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,
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
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,
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
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,
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
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”
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
. When DE WITT
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
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.
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
(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.)
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
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
(Editor's Note: - The following is taken from a set of mimeographed notes which DE WITT
As a prototype of a covariant theory DE WITT
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
are functions of the
's only. DE WITT
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
These will constitute the complete set of constraints
are certain coefficients. DE WITT
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
for certain coefficients , and if
where the are certain coefficients satisfying
The canonical momenta for the system are
gives immediately the set of constraints
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
The final expression for the energy function provides an illustration of Dirac's theorem
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
where denotes the Poisson bracket. In particular, the time rate of change of a is given by
's vanish their time derivatives must also vanish. This will happen automatically if the Poisson brackets
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,
The latter equations are simply the velocity constraints
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
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
Since the classical expressions for the
involve quantities which, in the quantum theory, do not commute, the factor ordering problem here makes its appearance. According to DE WITT
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
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,
which is just the Schrödinger equation.
Construction of the quantum form of
is more difficult than the construction of the quantum forms of
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
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
are arbitrary functions of the
's, the matrix (
) possessing, however, an inverse (
). These transformations, which DE WITT
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
-transformations, DE WITT
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
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
are then immediately recognizable as the structure constants
of a Lie group. The secondary constraints
The primary constraints
(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,
The operator remains invariant under point transformations which are restricted to the variables alone.
and any quantities constructed out of them are the true observables
One has only to require that
be independent of the
will then be a true observable
The gauge invariance of the quantum theory is now also immediately evident. In the “coordinate” system , , , gauge transformations are given simply by
There remains only the term
which is unknown. However this term, as Feynman
It is of interest to examine the Feynman quantization
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
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
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
is a suitable infinite normalization factor, the functional integration
is some true observable
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.
“(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
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
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
“In the following I wish to explain why I think that it is worthwhile to try Feynman's action method
I shall start with the simplest generally covariant theory known, that is electrodynamics, which is covariant with respect to gauge transformations.
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
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 above9 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
, but as Professor J. A. Wheeler
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
– “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.
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
I would like to close with a comment on the motivation for Feynman quantization
Feldquantisierung, lecture notes, Zürich, 1950-1951.
J. H. Van Vleck. Proc. Nat. Acad. Sci. 14, 178 (1928); C. Morette. Phys. Rev. 81, 848 (1951).
Can. J. Math. 2, 129 (1950).
Nuovo Cimento IV, 1445 (1956).
R.P. Feynman: Rev. Mod. Phys. 20, 367 (1948).
P. T. Matthews, A. Salam: Nuov. Cim 2, 120 (1955); W. K. Burton, A. H. DeBorde: Nuov. Cim 4, 254 (1956); P. W. Higgs: Nuov. Cim 6, 1263 (1956).
Feynman, op. cit., and Burton and DeBorde, op. cit..
Switching off of the electron charge is here assumed.
This does not mean that we have no use for other types of in intermediate calculations. In electrodynamics with truncated Lorentz-gauge action we may, for instance, calculate (21.2) with and that is very useful.
is not necessarily the classical action-functional for the system (also when such a functional exists), but it should go over into the classical function when .
In electrodynamics this question is easily answered: Every gauge-invariant functional is of the wanted type. If the gravitational field constant is switched on and off, the matter can to some extent be made equally simple in gravidynamics.
Burton and DeBorde, op. cit.; Higgs, op. cit..