F. J. BELINFANTE opened with an examination of the three-field problem: gravitational and electromagnetic fields plus the Dirac electron field. He considered first the “classical” theory, defined by him as the limit of the quantum theory as
so that gravitational and electromagnetic fields commute and spinor fields purely anticommute. The notational developments which he then proceeded to outline go something like this:
He then uses vierbeine
to describe the gravitational field, and introduces local
components independent of
for the spinor fields so that
arising from redundancy in the variables used to describe the spinor field.
In passing to the quantum theory one makes use of the modified Poisson bracket due to Dirac, which is defined by
The dynamical equations then become
The modified Poisson brackets for the spinor variables lead to some unwanted peculiarities which can be removed by redefining the spinor variables according to
BELINFANTE pointed out that in its present form his theory seems to be incovariant. This is related to the fact that the
do not vanish in the strict sense. (They only vanish “weakly,” in Dirac’s terminology.) However he proposed simply to bypass this problem for the time being, and, for the sake of being able to make practical computations, pass over to what he calls a “muddified theory,” i.e., a theory obtained by throwing in “mud.” In electrodynamics this is just Fermi’s procedure of adding a non-gauge invariant quantity to the Lagrangian. The first-class constraints then disappear and one must replace them by auxiliary conditions (e.g. the Lorentz condition). There is a certain arbitrariness here, since the forms of the auxiliary conditions depend on the precise form of the “mud” which has been thrown in. However, if the
’s are replaced by their expressions in terms of the
’s then the auxiliary conditions must reduce to the original constraints. BELINFANTE has made certain special choices for these conditions (e.g., De Donder
condition), based on convenience, and he hopes he can then do meaningful practical calculations, just as the Fermi theory was long used for practical calculations in electrodynamics before all the mathematical subtleties of various constraints were precisely understood. BELINFANTE has shown by explicit computation that the constraints of his “muddified” theory are conserved, and has calculated explicitly the
One can introduce annihilation and creation operators (for photons, gravitons, etc.), although the Fourier transformation procedure on which they are based is a non-covariant procedure.
BELINFANTE concluded by suggesting that a theory in which only “true variables” appear may be mathematically nice but somewhat impractical. From the point of view
of a scattering calculation, for example, there may be some truth even in an almost true “untrue variable.” In any event the “true” theory still eludes us at present.
Following BELINFANTE’s remarks, there was considerable discussion as to whether or not all true observables are necessarily constants of the motion in a generally
covariant theory. No progress was made on this question, however, and the answer is still up in the air as of this moment.
NEWMAN next reported on some work he has been doing to try to obtain the true observables by an approximation procedure. Instead of dealing directly with
the gravitational field he considered a “particle” Lagrangian of the form
for which the equations of motion are nonlinear but invariant under a transformation group analogous to the gauge group of electrodynamics or the coordinate transformation group of general relativity. The gravitational field is embraced by this example when
becomes transfinite. In a linear theory the true observables are easy to find. In NEWMAN’s approximation procedure the search for the next higher order terms (with respect to some expansion parameter) in the true observables is no worse than finding the exact expression in the linear case, and can actually be carried out, even when some of the constraints are quadratic in the momenta. In the case of the gravitational
field the true observables evidently become more and more nonlocal (i.e., involving higher order multiple integrals) at each higher level of approximation. NEWMAN could say nothing about the convergence of his expansion procedure.
BERGMANN remarked that NEWMAN’s procedure was quite different from the more common differential-geometric approach which is specifically tailored to the gravitational
case. He mentioned in this connection the work of Komar and Géhéniau on metric invariants constructed out of the curvature tensor. It is thought that these invariants have some close connection with the “true observables.”
MISNER advocated at this point that one simply forget about the true observables, at least as far as quantization is concerned. He suggested starting with a metricless, field-less space, defined simply in terms of quadruples of real numbers, then performing certain formal mathematical operations and finding the true observables later, if one desires them.
WHEELER remarked that all of these discussions lead to the conclusion that the problems we face are problems of the classical theory.
BERGMANN agreed, and expressed his conviction that once the classical problems are solved, quantization would be a “walk.”
WHEELER, however, still felt that in the Feynman quantization procedure the whole problem is already solved in advance.
GOLDBERG and ANDERSON concluded the afternoon session with a discussion of “Schwinger quantization,” that is, the procedure which uses a
-number Lagrangian as a starting point. In making variations of such a Lagrangian one must pay careful attention to the ordering of factors.
GOLDBERG outlined a problem of trying to find the unitary operators which generate various invariant transformations, within the subspace defined by the constraints.
ANDERSON pointed out some difficulties, connected with the factor ordering problem, of defining a unique
BELINFANTE suggested that, if an interaction representation could be found, at least some of the factor ordering problems might be avoided with the use of “Wick brackets” which reorder the annihilation and creation operators, even though these brackets, being defined with respect to a flat space, would have no generally covariant significance.