 99 continuous, 96 derivative, 9799 extremum of, 98 integral, 2427, 7071 integration by parts, 30, 116117 Laplacian, 99 linear, 97
G Generalized coordinate ntuple, 4 Generalized momentum ntuple, 8 Generalized velocity ntuple, 4 Generators, 19,37,48,6465,101 Green's function, causal, 75, 126 H
Haar measure, 2527, 69, 71, 110115 Hamilton canonical equations, 9
Hamilton canonical field equations, 57 Hamiltonian, 9, 57 Heisenberg commutation relations, 126 Heisenberg picture, 41, 85 Hypercharacteristic function, 128129 I
Interaction wave function, 131 Interaction wave functional, 79 Iteration solution, 29, 37, 74,79,131133 K
Kepler Hamiltonian, 46 L
Lagrangian, 16, 52, 54 density, 53 Legendre transform, 9, 10 Lie algebras, 1316, 4246, 6162, 8791 conjugate, 1415 Lie groups, 19, 101109 Lie product, 13 Lie quadratic identities, 19, 102 N
ntuple, 4 Normalization, 25, 27, 36, 70
o Observables, 12,41, 46, 60, 85 P Pauli neutrino, 4041, 84 Phase space, 18 Planck's constant, 24 Poisson bracket, 1213, 17, 60 Probability density, 22, 68 Propagation kernel, 2324, 6970
Index
151 Q
Quantum Hamiltonian, 37, 40, 79 Quasiclassical approximation, 3334, 77
Statistical mechanics, 127130 Structure constants, 19, 48, 102 Symmetry algebra, 21, 50, 66 Symmetry group, 21, 50, 66, 92 Symplectic group, 14
R
RayleighRitz procedure for functionalities, 146 Relativistic sumoverhistories, 118122 Renormalization, 93
s Smatrix, 80 Schrodinger equation, 3738, 7879 Semidirect sum, 14 Semigroup composition law, 23, 69 Space, 7, 97 State, 4, 9, 22, 52, 57, 68 ground, 39 Stationary states, 38, 8083, 142143, 145148
T Topological group, 26, 71
u Umatrix, 80
v Volterra expansion, 33, 99
w Wave function, 22, 34 Wave functional, 68 Wiener correlation function, 117 Wiener functional integral, 25, 117