F. J. BELINFANTE

He defines a “modernized” Poisson bracket with for Fermi-Dirac fields and for Bose-Einstein fields. He also distinguishes and , which is a refinement necessitated by factor ordering difficulties.

He then uses vierbeine to describe the gravitational field, and introduces local components independent of for the spinor fields so that

The canonical formalism can be developed following the method of Dirac^{1} There are 11 first-class primary constraints

arising from redundancy in the variables used to describe the spinor field.

One finds

where , and is canonically conjugate to , while with is integrated over space to give the energy-momentum (free-) fourvector, if one assumes vanishing fields and flat space-time at infinity.

In passing to the quantum theory one makes use of the modified Poisson bracket due to Dirac, which is defined by

with

and

One sets

The dynamical equations then become

where the symbol means place the factor where is taken out. The factor is given by

where the .

Also

the
and
being certain coefficients and
a function of the
’s,
’s, and their space derivatives only.
is quadratic in the
. The constraint expressions
and
here
are meant as functions of the
and
and therefore are expressions in the
partially linear in them. (Note that the constraints

Kennedy has verified the consistency of this scheme by direct computation of the modified Poisson brackets , , . They all reduce to linear combinations of the ’s and ’s alone.

The modified Poisson brackets for the spinor variables lead to some unwanted peculiarities which can be removed by redefining the spinor variables according to

One then finds

Thence, modified Poisson brackets defined by partial derivatives with respect to and equal ordinary (though modernized) Poisson brackets defined by partial derivatives with respect to and . Thus working with in terms of the and from the beginning makes it possible to quantize without ever mentioning modified Poisson brackets.

BELINFANTE

With the “muddified” theory

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

BELINFANTE

Following BELINFANTE’s

NEWMAN

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

BERGMANN

MISNER

SCHILLER

WHEELER

BERGMANN

WHEELER

GOLDBERG

GOLDBERG

ANDERSON

BELINFANTE