21 Quantum Gravidynamics

DE WITT opened the session by expressing the hope that one would soon be able to compute something in quantum gravidynamics. He felt that formal matters should be settled as soon as possible so that one could get down to physics. One of the most pressing problems to his mind was the question of what “measure” to use in the quantization of a non-linear theory, particularly in quantization by the Feynman method. In previous conversations with other conferees he had become aware of some differences of opinion on this point. In his view the appropriate measure or “metric” was in every case (including that of the gravitational field) already given by the Lagrangian of the system. As an illustration he considered a system described by a Lagrangian function of the form

where the may be functions of the ’s and the indices run from to . The fact that may be nondenumerably infinite is ignored. For an actual system the Lagrangian may possess other terms, but the essential difficulties are contained in the term considered. Inclusion of the other terms modifies the following discussion in no essential way.

If ( ) is a nonsingular matrix with inverse given by ( ) then the system possesses a Hamiltonian function given by

and the action

satisfies the Hamilton-Jacobi equations

According to DE WITT, the structure , which is already contained in the Lagrangian to be taken as the metric for the space of the ’s and provides the appropriate “measure” for a Feynman summation. Following Pauli¹ one may introduce a “classical kernel” of the form

where , and where the quantity is a determinant originally introduced by Van Vleck in an attempt to extend the WKB method to systems in more than one dimension.²

and satisfies an important conservation law:

which can be obtained by differentiating the first Hamilton-Jacobi equation with respect to and and multiplying by the inverse matrix . From the quantum viewpoint this law expresses conservation of probability.

It is easy to see that is an invariant under point transformations of the ’s. With the aid of the Hamilton-Jacobi equations one may show that it satisfies the differential equations

where for brevity we write , etc.; where the dot followed by indices denotes covariant differentiation with respect to either the or (as indicated by the context) and where the operator is defined by

DE WITT pointed out that if it were not for a certain peculiar phenomenon which occurs when the space of the ’s is curved, the operator could immediately be regarded as the Hamiltonian operator for the quantized system. In order to discuss this phenomenon some further development is necessary:

If one recalls that the classical action defines a canonical transformation by the equations

then one easily sees that the Van Vleck determinant is just the Jacobian involved in transforming from a specification of the classical path by means of the variables , , to a specification in terms of initial variables , . From the Hamilton-Jacobi equations one sees furthermore that the action may be expressed in the form

Therefore, noting that as the become infinite except when (for all ), one may write, for an arbitrary function ,

In order to evaluate this last expression one must evaluate the Van Vleck determinant. This is easily done by expanding the action about the point substituting it in the Hamilton-Jacobi equation. One finds, after a straightforward computation,

where

commas followed by indices denoting differentiation with respect to the ’s. Here a convention has been chosen so that the scalar is positive for a space of positive curvature.

Using the final expression for the Van Vleck determinant one infers

and

where is the invariant delta-function in -space. Referring to the differential equations satisfied by the classical kernel , one sees that it equals the true quantum transformation function of a system possessing the Hamiltonian operator

up to the first order in . That is,

where

satisfying the boundary condition

The “first order contact” between and is already sufficient to determine the behavior of wave packets for the quantized system. If the system has a Hamiltonian operator then its wave packets will more approximately along the classical paths of a classical system which has simply as Hamiltonian function. Conversely if the quantized system has Hamiltonian operator then the motions of its wave packets will be along the classical paths for a “classical” system which has the Hamiltonian function . Evidently there is an ambiguity here in the choice of Hamiltonian operator when the space is curved, and there is nothing in the classical theory to resolve it for us.

The Feynman formulation of quantum mechanics follows immediately from the order contact between and . Breaking the transformation function up into infinitely many pieces by means of the composition law

where , one may write

where . It is evident, from the expression for and the form of the expansion for the action , that the (in the limit) infinitely multiple integral receives significant contributions from the integrand only when the differences are of the order of or smaller. Therefore, if the symbol is used to denote equivalence as far as use in the infinitely multiple integral is concerned, one may write

since

and therefore

where is the action function for the system with classical Hamiltonian function

The Feynman formulation now becomes

and the dependence of the “sum-over-paths” on the metric is seen to occur through the invariant volume elements . This last expression is sometimes written in the symbolic form

where the symbol indicates that a “functional integration” is to be performed. In linear examples (i.e., linear equations of motion) the functional integration can often be easily performed without even referring to its rigorous definition as an infinitely multiple integral. This is not likely to be so simple in the present nonlinear case because of the occurrence of a variable metric in the invariant volume elements .

It will be noted that the last result is even more curious than the previous result obtained with the Pauli method using . In order to obtain the transformation function for a quantized system having Hamiltonian operator one must use, in the Feynman summation, the action corresponding to a classical system having Hamiltonian or Lagrangian . When DE WITT first discovered this phenomenon he thought that he had made an error in sign and that the occurrences of actually cancelled each other when one passed from the Pauli form on to the Feynman representation. J. L. Anderson later convinced him, however, of the reality of the phenomenon, by carrying out a computation patterned directly after Feynman's original paper.

Instead of stating the result in a symmetric manner one may also state it in the following forms: If the true classical action is used in the Feynman summation then one generates the transformation function satisfying

where

On the other hand, in order to generate the transformation function satisfying

one must use in the Feynman summation the action for a “classical” system possessing the Lagrangian function

and Hamiltonian function

FEYNMAN remarked that quantization of a system like is necessarily ambiguous when the space is curved. One must first imbed the space in higher dimensional space which is flat, since only in spaces which are flat (at least to a very high degree of approximation) do we have experiments to guide us to the appropriate form of the quantum theory. The original space must then be conceived of as the limiting form of a thin shell, the “particle” which moves in the original space being constrained to the shell by a very steep potential. But then the Schrödinger equation for the particle will depend on the precise manner in which the thickness of the shell tends to zero, and this is arbitrary.

DE WITT agreed, but pointed out that the manner in which the thickness of the shell tends to zero can be described by the addition of a simple “potential-like” term to the Lagrangian function appearing in the Feynman formulation. To illustrate this he considered the case of a particle constrained to move on the surface of an ellipsoid. He first took for the constraining shell the region between two confocal ellipsoids:

In order to eliminate, right at the start, the degree of freedom transverse to the shell - which must have no reality in the end, anyway - one may suppose that the wave function has a simple node on each ellipsoid and none in between (i. e., transverse ground state). Since we know that confocal ellipsoids form a separable system for Schrödinger equation, we may immediately factor out the transverse part of the wave function and at the same time subtract a constant term proportional to from the energy, where denotes the thickness of the shell at some point. The remaining part of the wave function then satisfies the Schrödinger equation with the simple operator appropriate to an ellipsoid. This corresponds to a Feynman formulation using the classical Lagrangian , for which the classical paths are attracted to the region of greatest positive curvature, the attraction being described by a potential . In the present case the region of greatest positive curvature is the “nose” of the ellipsoid and here the shell is thinnest. The three-dimensional wave function undergoes a “crowding” at this point. Since the transverse part of the wave function is constant, this crowding must be borne by the “active” two-dimensional part. The tendency of the amplitude to increase in the nose region is describable in classical terms as an attraction. A wave packet would actually display the effect of this attraction.

The Feynman formulation which starts with the classical Lagrangian , on the other hand, corresponds to the use of a shell of uniform thickness:

All points of the limiting ellipsoid are here weighted equally. DE WITT conjectured that the use of any other shells of varying thickness would quite generally be describable in terms of additions to the classical Lagrangian of potentials proportional to .

(Editor's Note: - DE WITT's argument is not entirely rigorous. He, of course, avoided discussion of a spherical shell since the curvature is then constant and has only the effect of uniformly shifting all energy levels. However, there is a difficulty for ellipsoidal systems in that the separation constants are not themselves separable, and hence the transverse part of the wave function is not rigorously factorable for arbitrary behavior of the non-transverse part. It may, however, be approximately factorable when the shell is thin.

Attention should also be called to the fact that the behavior of wave packets is not described by the Lagrangian appearing in the Feynman formulation but by the Lagrangian of the Pauli formulation, which is half-way between that of Feynman and the corresponding quantum form. Thus, when Feynman uses , Pauli uses to obtain the same quantum theory. Or when Feynman uses , Pauli uses . It is the Van Vleck determinant of the Pauli formulation which gives the key to the motion of wave packets, through its conservation law. Remembering that , one may write that conservation law in the form

If a wave packet is replaced by an ensemble of classical particles then gives a measure of the density of these particles at any time and place.)

WHEELER remarked that it had been suggested that an experimental determination of the Lagrangian to be used in the Feynman formulation in curved spaces might be achieved through observations on molecules which have “flags” on them - for example ethyl alcohol:

If some of the bonds are regarded as rigid (e.g., at room temperature) then the Lagrangian for such a molecule contains a metric corresponding to a space which is not flat. However, WHEELER pointed out that this system, in the last analysis, has its existence in a flat space - that the rigidity of the bonds is an idealization, and that, in fact, the binding forces themselves provide the specification of the “shell” on which this system is constrained to move. Hence such a system could tell us nothing about which Lagrangian to use, for example, in the Feynman quantization of the gravitational field where the ambiguity does arise.

ANDERSON pointed out that at any rate DE WITT's approach leaves no ambiguity in choice of metric. In fact, no other choice is possible if one requires . This relation will not generally be satisfied if one chooses a metric for -space which is unrelated to the appearing in the Lagrangian.

DE WITT then went on to discuss the second important and pressing problem which arises in the quantization of nonlinear systems, namely, the “factor ordering problem”: How should one order non-commuting operators to obtain appropriate quantum analogs of various classical equations?

When the matrix ( ) is non-singular the factor ordering Hamiltonian operator is easily solved. The Hermitian requirement on the momenta leads to the quantum representation requirement

in arbitrary curvilinear coordinates. Hence, in order that , one must write

The Hermitian character of is obvious from the symmetry of this expression.

For covariant theories, on the other hand, is singular, constraints are present, and the analysis becomes much more complicated. Moreover, since is singular it is not obvious immediately what “measure” to use in a Feynman formulation.

(Editor's Note: - The following is taken from a set of mimeographed notes which DE WITT distributed to the conferees. The actual exposition of it was broken up by many questions, mostly on the necessity of going through a careful layout of identities and constraints. These questions are best answered by the more complete uninterrupted presentation given below. One may add, to the purposes of an unambiguous derivation already mentioned in the previous paragraphs (finding the correct “factor ordering” and the “measure”), the search for the true observables of the theory.)

As a prototype of a covariant theory DE WITT considered a system possessing a Lagrangian function of the form

for which the equations of motion are

where . These equations of motion are, of course, invariant under point transformations of the 's among themselves as well as under “phase transformations” where is an arbitrary function of the 's, The “covariance” of the theory is described by an additional transformation group under which the equations of motion remain invariant, whose infinitesimal elements have the general form , where

The , , are certain definite functions of the 's, while the are arbitrary infinitesimal functions of the time and of the 's and any of their time derivatives. Such a transformation may be called a gauge transformation; in general relativity it is an infinitesimal coordinate transformation, the being the metric field variables.

The Lagrangian function must be altered under a gauge transformation by a total time derivative . It is not hard to see that must have the general form

where the , , , are functions of the 's only. DE WITT assumed that , as this simplifies the analysis in certain respects. This assumption is actually incorrect for the gravitational field, but is not expected to alter the main qualitative features of the analysis. (It will be discussed more fully at a later point.) By performing a gauge transformation on the Lagrangian and making a comparison with one finds that the following identities must be satisfied:

DE WITT called these identities of invariance.

It is the identity which shows that ( ) must be a singular matrix in a “gauge invariant” theory. This has the consequence that the momenta of the Hamiltonian formalism are not all independent. It also has the consequence that the initial conditions on the motion must be subject to constraints in order that the motion actually be able to make the action integral an extremum. Some, at least, of these constraints are obtained by multiplying the equations of motion by . Using identities of invariance one finds

These will constitute the complete set of constraints if their time derivatives to all orders vanish solely as a consequence of the equations of motion. If the time derivatives do not automatically vanish they must be made to vanish by adding extra constraints, whose time derivatives, in turn, must be made to vanish, etc.

DE WITT introduced a second set of identities by assuming that the form a complete and independent set of null eigenvectors of . One may then infer that

where the , , are certain coefficients. DE WITT called these identities of completeness.

A final set of identities are obtained from the group property of gauge transformations. Requiring that the difference between the results of applying two infinitesimal gauge transformations in different orders be also a gauge transformation (of the second infinitesimal order) one finds

where

only if

for certain coefficients , and if

where the are certain coefficients satisfying

DE WITT called these identities of integrability. It will be observed that they lead to the identifications

DE WITT then went on to remark that the above three groups of identities comprise all the identities that are available, or needed, for the study of the dynamical behavior of the system under consideration, and he called attention to the fact that they are obtained entirely within the framework of the classical theory. In developing the Hamiltonian formulation for the system DE WITT followed the method of Dirac³ which distinguishes between weak equations ( ) and strong equations ( ). Equations of motion and constraints (reduced to their lowest order) are weak equations. Equations of identity ( ) are strong equations. Also, the product of two weakly vanishing functions is strongly equal to zero.

The canonical momenta for the system are

Multiplication by gives immediately the set of constraints to which the momenta are subject:

Completing the set of vectors through the introduction of a set of independent vectors , and introducing also “inverses” , satisfying , one may write the “energy function” in the form

where

The final expression for the energy function provides an illustration of Dirac's theorem to the effect that the energy is always strongly equal to a function of the 's and 's, which may be called the Hamiltonian function, plus a linear combination of the 's with coefficients which depend on the velocities . In a theory without constraints the energy is identical with the Hamiltonian function. When constraints are present the two are only weakly equal.

If the coefficients multiplying the 's are completely undetermined by the equations of motion, and hence completely arbitrary, the 's are said to be “of the first class.” This point always requires special investigation. Using the equations of motion together with the constraints one may easily verify that the time rate of change of any function of the 's and 's is given by

where denotes the Poisson bracket. In particular, the time rate of change of a is given by

Since the 's vanish their time derivatives must also vanish. This will happen automatically if the Poisson brackets and vanish, at least weakly. If these Poisson brackets do not all vanish then the cannot all be completely arbitrary and the will be subject to additional constraints.

Using the relation together with identities of invariance, one finds, after a straightforward computation,

where

The latter equations are simply the velocity constraints reexpressed in terms of the momenta. The -equations, existing only in the canonical formalism, are sometimes known as “primary constraints.” The -equations, which follow from them, are then called “secondary constraints.” The 's as well as the 's must have vanishing time derivatives. This leads one to investigate also the Poisson brackets .

Using only the identities of invariance and completeness one finds

The vanishing (in the weak sense) of all these Poisson brackets means that there exist no further constraints and the 's are all of the first class. There still remains, however, one further Poisson bracket which it is necessary to examine, namely . Dirac has shown (op. cit.) that if this Poisson bracket does not vanish (at least weakly) then some of the canonical variables may be eliminated from the theory through the introduction of a new type of bracket, a modification of the ordinary Poisson bracket, which DE WITT has called the “Dirac bracket.” It is the Dirac bracket which corresponds to the commutator (or anticommutator) in the quantum form of the theory.

In the evaluation of identities of integrability are needed for the first time. One finds

Evidently Dirac brackets are here the same as ordinary Poisson brackets. Under these circumstances the 's are said to be “of the first class.” It is characteristic of any theory in which the constraints arise as a result of a gauge invariance principle that all of the 's and 's are of the first class.

DE WITT called attention to the fact that up to this point not all of the identities of integrability have been used. The classical theory, which is completed at this point, can in fact get along without the unused identities. In the quantum theory, however, the unused identities turn out to be crucial.

DE WITT first considered the and equations, which, in the quantum theory, are to be regarded as supplementary conditions on the state vector of the system:

Since the classical expressions for the and involve quantities which, in the quantum theory, do not commute, the factor ordering problem here makes its appearance. According to DE WITT the quantum analogs of and should be taken as

where denotes the anticommutator bracket. With the aid of the previously unused identities of integrability one may then show that

where denotes the commutator bracket and where the ordering of factors on the right is now important. Since the 's and 's all stand to the extreme right the corollaries

of the supplementary conditions are automatically satisfied. DE WITT pointed out that the appropriate quantum analog of a classical quantity may have one or more terms in it (such as the above) which are proportional to or powers of , and which do not appear in the corresponding classical expression (since ).

Similar considerations are involved in finding the quantum analog of the function . It is convenient to return for a moment to the classical theory: Since the 's and 's are all of the first class, the quantities appearing in the dynamical equation (i.e., for ) are completely arbitrary. One is at liberty to set them equal to arbitrary functions of the 's and 's through the addition of extra supplementary conditions:

The energy function is then strongly equal to

Furthermore, the dynamical equation may be replaced by

The conditions , are, of course, unaffected.

In the quantum theory the dynamical equation may be taken as

in which the quantum form of is written as above with the 's standing to the right. The time derivatives of the supplementary conditions, namely

will then be satisfied provided

The quantum form of will be chosen in such a way that these equations are automatically satisfied by virtue of the supplementary conditions. The supplementary conditions when combined with the dynamical equations for will generally imply

where indicates some suitable quantum analog for , and in this way the consistency of the quantum scheme can be checked for arbitrary choices of the .

It is to be noted that one is here working in the Heisenberg picture in which the state vector is time independent. The arbitrariness in the choice of the functions shows that there are many different Heisenberg pictures, all equally valid. They all lead, however, to a single unique Schrödinger picture:

One may arrive directly at the Schrödinger picture by starting, in the classical theory, from the “homogeneous velocity” formalism (Dirac, op. cit.) in which the time is introduced as a canonical variable with a conjugate momentum . This leads to a new equation, namely where . In the quantum theory, representation of puts the corresponding condition on the state vector in the form

which is just the Schrödinger equation.

Construction of the quantum form of is more difficult than the construction of the quantum forms of and for two reasons: (1) is quadratic rather than linear in the momenta. (2) No metric has yet been defined in -space. If there were no gauge transformation group for the system, would be nonsingular and would, as DE WITT had already pointed out, provide the natural metric. In the present case DE WITT suggested building up a nonsingular metric as a sort of superstructure with as a core, in the following manner:

matrix multiplication as indicated, where , are arbitrary functions of the 's. However, is assumed to have an inverse which is used to raise the indices , , , etc., while is used to raise the indices , , , etc. ( ) will then have an inverse given by

and a determinant given by

where , , . The metric may be used to make a special choice for the function , namely

corresponding to the choice

The equations of motion generated by this “Hamiltonian” may be derived from a Lagrangian function of the form

The quantum theory generated by this Lagrangian function is not identical with that which has been developed here, but becomes so upon the addition of the equations as supplementary conditions. This is a standard procedure in quantum electrodynamics in which these supplementary conditions become the “Lorentz conditions.”

A choice of metric having been adopted, one may now express the quantized momenta explicitly in the form

and one finds that the supplementary conditions become

a dot followed by an index denoting covariant differentiation with respect to the metric , with, however, the modification

being the Christoffel symbol defined by . A special significance possessed by the coefficients is here emphasized. The provide a kind of affine connection in terms of which one may define parallel displacements which are invariant not only under point transformations but also under transformations associated with the indices , , etc., of the form

where the are arbitrary functions of the 's, the matrix ( ) possessing, however, an inverse ( ). These transformations, which DE WITT called -transformations, leave all the identities of invariance, completeness, and integrability unchanged. The -transformation procedure may be extended to the indices , , , etc., by the definitions

where the , are also arbitrary functions of the 's, with ( ) possessing as an inverse ( ). The whole theory (in particular, the metric ) will then be invariant under -transformations provided

In order to show the invariance of the quantum theory under point, phase, gauge, and -transformations one does not actually have to use the differential representation of the momenta, convenient though it generally is. For example, the point transformation law for the momenta is

and, remembering that the indices etc., on the quantities , , , , , , etc., are all tensor indices under point transformations, one finds by straightforward computation that , , , provided one writes

where is a scalar function of the 's, as yet undetermined. Similarly, under -transformations one finds and

Since the 's and 's stand to the right the supplementary conditions remainunchanged: , . Invariance of the quantum theory under gauge transformations is not immediately apparent at this point, but becomes so subsequently. Phase transformations, on the other hand, are trivial:

or, if one is using the fixed differential representation for the momenta, one places the burden of keeping the theory phase-invariant on the state vector (or wave function) by the law

Having introduced -transformations, DE WITT then pointed out that the affinity satisfies the very important condition of being integrable. This means that one can carry out an -transformation which will make the vanish everywhere. The required transformation is given by the solutions of the simultaneous equations

The solubility of these equations is guaranteed by the identity of integrability . After this transformation has been carried out one has

which implies that there exists a point transformation such that

Making this point transformation, together with an additional -transformation

and letting the be given by

one arrives at a great simplification in the formalism. Using a simple replacement of indices ; , etc., to denote quantities in the new “coordinate system,” one has

Moreover, the identities of invariance and integrability take the forms

The identity implies that a phase transformation ( ) can be carried out which makes vanish. Assuming that it has already been performed and noting that

one may now write the supplementary condition in the forms

The first supplementary condition has the important result that the arbitrary quantities can now be actually eliminated entirely from the quantum theory, just as they are not needed in the classical theory. That is, having carefully built up a “superstructure” around the metric (or ), one then proceeds to throw it away. To do this one simply introduces a new wave function

This definition removes from the wave function that part of the metric density which refers exclusively to the , and corresponds to the use of as invariant volume element. The condition means that the new wave function does not depend on the variables . The are non-observable or “non-physical” variables of the system. This was, in fact, already true in the classical theory, since the can be made to undergo arbitrary changes by means of gauge transformations.

In the new representation by itself provides a natural metric in the reduced space of the . To see how this works for the Hamiltonian operator, one can multiply the Schrödinger equation

by . Using the explicit form for , one finds, after a straightforward calculation,

where the dots now denote covariant differentiation with respect to the metric , and where

The quantity is invariant under point transformations which are restricted to the variables alone.

The new Schrödinger equation may be written in the form

where

the explicit differential form for the momenta in the new representation being

The existence of the second supplementary condition suggests that there are further non-physical variables in addition to the . To show this one must first eliminate the term in from the supplementary condition. This can be done by carrying out a phase transformation satisfying

and at the same time satisfying

so as not to disturb the condition . The integrability of these equations is insured by the identities of invariance and integrability, as one may verify by straightforward computation, showing that the expressions for and vanish. One may then write

Next, by differentiating the identity one can show that

The coefficients are then immediately recognizable as the structure constants of a Lie group. The secondary constraints are, in fact, the infinitesimal generators of the group. In the general case the structure constants will be non-vanishing and the Lie group will be non-Abelian. This immediately leads one to various possibilities depending on the classification of the Lie group in question. As an illustration, suppose that the Lie group is the irreducible rotation group in dimensions. The indices must then range over different values. This is not however the number of further non-physical variables. A point transformation can be made such that the new variables describe the subspace in which the rotations actually take place. The quantity vanishes for the rotation group and the second supplementary condition is found to reduce to which, when analyzed, is seen to state simply that the wave function is invariant under rotations in the subspace of the . Therefore, in this case only further variables can be eliminated from the theory as non-physical. The wave function can still depend on the rotation-invariant combination as well as on the .

The primary constraints also, of course, generate a Lie group. But this group is necessarily Abelian. DE WITT assumed, for simplicity, that the group generated by the is also Abelian (so that ) and furthermore that the vectors are linearly independent of the and of each other. In this case, when , one may carry out a point transformation such that

(Here, in order to avoid confusion one must use a prime to distinguish indices related to the from those related to the .) In the new “coordinate system” one has

and the identities of invariance and integrability become

If the phase is adjusted so that

then one also has

Moreover, the supplementary conditions become

so that the , like the , are now “non-physical” variables.

The quantity is seen to be independent of the and to have a dependence on the which is no more than quadratic (since ). One may choose the origin of the unobservable “coordinates” in such a way that

where and are independent of the and . Furthermore, since and , one may write

where and are independent of the and . In order to complete the elimination of the from the theory, a natural metric for the reduced space of the must be introduced. DE WITT showed that it must be defined in the following manner:

in which use has been made of the relations , , and in which the matrix ( ) is assumed to have an inverse ( ).

With this definition one has and

where and . The determinant forms that part of the metric density which refers exclusively to the non-physical variables . It may be removed from the normalization of the wave function by introducing a new wave function

Since , , is independent of the and . It may becalled the “true physical wave function” of the system referring only to the “physical” variables . Its Schrödinger equation may be found by multiplying the Schrödinger equation by making use of the relations

One finds, after a straightforward computation,

where

The operator remains invariant under point transformations which are restricted to the variables alone.

The operators , and any quantities constructed out of them are the true observables of the theory. The theory is now reduced to its simplest terms. In this form it is quite easy to see that the Schrödinger equation is consistent with the constraints

One has only to require that be independent of the and . will then be a true observable (the true energy operator, depending only on the and ) and the wave function will remain independent of the and at all times.

The gauge invariance of the quantum theory is now also immediately evident. In the “coordinate” system , , , gauge transformations are given simply by

and hence

There remains only the term which is unknown. However this term, as Feynman pointed out previously, depends on how the system is constrained to a “shell” which passes over to the space of the in the limit, and this is arbitrary.

It is of interest to examine the Feynman quantization method in the present context. The path summation may be defined by breaking the time interval into small pieces and integrating over the physical variables only, using the measure defined by the metric . The variables may be chosen as arbitrary functions of the time (corresponding to the original arbitrariness in the ) and the original Lagrangian function may be used to compute the action, provided the variables are made to vary in time in such a way as to satisfy the velocity constraints which in present notation, become

or

For the Lagrangian function then becomes

which is precisely the form which gives rise to the Hamiltonian . One may therefore express the transformation function in the symbolic form

with the understanding that the functional integration is to be carried out over all paths between the points and which satisfy the velocity constraints with given, each path being assigned a weight according to the measuredefined by . (Here, potential-like terms proportional to , which may be added to , are ignored.)

It is not actually necessary to restrict the summation to the variables only. For example, one may work with the transformation function

which connects the values of the wave function at two different times. Since is independent of the for physical states, one may write

which implies

The expression on the right must be independent of the . Hence one may integrate over these variables provided one divides the result by an infinite normalization factor. Moreover, in the path summation expression for the choice of time-like behavior for the variables is immaterial. Hence one may write

where is a suitable infinite normalization factor, the functional integration being now carried out over all paths for which , at , respectively (all possible end-point values for the variables , being included in the summation) each path being given a weight according to the measure defined by metric density multiplied by the Jacobian of the transformation , , . In calculation of actual physical quantities one never needs to mention the normalization factor explicitly, since what is involved is always a ratio of the form

where is some true observable (function of the , alone), and the normalization factors in front of the functional integrals cancel.

DE WITT concluded his remarks with some comments about his simplifying assumption, made at the beginning, that the coefficients , occurring in the expression for , vanish. In the case of the Lagrangian for the gravitational field the do not vanish. This has the result that the secondary constraints are no longer linear in the momenta, but quadratic. In fact, it had been pointed out earlier in the conference that the quadratic term, in the gravitational case, is just the energy density for the gravitational field. This created a certain amount of perplexity, leading some of the participants to think that perhaps all true observables are simply constants of the motion. However, this problem is not yet settled. DE WITT pointed out that the discovery of the non-physical variables associated with the will be considerably more difficult in the case . The resulting quadratic dependence of the on the momenta means that the non-physical variables can no longer be found simply by a point transformation. A more complicated type of canonical transformation will be required.

DE WITT suggested that the use of parametrized space-like surfaces might help this situation. But he also pointed out that the equations like become variational differential equations in field equations in field theories, and that there is some uncertainty about the ease with which they can be solved in nonlinear contexts. The search for answers to these questions will provide a large program for the future.

FEYNMAN raised the question of the interest of the true observables. Commenting on Belinfante's remark that true observables are very useful to get meaningful answers, FEYNMAN proposed to formulate the problems not in terms of incident gravitons, photons, etc., but in terms of fermions or other sources of these incident and outgoing fields only, in an action-at-a-distance fashion.

LICHNEROWICZ remarked that the classical treatment may have the following geometrical interpretation:

“(1) It may be simpler to consider directly the configuration space-time of the system, i.e., to treat as a local coordinate. Let be the configuration space-time.

“(2) The geometrical frame is then the fiber bundle space of dimensions - say - consisting of the tangents of at all points.

“(3) The gauge transformations have, for infinitesimal generators, vector fields of . The precise hypothesis for the Lie algebra of the fields is the following: can be decomposed into a direct sum where is a tangent to the fibers. This decomposition is such that depends on the points of only, and depends linearly on the directions. defines a field of plane surfaces on which the pseudo-metric vanishes.

“(4) By introducing an extra dimension, one can have a Lagrangian corresponding to a Riemannian pseudo-metric (degenerate quadratic form), and it seems that the explicit calculations in the classical part could then be simplified.”

ANDERSON and LAURENT reported on the use and extension to the gravitational case of Feynman's method in its field theoretic form.

In order to investigate the question of what variables to employ in the Feynman integral in the gravitational theory, ANDERSON has looked at the corresponding problem in electrodynamics. “There the situation is somewhat simpler because of the linearity of the theory and consequently it is possible to see in a fairly straightforward fashion what one is to do in this case. Adopting the Feynman prescription in toto means that one must look for solutions to the field equations which are then in turn substituted into the Lagrangian for the electromagnetic field and integrate over the time interval.

The chief difficulty in this procedure rests in the fact, of course, that one of the equations of motion for the field is free of second time derivatives and therefore represents a restriction on initial and final values which one may impose on the field conditions. However, if one transforms to a set of field variables which include , , and the scalar field , then the equations of constraint do not depend on . In fact, the Lagrangian originally looks like

and the equations of constraint appear as

so in terms of , , equations of constraint look like

Thus solutions satisfying all field equations yield for

Thus the construction of the Feynman integral is then seen to be in terms of alone, and the measure is given uniquely by the Lagrangian - in this case simply 1.

If one were to extend this line of reasoning to the gravitational case, it is seen that one must look for a set of variables which have the property that they do not appear in the constraint equations and thus are capable of being given arbitrary initial and final values. The Feynman integral would then be over these variables and the measure would derive from the Lagrangian expressed in terms of these variables. As yet this is only a program of approach. However, one can see that it is possible to eliminate immediately as unphysical variables the which corresponds in the electromagnetic case to the scalar potentials. The elimination of the longitudinal part of the gravitational field is of course a much more difficult problem and has in no way been solved as yet, although it appears to be the central problem to the quantization.”

LAURENT showed how the Feynman action method works gauge-invariantly in quantum electrodynamics and then suggested an extension to gravitation theory whereby the use of “extended gauge-invariance” is forced upon the formalism. (The following is the reproduction of his notes. For a more complete exposition of these and related questions, the reader may consult a recent article of B. E. Laurent.⁴

“In the following I wish to explain why I think that it is worthwhile to try Feynman's action method in covariant theories. This is merely a sketch and does not pretend to be either complete or mathematically rigorous. It has been of great help to me to be able to discuss these problems with Professor O. Klein and Dr. S. Deser.

I shall start with the simplest generally covariant theory known, that is electrodynamics, which is covariant with respect to gauge transformations.

Feynman's well-known action principle⁵ has been used by several authors⁶ to calculate ratios of time-ordered vacuum matrix elements in non-covariant field theories. These authors have shown that such ratios may be written as follows:

Here is the action-functional for the fields (or field) and all stand for the field-functions in the space-time points . The integrals are functional integrals over the functions in all space-time, performed after a certain small imaginary term has been added to the action (as Burton and DeBorde show, this picks out a vacuum state). An extended class of operators may be constructed as sums of time-ordered products . To be general I shall in the following write instead of in (21.1). is here a functional of the fields .

In the usual Lorentz-gauge electrodynamics a vacuum propagator is to be written as follows:

A special device must here be used to obtain the wanted anticommuting properties of , ⁷

where is the Lorentz-metric and the Dirac-matrices.

(21.2) is exactly the ordinary vacuum propagator of electrodynamics if we restrict so that we have nothing but electrons and transversal photons in the initial and final states.⁸ may for instance have the following form:

Here is a normalizing factor

and .

The operator corresponding to (21.4) gives merely transversal photons in the initial and end states of the propagator.

Observe that in a gauge-transformation

in (21.4) is invariant. We shall assume throughout that our 's have this property.

Let us now re-form (21.2)

We have split up the integrals over all possible -functions in sums of integrals over part-regions, of which the first is around the -functions satisfying the Lorentz-condition. The other regions are obtained from the Lorentz-region by gauge-transformations. The regions cover completely the region of all -functions and they do not overlap. The part-regions can be made as narrow as we like.

Remembering that is invariant and that the gauge-transformation is linear we see that the following is true:

is the complete gauge-invariant action

is the Jacobian of the transformation from the -th region to the Lorentz-region. In this case is 1 but we shall keep it in in spite of that to be able to use the result later in gravitation theory.

We may now write

depends on the region only (when the region is narrow). In the Lorentz-region, . Hence

Further we see that

From this form it is evident that the propagator is invariant.

We see from this that under the supposition made about , that it must not change its form when is replaced by , the two demands are fulfilled that quantization shall, as in (21.10), be performed over the “real degrees of freedom” only and that the procedure shall be covariant as in (21.12).

According to the foregoing it is close at hand to guess that a propagator in a theory that is covariant with respect to linear transformation shall be written in the form (21.12), where and must be invariant in the sense mentioned above^{9 10} (in the general case we call it extended gauge-invariance). If in gravitation theory, for instance, is a functional of the only it must not change its form in the linear transformation ,where is an arbitrary (non-singular) transformation. We observe that a scalar does not necessarily fulfill the requirement, because, in general, it changes its functional form of in the transformation. If is a function as here described and the integration variables transform linearly, we see from the procedure followed in the electromagnetic case that we may always come from the covariant propagator of type (21.12) to the propagator of type (21.10) , where only so much of the field is quantized as there are “degrees of freedom.”¹¹

A difficulty in the gravitation case is that we do not know over which variables we ought to integrate, and which measure should be chosen in the functional integral.¹²

The choice of integration variables that seem most natural at first sight is perhaps that of the or , but as Professor J. A. Wheeler told me in a letter, a better choice is probably one which Dr. C. Misner has suggested, that is the coefficients in the transformation from the consideredmetric in a certain point to the Lorentz metric. However, the question cannot be regarded as solved yet.

(MISNER pointed out that this choice of variables gives an invariant measure which preserves the signature.)

Finally I should like to say a few words about how functionals of the type mentioned earlier may be constructed in gravidynamics.¹³ Integrals of scalar densities over all space-time is of this type but they do not seem to be of much use in this connection. Professor O. Klein has pointed out to me that the total energy-momentum pseudo-vector may be useful here, and in fact we see that it is of the correct type if we choose it in the following form:

(I do not think that this formula needs any explanation.)

is identically satisfied which gives (21.13) the desired property for -transformations which are identity-transformations on the boundaries of the integration-volume. Now all the propagators in the foregoing are limits when the space-time region involved becomes infinitely large¹⁴ and nothing hinders us from prescribing, for instance, the Lorentz-metric outside the region and on its boundaries. This means that every interesting transformation is of the type mentioned. In this case the propagator is dependent on the boundary conditions chosen in infinity.”

During the discussion which sandwiched and followed LAURENT's exposition, FEYNMAN asked which action was used in the gravitational case. To LAURENT's answer - - FEYNMAN raised the difficulties brought in by the presence in , of second derivatives.

BELINFANTE suggested that Laurent uses, in Feynman's sum over fields, the action from the muddified theory, together with the determinant derived from the muddified theory. However, it was pointed out that Laurent's proof of the equivalence of a “true” and a “muddified” treatment in the electromagnetic case may not hold in the gravitational case, because the determinant depends on the variables of integration.

FEYNMAN asked also what is the determinant in the path integral expressed in terms of the variables proposed by Misner (the variables defined by the equation )? More precisely, what is the expression of the determinant which gives a value to the path integral independent of the mesh introduced for its definition? This question remained unanswered but led to the discussion of the definition of Feynman's path integral in a generally covariant theory. Originally, Feynman's path integral was defined in terms of a time slicing as each time interval goes to zero. In the gravitational case, it is preferable not to single out the time, and to redefine Feynman's method accordingly.

WHEELER pointed out the various advantages of the latent variables (such as in electrodynamics) and of the true (or physical) variables (such as ). At present we have three schemes:

– “Sum over fields” expressed in terms of latent variables. The  infinities cancel out.

– Add a term to the Lagrangian so as to make the problem  non-singular.

– Use only physical variables.

The whole art is working back and forth between latent and true variables. It is important to show the equivalence of the various schemes and to show the equivalence of the various “slicings,” but it is a task yet to be accomplished.

BERGMANN summarized the session as follows: “It appears that there are two different methods of Feynman quantization - one principally represented by Bryce DeWitt and Jim Anderson, the other by Bertel Laurent and Stan Deser, who is unfortunately not with us this morning. The difference appears to be that one group wants to settle the question of covariant measure in the function space first and then integrate, whereas the other group prefers to integrate first and ask questions afterwards. To me as an innocent bystander it appears that one group knows what they are doing but don't know how to do it, and the other group is able to proceed immediately, but with some question as to the meaning of their results.

One word about why the two approaches appear to lead to mutually consistent results in electrodynamics: in electrodynamics the components of the metric tensor, that is the second derivative of the Lagrangian with respect to the velocities, are constants. It appears natural that in such a situation there should be no disagreement; but I feel very doubtful, without availability of further strong arguments, about anticipating a similar agreement in general relativity, where the same coefficients are complicated functions of the field variables.

I think it is obvious to all of us that the difference between the two approaches again amounts to an estimate of the relative importance of true observables vs. all dynamical variables. Therefore, permit me to give one more argument that indicates the probable importance of the true observables, or dynamical variables as John Wheeler has called them. In general relativity, it is sometimes useful, for reasons of convenience, to introduce parameters and thus to upgrade the ordinary coordinates into dynamical variables. Alternatively, it is sometimes desirable to introduce the so-called vierbeine (quadruples). If you do that without eventually going back to true observables, then you have a proliferation of variables that must eventually be represented by quantum operators, and I think if you try to do this you will go crazy.

I would like to close with a comment on the motivation for Feynman quantization as such. The proponents of Feynman quantization claim as its virtue that once a completely satisfactory classical theory is available, all they have to do is to follow an algorithm, and the task of quantization will be completed. I must admit to having some lingering doubts about the uniqueness of the procedure, in view of the fact that the integrals converge only conditionally, not absolutely; but presumably these doubts can be resolved. A more physical question is this: If you think for a moment about the classical Lagrangian for a free particle, we all know that there is a multiplicity of corresponding quantum theories, not all of which could have been produced by the unique device of Feynman quantization. There remains thus the possibility that we may have eventually to guess at a complete quantum theory rather than carry out the program in two steps, first classical theory then quantization. I do not think that we shall be able to resolve this question in the immediate future; hence I feel that we are completely justified in following both approaches - to work out methods of quantizing a given -number theory, and examining the intrinsic structure of a full-blown covariant -number theory.”

Footnotes