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:U C\B\ —> C" is an injective, holomorphic map, smooth up to U n B~[. If (j)(U n S2n~l) C 5 2 " - 1 then <j){UC\Bx CBi). If a real hypersurface in C n is transverse to the radial vector field then restricting the 1-form n
a
= Yyxidyj - yidxA 3= 1
to the hypersurface also defines a contact structure. This contact structure is generally different from the contact structure induced on a strictly pseudoconvex hypersurface from the complex structure of C". We finish this section by considering the intersection of the two types of examples. Let X be a real analytic manifold and let X be the complexification of X. Then X embeds into A" as a totally real submanifold. The tubular neighborhood theorem implies that there are diffeomorphisms from a neighborhood of X in X to a neighborhood of the zero section of T*X. (j>:X-> T*X which reduce to the identity on X. One can fix this diffeomorphism so that the induced complex structure on the image of / satisfies certain properties. Choose a real analytic Riemannian metric g on X. This defines a function pg(£) = ||£||2 on T*X. The map, can be chosen so that (a) cj>: X —» X is the identity. (b) With the complex structure induced on the image, T0'l<j>{X) = (^•T0'1^, we require that 1: (:r,£) —> (x, —£) is an anti-holomorphic map. (c) Imdpg =a = £ " = 1 iidxi. (d) (dd^r = 0, away from pj x (0). Let X£ = {pg < e 2 }. It follows from the second item above that the contact structure induced on bX€ by the complex structure agrees with the contact structure defined on bXe by the symplectic structure on T*X. The subsets Xe are called Grauert tubes. For more on this construction see 7 or 8 .
40
3. CR-functions and a generalization of Toeplitz operators On a CR-manifold there is a canonical differential operator, analogous to the d-operator denned by Btf := df rTo,iM •
(14) N
If $ = (<j>i,... ,(J>N) is an embedding of M into C then, $ is a CRembedding if and only if dbfa = 0 for alii = 1,..., N. It follows from the Leibniz formula that the kernel of db is an algebra. If M is the boundary of a domain X in C" then the ker db consists of the restrictions of holomorphic functions in X to M. More generally if M is an abstract, compact, strictly pseudoconvex CR-manifold which is the boundary of X, a compact, complex manifold with boundary, then the kerdt again consists of the restrictions to M of holomorphic functions on X. If dim M > 5 then such a compact, complex manifold with boundary always exists. Moreover the induced co-orientation of the contact field agrees with the given coorientation. If dim M = 3 then the existence of such a complex manifold is a very subtle question. Generically such a manifold does not exist, see 9 . A Stein space is a complex space with a very large family of holomorphic functions. Indeed it follows from results of Remmert that any Stein space admits a proper holomorphic embedding into CN, for some N. Grauert showed that a compact, complex manifold with a strictly pseudoconvex boundary is a "proper modification" of a Stein space. Prom work of Heunemann and Ohsawa it follows that a strictly pseudoconvex manifold with boundary X can be augmented along its boundary so that X C X is a relatively compact subset in a proper modification of a Stein space and M is therefore a proper hypersurface in X. If a function / £ ker 9 o n X then / f M is in ker db- Restricting a proper holomorphic embedding of X into C^ to M we obtain a CR-embedding of M into CN. In this case the algebra of CR-functions is large, containing the closure of the algebra generated by the coordinate functions of the embedding. Using work of Boutet de Monvel, Kohn and Harvey and Lawson, one can show that a compact, strictly pseudoconvex CR-manifold has a CR-embedding into CN, for some N if and only if it is the boundary of a strictly pseudoconvex, complex manifold with boundary, see 10, n and 12. We call such a CR-manifold embeddable. We can generalize the construction of a Toeplitz operator defined earlier on the boundary of the unit disk to an embeddable, strictly pseudoconvex CR-manifold. Suppose that M is such a manifold. The ker db is infinite dimensional, containing enough functions to separate points on M. Choose
41 a volume form on M, and let H2(M) denote the L2 closure of ker C°°(Kn), one would like to represent it as integration against a kernel. In a certain sense, this is always possible. Theorem 4.1 [Schwartz kernel theorem]. There exists a unique element KA
fc t-
^JK
X K.
)
such that
Af = JkAf in the weak sense, i.e. < Af, g >=< kA, f(x) ® g(y) > for every
f,g£C?(Rn).
42
Using the variables x, x — y we can represent this as a family of convolution operators: Af(x) = / kA(x,x-
y)f(y)dy
We make the following assumptions on the kernel kA • (a) kA(x, Z) a smooth function on M" with values in tempered distributions on E". (b) kA(x, z) is in C°°(E" x (E" \ {0})) - the only singularities are along the diagonal, i.e. where z = 0. (c) In virtue of our first assumption for each fixed x, kA(x,-) has a Fourier transform. Let a(x, £) be the partial Fourier transform of the kernel, that is > € Coo(En;«S'(En)).
a(a;,0 =< kA(x,z),e-*z
If we assume that the support of kA in z is compact then a(x, £) S C°°(En x R n ). Let (p 2 then we can write kA(x,
z) = 2 . If b(X, D) is the quantization of this symbol then it has the following properties: b
*'^=
A o b(X,D)
- I e * K N ( * ) , a n d b(X,D)
oA-Ie
*^{X).
Using the composition formula for pseudodifferential operators, we can actually find an operator B such that AoB-Ie where a-m(B) = b.
tf-~(X),
and B o A - I e * « " ( * ) .
48
A consequence of this almost invertibility is a criterion for a pseudodifferential operator to be Predholm. Proposition 4.3. If A £ \P£J,(X) is elliptic then A is a Fredholm operator as a map A : HS(X) -> Hs-m(X) for any s. It is now easy to show that, on a compact manifold of dimension at east 2, the range of a classical pseudodifferential projection has either finite dimension or finite co-dimension. Let P be an orthogonal projection, acting on functions which is a standard pseudodifferential operator. As it is bounded it must be an operator of order 0. Since P2 — P = 0 it would follow that ao(P)(ao(P)-l) = O. Hence ao(P) is either identically 0 or identically 1. If T-1, 7T2 : T*X -> R0. We define a parabolic action of M+ on T*X by letting Mx(O = ATTI(O + A2TT2(£) for £ e TX*X The trajectories look like parabolas with respect to (6,T"L) coordinates. Before we consider the compactification of T*X defined by this action, we examine the action in more detail. To understand this action, we use Theorem 2.2 to introduce Darboux coordinates. These are local coordinates, (t,x,y) on X such that: 9 = dt+ -(x-dy-y-dx),
the Reeb vector field is T = dt- We employ the following notation for covectors n
(&,£') = (£o,Ci,...,6n) ^^odt + Yl^jdxj
+£j+ndyj.
The projections in these coordinates are given by: " 1 1 TTi(&,£') = ]T(& + -yi(,o)dxi + fe+n - -Xi£0)dyi. The coefficients are linear functions on the cotangent space and therefore the symbols of vector fields,
ii + \yjto = e(\(dXi + \yA)),
(18)
Zj+n - 2 ^ 0 = cr(^{dyj - -Xjdt)). We denote these vector fields by: T:=dt,
Xr.= dXl + \dt,
Yr.= dyi -\xjdt.
(19)
50
The contact field is locally spanned by {X,,Y}}. These vector fields satisfy the following commutation relations:
[Xt,T] = [Yk,T] = 0, [Xj,Xk] = [Yj,Yk]=0, [Xj,Yk]=-8jkT. These relations define the Heisenberg Lie algebra. Integrating these vector fields gives the Heisenberg group structure on R 2n+1 . The addition is defined by (t, x, y) © (t1, x', y') = (t + t' + -{y1 • x - x' • y), x + x', y + y'). The parabolic dilation structure "twists" as the base point moves. By this we mean that there do not exist local coordinates (xo, • • • , #2n) such that, in terms of the linear coordinates, {/3j} defined on T*X by the local co-frame {dxo,... ,dx2n} the dilation structure takes the form Mx(x, /?) = (x; A2/J0l Aft,..., A/32n). This is a reflection of the non-integrability of the contact field. Instead we use the vector fields {T,Xj,Yk} to "flatten" the dilation structure. The symbols of these vector fields, {ao,..., f(y)dyd^.
(22)
We use this formula because Weyl quantization has better invariance properties than the left quantization used above. Definition 6.1. An operator A : C£°(X) -> C~°°(X) is a Heisenberg operator of order m on X if For any pair is a contact transformation then <j)*A~1* is again a Heisenberg operator. The reason for this is quite simple: If X is a diffeomorphism then the map * : T*X -» T*X has a smooth extension to HT*X if and only if <j> is a contact transformation. Observe that the symbols of order zero are defined, in a coordinate invariant way, as the set of smooth functions on a compact manifold with boundary. Exercise 10. Prove that a diffeomorphism <j>* : T*X —> T*X has a smooth extension to HT*X if and only if 0 is a contact transformation. If A G ty™(X) then it has local representations as in (22) where a € S™(U). Such a symbol has an asymptotic expansion in Heisenberg homogeneous terms. It can again be shown that the leading term in the expansion is well defined modulo lower order terms. We call this the Heisenberg principal symbol of A which we denote by a^(A). In the manifold context we interpret the ^-variable as the linear coordinate on the fibers of T*X induced by the local coordinates x. To better understand the link between the class of Heisenberg operators and the Heisenberg group we change variables in (22) as follows: For each x define the (2n + 1) x (2n + 1) matrix
(
i _i 1
i\
2 2\
0 1 0 0 0 1/
where x = (xo,x,x),y = (yo,y,y), and x,x,y,y G R n . Observe that L " 1 = L-x. If © denote the Heisenberg addition operation then x®y~l = Lx(x-y).
53
Now formula (22) reads
aw(X,D)f = =
fL{^,tiy C°°(X- A°b'q).
The rib-operator on (0, g)-forms is defined by D6c* = (8;8b + 8b8*b)a.
60 A calculation shows that P i=i
HP
je/3
+ lower order terms, see Beals and Greiner. 16 Using the coordinates introduced above, we obtain that, at the center x of this coordinate system, D a
* = -\ E 0 L > 2 + Y f + ~^n - 2M = /o
ranged' is the identity up to a compact error. Using a similar construction leads to an approximate right inverse. This proves the proposition. • Definition 9.5. The relative index is defined to be the Fredholm index of the restriction R-Ind(S, S') = Ind(S'S : range S -> range S'). The relative index for generalized Szego projectors satisfies a co-cycle relation: Proposition 9.2. If S, S', S" are generalized Szego projectors at level zero then then R-Ind(S, S") = R-Ind(S,5") + R-Ind(S", S").
(36)
Remark 9.3. Relative indices are a basic tool for relating one kind of index problem to another. The relative index labels the path components of the space of generalized Szego projections. That is two generalized Szego projectors have relative index zero if and only if there is a smooth path through generalized Szego projectors from one to the other. This result is proved in 19. The notion of relative indices appears in the literature on index formulae for boundary value problems. In the context of boundary value problems, the projections are classical pseudodifferential operators often assumed to have the same principal symbol. The relative index was introduced in the context of the Szego projectors defined by integrable CRstructures in 9 and in the generality considered here in 19. We now define generalized Toeplitz operators. Definition 9.6. A generalized Toeplitz operator is an operator of the form
TAf = SASf where S is a generalized Szego projector and A € y™(M). Such operators were considered in Boutet de Monvel-Guillemin 18, but A was assumed to be a classical pseudodifferential operator. As we shall see this is, up to lower order terms, the same class of operators. As before for generalized Szego projectors, we can define generalized Toeplitz operators at level k and to level N.
68
Definition 9.7. A generalized Toeplitz operator at level A; is a operator of the form
TAf = SASf where S is a generalized Szego projector at level k and A E \I>™(M). Definition 9.8. A generalized Toeplitz operator to level N is a. operator of the form TAf = SASf where 5 is a generalized Szego projector to level N and A E ^ ^ ( M ) . In the sequel the unmodified term "generalized Toeplitz operator" refers to a generalized Toeplitz operator at level 0. In our formulation it is very easy to see that Toeplitz operators form an algebra: TATB = SASSBS = S(ASB)S. Since ASB G #*(X) if A,B G $*H{X) the claim follows immediately. Proposition 9.3. Let S be the generalized Szego projection (at level 0) and let A be an element of \Er°, then there exists a smooth function a such that SAS-SMaSeV-^M). Here Ma is the multiplication operator Maf = af. The function a is given by a(x) = J
so#aS(A)(+)#sOLjn.
Hi The symbol So is denned in (35). If A is a classical pseudodifferential operator then a is just the principal symbol of A restricted to the positive contact direction. If a is a smooth function then we denote the operator SMaS by Ta. For a, a smooth function, and B G *f>™(X) a straightforward computation shows that a^(MaB) = a" (BMa) = aa^B). This in turn implies that X be a complex vector bundle over a contact manifold. A generalized Szego projection, at level 0, acting on the sections of E is any projection operator SE € ^HO^! E) s u c n ^ ^ GQ{SE)
= soIdE.
Here SQ is the symbol of a scalar generalized Szego projector. Similarly, we define generalized Szego projections at level k and to level N acting on sections of E as projection operators SE G ^(X;E) with the following properties SE is a projection at level k if cr£ (SE) = Sfc ® Id# N
SE is a projection to level N if ^ ( S B ) = ( ^ Sk) Ids . l
Here Sk denotes the symbol of a scalar, generalized Szego projector at level k. Once again for each family of generalized Szego projectors acting on sections of E there is a corresponding family of Toeplitz operators TA = SEASE where A G $£(M;£). Before considering index formulae for Heisenberg operators we first review the Atiyah-Singer theorem for classical, elliptic pseudodifferential operators. As above we let Y be a compact manifold of dimension n and E, F be vector bundles over Y. Let P e V0(Y;E;F). The principal symbol of P, 0o(P) is a section of Hom(7r*i?) n*F), homogeneous of degree 0, i.e. ao(P)(Y,XO = MP)(x,Z)
for A G R+.
The operator is elliptic if the homomorphism a0(P)(x,0:Ex-^Fx is invertible for all nonzero £. In this case P : L2(Y;E) —> L2{Y;F) is a Fredholm operator. As usual we define its analytic index to be Ind(P) = dimkerP - dimcokerP.
72
The Atiyah-Singer index theorem identifies this index with a topological invariantjconstructed from the triple [E,F,ao(P)}. Let T*Y be one point fiber compactification of T*Y, that is we add one point to each fiber of T*Y. The compactified space T?:T*Y
—>Y
n
is a fiber bundle with fiber S . It is useful to have a second description of this space. Fix a Riemannian metric g on T*Y and define the co-ball bundle B*X = {(x,Z)eT*Y : ||£|| C°°(Y; A°'oMY). The index of this operator is called the Todd genus of Y\ it is expressed as a cohomological pairing on Y as ind(a c ) = (Td(T1-°y),[y]>. If E —» Y is a complex vector bundle then, by choosing an Hermitian connection on E one defines a twisted, Spin-C Dirac operator 5E, acting on C°°{Y\ E®S(Y)). Again there is a formula for Ind(9j5) a s cohomological pairing on Y : Ind(3B) = (Ch(£) A TdCT^Y), [Y]).
(39)
Here, as above, Ch(jB) is the Chern character of the complex vector bundle E —> Y. While formula 38 looks quite similar to this formula, the latter formula is much simpler because: (1) The integration takes place on Y.
74
(2) Only the Chern character of the bundle E —> Y appears, the symbol of the Dirac operator does not appear explicitly. This special case of the Atiyah-Singer theorem suffices for most of our applications. It is useful to have an explicit formula for the Chern character. Using the Chern-Weil theory it can be expressed in terms of the curvature of a connection defined on E, see 20 and 22 . Let E be an hermitian, complex vector bundle and V B a unitary connection on E. The curvature of the connection is KE = V^; in terms of a local trivialization of E it is represented by a matrix of 2-forms QE- Using the transformation formula for QE> under a change of local framing, as well as the invariance properties of the trace powers it is not difficult to show that
Ch(£)=Tr(exP[^-fiEj). is globally defined. The Bianchi identity implies that it is closed. This sum of forms is a representative of the Chern character of E. Remark 10.2. Let X be a contact manifold with contact field 7i. A choice of almost complex structure on the fibers of Tt defines the bundles Ab'9X. The Whitney sum
q=Q
defines a Spin-C structure on X. The 9&-operator is defined by (14), though it does not define a complex unless the almost complex structure comes from an integrable CR-structure. In any case the operator 3c = Bt + 8*b : C°°(X; Sx) — C°°(X; Sx) is a self adjoint Spin-C Dirac operator. 11. The Boutet de Monvel index formula Let X denote a compact, contact manifold. We would like to prove a result, analogous to (38), for elliptic operators in ^^(X;E). A problem arises at the beginning of the discussion: The condition for a Heisenberg operator A to be elliptic is that its principal symbol cr^A) be invertible in the isotropic algebra. This is not a pointwise condition in T*X but rather that there exist a symbolCT"m (B) so that a^(A)(±)#±alm(B)(±)
= a"_m(B)(±)#±a^(A)(±) = Id.
75
This condition is global in each fiber of the cotangent space. Consider the following example. Let S be a generalized Szego projection (at level 0) and /x be a complex valued function on X. The operator AM = Id +(iS is Predholm if and only if /x does not assume that value — 1. Let
A simple calculation shows that
BllAll = Id +
^-[S,n]S.
As noted above [[i,S] € ^^(X), thus B^ is a left inverse up to a compact error. A similar calculation shows that B^ is also a right inverse, up to a compact error. On the other hand, in appropriate coordinates the principal symbol of A^ is given by forCTo oo. Theorem 11.1 only gives the formula for the index of a Toeplitz operator at level 0. In fact we can replace SE with a Szego projector at level 0 acting on a larger vector bundle. To accomplish this, and thereby compute the index of the approximating Toeplitz operators, we need to understand the symbols {sk • k > 0} in greater detail. Indeed, it suffices to do this
79
analysis in the model case of IR2n with its standard symplectic and complex structures. The transition to the contact manifold case follows exactly as in the construction of the bundle of isotropic algebras. 13. The structure of the higher eigenprojections If u) is the standard symplectic structure on R 2n , then the #-product on isotropic symbols is given by the oscillatory integral
a#b{w) = - ^ f[a(w + u)b{w + v)e2iu^v)dudv. If a and b belong to Schwartz class then this is an absolutely convergent integral. If a G 5^ o (K 2n ) and b £ 5^o'(lR2n) then a#b £ 5™o+m'(R2n) and it has the following asymptotic expansion: a h
*
~ £ ^ T T ^ -Dn-Dy fc>o K-
Di)ka{x,Z)b{y,r,)\x=yt^v.
(41)
It is well known that if either a or 6 is a polynomial then this sum is finite and gives an exact formula for a#b, see 3 . The differential operator appearing in this formula can be re-expressed in terms of the standard (1,0)- and (0, l)-vector fields on K 2n : Dx-D^-DyD^
1
-[{DX
- i£>£) • (Dy + iDn) - (Dx + iDtf • (Dy - iDn)\
= dz • dn, - dz • dw. Prom this formula it is apparent that if a and b are both holomorphic (or both anti-holomor-phic) polynomials, then only the k = 0 term in (41) is non-zero. This shows that the holomorphic and antiholomorphic polynomials are subalgebras in the #-product structure. In fact the #-product and ordinary pointwise product agree on these subalgebras. Let Zj = Xj — i£j and 2j = Xj + i£j. Let 2]t denote the set of multi-indices of length k and Wk and Wk denote linear spans of {za : a S Ik) and {za : a\ S Ik} respectively. Note that as vector spaces Wk ~ Wk#s0 and Wk ~ SoifWk-
(42)
Given the symplectic form LJ we define a linear functional on »S(K2n) by setting Tr(a) / aw". R2n
(43)
80
Exercise 25. If a € <S(R2n) show that qw{a) is a trace class operator and Tr(a) = Tr(qw(a)). Exercise 26. Show that if a, b £ 5(K 2n ) then prove directly that Tr(a#b) = Tr(b#a). This shows that Tr behaves like a trace on this algebra. Using the trace functional we define inner products on the vector spaces W k: Y be a second complex vector bundle and Tt the holomorphic representatives of the restrictions of n*(F) and SC:F the classical Szego projector onto boundary values of holomorphic sections of J-c. Finally let A be a classical elliptic pseudodifferential operator: A:COO(Y;E)-*COO{Y;F). For purposes of index computations there is no loss in generality in assuming that A is of order zero, let a0 eC°°(T*Y\{0};hom(ir*(E),iT*(Y))) denote
87 its principal symbol. We define a Toeplitz operator: T£,ao = S€,FMaoSe,E
: nn(bXe;£e)
—» fi"(6*£; JF£).
This is slightly different from what we considered before as we now allow the operator to act between sections of different vector bundles. On the other hand, the Szego projectors are now defined by integrable, almost complex structures. Using results in 2 8 one can show that for e < ei A€ = GetFTeiaoGl
E
is a pseudodifferential operator with the same principal symbol as A. As GCIE and Ge,F are isomorphisms we obtain:
Theorem 14.1. If E —> Y and F —> Y are complex vector bundles and A s ^ ° N ( y ; E, F) is an elliptic operator with principal symbol ao then for sufficiently small e we have lnd(A) = Ind(T tlO0 ). As before Ind(T 6O0 ) can also be identified with a relative index Ind(T£,O0) = R-Ind(S £ ,.E,ao ll5 e,Fao)' The theorem therefore has the satisfying philosophical consequence of identifying the indices of all elliptic operators with relative indices of generalized Szego projectors. A cohomological formula for Ind(T £]ao ) leads immediately to a formula for Ind(A). For the case E = F such a formula is given in (40). The general case has been considered by Leichtnam, Nest and Tsygan. 2 9 .
15. The contact degree and index of FIOs In this section (X, 7i) denotes a compact, contact manifold. Recall that a diffeomorphism T/>, of X is a contact map if "4>*HX = Hijt^x). We call an orientation preserving contact diffeomorphism which preserves denote the co-orientation of H a contact transformation. Let M(X,H) isotopy classes of contact transformations; this is the contact mapping class group. As a final application of the ideas presented above we define a homomorphism c-deg : M (X, H) —> Z and give a formula for c-deg which is analogous to (40). If X = S*Y for a compact manifold Y then, using a construction similar to that used in the previous section, we can relate
88
this integer to the index of a certain class of Predholm Fourier integral operators. Most of these results can be found in 19. Choose an adapted almost complex structure J, for the fibers of H. Let S denote a generalized Szego projector with principal symbol defined by the field of vacuum states for the corresponding field of harmonic oscillators. If ip is a contact transformation then is also a generalized Szego projector, though its principal symbol is in general different from that of S. We define the contact degree as follows c-deg(ip) = R-Ind(5,5v,). Using the stability properties of the Predholm index described in Lemma 1.1, it follows immediately that c-deg(ip) depends only on the equivalence class of ip in M(X, Ti). Using the co-cycle relation (36) one can prove that c-deg(f/>) does not depend on the choice of S and that if ip and 4> are two contact transformations then c-deg('0 o (j>) = c-deg(ip) + c-deg(). In other words c-deg : M(X,H) —> Z is a homomorphism. Exercise 28. Prove this statement. The next order of business is to find a formula for c-deg(V'). For an arbitrary pair S, S" of generalized Szego projectors, the relative index is a very delicate invariant and the value of R-Ind(5, S") is difficult to compute. However in the special case that S' = S^ the relative index is much more robust and is essentially a topological invariant. We use a construction similar to that used to relate the index of a Toeplitz operator to the index of a Dirac operator on a mapping torus. Define the mapping torus Z^Xx[0,l]/(i,0)~(^),l). Since ip is a contact transformation the bundle H lifts to define a codimension 2 subbundle Hip C TZ^. This bundle has a conformal symplectic structure and therefore we can define an almost complex structure J on H^. If t denotes the parameter and T a vector field tangent to the fibers of Z^, —> Sl transverse to H^ © M.dt then we can extend the almost complex structure defined on H.^ to an almost complex structure denned on TZ^ by letting JT = dt.
89 The almost complex structure on TZ$ defines a canonical Spin-C structure on Z^. Let 9,/, denote the Spin-C Dirac operator defined by a choice of hermitian structure. We think of 9,/, as acting from even to odd spinors. In 19 the following result is proved Theorem 15.1. If ip is a contact transformation of(X,7i) then c-degip =Ind(cV).
(53)
To prove this result we introduce a 'resolution' of the range of S which is an acyclic differential complex very similar to the db-complex. Recall however that the almost complex structure J, on % is not required to be integrable. This complex defines a self adjoint, Fredholm Heisenberg pseudodifferential operator Do- The same construction applied to S^ leads to the operator D\ = (ip-1)*Doip* which is therefore isospectral to Do. These operators can be put into a continuous family Dt. Since the ends are isospectral, the spectral flow of the family is well defined; denote it by sfQDt]). The first step is to show that R-Ind(5,5^) = sf([A])The next problem is to deform the family [Dt] to a family of Spin-C Dirac operators [9t] through Fredholm operators. Technically this is the most challenging step. This is because the operators in the family [Dt] while Fredholm are not even Heisenberg elliptic, whereas Dirac operators are classical elliptic differential operators. To control this very singular perturbation problem we introduction an extension of the Heisenberg calculus which includes both the classical calculus and the Heisenberg calculus as subalgebras. Once this is accomplished, it follows from the stability of the Fredholm index that sf([Z?t]) = sf([9t]). It is then a fairly standard result that sf([9t]) = Ind(8tf). For the case that the dimX = 3 one can use the Atiyah-Singer index theorem and Hirzebruch signature formula to obtain a simple explicit formula for c-deg(V'). In this case
90 Here 11 denotes a trivial line bundle. Using formula (18) in that
19
we deduce
Ind(o>) =4 P l (T^)[^]
\ (54) sig Z 12 ( ^)' where p\ is the first Pontryagin class and sig(i^) is the signature of the 4-manifold Z^. Let P : Z^ —-> S1 be the canonical projection and set =
Zo =
P-^O.TT])
and Zi = P " 1 ^ * - ] ) .
The manifolds with boundary ZQ and Z\ are diffeomorphic and, as oriented manifolds,
z,p ~ z 0 [_J —ZW boundary
that is we reverse the orientation of Z\. Let sig(Zo) = sig(Zi) denote the index of the non-degenerate pairing defined by the cup product on the image of H2(Zi,bZi) in H2(Zi). The Novikov addition formula states that sig(^) = sig(Z o )-sig(Zi) = O, 30 31
see , . This proves the following result Theorem 15.2. If X is a 3-dimensional contact manifold then the contact degree is the zero homomorphism. Remark 15.1. I would like to thank Rafe Mazzeo for pointing out the connection, for this case, between the contact degree and the signature, and Dennis Sullivan for telling me about the Novikov addition formula. Remark 15.2. In 9 the relative index is defined for a pair of embeddable CR-structures denned on a 3-dimensional contact manifold. There it is shown that the index descends to define an invariant on the "Teichmuller" space of CR-structures. This is the space of equivalence classes of complex structures on the contact field where two structures are equivalent if one is the push forward of the other by a contact map isotopic to the identity. The contact mapping class group acts on this Teichmuller space. Theorem 15.2 implies that the relative index actually descends to define an invariant on the moduli space itself. As a second application of Theorem 15.1 we show that the contact degree is related to the index of a certain class of Fourier integral operators. For
91 this application we need to restrict X = S*Y, for Y a compact manifold. If V : S*Y —> S*Y is a contact transformation then it defines a conic Lagrangian submanifold of T*Y\ {0} x T*Y\ {0}. To see this identify S*Y with a unit cosphere bundle and extend the map ip : S*Y —» S*Y to be homogeneous of degree 1. Denote this extension by \I>. It is a canonical transformation of the cotangent bundle, its graph, A,/, is therefore a conic, Lagrangian submanifold. Such a submanifold defines a class of Fourier integral operators, see 32 or 33 . Roughly speaking an operator belongs to this class if the wave front set of its Schwartz kernel is contained in A,/,. For example, pseudodifferential operators are among the Fourier integral operators defined by the identity map. Let {Xe} denote the Grauert tubes introduced in the previous section. As noted in section 1, S*Y is contact equivalent to bXc for any e < eoTherefore a contact transformation of S*Y can be thought of as defining a contact transformation of bXe. Let ip : bXe —> bXe be such a map. We define an operator on C°°(Y) by setting F^u = Gcip*G*{u). This is a Fredholm Fourier integral operator associated to the Lagrangian submanifold A,/,. In 19 the index of this operator is computed. Theorem 15.3. If ip : S*Y —> S*Y is a contact transformation then Ind(i^) = c-deg(ip). As a corollary of this result and Theorem 15.2 we have Corollary 15.1. If dim Y = 2 then Ind(F,p) = 0 for every contact transformation. The detailed proofs of these results are in
19
and 1.
Remark 15.3. Thus far no example of a contact manifold X and contact transformation ip '• X —> X such that c-deg(V') ^ 0 has been found. Acknowledgments I would like to thank the organizers of the summer school at CIRM in Marseilles, B. Coupet, J. Merker and A. Shukov as well as the the organizers of the meeting in Woods Hole, Nils Tongring and Dennis Sullivan for giving me the opportunity to present these lectures. I would also like to thank Andy Solow for making the Math meetings in Woods Hole a possibility. I am most grateful to Hyunsuk Kang
92 for providing a careful transcript of the original lectures. Finally I would like to thank my collaborators, Richard Melrose and Gerardo Mendoza for allowing me to write this expository account of our joint work. The research described in these notes was partially supported by the National Science Foundation. References 1. Charles L. Epstein and Richard Melrose. The Heisenberg algebra, index theory and homology. preprint, 2003. 2. Fritz Noether. Uber eine Klasse singularer Integralgleichungen. Math. Ann., 82:42-63, 1921. 3. L. Hormander. The Analysis of Linear Partial Differential Operators, volume 3. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985. 4. Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek. Classes of Linear Operators, vol. I and II. Birkhauser, Basel-Boston-Berlin, 1990. 5. P.D. Lax. On factorization of matrix valued functions. Comm. Pure Appl. Math., 29:683-688, 1976. 6. V.I. Arnold. Mathematical Methods in Classical Mechanics, volume 60 of GTM. Springer Verlag, Berlin and New York, 1978. 7. Laszlo Lempert and Robert Szoke. Global solutions of the homogeneous complex Monge-Ampre equation and complex structures on the tangent bundle of riemannian manifolds. Math. Ann., 290:689-712, 1991. 8. V.W. Guillemin and M. Stenzel. Grauert tubes and the homogeneous MongeAmpere equation. J. Differential Geom., 1991. 9. Charles L. Epstein. A relative index on the space of embeddable CRstructures, I, II. Annals of Math., 147:1-59, 61-91, 1998. 10. L. Boutet de Monvel. Integration des equations Cauchy-Riemann induites formelles. Seminar Goulaouic-Lions-Schwartz, pages IX.1-IX.13, 1974-75. 11. J.J. Kohn. The range of the tangential Cauchy-Riemann operator. Duke J., 53:525-545, 1986. 12. F. Reese Harvey and H. Blaine Lawson. On the boundaries of complex analytic varieties. Ann. of Math. (2), 106:223-290, 1977. 13. J.J. Kohn and L. Nirenberg. On the algebra of pseudo-differential operators. Comm. Pure Appl. Math., 18:269-305, 1965. 14. J.J. Kohn and L. Nirenberg. Non-coercive boundary value problems. Comm. Pure Appl. Math., 18:443-492, 1965. 15. M.E. Taylor. Noncommutative microlocal analysis, part I, volume 313 of Mem. Amer. Math. Soc. AMS, 1984. 16. R. Beals and P. Greiner. Calculus on Heisenberg Manifolds, volume 119 of Annals of Mathematics Studies. Princeton University Press, 1988. 17. M.E. Taylor. Noncommutative harmonic analysis, volume 22 of Mathematical Surveys and Monographs. AMS, Providence, R.I., 1986. 18. L. Boutet de Monvel and V. Guillemin. The spectral theory of Toeplitz operators, volume 99 of Ann. of Math. Studies. Princeton University Press, 1981.
93 19. Charles L. Epstein and Richard Melrose. Contact degree and the index of Fourier integral operators. Math. Res. Letters, 5:363-381, 1998. 20. Friedrich Hirzebruch. Topological methods in algebraic geometry, volume 131 of Grundlehren der mathematishen Wissensckaften. Springer Verlag, 1978. 21. H. Blaine Lawson Jr. and Marie-Louise Michelson. Spin Geometry, volume 38 of Princeton Mathematical Series. Princeton University Press, 1989. 22. S. S. Chern. Complex Manifolds Without Potential Theory. Van Nostrand Reinhold Co., New York, 1967. 23. L. Boutet de Monvel. On the index of Toeplitz operators of several complex variables. Invent. Math., 50:249-272, 1979. 24. N. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Operators, volume 298 of Grundlehren der mathematischen Wissenschaften. SpringerVerlag, Berlin Heidelberg New York, 1992. 25. Charles L. Epstein and Richard Melrose. Shrinking tubes and the d-Neumann problem, preprint, 1990. 26. Raul Tataru. Adiabatic limit and SzegS projections. MIT PhD Thesis, 2003. 27. J. Leiterer. Holomorphic vector bundles and the Oka-Grauert principle. In S.G. Gindikin and G.M. Khenkin, editors, Several Complex Variables, IV, volume 10 of Encyclopedia of Mathematical Sciences, chapter 2. Springer Verlag, 1990. 28. V.W. Guillemin. Toeplitz operators in n dimensions. Int. Eq. Op. Theory, 7:145-205, 1984. 29. Eric Leichtnam, Ryszard Nest, and Boris Tsygan. Local formula for the index of a fourier integral operator, to appear JDG, pages 1-25, 2002. 30. M.F. Atiyah, V.K. Patodi, and I.M. Singer. Spectral asymmetry and Riemannian geometry, I. Math. Proc. Camb. Phil. Soc, 77:43-69, 1975. 31. M.F. Atiyah and I.M. Singer. The index of elliptic operators, III. Ann. of Math., 87:546-604, 1968. 32. L. Hormander. Fourier integral operators, I. Ada Math., 127:79-183, 1971. 33. J. Briining and V.W. Guillemin (Editors). Fourier integral operators. Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1994.
94
BIOLOGIC II
LOUIS H. KAUFFMAN Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan St., Chicago IL 60607-7045, U.S.A. E-mail: [email protected] In this paper we explore the boundary between biology, topology, algebra and the study of formal systems (logic).
1. Introduction This paper concentrates on relationships of formal systems with biology. In particular, this is a study of different forms and formalisms for replication. It is a sequel to Kauffman10 and contains much of the material in that paper plus new material about projectors in the Temperley Lieb algebra. In living systems there is an essential circularity that is the living structure. Living systems produce themselves from themselves and the materials and energy of the environment. There is a strong contrast in how we avoid circularity in mathematics and how nature revels in biological circularity. One meeting point of biology and mathematics is knot theory and topology. This is no accident, since topology is indeed a controlled study of cycles and circularities in primarily geometrical systems. In this paper we will discuss DNA replication, logic and biology, the relationship of symbol and object, the emergence of form. It is in the replication of DNA that the polarity (yes/no, on/off, true/false) of logic and the continuity of topology meet. Here polarities are literally fleshed out into the forms of life. We shall pay attention to the different contexts for the logical, from the mathematical to the biological to the quantum logical. In each case there is a shift in the role of certain key concepts. In particular, we follow the notion of copying through these contexts and with it gain new insight into the role of replication in biology, in formal systems and in the quantum
95 level (where it does not exist!). In the end we arrive at a summary formalism, a chapter in boundary mathematics (mathematics using directly the concept and notation of containers and delimiters of forms - compare Bricken & Gullichsen3 and Spencer-Brown11) where there are not only containers , but also extainers >< - entities open to interaction and distinguishing the space that they are not. In this formalism we find a key for the articulation of diverse relationships. The boundary algebra of containers and extainers is to biologic what boolean algebra is to classical logic. Let C = < > and E =>< then EE = > < > < = > C < and CC = < > < > = < E > Thus an extainer produces a container when it interacts with itself, and a container produces an extainer when it interacts with itself. The formalism of containers and extainers is a chapter in the foundations of a symbolic language for shape and interaction. With it, we can express the form of DNA replication succinctly as follows: Let the DNA itself be represented as a container DNA = < > .
(1)
We regard the two brackets of the container as representatives for the two matched DNA strands. We let the extainer E =>< represent the cellular environment with its supply of available base pairs (here symbolized by the individual left and right brackets). Then when the DNA strands separate, they encounter the matching bases from the environment and become two DNA's. D N A = < > — > < £ • > — > < > < > = DNA DNA.
(2)
Life itself is about systems that search and learn and become. Perhaps a little symbol like E —X with the property that EE =>< produces containers and retains its own integrity in conjunction with the autonomy of (the DNA) could be a step toward bringing formalism to life. These concepts of concatenation of extainers and containers lead, in Section 6, to a new approach to the structure of and generalizations of the Temperley Lieb algebra. In this Section we discuss how projectors in the Temperley Lieb algebra can be regarded as topological/algebraic models of self-replication, and we take this point of view to characterize multiplicative elements P of the Temperley Lieb algebra such that PP = P. What emerges here is a topological view of self-replication that is different in principle from the blueprint-driven self-replications of logic and from the environmentally driven self-replication described above as an abstraction
96
of DNA action. This topological replication is a direct descendant of the fact that you can get two sticks from one stick by breaking it in the middle. Here we obtain more complex forms by allowing topological deformation of the stick before it is broken, but to see how this works the reader should go to Section 6.1. 2. Replication of DNA We start this essay with the question: During the replication of DNA, how do the daughter DNA duplexes avoid entanglement? In the words of John Hearst6, we are in search of the mechanism for the "immaculate segregation." This question is inevitably involved with the topology of the DNA, for the strands of the DNA are interwound with one full turn for every ten base pairs. With the strands so interlinked it would seem impossible for the daughter strands to separate from their parents. A key to this problem certainly lies in the existence of the topoisomerase enzymes that can change the linking number between the DNA strands and also can change the linking number between two DNA duplexes. It is however, a difficult matter at best to find in a tangled skein of rope the just right crossing changes that will unknot or unlink it. The topoisomerase enzymes do just this, changing crossings by grabbing a strand, breaking it and then rejoining it after the other strand has slipped through the break. Random strand switching is an unlikely mechanism, and one is led to posit some intrinsic geometry that can promote the process. In Kauffman6 there is made a specific suggestion about this intrinsic geometry. It is suggested that in vivo the DNA polymerase enzyme that promotes replication (by creating loops of single stranded DNA by opening the double stranded DNA) has sufficient rigidity not to allow the new loops to swivel and become entangled. In other words, it is posited that the replication loops remain simple in their topology so that the topoisomerase can act to promote the formation of the replication loops, and these loops once formed do not hinder the separation of the newly born duplexes. The model has been to some degree confirmed Zechiedrich, et al.12 The situation would now appear to be that in the first stages of the formation of the replication loops Topo I acts favorably to allow their formation and amalgamation. Then Topo II has a smaller job of finishing the separation of the newly formed duplexes. In 1 we illustrate the schema of this process. In this Figure we indicate the action of the Topo I by showing a strand being switched in between two replication loops. The action of Topo II is only
97
stated but not shown. In that action, newly created but entangled DNA strands would be disentangled. Our hypothesis is that this second action is essentially minimized by the rigidity of the ends of the replication loops in vivo.
I replication loops
)'/
\ r
i 7
\
top
°'* r i
ytopo n \ > o o o o c
^'"'•Wi'-W vi V \ x \ *>
\r
'
DNA
to on )l Vs. P *"Nr-v^>v-Nr-x-^r It \\ — TOOOwCX DNA >^^y Figure 1. DNA Replication
In the course of this research, we started thinking about the diagrammatic logic of DNA replication and more generally about the relationship between DNA replication, logic and basic issues in the foundations of mathematics and modelling. The purpose of this paper is to explain some of these issues, raise questions and place these questions in the most general context that we can muster at this time. The purpose of this paper is there-
98 fore foundational. It will not in its present form affect issues in practical biology, but we hope that it will enable us and the reader to ask fruitful questions and perhaps bring the art of modelling in mathematics and biology forward. To this end we have called the subject matter of this paper "biologic" with the intent that this might suggest a quest for the logic of biological systems or a quest for a "biological logic" or even the question of the relationship between what we call "logic" and our own biology. We have been trained to think of physics as the foundation of biology, but it is possible to realize that indeed biology can also be regarded as a foundation for thought, language, mathematics and even physics. In order to bring this statement over to physics one has to learn to admit that physical measurements are performed by biological organisms either directly or indirectly and that it is through our biological structure that we come to know the world. This foundational view will be elaborated as we proceed in this paper.
3. Logic, Copies and DNA Replication In logic it is implicit at the syntactical level that copies of signs are freely available. In abstract logic there is no issue about materials available for the production of copies of a sign, nor is there necessarily a formalization of how a sign is to be copied. In the practical realm there are limitations to resources. A mathematician may need to replenish his supply of paper. A computer has a limitation on its memory store. In biology, there are no signs, but there are entities that we take as signs in our description of the workings of the biological information process. In this category the bases that line the backbone of the DNA are signs whose significance lies in their relative placement in the DNA. The DNA itself could be viewed as a text that one would like to copy. If this were a simple formal system it would be taken for granted that copies of any given text can be made. Therefore it is worthwhile making a comparison of the methods of copying or reproduction that occur in logic and in biology. In logic there is a level beyond the simple copying of symbols that contains a non-trivial description of self-replication. The schema is as follows: There is a universal building machine B that can accept a text or description x (the program) and build what the text describes. We let lowercase
99
x denote the description and uppercase X denote that which is described. Thus B with x will build X. In fact, for bookkeeping purposes we also produce an extra copy of the text x. This is appended to the production X as X, x. Thus B, when supplied with a description x, produces that which x describes, with a copy of its description attached. Schematically we have the process shown below. B, x —> B, x; X, x
(3)
Self-replication is an immediate consequence of this concept of a universal building machine. Let b denote the text or program for the universal building machine. Apply B to its own description. B,b—>B,b;B,b
(4)
The universal building machine reproduces itself. Each copy is a universal building machine with its own description appended. Each copy will proceed to reproduce itself in an unending tree of duplications. In practice this duplication will continue until all available resources are used up, or until someone removes the programs or energy sources from the proliferating machines. It is not necessary to go all the way to a universal building machine to establish replication in a formal system or a cellular automaton (see the epilogue to this paper for examples). On the other hand, all these logical devices for replication are based on the hardware/software or Object/Symbol distinction. It is worth looking at the abstract form of DNA replication. DNA consists in two strands of base-pairs wound helically around a phosphate backbone. It is customary to call one of these strands the "Watson" strand and the other the "Crick" strand. Abstractly we can write DNA =< W\C >
(5)
to symbolize the binding of the two strands into the single DNA duplex. Replication occurs via the separation of the two strands via polymerase enzyme. This separation occurs locally and propagates. Local sectors of separation can amalgamate into larger pieces of separation as well. Once the strands are separated, the environment of the cell can provide each with complementary bases to form the base pairs of new duplex DNA's. Each strand, separated in vivo, finds its complement being built naturally in the environment. This picture ignores the well-known topological difficulties
100
present to the actual separation of the daughter strands. The base pairs are AT (Adenine and Thymine) and GC (Guanine and Cytosine). Thus if < W | = < .. .TTAGAATAGGTACGCG... |
(6)
\C > = \...AATCTTATCCATGCGC...
(7)
Then >.
Symbolically we can oversimplify the whole process as < W\ + E —>< W\C >= DNA E+\C>—><W\C>=DNA < W\C>—-*< W\ + E + \C>=< W\C>< W\C>
(8) (9) (10)
Either half of the DNA can, with the help of the environment, become a full DNA. We can let E —> \C >< W\ be a symbol for the process by which the environment supplies the complementary base pairs AG, TC to the Watson and Crick strands. In this oversimplification we have cartooned the environment as though it contained an already-waiting strand \C > to pair with < W\ and an already-waiting strand < W\ to pair with \C > . In fact it is the opened strands themselves that command the appearance of their mates. They conjure up their mates from the chemical soup of the environment. The environment E is an identity element in this algebra of cellular interaction. That is, E is always in the background and can be allowed to appear spontaneously in the cleft between Watson and Crick: < W\C >—>< W\\C >^< W\E\C> ((10) —4<W\\C><W\\C>—+<W\C><W\C> ' This is the formalism of DNA replication. Compare this method of replication with the movements of the universal building machine supplied with its own blueprint. Here Watson and Crick ( < W\ and \C > ) are each both the machine and the blueprint for the DNA. They are complementary blueprints, each containing the information to reconstitute the whole molecule. They are each machines in the context of the cellular environment, enabling the production of the DNA. This coincidence of machine and blueprint, hardware and software is an important difference between classical logical systems and the logical forms that arise in biology.
101 4. Lambda Algebra - Replication Revisited One can look at formal systems involving self-replication that do not make a distinction between Symbol and Object. In the case of formal systems this means that one is working entirely on the symbolic side, quite a different matter from the biology where there is no intrinsic symbolism, only our external descriptions of processes in such terms. An example at the symbolic level is provided by the lambda calculus of Church and Curry2 where functions are allowed to take themselves as arguments. This is accomplished by the following axiom. Axiom for a Lambda Algebra: Let A be an algebraic system with one binary operation denoted ab for elements a and b of A. Let F(x) be an algebraic expression over A with one variable x. Then there exists an element a of A such that F(x) = ax for all x in A. An algebra (not associative) that satisfies this axiom is a representation of the lambda calculus of Church and Curry. Let b be an element of A and define F(x) — b(xx). Then by the axiom we have a in A such that ax = b(xx) for any x in A. In particular (and this is where the "function" becomes its own argument) aa = b(aa).
(12)
Thus we have shown that for any b in A, there exists an element x in A such that x = bx. Every element of A has a "fixed point." This conclusion has two effects. It provides a fixed point for the function G(x) = bx and it creates the beginning of a recursion in the form aa = b{aa) = b(b(aa)) = b{b{b(aa))) = ...
(13)
The way we arrived at the fixed point aa was formally the same as the mechanism of the universal building machine. Consider that machine: B,x—>X,x
(14)
We have left out the repetition of the machine itself. You could look at this as a machine that uses itself up in the process of building X. Applying B to its own description b we have the self-replication B,b—>B,b.
(15)
102
The repetition of x in the form X, x on the right hand side of this definition of the builder property is comparable with ax = b(xx)
(16)
with its crucial repetition as well. In the fixed point theorem, the arrow is replaced by an equals sign! Repetition is the core of self-replication in classical logic. This use of repetition assumes the possibility of a copy at the syntactic level, in order to produce a copy at the symbolic level. There is, in this pivot on syntax, a deep relationship with other fundamental issues in logic. In particular this same form of repetition is in back of the Cantor diagonal argument showing that the set of subsets of a set has greater cardinality than the original set, and it is in back of the Godel Theorem on the incompleteness of sufficiently rich formal systems. The pattern is also in back of the production of paradoxes such as the Russell paradox of the set of all sets that are not members of themselves. There is not space here to go into all these relationships, but the Russell paradox will give a hint of the structure. Let "ab" be interpreted as "b is a member of a". Then RX = -(XX). Substituting R for X we obtain RR = ->(RR), which says that R is a member of R exactly when it is not the case that R is a member of R. This is the Russell paradox. From the point of view of the lambda calculus, we have found a fixed point for negation. Where is the repetition in the DNA self-replication? The repetition and the replication are no longer separated. The repetition occurs not syntactically, but directly at the point of replication. Note the device of pairing or mirror imaging. A calls up the appearance of T and G calls up the appearance of C. < W\ calls up the appearance of \C > and \C > calls up the appearance of < W\. Each object O calls up the appearance of its dual or paired object O*. O calls up O* and O* calls up O. The object that replicates is implicitly a repetition in the form of a pairing of object and dual object. OO* replicates via O —» OO*
(17)
O* —+ OO*
(18)
103
whence OO* —> O O* —• OO* OO*.
(19)
The repetition is inherent in the replicand in the sense that the dual of a form is a repetition of that form.
5. Quantum Mechanics We now consider the quantum level. Here copying is not possible. We shall detail this in a subsection. For a quantum process to copy a state, one needs a unitary transformation to perform the job. One can show, as we explain in the last subsection of this section, that this cannot be done. There are indirect ways that seem to make a copy, involving a classical communication channel coupled with quantum operators (so called quantum teleportation13). The production of such a quantum state constitutes a reproduction of the original state, but in these cases the original state is lost, so teleportation looks more like transportation than copying. With this in mind it is fascinating to contemplate that DNA and other molecular configurations are actually modelled in principle as certain complex quantum states. At this stage we meet the boundary between classical and quantum mechanics where conventional wisdom finds it is most useful to regard the main level of molecular biology as classical. We shall quickly indicate the basic principles of quantum mechanics. The quantum information context encapsulates a concise model of quantum theory: The initial state of a quantum process is a vector \v > in a complex vector space H. Observation returns basis elements (3 of H with probability | \2/
(20)
where < v \w >= v*w with v* the conjugate transpose ofv. A physical process occurs in steps \v >—> U\v >= \Uv > where U is a unitary linear transformation. Note that since < Uv \Uw >=< v\w > when U is unitary, it follows that probability is preserved in the course of a quantum process. One of the details for any specific quantum problem is the nature of the unitary evolution. This is specified by knowing appropriate information about the classical physics that supports the phenomena. This information
104 is used to choose an appropriate Hamiltonian through which the unitary operator is constructed via a correspondence principle that replaces classical variables with appropriate quantum operators. (In the path integral approach one needs a Lagrangian to construct the action on which the path integral is based.) One needs to know certain aspects of classical physics to solve any given quantum problem. The classical world is known through our biology. In this sense biology is the foundation for physics. A key concept in the quantum information viewpoint is the notion of the superposition of states. If a quantum system has two distinct states \v > and \w >, then it has infinitely many states of the form a\v > +b\w > where a and b are complex numbers taken up to a common multiple. States are "really" in the projective space associated with H. There is only one superposition of a single state \v > with itself. Dirac5 introduced the "bra-(c)-ket" notation < A\B >= A*B for the inner product of complex vectors A,B £ H. He also separated the parts of the bracket into the bra < A | and the ket \B > . Thus < A\B >=< A\ \B> .
(21)
In this interpretation, the ket \B > is identified with the vector B G H, while the bra < A | is regarded as the element dual to A in the dual space H*. The dual element to A corresponds to the conjugate transpose A* of the vector A, and the inner product is expressed in conventional language by the matrix product A*B (which is a scalar since B is a column vector). Having separated the bra and the ket, Dirac can write the "ket-bra" \A > < B | = AB*. In conventional notation, the ket-bra is a matrix, not a scalar, and we have the following formula for the square of P = \A >< B | :
P2 = \A >< B \\A >< B | = A(B*A)B* = {B*A)AB* =< B \A > P. (22) Written entirely in Dirac notation we have
P2 = \A >< B\\A >< B\ = \A >< B\A>} is an orthonormal basis for H, and P, = |Cj >< Ci|, then for any vector \A > we have |A >=< Cx \A > | d > + • • • + < Cn \A > \Cn > .
(25)
Hence < B |A >=< d\A >< B \d >+•••+< Cn\A >< B \Cn > = + ---+< Cn \A > =< B | [|d >< Ci | + • • • + |C n >< Cn |] |4 > =.
. .
[M)
We have written this sequence of equalities from < j B | A > t o < B | l | j 4 > to emphasize the role of the identity E£=1flfe = E ] J = 1 | C f c x C 7 f c | = l
(27)
so that one can write
< B \A >=< B 11 \A >=< B IEJUIC* X Ck \\A > = Eg=1.
l ; (27)
In the quantum context one may wish to consider the probability of starting in state \A > and ending in state \B > . The square of the probability for this event is equal to | < B \A > | 2 . This can be refined if we have more knowledge. If it is known that one can go from A to C* (i = 1, • • • , n) and from Ci to B and that the intermediate states \d > are a complete set of orthonormal alternatives then we can assume that < C, \d >= 1 for each i and that Ej|Cj >< d | = 1. This identity now corresponds to the fact that 1 is the sum of the probabilities of an arbitrary state being projected into one of these intermediate states. If there are intermediate states between the intermediate states this formulation can be continued until one is summing over all possible paths from A to B. This becomes the path integral expression for the amplitude .
5.1. Quantum Formalism and DNA
Replication
We wish to draw attention to the remarkable fact that this formulation of the expansion of intermediate quantum states has exactly the same pattern as our formal summary of DNA replication. Compare them. The form of
106 DNA replication is shown below. Here the environment of possible base pairs is represented by the ket-bra E = \C >< W |.
<W\O-^<W\\C>—*<W\E\C>
—•<w\\c><w\\c>—+ <w\c><w\c>
. .
{
'
v
;
Here is the form of intermediate state expansion.
—>-^ ^^Zk. We compare E = \C><W\
(31)
1 = Efc \Ck X Ck |.
(32)
and
That the unit 1 can be written as a sum over the intermediate states is an expression of how the environment (in the sense of the space of possibilities) impinges on the quantum amplitude, just as the expression of the environment as a soup of bases ready to be paired (a classical space of possibilities) serves as a description of the biological environment. The symbol E = \C >< W | indicated the availability of the bases from the environment to form the complementary pairs. The projection operators \Ci X Ci | are the possibilities for interlock of initial and final state through an intermediate possibility. In the quantum mechanics the special pairing is not of bases but of a state and a possible intermediate from a basis of states. It is through this common theme of pairing that the conceptual notation of the bras and kets lets us see a correspondence between such separate domains.
5.2. Quantum Copies are not Possible Finally, we note that in quantum mechanics it is not possible to copy a quantum state! This is called the no-cloning theorem of elementary quantum mechanics13. Here is the proof: Proof of the No Cloning Theorem: In order to have a quantum process make a copy of a quantum state we need a unitary mapping U : H®H —> H H where H is a complex vector space such that there is a fixed state \X > S H with the property that U(\X>\A>)
= \A>\A>
(33)
107 for any state \A>e H. (\A> \B > denotes t h e tensor product \A> ®\B > .) Let
T{\A >) = U(\X > \A >) = \A > \A > .
(34)
Note that T is a linear function of \A > . Thus we have IT\O>= | 0 > | 0 > = |00>,
(35)
T | l > = |1 > | 1 > = |11 >,
(36)
T(Q|0 > +/3\l >) = (a|0 > +/J|l >)(a|0 > +/3|1 >).
(37)
But T(a|0> +/3|l >) = a|00 >+/?|11 > .
(38)
a|00 > +p\ll > = (a|0 > +/3|1 >)(a|0 > +/3|1 >) = a 2 |00 > +/? 2 |H > +a/?|01 > +/3a|10 >
,„„, ^ ;
Hence
From this it follows that a/? = 0. Since a and /? are arbitrary complex • numbers, this is a contradiction. The proof of the no-cloning theorem depends crucially on the linear superposition of quantum states and the linearity of quantum process. By the time we reach the molecular level and attain the possibility of copying DNA molecules we are copying in a quite different sense than the ideal quantum copy that does not exist. The DNA and its copy are each quantum states, but they are different quantum states! That we see the two DNA molecules as identical is a function of how we filter our observations of complex and entangled quantum states. Nevertheless, the identity of two DNA copies is certainly at a deeper level than the identity of the two letters "i" in the word identity. The latter is conventional and symbolic. The former is a matter of physics and biochemistry.
6. Mathematical Structure and Topology We now comment on the conceptual underpinning for the notations and logical constructions that we use in this paper. This line of thought will lead to topology and to the formalism for replication discussed in the last section.
108 Mathematics is built through distinctions, definitions, acts of language that bring forth logical worlds, arenas in which actions and patterns can take place. As far as we can determine at the present time, mathematics while capable of describing the quantum world, is in its very nature quite classical. Or perhaps we make it so. As far as mathematics is concerned, there is no ambiguity in the 1 + 1 hidden in 2. The mathematical box shows exactly what is potential to it when it is opened. There is nothing in the box except what is a consequence of its construction. With this in mind, let us look at some mathematical beginnings. Take the beginning of set theory. We start with the empty set <j> = { } and we build new sets by the operation of set formation that takes any collection and puts brackets around it: abed —> {a,b,c,d}
(40)
making a single entity {a, b, c, d} from the multiplicity of the "parts" that are so collected. The empty set herself is the result of "collecting nothing". The empty set is identical to the act of collecting. At this point of emergence the empty set is an action not a thing. Each subsequent set can be seen as an action of collection, a bringing forth of unity from multiplicity. One declares two sets to be the same if they have the same members. With this prestidigitation of language, the empty set becomes unique and a hierarchy of distinct sets arises as if from nothing. — { }— {{}}-^{{}>{{}}}—>••• (41) All representatives of the different mathematical cardinalities arise out of the void in the presence of these conventions for collection and identification. We would like to get underneath the formal surface. We would like to see what makes this formal hierarchy tick. Will there be an analogy to biology below this play of symbols? On the one hand it is clear to us that there is actually no way to go below a given mathematical construction. Anything that we call more fundamental will be another mathematical construct. Nevertheless, the exercise is useful, for it asks us to look closely at how this given formality is made. It asks us to take seriously the parts that are usually taken for granted. We take for granted that the particular form of container used to represent the empty set is irrelevant to the empty set itself. But how can this
109 be? In order to have a concept of emptiness, one needs to hold the contrast of that which is empty with "everything else". One may object that these images are not part of the formal content of set theory. But they are part of the formalism of set theory. Consider the representation of the empty set: { }. That representation consists in a bracketing that we take to indicate an empty space within the brackets, and an injunction to ignore the complex typographical domains outside the brackets. Focus on the brackets themselves. They come in two varieties: the left bracket, {, and the right bracket, }. The left bracket indicates a distinction of left and right with the emphasis on the right. The right bracket indicates a distinction between left and right with an emphasis on the left. A left and right bracket taken together become a container when each is in the domain indicated by the other. Thus in the bracket symbol { }
(42)
for the empty set, the left bracket, being to the left of the right bracket, is in the left domain that is marked by the right bracket, and the right bracket, being to the right of the left bracket is in the right domain that is marked by the left bracket. The doubly marked domain between them is their content space, the arena of the empty set. The delimiters of the container are each themselves iconic for the process of making a distinction. In the notation of curly brackets, { , this is particularly evident. The geometrical form of the curly bracket is a cusp singularity, the simplest form of bifurcation. The relationship of the left and right brackets is that of a form and its mirror image. If there is a given distinction such as left versus right, then the mirror image of that distinction is the one with the opposite emphasis. This is precisely the relationship between the left and right brackets. A form and its mirror image conjoin to make a container. The delimiters of the empty set could be written in the opposite order: }{. This is an extainer. The extainer indicates regions external to itself. In this case of symbols on a line, the extainer }{ indicates the entire line to the left and to the right of itself. The extainer is as natural as the container, but does not appear formally in set theory. To our knowledge, its first appearance is in the Dirac notation of "bras" and "kets" where Dirac takes an
110 inner product written in the form < B\A > and breaks it up into < B | and \A > and then makes projection operators by recombining in the opposite order as \A >< B |. See the earlier discussion of quantum mechanics in this paper. Each left or right bracket in itself makes a distinction. The two brackets are distinct from one another by mirror imaging, which we take to be a notational reflection of a fundamental process (of distinction) whereby two forms are identical (indistinguishable) except by comparison in the space of an observer. The observer is the distinction between the mirror images. Mirrored pairs of individual brackets interact to form either a container C = {}
(43)
E =}{.
(44)
or an extainer
These new forms combine to make: CC = {}{} = {£}
(45)
EE=}{}{=}C{.
(46)
and
Two containers interact to form an extainer within container brackets. Two extainers interact to form a container between extainer brackets. The pattern of extainer interactions can be regarded as a formal generalization of the bra and ket patterns of the Dirac notation that we have used in this paper both for DNA replication and for a discussion of quantum mechanics. In the quantum mechanics application {} corresponds to the inner product < A\B >, a commuting scalar, while }{ corresponds to \A >< B |, a matrix that does not necessarily commute with vectors or other matrices. With this application in mind, it is natural to decide to make the container an analog of a scalar quantity and let it commute with individual brackets. We then have the equation EE =}{}{=}C{= C}{= CE.
(47)
By definition there will be no corresponding equation for CC. We adopt the axiom that containers commute with other elements in this combinatorial algebra. Containers and extainers are distinguished by this property. Containers appear as autonomous entities and can be moved about. Extainers are open to interaction from the outside and are sensitive to their
111 surroundings. At this point, we have described the basis for the formalism used in the earlier parts of this paper. If we interpret E as the "environment" then the equation }{= E = 1 expresses the availability of complementary forms so that
0 — {E} — {}{}
(48)
becomes the form of DNA reproduction. We can also regard EE = {}E as symbolic of the emergence of DNA from the chemical substrate. Just as the formalism for reproduction ignores the topology, this formalism for emergence ignores the formation of the DNA backbone along which are strung the complementary base pairs. In the biological domain we are aware of levels of ignored structure. In mathematics it is customary to stop the examination of certain issues in order to create domains with requisite degrees of clarity. We are all aware that the operation of collection is proscribed beyond a certain point. For example, in set theory the Russell class R of all sets that are not members of themselves is not itself a set. It then follows that {R}, the collection whose member is the Russell class, is not a class (since a member of a class is a set). This means that the construct {R} is outside of the discourse of standard set theory. This is the limitation of expression at the "high end" of the formalism. That the set theory has no language for discussing the structure of its own notation is the limitation of the language at the "low end". Mathematical users, in speaking and analyzing the mathematical structure, and as its designers, can speak beyond both the high and low ends. In biology we perceive the pattern of a formal system, a system that is embedded in a structure whose complexity demands the elucidation of just those aspects of symbols and signs that are commonly ignored in the mathematical context. Rightly these issues should be followed to their limits. The curious thing is what peeks through when we just allow a bit of it, then return to normal mathematical discourse. With this in mind, lets look more closely at the algebra of containers and extainers. Taking two basic forms of bracketing, an intricate algebra appears from their elementary interactions:
112 E = >
, [>[=[>G HH=] E G F = >[][=[]
