MATHEMATICAL PHYSICS: RELATIVITY, GRAVITATION, AND

MATHEMATICAL PHYSICS: RELATIVITY, GRAVITATION, AND
FIELD THEORY
COVARIANT QUANTIZATION OF NONLINEAR FIELD THEORIES
PETER
G.
BERGMANN
Sponsored by the ONR
In three preceding papers, the classical canonical formulation of covariant
field theories has been developed in general terms and carried out for the general theory of relativity. [P. G. Bergmann, The Physical Review vol. 75 (1949)
p. 680; P. G. Bergmann and J. H. M. Brunings, Reviews of Modern Physics
vol. 21 (1949) p. 480; Bergmann, Penfield, Schiller, and Zatzkis, The Physical
Review vol. 78 (1950) p. 329.] These papers approach the problem of quantization without solving it completely. The purpose of this paper is to present a
complete and self-consistent scheme of quantization of a covariant, nonlinear
field theory.
That the commutation relations between canonical variables are covariant
has been shown previously [P. G. Bergmann and J. H. M. Brunings, loc. cit.],
The remaining difficulty was to find operators satisfying both the commutation
relations and the algebraic constraints (true identities [L. Roscnfeld, Annales
de l'Institut Henri Poincaré vol. 2 (1932) p. 25]) typical for covariant and gauge
invariant theories. It is well known that no observable appearing in a commutation relation with a c-number right-hand side possesses normalizable eigenfunctions; therefore, no state vector satisfying the algebraic constraints can be normalized in a linear vector space composed of both the state vectors satisfying the
constraints and those disobeying them. However, normalization is possible in
the subspace of those state vectors obeying all the constraints imposed in a
given theory. In a typical field theory, the number of these constraints is always
transfinite.
The following procedure is self-consistent and invariant. Through a suitable
canonical transformation, all the constraints are made canonical variables,
preferably momentum densities. If these constraints include the Hamiltonian
constraint [P. G. Bergmann and J. H. M. Brunings, loc. cit.], the remaining
variables are subject solely to the commutation relations. The usual condition
represented by the Schrödinger equation is included among the constraints.
In the typical wave-mechanical representation, the state vector is then a normalizable functional of all the "coordinate" variables, except those conjugate to the
constraints, and except for its normalizability arbitrary. Once the theory has
been constructed in this representation, the most general representation can be
obtained by carrying out an arbitrary canonical (unitary) transformation, involving all the variables including the constraints.
651
652 SECTION V. MATHEMATICAL PHYSICS AND APPLIED MATHEMATICS
The next step in this investigation will be* the application of this general
formalism to specific problems.
SYRACUSE UNIVERSITY,
SYRACUSE, N. Y., U. S. A.
GRAVITATIONAL SHIFT IN THE SOLAR SPECTRUM
A. J. COLEMAN
This paper studies the interaction between a Schwarzschild gravitational field
and the electric field of the nucleus of a hydrogen-like atom. It is known that
within the framework of general relativity an invariant form of quantum mechanics leads, in a first approximation, to the formula for the spectral shift
which Einstein deduced merely by treating the atom as a symmetric clock. Carrying the quantum-mechanical discussion to a higher approximation reveals a
perturbation of the Coulomb potential of the nucleus which is directionalized
with respect to the gravitational field. This new term is of the correct order of
magnitude and general character to explain the limb-effect, that is, the fact, known
for more than forty years and hitherto unexplained, that the shift in lines of
the solar spectrum is different at the edge and center of the sun's disc. It may
possibly also explain the discrepancies between the spectral shifts at the center
predicted by Einstein and the recent observations of M. G. Adam.
UNIVERSITY OF TORONTO,
TORONTO, ONT., CANADA.
LA RECESSION DES NÉBULEUSES EXTRA-GALACTIQUES
P. DRUMAUX
L'étude mathématique de la gravitation, faite à la lumière de la relativité,
montre que le monde cosmique qui nous environne, à savoir l'ensemble des
nébuleuses extra-galactiques, ne correspond pas à l'idée qu'on s'en était fait.
Les nébuleuses n'ont pas seulement une vitesse radiale de recession mais en
outre une vitesse transversale de même ordre de grandeur. Il faut d'autre part
faire une distinction entre le mouvement relatif des nébuleuses par rapport à
la Voie lactée et le mouvement général d'entraînement de l'ensemble des nébuleuses, y compris la Voie lactée, sous l'effet du champ gravifique cosmique dans
lequel cet ensemble est plongé. Le calcul montre que la vitesse de ce mouvement
d'entraînement est de l'ordre de 100,000 km/sec.
Le calcul conduit d'autre part à la connaissance des trajectoires des nébuleuses
qui sont soit des spirales gauches elliptiques, soit des courbes exponentielles
apériodiques. La détermination astronomique de ces trajectoires résultera de
RELATIVITY, GRAVITATION, AND FIELD THEORY
653
la mesure des effets Doppler et des magnitudes apparentes pour des paires de
nébuleuses situées dans la même direction et cela pour au moins six directions
différentes.
Le calcul montre qu'il est alors possible de déterminer les vitesses transversales et aussi les trajectoires à condition que les mesures puissent se faire avec
haute précision.
Or l'incertitude actuelle dans les magnitudes absolues est un grand obstacle
à surmonter. Un premier pas serait fait en mesurant les écarts dans les magnitudes absolues d'une nébuleuse à l'autre et cela paraît possible en opérant sur
deux nébuleuses situées dans la même direction et ayant approximativement
même effet Doppler ainsi que même type spectral. On peut alors mesurer les
écarts susdits tout en ignorant les magnitudes absolues elles-mêmes. La connaissance de ces écarts permettrait la mesure des rapports de distance de nébuleuses situées dans une même direction, ce qui conduirait à la détermination
des trajectoires.
UNIVERSITY OF GHENT,
GHENT, BELGIUM.
AN EXPANSION OF A FOUR-DIMENSIONAL PLANE WAVE
IN TERMS OF EIGENFUNCTIONS
KATHLEEN SARGINSON
The problem is to find the expansion of a four-dimensional plane wave
exp i{$-r — Et} in terms of those solutions of the scalar relativistically invariant wave equation D V = &V in which R, a, 6, 0, where r = R cosh a, t = R sinh a
and r, 6, cj> are ordinary spherical polar coordinates, are taken as independent
variables. The solution of the wave equation in terms of these coordinates may
be expressed in terms of associated Legendre functions and Bessel functions, the
orders of the appropriate functions being determined by the condition of quadratic integrability.
The nature of the expansion of the plane wave depends on whether (r, t),
(p, E) are space-like vectors or time-like vectors. When (r, t) and (p, E) are
both space-like, the expansion consists of a finite sum of terms together with
an integral. When either (r, t) or (p, E) is time-like, or when both (r, t) and
(p, E) are time-like, the finite sum is not present in the expansion, and this consists of an integral only.
SOMERVILLE COLLEGE,
OXFORD, ENGLAND.
654 SECTION V. MATHEMATICAL PHYSICS AND APPLIED MATHEMATICS
ON THE INTERPRETATION OF T H E PARAMETERS OF THE
PROPER LQRENTZ GROUP
EDWARD JAY SCHREMP
The six essential parameters of the proper Lorentz group are commonly known
to be equivalent to two vectors in ordinary Euclidean 3-space, one vector representing a spatial rotation, and the other representing the characteristic relative
velocity of a rotation-free Lorentz transformation. (What are here provisionally
pictured as vectors are actually specialized forms of the quaternions described
below.) Unless these two vectors are collinear, however, their specification is not
altogether unique, because of the familiar properties of noncommutativity of
the transformations' which they represent. The non-uniqueness which thus characterizes this physical mode of interpretation would seem to reflect a certain want
of perfection.
There are, of course, alternative mathematical representations of these parameters wherein this non-uniqueness may be removed. The purpose of this note is
to single out one particular geometrical interpretation of these parameters which
appears to be of intrinsic physical interest.
This geometrical interpretation depends upon the well-known fact that any
proper Lorentz matrix L is expressible in the form
L = QQ = QQ,
where Q is a suitable complex matrix, and Q is the corresponding complex conjugate matrix. It then turns out that one may regard every matrix Q as a representation of a complex quaternion
Oi
1 i
2 •
3
q = e0q + eiq + e2q + e^q
whose norm is of unit magnitude. Accordingly, the three ratios q1/q°, q2/q°, qZ/q°
constitute a set of three complex essential parameters of the proper Lorentz
group. The geometrical interpretation of these parameters is then evidently
equivalent to the geometrical interpretation of a general complex quaternion.
As in the case of real quaternions, the geometrical interpretation of complex
quaternions involves a consideration not only of the quaternion group itself
but also of the adjoint group of the quaternion group. The complex quaternion
group defines, with its reciprocal group, a complex 3-dimensional group space
with homogeneous coordinates (q°, q, q, q); while its adjoint group defines a
complex 3-dimensional vector space which is an immediate generalization of the
Euclidean 3-space of ordinary experience. In this complex vector space, there is
a unique geometrical interpretation of a general proper Lorentz transformation.
In the terminology appropriate to this geometry, such a transformation would
be called a complex rotation about a complex direction.
The terminology and the properties of the two foregoing complex geometries
have been systematized in the present work, following the principle that these
geometries are specializations of the general complex projective geometry of
RELATIVITY, GRAVITATION, AND FIELD THEORY
655
three dimensions. In the course of this study it has appeared that to every individual geometrical concept there belongs an intrinsic physical content, the
physical ideas of time and motion being reflected in the complex character of
the geometry.
NAVAL RESEARCH LABORATORY,
WASHINGTON, D. C , U. S. A.
EMPTY SPACE-TIMES ADMITTING THREE-PARAMETER GROUPS
OF MOTION
A. H.
TAUB
An empty space-time is a four-dimensional Riemannian space of signature
{+ — ) which has a vanishing Ricci tensor and no singularities in the metric
tensor. The existence of empty space-times admitting three-parameter groups
of motion which are not flat spaces is shown. In special coordinate systems the
metric tensor of these spaces has components which are not bounded for all
values of the spatial coordinates (including infinite ones). However, it is unlikely
that this is due to an "essential" singularity in the metric tensor. These singularities are to be attributed to the coordinate system used since the hypersurfaces on
which they occur have transitive groups operating on them. It is further shown
that a space-time which admits the three-parameter group of Euclidean translations, which has a vanishing Ricci tensor, and which has the further property
that the curvature tensor is not singular along the time axis is a flat space. The
Einstein field equations for vanishing stress-energy tensor are integrated under
the assumption that the space-time admits a three-parameter group of motions
with minimum invariant varieties consisting of two-dimensional surfaces of
constant curvature. There are three such cases, the Schwarzchild one and two
others. It is shown that all three cases have many common properties. In particular, all three are static.
UNIVERSITY OF ILLINOIS,
URBANA, I I I . , U. S. A.
INTEGRAL RELATIONSHIPS BETWEEN NUCLEAR QUANTITIES
ENOS E. WITMER
One-eleventh of the electron mass appears to be the natural unit of mass for
the masses of nuclei. This unit we designate the prout. We [Enos E. Witmer,
The Physical Review vol. 78 (1950) p. 641] have introduced the hypothesis
that the masses of all those nuclei in the ground state, which are not subject to
ß-decay, are an integral number of prouts. Furthermore some of the nuclei
656 SECTION V. MATHEMATICAL PHYSICS AND APPLIED MATHEMATICS
subject to ß-decay appear to follow this integral rule. It is a consequence of this
that a large class of nuclear reaction energies should be an integral number of
prouts. It is only in the case of the lighter nuclei that some of the masses and
reaction energies are known with sufficient accuracy to test this hypothesis.
However, the agreement in this case is as good as can be expected.
The best tables of nuclear reaction energies and nuclear masses are probably
those in a recent article [Tollestrup, Fowler, and Lauritsen, The Physical Review vol. 78 (1950) p. 372]. The reaction energies in this table conform to our
rule quite well, taking account of the probable errors. Furthermore almost all
of these reaction energies approximate an even number of prouts.
Using this and other data we have made a table of nuclear masses in prouts
up to Ne20 and a few beyond that. We find that all except two of these nuclear
masses are an even number of prouts. A number of them contain many powers
of 2 as factors. Thus in 24 nuclear masses there are a total of 55 powers of 2.
In particular, all masses of stable nuclei consisting of 2n protons and 2n neutrons
are divisible by 4 when expressed in prouts.
We take the number of prouts in the nucleus of 0 16 to be 320616. It follows
from this that 1 prout is 0.00004989024831 m.u. Also 1 prout is 46.453 kev. The
isotopie masses of n1, H1, and H2 recomputed by these ideas are 1.0089804,
1.0081322, and 2.0147179 m.u. respectively.
A number of nuclear magnetic moments are represented by the formula
k
0-^)
where k is an integer.
These results are in accord with our ideas of the importance of integers in
the nuclear domain.
UNIVERSITY OP PENNSYLVANIA,
PHILADELPHIA, PA., U. S. A.