Geometrical Formulation of Quantum Mechanics
Abstract
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a Kähler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics which, although equivalent to the standard algebraic formulation, has a very different appearance. In particular, states are now represented by points of a symplectic manifold (which happens to have, in addition, a compatible Riemannian metric), observables are represented by certain realvalued functions on this space and the Schrödinger evolution is captured by the symplectic flow generated by a Hamiltonian function. There is thus a remarkable similarity with the standard symplectic formulation of classical mechanics. Features—such as uncertainties and state vector reductions—which are specific to quantum mechanics can also be formulated geometrically but now refer to the Riemannian metric—a structure which is absent in classical mechanics. The geometrical formulation sheds considerable light on a number of issues such as the second quantization procedure, the role of coherent states in semiclassical considerations and the WKB approximation. More importantly, it suggests generalizations of quantum mechanics. The simplest among these are equivalent to the dynamical generalizations that have appeared in the literature. The geometrical reformulation provides a unified framework to discuss these and to correct a misconception. Finally, it also suggests directions in which more radical generalizations may be found.
I Introduction
Quantum mechanics is probably the most successful scientific theory ever invented. It has an astonishing range of applications—from quarks and leptons to neutron stars and white dwarfs—and the accuracy with which its underlying ideas have been tested is equally impressive. Yet, from its very inception, prominent physicists have expressed deep reservations about its conceptual foundations and leading figures continue to argue that it is incomplete in its core. Time and again, attempts have been made to extend it in a nontrivial fashion. Some of these proposals have been phenomenological (see, e.g., [1, 2, 3]), aimed at providing a ‘mechanism’ for the state reduction process. Some have been more radical, e.g., invoking hidden variables (see, e.g., [4]). Yet others involve nonlinear generalizations of the Schrödinger equation [5, 6, 7]. Deep discomfort has been expressed at the tension between objective descriptions of happenings provided by the spacetime geometry of special relativity and the quantum measurement theory [8, 9]. Further conceptual issues arise when one brings general relativity in to picture, issues that go under the heading of ‘problem of time’ in quantum gravity [10, 11, 12]. Thus, while there is universal agreement that quantum mechanics is an astonishingly powerful working tool, in the ‘foundation of physics circles’ there has also been a strong sentiment that sooner or later one would be forced to generalize it in a profound fashion [13, 14, 15].
It is often the case that while an existing theory admits a number of equivalent descriptions, one of them suggests generalizations more readily than others. Furthermore, typically, this description is not the most familiar one, i.e., not the one that seems simplest from the limited perspective of the existing theory. An example is provided by Cartan’s formulation of Newtonian gravity. While it played no role in the invention of the theory (it came some two and a half centuries later!) at a conceptual level, Cartan’s framework provides a deeper understanding of Newtonian gravity and its relation to general relativity. A much more striking example is Minkowski’s geometric reformulation of special relativity. His emphasis on hyperbolic geometry seemed abstract and abstruse at first; at the time, Einstein himself is said to have remarked that it made the subject incomprehensible to physicists. Yet, it proved to be an essential stepping stone to general relativity.
The purpose of this article is to present, in this spirit, a reformulation of the mathematical framework underlying standard quantum mechanics (and quantum field theory). The strength of the framework is that it is extremely natural from a geometric perspective and succinctly illuminates the essential difference between classical and quantum mechanics. It has already clarified certain issues related to the second quantization procedure and semiclassical approximations [16]. It also serves to unify in a coherent fashion a number of proposed generalizations of quantum mechanics; in particular, we will see that generalizations that were believed to be distinct (and even incompatible) are in fact closely related. More importantly, this reformulation may well lead to viable generalizations of quantum mechanics which are more profound than the ones considered so far. Finally, our experience from seminars and discussions has shown that ideas underlying this reformulation lie close to the heart of geometrically oriented physicists. It is therefore surprising that the framework is not widely known among relativists. We are particularly happy to be able to rectify this situation in this volume honoring Engelbert and hope that the role played by the Kähler geometry, in particular, will delight him.
Let us begin by comparing the standard frameworks underlying classical and quantum mechanics. The classical description is geometrical: States are represented by points of a symplectic manifold , the phase space. The space of observables consists of the (smooth) realvalued functions on this manifold. The (ideal) measurement of an observable in a state yields simply the value at the point ; the state is left undisturbed. These outcomes occur with complete certainty. The space of observables is naturally endowed with the structure of a commutative, associative algebra, the product being given simply by pointwise multiplication. Thanks to the symplectic structure, it also inherits a Liebracket—the Poisson bracket. Finally to each observable is associated a vector field called the Hamiltonian vector field of . Thus, each observable generates a flow on . Dynamics is determined by a preferred observable, the Hamiltonian ; the flow generated by describes the time evolution of the system.
The arena for quantum mechanics, on the other hand, is a Hilbert space . States of the system now correspond to rays in , and the observables are represented by selfadjoint linear operators on . As in the classical description, the space of observables is a real vector space equipped with with two algebraic structures. First, we have the the Jordan product—i.e., the anticommutator—which is commutative but now fails to be associative. Second, we have ( times) the commutator bracket which endows the space of observables with the structure of a Lie algebra. Measurement theory, on the other hand, is strikingly different. In the textbook description based on the Copenhagen interpretation, the (ideal) measurement of an observable in a state yields an eigenvalue of and, immediately after the measurement, the state is thrown into the corresponding eigenstate.specific outcome can only be predicted probabilistically. As in the classical theory, each observable gives rise to a flow on the state space. But now, the flow is generated by the 1parameter group and respects the linearity of . Dynamics is again dictated by a preferred observable, the Hamiltonian operator .
Clearly, the two descriptions have several points in common. However, there is also a striking difference: While the classical framework is geometric and nonlinear, the quantum description is intrinsically algebraic and linear. Indeed, the emphasis on the underlying linearity is so strong that none of the standard textbook postulates of quantum mechanics can be stated without reference to the linear structure of .
From a general perspective, this difference seems quite surprising. For linear structures in physics generally arise as approximations to the more accurate nonlinear ones. Thus, for example, we often encounter nonlinear equations which correctly capture a physical situation. But, typically, they are technically difficult to work with and we probe properties of their solutions through linearization. In the present context, on the other hand, it is the deeper, more correct theory that is linear and the nonlinear, geometric, classical framework is to arise as a suitable limiting case.
However, deeper reflection shows that quantum mechanics is in fact not as linear as it is advertised to be. For, the space of physical states is not the Hilbert space but the space of rays in it, i.e., the projective Hilbert space . And is a genuine, nonlinear manifold. Furthermore, it turns out that the Hermitian innerproduct of naturally endows with the structure of a Kähler manifold. Thus, in particular, like the classical state space , the correct space of quantum states, , is a symplectic manifold! We will therefore refer to as the quantum phase space. Given any selfadjoint operator , we can take its expectation value to obtain a real function on . It is easy to verify that this function admits an unambiguous projection to the projective Hilbert space . Recall, now, that every phase space function gives rise to a flow through its Hamiltonian vector field. What then is the interpretation of the flow ? It turns out [17] to be exactly the (projection to of the) flow defined by the Schrödinger equation (on ) of the quantum theory. Thus, Schrödinger evolution is precisely the Hamiltonian flow on the quantum phase space!
As we will see, the interplay between the classical and quantum ideas stretches much further. The overall picture can be summarized as follows. Classical phase spaces are, in general, equipped only with a symplectic structure. Quantum phase spaces, , on the other hand, come with an additional structure, the Riemannian metric provided by the Kähler structure. Roughly speaking, features of quantum mechanics which have direct classical analogues refer only to the symplectic structure. On the other hand, features—such as quantum uncertainties and state vector reduction in a measurement process—refer also the Riemannian metric. This neat division lies at the heart of the structural similarities and differences between the (mathematical frameworks underlying the) two theories.
Section II summarizes this geometrical reformulation of standard quantum mechanics. We begin in II.1 by showing that the quantum Hilbert space can be regarded as a (linear) Kähler space and discuss the roles played by the symplectic structure and the Kähler metric. In II.2, we show that one can naturally arrive at the quantum state space by using the BergmannDirac theory of constrained systems. (This method of constructing the quantum phase space will turn out to be especially convenient in section III while analyzing the relation between various generalizations of quantum mechanics.) Section II.3 provides a selfcontained treatment of the various issues related to observables—associated algebraic structures, quantum uncertainty relations and measurement theory—in an intrinsically geometric fashion. These results are collected in section II.4 to obtain a geometric formulation of the postulates of quantum mechanics, a formulation that makes no reference to the Hilbert space or the associated linear structures. In practical applications, except while dealing with simple cases such as spin systems, the underlying quantum phase space is infinitedimensional (since it comes from an infinitedimensional Hilbert space ). Our mathematical discussion encompasses this case. Also, in the discussion of measurement theory, we allow for the possibility that observables may have continuous spectra.
In section III, we consider possible generalizations of quantum mechanics. These generalizations can occur in two distinct ways. First, we can retain the original kinematic structure but allow more general dynamics, e.g., by replacing the Schrödinger equation by a suitable nonlinear one (see, e.g., [5]). In section III.1 we show that the geometrical reformulation of section II naturally suggests a class of such extensions which encompasses those proposed by Birula and Mycielski [5] and by Weinberg [7]. In section III.2 we consider the possibility of more radical extensions in which the kinematical set up itself is changed. Although (to our knowledge) there do not exist interesting proposals of this type, such generalizations would be much more interesting. In particular, it is sometimes argued that the linear structure underlying quantum mechanics would have to be sacrificed in a subtle but essential way to obtain a satisfactory quantum theory of gravity and/or to cope satisfactorily with the ‘measurement problem’ [13]. To implement such ideas, the underlying kinematic structure will have to be altered. A first step in this direction is to obtain a useful characterization of the kinematical framework of standard quantum mechanics. Section III.2 provides a ‘reconstruction theorem’ which singles out quantum mechanics from its plausible generalizations. The theorem provides powerful guidelines: it spells out directions along which one can proceed to obtain a genuine extension.
Section IV is devoted to semiclassical issues. Consider a simple mechanical system, such as a particle in . In this case, the classical phase space is sixdimensional while the quantum phase space is infinitedimensional. Is there a relation between the two? In section IV.1 we show that the answer is in the affirmative: is a bundle over . Furthermore, the bundle is trivial. Thus, through each quantum state , there is a crosssection, i.e. a copy of . It turns out that the quantum states that lie on any one crosssection are precisely the generalized coherent states [18, 20, 21]. In the remainder of section IV, we use this interplay between and to discuss the relation between classical and quantum dynamics. Section IV.2 is devoted to the correspondence in terms of Ehrenfest’s theorem while IV.3 discusses the problem along the lines of the WKB approximation. Somewhat interestingly, it turns out that WKB dynamics is an example of generalized dynamics of the Weinberg type [7].
Our conventions are as follows. If the manifold under consideration is infinitedimensional, we will assume that it is a Hilbert manifold. (Projective Hilbert spaces are naturally endowed with this structure; see, e.g., [16].) Riemannian metrics and symplectic structures on these manifolds will be assumed to be everywhere defined, smooth, strongly nondegenerate fields. (Thus, they define isomorphisms between the tangent and cotangent spaces at each point). In detailed calculations we will often use the abstract index notation of Penrose’s [22, 23]. Note that, in spite of the appearance of indices, this notation is welldefined also on infinitedimensional manifolds. (For example, if denotes a contravariant vector field; the subscript does not refer to its components but is only a label telling us that is a specific type of tensor field, namely a contravariant vector field. Similarly, is the function obtained by the action of the 1form on the contravariant vector field .) Finally, due to space limitation, we have not included detailed proofs of several technical assertions; they can be found in [16]. Our aim here is only to provide a thorough overview of the subject.
This work was intended to be an extension of a paper by Kibble [17] which pointed out that the Schrödinger evolution can be regarded as an Hamiltonian flow on . However, after completing this work, we learned that many of the results contained in sections II and III.2 were obtained independently by others (although the viewpoints and technical proofs are often distinct.) In 1985, Heslot [24] observed that quantum mechanics admits a symplectic formulation in which the phase space is the projective Hilbert space. That discussion was, however, restricted to the finitedimensional case and did not include a discussion of the role of the metric, probabilistic interpretation and quantum uncertainties. Anandan and Aharonov [25] rediscovered some of these results and also discussed some of the probabilistic aspects. This work was also restricted to finitedimensional systems and focussed on the issue of evolution. Similar observations were made by Gibbons [26] who also discussed density matrices (which are not considered here) and raised the issue of characterization of quantum mechanics (which is resolved in section IV.2). An essentially complete treatment of the finitedimensional case was given by Hughston [27]. (This work was done in parallel to ours. However, it also contains some proposals for mechanisms for state reduction [28] which are not discussed here.) The only references (to our knowledge) which treat the infinitedimensional case are [29, 30], which also discuss the issue of characterization of standard quantum mechanics. Finally, since the geometric structures that arise here are so natural, it is quite possible that they were independently discovered by other authors that we are not aware of.
Ii Geometric formulation of quantum mechanics
The goal of this section is to show that quantum mechanics can be formulated in an intrinsically geometric fashion, without any reference to a Hilbert space or the associated linear structure. We will assume that the reader is familiar with basic symplectic geometry.
ii.1 The Hilbert space as a Kähler space
Let us begin with the standard Hilbert space formulation of quantum mechanics. In this subsection we will view the Hilbert space as a Kähler space and examine the role played by the associated symplectic structure and the Riemannian metric. This discussion will serve as a stepping stone to the analysis of the quantum phase space in section II.2.
The similarities between classical and quantum mechanics can be put in a much more suggestive form with an alternative, but equivalent, description of the Hilbert space. We view as a real vector space equipped with a complex structure . The complex structure is a preferred linear operator which represents multiplication by ; hence . Initially, this change of notation seems rather trivial; the element which is typically written is now denoted and (external) multiplication of vectors by complex numbers is not permitted. However, this slight change of viewpoint will come with a reward—a symplectic formulation of quantum mechanics.
Since is now viewed as a real vector space, the Hermitian innerproduct is slightly unnatural. We therefore decompose it into real and imaginary parts,
(1) 
(The reason for the factors of will become clear shortly.) Properties of the Hermitian innerproduct imply that is a positive definite, real innerproduct and that is a symplectic form, both of which are strongly nondegenerate. Moreover, since , one immediately observes that the metric, symplectic structure and complex structure are related as
(2) 
That is, the triple equips with the structure of a Kähler space. Therefore, every Hilbert space may be naturally viewed as a Kähler space.
Next, by use of the canonical identification of the tangent space (at any point of ) with itself, is naturally extended to a strongly nondegenerate, closed, differential 2form , which we will denote also by . Any Hilbert space is therefore naturally viewed as the simplest sort of symplectic manifold, i.e., a phase space. The inverse of may be used to define Poisson brackets and Hamiltonian vector fields. As we are about to see, these notions are just as relevant in quantum mechanics as in classical mechanics.
ii.1.1 The symplectic form
In classical mechanics, observables are realvalued functions, and to each such function is associated a corresponding Hamiltonian vector field. In quantum mechanics, on the other hand, the observables themselves may be viewed as vector fields, since linear operators associate a vector to each element of the Hilbert space. However, the Schrödinger equation, which in our language is written as motivates us to associate to each quantum observable the vector field
(3) 
This Schrödinger vector field is defined so that the timeevolution of the system corresponds to the flow along the Schrödinger vector field associated to the Hamiltonian operator. (Note that, if the Hamiltonian is unbounded, it is only densely defined and so is the vector field. The (unitary) flow, however, is defined on all of . See, e.g., [31].)
Natural questions immediately arise. Let be any bounded, selfadjoint operator on . Is the corresponding vector field, , Hamiltonian on the symplectic space ? If so, what is the realvalued function which generates this vector field? What is the physical meaning of the Poisson bracket? In particular, how is it related to the commutator Lie algebra?
The answers to these questions are remarkably simple. As we know from standard quantum mechanics, generates a oneparameter family of unitary mappings on . By definition, is the generator of this oneparameter family and therefore preserves both the metric and symplectic form . It is therefore locally Hamiltonian, and, since is a linear space, also globally Hamiltonian! In fact, the function which generates this Hamiltonian vector field is of physical interest; it is simply the expectation value of .
Let us see this explicitly. Denote by the expectation value function,
(4) 
We will continue to use this notation; expectation value functions will be denoted by simply “unhatting” the corresponding operators. Now, if is any tangent vector at , then^{1}^{1}1With our conventions for symplectic geometry, the Hamiltonian vector field generated by the function satisfies the equation , and the Poisson bracket is defined by .
(5)  
where we have used the selfadjointness of , Eq. (2) and the definition of . Therefore, the Hamiltonian vector field generated by the expectation value function coincides with the Schrödinger vector field associated to . As a particular consequence, the time evolution of any quantum mechanical system may be written in terms of Hamilton’s equation of classical mechanics; the Hamiltonian function is simply the expectation value of the Hamiltonian operator. Schrödinger’s equation is Hamilton’s equation in disguise!
Next, let and be two quantum observables, and denote by and the respective expectation value functions. It is natural to ask whether the Poisson bracket of and is related in a simple manner to an algebraic operation involving the original operators. Performing a calculation as simple as Eq. (5), one finds that
(6) 
Notice that the quantity inside the brackets on the right side of Eq. (6) is precisely the quantum Lie bracket of and . The algebraic operation on the expectation value functions, which is induced by the commutator bracket is exactly a Poisson bracket! Note that this is not Dirac’s correspondence principle; the Poisson bracket here is the quantum one, determined by the imaginary part of the Hermitian innerproduct.
The basic features of the classical formalism appear also in quantum mechanics. The Hilbert space, as a real vector space, is equipped with a symplectic form. To each quantum observable is associated a realvalued function on , and the timeevolution is determined by the Hamiltonian vector field associated to a preferred function. Moreover, the Lie bracket of two quantum observables corresponds precisely to the Poisson bracket of the corresponding functions.
ii.1.2 Uncertainty and the real innerproduct
Let us now examine the role played by the metric . Clearly, enables us to define a real innerproduct, between any two Hamiltonian vector fields and . One may expect that this innerproduct is related to the Jordan product in much the same way that the symplectic form corresponds to the commutator Lie bracket. It is easy to verify that this expectation is correct. Operating just as in Eq. (6), we obtain
(7) 
The operation defined by the first equation above will be called the Riemann bracket of and . Up to the factor of , the Riemann bracket of and is simply given by the (real) innerproduct of their Hamiltonian vector fields, and corresponds precisely to the Jordan product of the respective operators.
Since the classical phase space is, in general, not equipped with a Riemannian metric, the Riemann product does not have an analogue in the classical formalism; it does, however, admit a physical interpretation. In order to see this, note that the uncertainty of the observable at a state with unit norm is given by
(8) 
Thus, the uncertainty of an operator , when written in terms of the expectation value function , involves the Riemann bracket. Moreover, this expression for the uncertainty is quite simple.
In fact Heisenberg’s famous uncertainty relation also assumes a nice form when expressed in terms of the expectation value functions. It is very wellknown that the familiar uncertainty relation between two quantum observables may be written in a slightly stronger form (see, e.g.,[32]):
(9) 
where is the nonlinear operator defined by
so that is orthogonal to , if .
ii.2 The quantum phase space
We have seen that to each quantum observable is associated a smooth realvalued (expectation value) function on the Hilbert space. Further, the familiar operations involving quantum operators correspond to simple “classical looking” operations on the corresponding functions. These observations suggest a formulation of standard quantum mechanics in the language of classical mechanics. However, there are two difficulties. First, although the Hilbert space is a symplectic space, because two state vectors related by multiplication by any complex number define the same state, it is not the space of physical states, i.e., the quantum analog of the classical phase space. Second, the description of the measurement process in a manner intrinsic to the Kähler structure on turns out not to be natural.
The true state space of the quantum system is the space of rays in the Hilbert space, i.e., the projective Hilbert space, which we shall denote . It should not be surprising that is a Kähler manifold, and hence, in particular, a symplectic manifold. After all, for the special case in which is , is the complex projective space —the archetypical Kähler manifold. In this section, we present a particularly useful description of the projective Hilbert space which is valid for the infinitedimensional case and which illuminates the role of its symplectic structure. These developments will enable us to handle the above complications and allow an elegant geometric formulation of quantum mechanics.
ii.2.1 Gauge reduction
The standard strategy to handle the ambiguity of the state vector is to consider only those elements of the Hilbert space which are normalized to unity. We will adopt this approach by insisting that the only physically relevant portion of the Hilbert space is that on which the constraint function,
(11) 
vanishes. The attitude adopted here is one in which the the constraint surface—the unit sphere, with respect to the Hermitian innerproduct—is the only portion of the Hilbert space which is accessible to the system. The rest of the Hilbert space is often quite convenient, but is viewed as an artificial element of the formalism.
Let us consider the above restriction from the point of view of the BergmannDirac theory of constrained systems (see, e.g., [33]). In other words, we will pretend, for a moment, that we are dealing with a classical theory with the constraint . First notice that since the timeevolution preserves the constraint surface, no further (secondary) constraints arise. Since we have only a single constraint, it is trivially of firstclass in Dirac’s terminology; i.e., the constraint generates a motion which preserves the constraint surface ().
Recall that to every firstclass constraint on a Hamiltonian system is associated a gauge degree of freedom; the associated gauge transformations are defined by the flow along the Hamiltonian vector field generated by the constraint function. In our case, the gauge directions are simply given by
(12) 
where denotes the LeviCività derivative operator. For later convenience, let us define
(13) 
Notice that is the generator of phase rotations on . Therefore, the gauge transformations generated by the constraint are exactly what they ought to be; they represent the arbitrariness in our choice of phase!
Thus, we now see the relevance of the description in terms of constrained Hamiltonian systems. By taking the quotient of the constraint surface of any constrained system by the action of the gauge transformations, one obtains the true phase space of the system—often called the reduced phase space. The projective Hilbert space may therefore be interpreted as the “reduced phase space” of our constrained Hamiltonian system. In order to emphasize both its physical role and geometric structure, we will refer to the projective Hilbert space as the quantum phase space. One can explicitly show that if is infinitedimensional, is an infinitedimensional Hilbert manifold [16].
As the terminology suggests, any reduced phase space is equipped with a natural symplectic structure. This fact may be seen as follows. Denote by and the inclusion mapping and projection to the quantum phase space, respectively. By restricting the symplectic structure to the constraint surface, one obtains a closed 2form on . This 2form is degenerate, but only in the gauge direction. Fortunately, since gauge transformations are defined by the Hamiltonian vector field generated by the constraint function, is constant along its directions of degeneracy. As a result, there exists a symplectic form on whose pullback via agrees precisely with . This is a standard construction in the theory of systems with first class constraints; the only novelty here lies in its application to ordinary quantum mechanics. Finally, we note that this result applies to the typical case of interest, in which the original Hilbert space is infinitedimensional [16]. The symplectic structure is then a smooth, strongly nondegenerate field (i.e., defines an isomorphism from the tangent space to the cotangent space at each point.)
Before discussing the geometry the quantum phase space, we should point out that the viewpoint adopted in this section in fact generalizes quantum mechanics, but in a very trivial way. In section II.1 we observed that each quantum observable defines a realvalued function on the entire Hilbert space, and that the quantum evolution is given by the Hamiltonian flow defined by a preferred function. In viewing a quantum system as a constrained Hamiltonian system, we must concede that it is only the constraint surface that is of physical relevance. In particular, the restriction of the expectation value function contains all gaugeinvariant information about the observable; one may extend off the constraint surface in any desirable manner without affecting the corresponding flow on the projective space. The particular extensions defined by expectation values of (bounded) selfadjoint operators may be viewed as mere convention. We will make use of this point in section III.1.
ii.2.2 Symplectic geometry
Our method of arriving at the quantum phase space by the reduced phase space construction of constrained systems immediately suggests further definitions and constructions.
Recall that to each bounded, selfadjoint operator on , we have associated the function on the Hilbert space. In fact, we may go a short step further. First, let us restrict the expectation value function to the constraint surface, thereby obtaining the function . is clearly gaugeinvariant (i.e. independent of phase), and therefore defines the function for which . Therefore, to each quantum observable is associated a smooth, realvalued function on the quantum phase space. The functions obtained in this manner will represent the observables in the geometric formulation of quantum mechanics. Let us therefore make
Definition II.1
Let be a smooth function on . If there exists a bounded, selfadjoint operator on for which , then is said to be an observable function.
Note that we consider the set of quantum observables to consist of the bounded selfadjoint operators on . At first sight this appears to be a severe restriction. However, further reflection shows that it is not. In any actual experiment, one deals only with a finite range of relevant parameters and hence in practice one only measures observables of the type considered here. Thus, there is by definition a onetoone correspondence between quantum observables and the observable functions on . As we will see below, the set of observable functions is a very small subset of the entire function space.
A natural question arises: What is the relationship between the Hamiltonian vector fields (on ) and (on )? Given any point , we may pushforward the vector to obtain a tangent vector at . Since is gaugeinvariant, it commutes with ; therefore
As a consequence, is “constant along the integral curves of ”. Thus, by pushingforward at each point of , one obtains a welldefined (smooth) vector field on all of . As is known to those familiar with the analysis of constrained systems, this vector field is also Hamiltonian; in fact, it agrees precisely with . The flow on , which is induced by the Schrödinger vector field of corresponds exactly to the Hamiltonian flow determined by the observable function .
Next, consider the Poisson bracket defined by the reduced symplectic structure . Let be expectation value functions of two quantum observables and denote by the corresponding observable functions on . As a consequence of the above result,
(14) 
Therefore, the Poisson bracket defined by exactly reflects the commutator bracket on the space of quantum observables.
In summary, to each quantum observable is associated a realvalued function on the quantum phase space. The Schrödinger vector field determined by determines a flow on ; this flow is generated by the Hamiltonian vector field associated to the observable function . Further, the mapping is onetoone and respects the Lie algebraic structures provided by the commutator and Poisson bracket on , respectively.
ii.3 Riemannian geometry and measurement theory
Any quantum mechanical system may be described as an infinitedimensional Hamiltonian system. However, the structure of the quantum phase space is much richer than that of classical mechanics. is also equipped with a natural Riemannian metric. As we will see, the probabilistic features of quantum mechanics are conveniently described by the Riemannian structure.
The quantum metric may be described in much the same way as the symplectic structure. The restriction of to the unit sphere is a strongly nondegenerate Riemannian metric on . Recall that the gauge generator is (up to the constant factor of ) the Schrödinger vector field associated to the identity operator. Since any Schrödinger vector field preserves the Hermitian innerproduct, it preserves both the symplectic structure and the metric . As a consequence, is a Killing vector field on ;
(15) 
Therefore, may also be described as the Killing reduction[34] of with respect to the Killing field .
A manifold which arises in this way is always equipped with a Riemannian metric of its own.^{2}^{2}2One must require that the integral curves of the Killing vector field do not come arbitrarily close to one another. This condition is satisfied in our case. Although is “constant” on the integral curves of , it is not degenerate in that direction. However, by subtracting off the component in the direction of ,
(16) 
we obtain a symmetric tensor field which agrees with when acting on vectors orthogonal to , is constant along and is degenerate only in the direction of . Therefore defines a strongly nondegenerate Riemannian metric on . It is a simple matter to verify that , when combined with the symplectic structure , equips the quantum phase space with the structure of a Kähler manifold.
ii.3.1 Quantum observables
According to Def. II.1, quantum mechanical observables may be represented by realvalued functions on the quantum phase space. Unfortunately, these functions have still been defined in terms of selfadjoint operators on the Hilbert space. Our goal, however, is a formulation of quantum mechanics which is intrinsic to the projective space; we wish to avoid any explicit reference to the underlying Hilbert space. We now explain how this deficiency may be overcome.
Since the Schrödinger vector field generates a oneparameter family of unitary transformations on , preserves not only the symplectic structure , but the metric as well; is also a Killing vector field. This fact also holds for the corresponding observable function ; the Hamiltonian vector field associated to any observable function is also a Killing vector field on . We will see that it is this property which characterizes the set of observable functions on the quantum phase space.
Let us begin by recalling a general property of Killing vector fields. Since the calculations are somewhat involved, we will now use Penrose’s abstract index notation (which, as already pointed out, is meaningful also on infinitedimensional Hilbert manifolds.) Let be any Killing vector field on . Then, by definition, , where denotes the (LeviCività) derivative operator associated to the metric . Therefore, is necessarily skewsymmetric. As one can easily verify,[35] satisfies the identity
(17) 
(Our conventions are such that for any 1form , .) As a consequence, the Killing vector field is completely determined by its value and first covariant derivative at a single point. (See [36] or Appendix C of Ref. [35] for a discussion of this useful fact.)
Now suppose that the above Killing vector field is generated by the observable function . (Below, it will be understood that .) Since
(18) 
satisfies the additional property that is symmetric. (It then defines a bounded, skewselfadjoint operator on each tangent space.) By considering the coupled differential equations:
(19)  
(20)  
(21) 
we see [16] that any observable function is completely determined by its value and first two derivatives at a single point! This fact motivates
Definition II.2
For each point , let consist of all triples, , where is a real number, is a covector at , and is a 2form at for which . We call the algebra of symmetry data at .
Thus, any observable function determines an element ( provides the value of at ), and is completely determined by this symmetry data. The algebra of quantum observables is then isomorphic to a subset of . The converse to this statement is provided by [16]:
Theorem II.1
For any element of , there exists an observable function such that , and .
Therefore, the space of observable functions is isomorphic to the algebra of symmetry data for any . We will utilize this result in section III.2. For the purposes of this section, however, the main use of the above theorem is the following. Recall that the Hamiltonian vector field of any observable function is a Killing vector field. Conversely, let be any smooth, realvalued function on for which is also a Killing vector field. Of course, the value and first two covariant derivatives of at determine an element of . Therefore, by Thm. II.1, is an observable function. This important result is expressed in
Corollary 1
A smooth function is observable if and only if its Hamiltonian vector field is also Killing.
Therefore, as in classical mechanics, the space of quantum observables is isomorphic to the space of smooth functions on the phase space whose Hamiltonian vector fields are infinitesimal symmetries of the available structure. However, now the available structure includes not just the symplectic structure but also the metric. Hence, unlike in classical mechanics, this function space is an extremely small subset of the set of all smooth functions on . This should not be terribly surprising; as an example consider the finitedimensional case, for which the function space is infinitedimensional while the algebra of Hermitian operators is finitedimensional.
ii.3.2 Quantum uncertainty
In analogy with our considerations of the symplectic geometry of , let us consider the Riemann bracket defined by the quantum metric . We denote the Riemann bracket of by
(22) 
If and correspond to the expectation value functions , then, by Eq. (16),
(23) 
where we have used the fact that for any observable ,
(24) 
Therefore, the observable function which corresponds to the Jordan product of and is given not by the Riemann bracket of and , but by the quantity
(25) 
where we have utilized a minor notational abuse to emphasize the relationship with Eq. (7). The above quantity will be called the symmetric bracket of and .
Note that the Riemann bracket of two observables, while not necessarily an observable itself, is a physically meaningful quantity; it is simply the quantum covariance function (see the comment following Eq. (10)). In particular, is exactly the squared uncertainty of at the quantum state labeled by ;
(26) 
In order to obtain a feeling for the physical meaning of the quantum covariance, notice that the uncertainty relation of Eq. (10) assumes the form
(27) 
As one can easily show, the standard uncertainty relation
(28) 
is saturated at the state if and only if and , where is the complex structure on (compatible with and ). The quantum covariance therefore measures the “coherence” of the state with respect to the observables and .
Let us conclude this subsection by reproducing a result of Anandan and Aharanov[25]. Suppose that the Hamiltonian operator of a quantum system is bounded and let be the corresponding observable function. By definition of the Riemann bracket, the uncertainty of is given by . Therefore, apart from the constant coefficient, the uncertainty in the energy is exactly the length of the Hamiltonian vector field which generates the timeevolution. Thus, the energy uncertainty can be thought of as ‘the speed with which the system moves through the quantum phase space’; during its evolution, the system passes quickly through regions where the energy uncertainty is large and spends more time in states where it is small.
ii.3.3 The measurement process
Some of the most significant aspects of quantum mechanics involve the measurement process and, without a complete geometric description of these issues, our program would be incomplete. In this subsection we will sketch the desired geometric description, including the case when the spectrum of the operator is continuous. A more complete discussion may be found in [16].
Let be an arbitrary normalized element of the Hilbert space, and let be its projection to . Of obvious interest, in the context of measurement, is the function . Since is independent of the phase of it defines a function on via
(29) 
If the quantum system is in the state labeled by when a measurement is performed, the relevant quantum mechanical probability distribution is determined by the function . We therefore desire a description of this function which does not rely explicitly on the underlying Hilbert space. This is provided by
Theorem II.2
Given arbitrary points there exists a (closed) geodesic which passes through and . Further, , where denotes the geodesic separation of and .
A few comments regarding Thm. II.2 are in order. First, if and are nonorthogonal, then the abovementioned geodesic is unique, up to reparameterization. Next, since all geodesics on are closed, the geodesic distance is, strictly speaking, illdefined. Due to the periodicity of the cosine function, however, the “transition amplitude function”, , is insensitive to this ambiguity. For the sake of precision, by , we will mean the minimal geodesic distance separating and .
Suppose one is dealing with the measurement of an operator with discrete, nondegenerate spectrum, and let be the corresponding observable function. Each eigenspace of is one complexdimensional, and therefore determines a single point of . Denote these eigenstates by . Suppose the system is in the state labeled by the point when an ideal measurement of is performed. We know that the system will ‘collapse’ to one of the states . Theorem II.2 provides the corresponding probabilities. It is interesting to notice that the probability of collapse to an eigenstate is a monotonically decreasing function of the geodesic separation of and ; the system is more likely to collapse to a nearby state than a distant one.
We now have a description of the probabilities associated with the measurement process, but there are two deficiencies to be remedied. The eigenstates above have been defined in terms of the algebraic properties of the operator . For our program to be complete, we require a definition of these eigenstates which does not refer to the Hilbert space explicitly. Next, the above discussion was limited to the measurement of an observable with discrete, nondegenerate spectrum. We will describe the generic situation in two steps. First, we consider the measurement of observables with discrete, but possibly degenerate, spectra. This will require the aforementioned description of the eigenstates. We will then be prepared to consider measurement of observables with continuous spectra.
Let us first examine the notions of eigenstates and eigenvalues of an observable operator . Let and be the expectation value and corresponding observable function, respectively. A vector is an eigenstate of iff , for some (real) . Alternatively, by Eq. (3),
(30) 
That is, is an eigenstate of if and only if the Hamiltonian vector field, is vertical (i.e., purely gauge) at . This will be the case if and only if vanishes at ; is then a critical point of the function . Evidently, the corresponding eigenvalue is exactly the (critical) value of at . In summary:
Definition II.3
Let be an observable function. Critical point of are called eigenstates of . The corresponding critical values are called eigenvalues.
We now consider the measurement of an observable whose spectrum is discrete but possibly degenerate (of course, the “spectrum of ” coincides, by definition, with the spectrum of the corresponding operator ). Let be a degenerate eigenvalue of , and denote by the associated eigenspace of . Associated to this eigenspace is a submanifold, , of , which we shall call the eigenmanifold associated to . Suppose that the system is prepared in the state labeled by and let . The postulates of ordinary quantum mechanics assert that measurement of will yield the value with probability , where is the projector onto the relevant eigenspace of . From the above considerations, we know that this probability may be expressed in terms of the geodesic separation of the points and . We will denote the latter point by .
What sets the point apart from all other elements of ? We need only notice that for any point ,
(31) 
Therefore, of all elements with unit normalization, that which maximizes the quantity is simply , i.e., that to which the state will ‘collapse’ in the event that measurement of yields the value . Therefore, by Thm. II.2, is simply that point of which is nearest !
We may now describe the measurement of an observable with discrete spectrum as follows. Suppose that immediately prior to measurement of , the system is in the state . Denote by the critical values of and, by the corresponding eigenmanifolds. Interaction with the measurement device causes the system to be projected to one of the eigenmanifolds, say . “Realizing that it collapsed” to the state , the system returns what is knows to be the value of the observable under consideration, i.e., . Of course, the probability that measurement causes reduction to is given by , where, denotes the minimal geodesic separation of and the submanifold .
Now let us study the generic case. Let be any observable operator on , the spectrum of which is allowed to be continuous. We first need a definition of the spectrum of in terms of the corresponding observable function . Recall the standard definition [37]: is an element of the spectrum if and only if the operator is not invertible. Equivalently, iff given any positive , such that . This condition guarantees that, to arbitrary precision, is an approximate eigenvalue of .
Using Eqs. (25) and (26), we may write the (square of the) above quantity as
(32) 
This equation allows us to define the spectrum of in terms of the function ;
Definition II.4
The spectrum of an observable consists of all real numbers for which the function is unbounded.
Of course, a point at which corresponds to an eigenstate of .
The next step is a description of the spectral projection operators. Let be a closed subset of the spectrum of , and denote by the projection operator associated to and . In analogy with the above, put , and let denote the projection of to . Note that the set —the analogue of the eigenspace above— actually is the eigenspace of corresponding to the eigenvalue . Therefore, we have consists of the critical points of an expectation value function, associated to the critical value . Unfortunately, this expectation value function is not directly expressible in terms of and . We must look for an alternative description of the submanifold .
In the representation defined by the operator , elements of have support on . Therefore, iff , where denotes the image of under the map . Recall that is the (projection to of the) expectation value of . In general, the expectation value of projects to the fold symmetric product . Therefore, we have
(33) 
where there are factors of occurring above.
Having obtained a description of , we may now define the spectral projections in a manner intrinsic to the projective space. By precisely the same reasoning surrounding Eq. (31), maps a point to that element of which is nearest .
The measurement process may then be described as follows. Suppose the quantum system is in the state labeled by the point at the instant an experimenter performs a measurement of the observable . Following the rules of quantum mechanics, she “asks the system” whether the value of lies in —a closed subset of , which she is free to choose. The experimental apparatus drives the system to one of two states—either or , where is the (closure of the) complement, in , of . The system is reduced to the former with probability
in this event, the experiment yields the positive result (). Having precisely prepared the system in the state , the experimenter may then infer the value of the observable . The probability of reduction to the latter state is obtained by replacing by above.
Note that this description encompasses all measurement situations. In the event that the spectrum of is discrete and nondegenerate, the experimenter may choose to let contain the single eigenvalue . Moreover, she may measure all of the projections simultaneously. In this way, one recovers the first familiar description of the measurement process. Note also that, while the above discussion of the spectral projections may seem complicated and somewhat unnatural at first, the definition of the spectral projection operators on the Hilbert space has the same features. (Indeed, most text books simply skip this technical discussion.) This is simply one of the technical complications that the geometric formalism inherits from the Hilbert space framework.
To conclude, we wish to emphasize that the topic of our discussion has been ordinary quantum mechanics. We have just restated the wellknown quantum mechanical formalism in a language intrinsic to the true space of states—the quantum phase space, ; no new ingredients have been added to the physics. A particularly attractive feature of the formalism, however, is the fact that slight modifications of the standard picture naturally present themselves. For example, using many of these geometric ideas, Hughston[28] has explored a novel approach to the measurement problem in terms of stochastic evolution.
ii.4 The postulates of quantum mechanics
Let us collect the results obtained in the first three subsections.
We have formulated ordinary quantum mechanics in a language which is intrinsic to the true space of quantum states—the projective Hilbert space . As in classical mechanics, observables are smooth, realvalued functions which preserve the kinematic structure. Being a Kähler manifold, is a symplectic manifold. The role of the quantum symplectic structure is precisely that of classical mechanics; it defines both the Lie algebraic structure on the space of observables and generates motions including the timeevolution.
There are, however, two important features of quantum mechanics which are not shared by the classical description. First, the phase space is of a very particular nature; it is a Kähler manifold and, as we will see in section III.2, one of a rather special typenamely one of constant holomorphic sectional curvature.^{3}^{3}3In this sense, the quantum framework is actually a special case of the classical one! The second difference lies in the probabilistic aspects of the formalism, which is itself intimately related to the presence of the Riemannian metric. More generally, this metric describes those quantum mechanical features which are absent in the classical theory—namely, the notions of uncertainty and state reduction. For example, the transition probabilities which arise in quantum mechanics are determined by a simple function of the geodesic distance between points of the phase space.
These results are most easily summarized by stating the postulates in the geometric language:

Physical states: Physical states of the quantum system are in onetoone correspondence with points of a Kähler manifold , which is a projective Hilbert space.^{4}^{4}4We will see in section III.2 that quantum phase spaces can be alternatively singled out as Kähler manifolds which admit maximal symmetries. This provides an intrinsic characterization without any reference to Hilbert spaces.

Kähler evolution: The evolution of the system is determined by a flow on , which preserves the Kähler structure. The generator of this flow is a densely defined vector field on .

Observables: Physical observables are represented by realvalued, smooth functions on whose Hamiltonian vector fields preserve the Kähler structure.

Probabilistic interpretation: Let be a closed subset of the spectrum of an observable , and suppose the system is in the state corresponding to the point . The probability that measurement of will yield an element of is given by
(34) where is the point, closest to , in the space , defined bye Eq. (33).

Reduction, discrete spectrum: Suppose the spectrum of an observable is discrete. This spectrum provides the set of possible outcomes of the ideal measurement of . If measurement of yields the eigenvalue , the state of the system immediately after the measurement is given by the associated projection, , of the initial state .

Reduction, continuous spectrum: A closed subset of the spectrum of determines an ideal measurement that may be performed on the system. This measurement corresponds to inquiring whether the value of lies in . Immediately after this measurement, the state of the system is given by or , depending on whether the result of the measurement is positive or negative, respectively.
In the last postulate, is the closure of the complement, in the spectrum of of the set . Although the first “reduction postulate” is a special case of the second, both have been included for comparison with standard textbook presentations.
To conclude, although it is not obvious from textbook presentations, the postulates of quantum mechanics can be formulated in an intrinsically geometric fashion, without any reference to the Hilbert space. The Hilbert space and associated algebraic machinery provides convenient technical tools. But they are not essential. Mathematically, the situation is similar to the discussion of manifolds of constant curvature. In practice, one often establishes their properties by first embedding them in (equipped with a flat metric of appropriate signature). However, the embedding is only for convenience; the object of interest is the manifold itself. There is also a potential analogy from physics, alluded to in the Introduction. Perhaps the habitual linear structures of quantum mechanics are analogous to the inertial rest frames in special relativity and the geometric description summarized here, analogous to Minkowski’s reformulation of special relativity. Minkowski’s description paved the way to general relativity. Could the geometric formulation of quantum mechanics lead to a more complete theory one day?
Iii A unified framework for generalizations of quantum mechanics
There are three basic elements of the quantum mechanical formalism which may be considered for generalization: the state space, the algebra of observables and the dynamics. The framework developed in section II suggests avenues for each of these. First, while it is not obvious how one might “wiggle” a Hilbert space, one may generalize the quantum phase space by considering, say, the class of all Kähler manifolds . The geometric language also suggests an obvious generalization of the space of quantum observables: one might consider the space of all smooth, realvalued functions on the phase space. Finally, whether or not one chooses either of these extensions, one may consider generalized dynamics which, as in classical mechanics, preserves only the symplectic structure. Thus, one might require the dynamical flow to preserve only the symplectic structure, and not necessarily the metric.
While each of these structures may be extended separately, they are intimately related and construction of a complete, consistent framework is a highly nontrivial task. Thus, for example, if one allows all Kähler manifolds as possible quantum phase spaces, is seems very difficult to obtain consistent probabilistic predictions for outcomes of measurements. More generally, the problem of systematically analyzing viable, nontrivial generalizations of the the kinematic structure would be a major undertaking, although the payoff may well be exceptional. Modification of dynamics, on the other hand, is easier at least in principle. Therefore, we will first consider these in section III.1 and return to kinematics in section III.2.
iii.1 Generalized dynamics
Let us then suppose that we continue to use a projective Hilbert space for the quantum phase space, and let dynamics be generated by a preferred Hamiltonian function. However, let us only require that timeevolution should preserve the symplectic structure (as in classical mechanics), and not necessarily the metric . From the viewpoint of the geometric formulation, this is the simplest and most obvious generalization of the standard quantum dynamics.^{5}^{5}5Recall that the Hamiltonian need not be an observable function in the sense of section II.2. We could extend the kinematical set up as well and regard any smooth function on as an observable function. (This would be analogous to Weinberg’s [7] proposal which, however, was made at the level of the Hilbert space rather than the quantum phase space .) We have refrained from doing this because the required extension of measurement theory is far from obvious. The idea is reminiscent of the “nonlinear Schrödinger equations” that have been considered in the past. Therefore, it is natural to ask if these there is a relation between the two. We will see that the answer is in the affirmative. Furthermore, the geometric framework provides a unified treatment of these proposals and makes the relation between them transparent, thereby enabling one to correct a misconception.
Let us begin by defining the the class of Hamiltonians we now wish to consider. will consist of densely defined functions on satisfying the following properties: i) is smooth on its domain of definition; and ii) the Hamiltonian vector field it defines generates a flow on all of . In particular, The Hamiltonian functions we considered in section II—expectation values of a possibly unbounded selfadjoint operator—belong to but they constitute only a ‘small subset’ of .
The existing proposals of nonlinear dynamics refer to flows in the full Hilbert space rather than in the quantum phase space . To compare the two, we need to lift our flows to . Let us begin by recalling that it is natural to regard as a reduced phase space, resulting from the first class constraint on (see Eq. 11). Therefore, a function on admits a natural lift to , the unit sphere in . This function on is constant along the integral curves of the vector field which generates phase rotations; . Denote the space of these lifts by .
To discuss dynamics on , we need to extend these functions^{6}^{6}6Strictly speaking, we only need to extend the dynamical flow off . However, to compare our results with Weinberg’s [7] we need to consider extensions of Hamiltonians. In discussions on generalized dynamics [7, 5], the issue of domains of definition of operators is generally ignored. Our treatment will be at the same level of rigor. In particular, we will ignore the fact that our Hamiltonian functions and vector fields are only densely defined. off . From the reduced phase space viewpoint, the extension is completely arbitrary. For, we can construct the Hamiltonian vector field on generated by any extension . The restriction to of this vector field does depend on the extension but the the horizontal part of the restriction—i.e., the part orthogonal to —does not. Hence, the projection of the vector field to agrees with , irrespective of the choice of the initial extension. Thus, the generalized dynamics generated by a given Hamiltonian function on can be lifted to a whole family of flows on , all of which, however, evolve the physical quantum states—elements of —in the same way. Because of this, apparently distinct proposals for nonlinear evolutions on can in fact be physically equivalent. This point is rather trivial from the viewpoint of geometric quantum mechanics. The reason for our elaboration is that—as we will see below—it has not been appreciated in the Hilbert space formulations.
While the extension off of elements of is completely arbitrary, one can use the standard quantum mechanical framework to select a specific rule. Consider, to begin with, a bounded, selfadjoint operator on , and let be the restriction to of the corresponding expectation value function. There is then an obvious extension of to all of : set (which we denoted by in section II.1). One can restate this rule as:
(35) 
The advantage is that this equation may now be used to extend any element of to all of . Note that, with this preferred extension, the Hamiltonian vector field is homogeneous of degree one on :
(36) 
Hence, the flow on which is generated by is homogeneous, but fails to be linear unless is the restriction to of the expectation value function defined by a selfadjoint operator. Next, it is easy to verify that these strongly commute with the constraint function . Hence the flow along preserves the constraint. Therefore, the specific extension considered above has the property that the corresponding flow preserves not only the symplectic structure but also the norm on . However, unless is an observable function, it does not preserve the metric .
Note that, even if we consider just these preferred extensions, the set of possible Hamiltonian functions on has been extended quite dramatically. To see this, consider the case when is finitedimensional. Then, the class of Hamiltonian functions on allowed in standard quantum mechanics forms a finitedimensional real vector space; it is just the space of expectation value functions constructed from selfadjoint operators. The space , on the other hand, is infinitedimensional since its elements are in onetoone correspondence with smooth functions on . And each element of admits an unique extension to via Eq. (35). It is natural to ask if one can do something ‘inbetween’. Can we impose more stringent requirements to select a class of potential Hamiltonians which is larger than that of observable functions of section II but smaller than ? For example, one might imagine looking for the class of functions on whose Hamiltonian flows preserve not just the norms but also the innerproduct. It turns out, however, that this class consists precisely of the expectation value functions defined by selfadjoint operators; there is no such thing as a nonlinear unitary flow on [16]. Despite the magnitude of our generalization, it seems to be the only available choice.
We are now ready to discuss the relation between this generalization and those that have appeared in the literature. Note first that Eq. (35) implies that there is a onetoone correspondence between elements of and smooth functions on which are gaugeinvariant (i.e. insensitive to phase) and homogeneous of degree two. (If is viewed as a vector space over complex numbers, this corresponds to homogeneity of degree one in both and .) It turns out that this is precisely the class of permissible Hamiltonians that Weinberg [7] was led to consider while looking for a general framework for nonlinear generalizations of quantum mechanics. Let us therefore call functions on satisfying Eq. (35) Weinberg functions, and denote the space of these functions by . Our discussion shows that there is a onetoone correspondence between smooth functions on the projective Hilbert space and Weinberg functions on the punctured Hilbert space ;the homogeneity restriction simply serves to eliminate the freedom in the extension of the function on . Thus the extension of quantum dynamics that is immediately suggested by the geometrical framework reproduces key features of Weinberg’s proposal.
There are, however, considerable differences in the motivations and general viewpoints of the two treatments. In particular, Weinberg works with the Hilbert space (and, without explicitly saying so, sometimes with ). However, he does assume at the outset that elements and of define the same physical state of the quantum system for all complex numbers . Thus, although it is not explicitly stated, his space of physical states is also . Therefore, it is possible to translate his constructions to the the geometric language. As we will see below, the geometric viewpoint is often clarifying.
Next, let us consider two specific examples of nonlinear dynamics that have been considered in the literature. Each of these involves the nonrelativistic mechanics of a point particle moving in and the Hilbert space consists of squareintegrable functions on . Therefore, it will be useful to reinstate a complex notation for the remainder of this subsection. In this notation, an element of is a realvalued function of both and its conjugate, which is homogeneous of degree one in each argument. The Hamiltonian vector field generated by such a function corresponds to
(37) 
The simplest example of this type is provided by is the socalled “nonlinear Schrödinger equation”. This equation is given by
(38) 
where is the standard Hamiltonian operator describing a nonrelativistic particle under the influence of a conservative force with potential . Note that the quantity is the modulus of , not the norm of the statevector . It is easy to verify that (38) induces a flow on . This flow is Hamiltonian and the generating function is the projection to of the function on , where,
(39) 
Thus, Eq. (38) is indeed a specific example of our generalized dynamics.
Note, however, that the dynamical vector field on defined directly by the nonlinear Schrödinger equation is given by
(40) 
where . Clearly, it is not homogeneous in the sense of Eq. (36), and hence is not generated by a Weinberg function. Therefore, Weinberg was led to state that the “results obtained by the mathematical studies of this equation are unfortunately of no use” to the generalization he considered. However, we just saw that the nonlinear Schrödinger equation does correspond to generalized dynamics on and is therefore of ‘Weinberg type’. Hence, the statement in quotes is somewhat misleading.
Let us elaborate on this point. If we first focus on physical states, what matters is just the projected flow on . This in turn is completely determined by the restriction of to . Therefore, we may feel free to ignore the behavior of off of . Of course, to construct a vector field on which projects to the relevant one on , we can extend arbitrarily and compute the associated Hamiltonian vector field. In particular, we may extend it in the way which Weinberg would suggest:
(41) 
Thus, we have seen explicitly that the flow on which is defined by may also be described by a Weinberg function! The emphasis on the true space of states of the geometric treatment clarifies this point which seems rather confusing at first from the Hilbert space perspective.
The nonlinear Schrödinger equation is a fairly simple example since the generating function is itself homogeneous (but of the “wrong” degree to be a Weinberg function). Let us now consider an example which is more sophisticated.
In an effort to address the problem of combining systems which are subject to a nonlinear equation of motion, BialynickiBirula and Mycielski were led to a logarithmic equation of motion [5]. They began with a general equation of motion of the form
(42) 
and showed that physical considerations, particularly the requirement that term should not introduce interactions between otherwise noninteracting subsystems, imposes severe restrictions the functional form of . The only possibility is to have: for some constants and . (For details see [5]). Choosing units^{7}^{7}7The constant is a length scale of no physical significance, since it may be altered by addition of a constant to the Hamiltonian operator. with respect to which , the vector field along which the system evolves may be written as
(43) 
where . Again, the extra term may be seen to be Hamiltonian; , where
(44) 
Since is phaseinvariant, this is also an example of our generalized dynamics.
Again, we can carry out the procedure used for the nonlinear Schrödinger equation to see that the corresponding motion on the projective space may be described by use of a Weinberg function. It is not difficult to show that the corresponding homogeneous function is given by
(45) 
Therefore, the logarithmic equation induces a flow on