Volume 7, Issue 5, May 1966
Index of content:

Generalized PhaseSpace Distribution Functions
View Description Hide DescriptionA set of quasiprobability distribution functions which give the correct quantum mechanicalmarginal distributions of position and momentum is studied. The phasespace distribution does nothave to be bilinear in the state function. The Wigner distribution is a special case. A general relationshipbetween the phasespace distribution functions and the rule of associating classical quantitiesto quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of thegeneralized phasespace distribution are considered.

The Invariance of the Vacuum is the Invariance of the World
View Description Hide DescriptionThe following theorem is demonstrated: If the vacuum is invariant under the group generated bythe space integral of the time component of a local vector current, then the Hamiltonian is invariantalso. A similar theorem holds for the group generated by the space integral of a space component ofa local axialvector current.

Adiabatic Invariants and the “Third” Integral
View Description Hide DescriptionIn a general case of Hamiltonian systems of n degrees of freedom, depending periodically ontime, n formal “third” integrals of motion are found. Their application in finding boundaries for theorbits is illustrated in a special case. Then a comparison is made between these integrals and theadiabatic invariants. Both are series expansions but the small parameter used is of different characterin each case. This is shown explicitly in a simple example and the relative accuracy of the twoexpansions is discussed.

Failure of the Mayer Irreducible Cluster Theorem with Wave Mechanics
View Description Hide DescriptionThe wave mechanical form of the cluster integrals in the activity expansion for the pressure ofnonideal gas is written in the Wiener functional integral form. The various pieces of the nthordercluster integral may then be easily expanded around the classical limit [WignerKirkwood (WK)expansion] using the procedure of Gel'fand and Yaglom. It is shown that the reducible diagrams failto factor at , and thus the Mayer theorem that only irreducible diagrams come into the virialcoefficients of the density expansion of the pressure is valid only through . As an example, theWK expansion of the third virial coefficient is worked out to . It is shown also how the functionalintegral formatism may be used to expand quantum statistical mechanical perturbation theory aboutthe classical limit.

Adiabatic Switching in the Schrödinger Theory of Scattering
View Description Hide DescriptionThe timedependent Schrödinger theory of scattering is studied rigorously with the potential Vreplaced by . Sufficient conditions are given that the Møller wavematrices be obtainablefrom this theory as . The conjecture that this theory can be used to define a reasonable Smatrixwhen the do not exist is shown to be false for the Coulomb potential. It is remarked that theGoldbergerGellMann switching procedure also breaks down in this case.

Structure of the Crossing Matrix for Arbitrary Internal Symmetry Groups
View Description Hide DescriptionSome properties of crossing matrices are deduced which are independent of the particular symmetrygroup from which the crossing matrices are derived. In particular, a, factorization of any elasticcrossing matrix, analogous to the factorization of a rotation matrix in n dimensions is found, and theconnection between, crossing and unitary matrices elucidated. In the special case of the crossingmatrix for elasticscattering of particles which transform as representations of , as well as theusual consistency requirement that each row should sum to unity, a new consistency requirement onthe elements in a given column is proved. As a byproduct of this work, a possibly new quadraticidentity for Racah coefficients is exhibited.

Generalized Spherical Functions for the Noncompact Rotation Groups
View Description Hide DescriptionSpecial representations of arbitrary noncompact rotation groups labeled by one independentCasimir operator are considered. The explicit construction of the corresponding generalized sphericalfunctions is given and the properties of these representations are discussed in detail.

DirectProduct Representations of the Canonical Commutation Relations
View Description Hide DescriptionWe consider directproduct representations of the canonical commutation relations. An irreduciblerepresentation is defined on each of the incomplete directproduct spaces (IDPS) of von Neumann.We prove that two such representations are unitarily equivalent if and only if the correspondingIDPS are weakly equivalent, for which simple analytic tests exist. The matrix elements of theserepresentations, coupled with a FriedrichsShapiro type of integral, fulfill group orthogonalityrelations. This classification into unitary equivalence classes also applies to directproduct representationsof the canonical anticommutation relations.

Conditions for the Existence of Closed Solutions by the Normal Ordering Method
View Description Hide DescriptionIt is shown that the normal ordering method of solving boson operator equations leads to closedsolutions of the Schrödinger and density operator equations only in certain cases. In particular, anecessary condition for the existence of solutions of the form with G a finite multinomialis that the Hamiltonian be of the general form . Approximate solutions may be obtained for the density operator for arbitraryHamiltonians.

The InitialValue Problem for a Slab. I. Anisotropic Scattering
View Description Hide DescriptionIn this paper, the spectral properties of the Boltzmann operator describing the transport of monoenergeticneutrons with anisotropic scattering in a slab are considered. The HilleYosida theorem isapplied to obtain the semigroup of operators solving the initialvalue problem.

The InitialValue Problem for a Slab. II. Nonuniform Slab
View Description Hide DescriptionThis paper deals with the timedependent neutron transport in a plane slab of material withvariable nuclear properties. The onevelocity theory and the isotropic scattering are assumed. Thespectral properties of the corresponding Boltzmann operator are found, and the initialvalue problemis solved.

The Physical Regions of ManyParticle Processes
View Description Hide DescriptionLorentz invariant expressions, in the form of determinantal conditions, are derived for the physicalregions of manyparticle processes. They are explicitly solved in the case of fiveparticle processesand the solutions are exhibited in the planes of pairs of the five independent kinematic invariants.

Determination of Optical Field Correlations from Photon Counts
View Description Hide DescriptionIt is shown that the offdiagonal matrix elements of an operator on an Nboson space are determinedby diagonal matrix elements in different symmetrized product bases. The motivating applicationis that optical field correlations are determined by moments of the numbers of photons in differentmodes.

Ray Representations of Finite Nonunitary Groups
View Description Hide DescriptionAn extension of the ray representation theory is formulated to embrace nonunitary groups. Thecoray representations are obtained by the ray representations of its unitary subgroup. Theorems ofcoray representations are stated. The usefulness of the formalism is discussed.

The Implications of Unitarity on Possible Higher Symmetries
View Description Hide DescriptionUsing the covariant spinor formalism for higherspin particles, it is shown that the only combinationof the internal symmetry group and the Lorentz group that is compatible with the unitarity conditionis the direct product. Three theorems are proved which severely limit the type of symmetries that areconsistent with the unitarity condition.

Note on the BondiMetznerSachs Group
View Description Hide DescriptionIt is shown that, in spacetimes which are asymptotically flat, there are reasonable physical restrictionsthat allow one to impose coordinate conditions (in addition to the usual Bonditype conditions)which restrict the allowed coordinate group to a subgroup of the BondiMetznerSachsgroup. This subgroup is isomorphic to the improper orthochronous inhomogeneous Lorentz group.

Spectrum of the S Operator
View Description Hide DescriptionThe structure of a scattering operator which describes events above a threebody and below afourbody threshold has been studied. It is shown that one can approximate it arbitrarily closely inthe norm as the sum of two operators, each having a simple form. Using this decomposition, one obtainsa general form for the spectral decomposition of this scattering operator.

Exchange of Massive Particles in the BetheSalpeter Equation
View Description Hide DescriptionThe highenergy behavior of the absorptive part A(s) of the forward scattering amplitude in the laddergraph approximation can be obtained from a homogeneous integral equation. If A(s)grows as , α is the highest characteristic value of this equation. When the mass μ of the exchangedparticles is nonzero, the highg behavior of α is radically different from that in the case of scalarphoton exchange. For small g, , where a is independent of μ. The dependenceof b on μ has been calculated.

Properties of Quantum Statistical Expectation Values
View Description Hide DescriptionThe semigroup of statistical operators, and the unitary group of time translation operators generatedby the same Hamiltonian of a nonrelativistic fermion field, are naturally imbedded in a holomorphichalfplane semigroup. The statistical expectation values of products of timedependentoperators are then boundary values of holomorphic functions which allow a Bochner integral representationwith a Cauchy kernel. The general timetemperaturedependent Green's functions permit aconcise spectral representation. It is suggested that a thermodynamic perturbation theory shouldtreat the Heisenberg and Bloch equations simultaneously in terms of a perturbation theory for holomorphichalfplane semigroups generated by semibounded selfadjoint Hamiltonians.

Rigid Motions in Einstein Spaces
View Description Hide DescriptionThe dyadic formulation of general relativity is used systematically to discuss rigid congruences inEinstein spacetime. For spacetime of uniform curvature, the quotient space metrics of rotatingand accelerating rigid bodies are obtained. For Einstein spacetime of nonuniform curvature, allirrotational, nonisometric, rigid motions are explicitly displayed. They have one degree of freedom,and occur only in degenerate static metrics of Class B. Rotating rigid congruences in Einstein spacetimeof nonuniform curvature are shown to have no degrees of freedom. Their evolution is in factfound to be governed by a complete set of 14 firstorder total differential equations, linear in the timederivatives of the dyadic variables. Such rotating motions are shown further to be constrained by a setof algebraic conditions, and the implication of this for the validity of the HerglotzNoether theoremin Einstein spacetime is discussed.