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Mech. Anal. 59,
Geometrization of Quantum Mechanics T. W. B. Kibble Blackett Laboratory, Imperial College, London SW7 2BZ, England
Abstract. Quantum mechanics is cast into a classical Hamiltonian form in terms of a symplectic structure, not on the Hilbert space of state-vectors but on the more physically relevant infinite-dimensional manifold of instantaneous pure states. This geometrical structure can accommodate generalizations of quantum mechanics, including the nonlinear relativistic models recently proposed. It is shown that any such generalization satisfying a few physically reasonable conditions would reduce to ordinary quantum mechanics for states that are "near" the vacuum. In particular the origin of complex structure is described.
Funct. Anal. 29, operators with
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ticle Schrodinger 710-713 (1961) i, 684-689 (1962) ticle Schrodinger and four particle lath. Anal. Appl. rsis of operators. I.
math. Phys. 27.
)tentials and the '73) ems. Math. Ann. em. (to appear)
1. Introduction Geometrical ideas, especially symplectic structures, have come to play an increasingly important role in classical mechanics [I, 21. The geometry of the classical phase space r also underlies the geometrical quantization programme of Souriau [3,4] (see also Kostant [5]). Moreover it is known that quantum dynamics can be expressed in terms of a Hamiltonian structure on the Hilbert space X of statevectors, where the imaginary part of the scalar product defines a symplectic structure [6]. However if one is seeking an axiomatic basis for quantum mechanics, it seems better to start from structures of direct physical significance, as in the operational approach of Haag and Kastler [7] and others [8,9] or the work on the geometry of quantum logics [lo, 111. It is pointed out in Sect. 2 that this can be achieved by a slight modification of the Hamiltonian formalism. We have to consider not the Hilbert space X itself but the manifold C of "instantaneous pure states" which is (essentially but not quite) a projective Hilbert space. This formalism is closely akin to the work of Mielnik [I21 on the geometry of the space of quantum states. It provides a convenient framework within *which to discuss possible generalizations of quantum mechanics. In particular it can readily accommodate the relativistic model theories proposed in a recent paper [13], or at least a large class of them. It is
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worth noting that by formulatitig the theory on C rather than 2 we automatically ensure that it satisfies the scaling property shown in 1131 to be necessary for a consistent mcasuremcnt tlicory (scc also [ I 41). From tliis point of view, thc csscntial diffcrcncc bctwccn classical and quantum mechanics lies not in the set of states (save for the infinite dimensionality) nor in the dynamic evolution, but rather in the choice of the class of observables, which is far more restricted in quantum than in cl:~ssicalmcchan~cs. One motivation for tlie present work is the possibility that it might be of use in the unification of quantum mechanics with general relativity. The idea that this unification must bc an essentially geometric one, so long clianipioned by Einstein in Iiis search for a unified theory, has recently been coming back into favour. It seems natural tliat as a prerequisite quantum mechanics itself should be cast in geometrical language. Moreovcr there are good reasons for seeking to generalize it, to free it from tlie restrictions of linearity just as general relativity has freed space-time from tlic limitations of flatness 112, 151. The geometrical structure described in the present paper can easily be generalized to allow tlie space of quantum statcs to be an arbitrary infinite-dimensional symplcctic manifold. Obviously, any viable generalization of quantum mechanics must reduce to tliat theory for a widc rangc of phenomena. Onc of tlie main aims of this paper is to discuss how t h ~ might s liappcn. I shall show, on tlie basis of some ratlier natural physical assuniptions, tliat all thc main features of conventional quantum mechanics would cmerge naturally for states that are in a suitable sensc near tlie vacuum, near enough to be represented by vectors in the tangent space. I n Sects. 2 and 3 we recall the main features of symplectic structures and Hamiltonian dynamics, and introduce the quantum phase space C. For conventional quantum mechanics, C is essentially a projective conlplex Hilbert space; more precisely, a dense subspace thereof. However the formalism can be applied to a much more general case, in which C is a real infi~iite-dirne~isio~ial manifold with (weak) symplectic structure. Tlie dynamics is discussed in Sect. 3 in terms of a Hamiltonian function E on C. For ordinary quantum mechanics E is the expectation value of tlie Hamiltonian operator, and the evolution equation reproduces Sclirodinger's equation. The concept of a symmetry is examined in Sect. 4 witli particular emphasis on the space-time symmetries associated with the Poincare group. Although an eventual aim is tlie unificatio~iof quantum theory with gravity, for the present a flat space-timc is assumcd. So too is the existence of a unique vacuum state u. The tangent space T, to C at the vacuum plays an important role. The aim of Sect. 5 is to show tliat for states close enough to the vacuum to be represented by vectors in the tangent space, the formalism necessarily reduces to ordinary linear quantum mechanics. In particular, I shall show how, although C is a rcal manifold, the complcx structure of quantum mcclianics would naturally appear on the tangent space. The conclusions are summarized in Sect. 6 and a number of unresolved questions described. 2. The Quantum Phase Space Axiomatic treatments of quantutn mecha~iicsoften begin with . . the set of all mixed ,, -- . . -- . . ,
elements of this convex set. However we shall deal exclusively with the set .Y of pure states, and assume at the outset that all others call be expl-cssed as (presumably countable) mixturcs ol' tlicm. Morcovcr instcad of tlic I-lciscnbcrg-picture approach wc shall usc the Schrodinger picture. We assume that, with respect to a given choice of time axis each pure state can be represented by a path, or "history" in a space C of ~nslnnlancouspure states". For convcnicllcc, wc rcscl-vc thc tcrni slale for an ,L.
element of C; elements of 5" will be called histo~.ies. Of course, tlie dynamics establishes a one-to-one correspondence between clemcnts of C at a spccificd time and elements of Y. Although tlie formalism will be more general, i t will be useful to begin by discussing the structure of 2' in the special case of standard quantum mechanics. Indeed this is the only space we shall actually need in this paper because the generalized ~nodelsconsidered use the same set of instantaneous states and differ from the standard theory only in tlic dynamics. Later, however, more general theories will be considered. In classical mechanics, C is of course tlie phasc space f, normally a finitedimensional manifold. but in quantum mechanics it is infinite-climcnsional. Essentially it is a projective Hilbcrt space, thc set of rays in a complex Hilbcrt space .Yf of statc vectors. I-Iowevcr tlicrc arc technical problc~nsassociated with the fact thal the Hamiltonian operator H is unbounded. We have to choosc between two alternative fornialisms, each with its peculiar advantages and disadvantages. 01ic is to work with the rays of .Y? itsclf and accept that the Hamiltonian vcctor field which spccifics tlie dynamics is defined only on a dense subspace [6]. Tlie othcr, which we shall use here, is to work fro~nthe start with a dense subspace .Xof .Ye equipped witli a finer topology that makes H continuous. The chief drawback of this second alternative is that C possesses only a weak symplectic structure (see below). I f we werc trying to prove existence theorems for solutions of the time-evolution equation, this might be a major defect, but for our present purposes it is not i~ilportant. Let us assume then that there is a dense linear subspacc .X of .Yf which forms a common invariant donlain for all the operators we wish to consider, including in particular the Hamiltonian and otl~crsymmetry generators. For example, in a nonrelativistic ~iiany-bodytheory we may tale .X to be the subspace of states whose wave functions belong to sollie test-function space, say the Schwartz space Y (see for instance [18]). I n a field theory wc miglit take it to be tlie subspace generated by applying some algebra of observables to the vacuuni state. Physically, it is only states in this subspacc that we can actually prepare, so ,;//' is of more direct physical relevance tha~i.K The space .X is assumed to bc cquippcd with a topology fincr than that ol' .w; defined for example by a countable family of norlns [18]. whicli makes H and the other sym~iietrygenerators continuous on .YL I t is possible to do this in such a way that .X becoliles a Frtchet-Schwartz space. Now let .Tobe the set of all nonzero vectors in .;//. and let C" be tlic multiplicative group of all nonzero complex numbcrs. '1-Iicn we chuosc as our standard quantum phase space the projective sl7acc C = . f l o i ( r O . whicli \aT(, c h : ~ l l
T. W. B. Kibble
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For the general theory, all we shall assume is that C is a real infinitedimensional paracompact manifold modelled on some FrCchet-Schwartz space K What this means [L9J is that C is a Hausdorff topological space that can be covered by a countable family of open sets U , on each of which a homeo~norphism 4, is defined to an open set in !; and that wlie~icverU,nUB+O, the map 4,4l,' restricted to +p(U,nU,,) is twice continuously differentiable. I n a general infinitedimensional non-Banacl~space, the definition of differentiability is quite problematic [20]. However in the particular case of Frechet-Schwartz spaces, there is a perfectly serviccablc dcfinition [21] which allows the construction of Ck manifolds (see also [22. 231). It might not be unreasonable to require that C be a C" manifold, but we shall not need that assu~nptionhere. I n addition to its manifold structure, C is required to have a weak symplectic structure. This mcans [6] that there is defincd on C a closed two-form w wliich is weakly tion-degenerate in the sense that if o,(X, Y)=O for all tangent vectors YE T,C,then X =O. This for111can be used to "lower" indices [I]. It defines a map X+.Xb from the tangent bundle T C to the cotangent bundle P C : if X is any tangent vector (or vector field) the corresponding cotangent vector (or differential one-form) is Xb=ixo,
(1)
where ix is the interior product (evaluation function) defined by ixw(Y)=w(X, Y) . (Note that therc is a difference of a factor 2 between co~iventionsused here and in references [I, 2, 61. 1 follow the co~iventionsof Choquet-Bruhat et al. [24]). The map X+XQs injective. However, w is only weakly nondegencrate in the sense that X+.XDis not in general bijective. There may be one-forms that are not images of any vector field. I n the special case of ordinary quantum mechanics, w is defined in terms of the inner product. (As well its finer topology .T retains the inner product ( ., .) of 2.) To define o we need to introduce a local coordinate system in C. Let IT be the canonical projection of ,Toonto C. Thus if u is any nonzero vector in X,nu=u denotes the corresponding state. Let v be any normalized vector in .j/L Then in a neighbourhood of n v ~ Cwe can represent states uniquely by vectors u in the hyperplane
Geometrization
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This two-form w is obviously skew-symmetric and wcakly nondegcnerate, But while w is readily seen to be closed, it is not exact. (1t is easy to construct a closed \
surface over which its integral is nonzero, although locally ( I ) = - d O with
This is another interesting difference between quantum and classical mechanics. Classically, the phase space has the structure of a cotangent bundle and c u = -dO where O is the canonical one-fonn O=Cpdy. No such form exists globally on the quantum phase space C. 3. Dynamics on the Quantum P l ~ a s cSpace
Let us now consider the specification of dynaniics on C , assumed to be a real CZ Frichet-Schwartz manifold equipped with a weak symplectic structure. The dynamics is described by a Hamilionian flow, that is to say a oneparameter group of diffeomorphis~nss, :C - t C (with s,, = 1 and s,+, = s, +s,) which preserve the symplectic structure, i.e. z:w=w.
(4) The z, depend continuously and differentiably [21] on r and can thus bc written
z , = c x p ~ K, where K is a C' vector field on C , which of coursc also leaves invariant the symplectic structure :
where L, is the corresponding Lie derivative. (Had we adopted the alternative approach ~nentioncdin Sect. 2 we could not have required T, to be differentiable in t and would have had to allow K to be defined only on a dense subset of C, see [61.) A history, i.e. an element of .C/: is an integral curvc of this flow in C, a curvc c everywhere tangent to K : --
dt - K c , , , ,
Thus we have a ho~neomorphism4 from this neighbourhood to an open set in ZV. We then define o) at nv by (2) c.,(X, Y)= 2 Im ((I)&, d)* Y) for any X, YET,,C, or more generally at any point in this coordinate patch by
or equivalently
By virtue of the identity L,=i,d+di, from (5) and the definition (1) illat d(Kb)=O,
where X, YE T,Z and P,, is the orthogonal projector onto the subspace normal to u.
i.e. K b is closed.
and the fact that w is closed, it follows (8)
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In standard quantum mechanics, the space C defined in Sect. 2 is a simply connected paracompact manifold, and so its first de Rhain col~omologygroup is trivial. (De Rliam's tlieorc~iiin general is hard to prove but here we need only the comparatively straightforward result that a closed form in such a space is exact.) This is an important fcaturc, bccausc i t mcans that from (8) we can conclude that K" is exact, i.e. that a Hamiltonian function cxists. We shall assume that tlie same is true in any gcncralizcd quantum theory wc consider, though it might iiidccd be intriguing to consider a theory witli a non-simply-conliected quantum phase spacc, a toroidal space say. With this assumptio~lit follows from (8) that there exists a Hamiltoiiiali function E, ~uliqueup to a constant, such that Kb=dE .
(9)
I t follows that any liistory is a solution of Halllilton's equations
Of course the function E uniquely determines tlie vector field K and therefore the one-parameter group z,. What is I I I U C ~ less obvious is whether, given E, any such flow exists, or ill other words to prove an existence theorem for solutions of the differential Eq. (10). Wc shall not attempt to treat this problem here (see [6]). 111ordinary quantum mechanics, the functio~iE is identified with the expectation valuc of thc Hamiltonian opcrator, E(u) = (H),
E
(44 H4u) ( 4 ~ 7$10
Geomctrization
195
model of a scalar field 4. Let H be the Hamiltonia~loperator for the usual linear quantum field theory with d4 i~~teraction. Let us take E(u) = (H)"
+
$ 3 L S
d3~
