H. E. Moses

The Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis

The irreducible ray representations of the proper, orthochronous, inhomogeneous Lorentz group were originally given by Wigner in terms of a basis in which the energy and linear momenta are diagonal. In the present paper we show how the infinitesimal generators of the irreducible representations act on a basis in which the energy, the square of the angular momentum, the component of the angular momentum along the z axis, and the helicity (or circular polarization) are diagonal.We consider representations corresponding to particles of nonzero mass, and any spin and of zero mass and finite spin. The continuous‐spin case is to be treated in a later paper.

Reduction of Reducible Representations of the Infinitesimal Generators of the Proper, Orthochronous, Inhomogeneous Lorentz Group

In the present paper we show explicitly how to reduce reducible representations of the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group. We first construct a basis in which all the reducible representations are expressed as integrals of representations over masses and in which the infinitesimal generators act as in the Foldy‐Shirokov realization for nonzero masses and in the Lomont‐Moses realization for zero masses. However, the spin operators which appear in the Foldy‐Shirokov realizations and the generators of the Euclidean group which appear in the Lomont‐Moses realizations are reducible in general. Thus reducible representations are only partially reduced in this basis. On carrying out the reduction of the spin operators and the generators of the Euclidean group, we introduce a second basis such that the reducible representations are completely reduced. By changing the emphasis slightly, the methods of the present paper can be used to obtain the irreducible representations of the generators. One of us (H. E. M.) has already used the methods of the present paper to reduce the electromagnetic vector potential and, in papers which follow the present one, will show how to reduce wavefunctions in general and will also derive the Clebsch‐Gordan expansion for the direct product of two massless representations of finite spin and the same sign of energy. From the work of Mautner and Mackey it is known that every reducible unitary ray representation of the proper, orthochronous, inhomogeneous Lorentz group can be reduced to a direct integral of the irreducible representations. The techniques of the present paper thus enable us to carry out the reduction explicitly.

Transformation from a Linear Momentum to an Angular Momentum Basis for Particles of Zero Mass and Finite Spin

The infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In the present paper we show how the two bases are related for representations corresponding to zero mass and any (finite) spin. It will be shown how this relation enables one to integrate the infinitesimal generators in the angular momentum basis and thereby permits one to show how the angular momentum of a particle changes under the inhomogeneous Lorentz group. In particular, we study the way that the angular momentum of a massless particle of any spin appears in translated and moving frames of reference.

REPRESENTATIONS OF THE INHOMOGENEOUS LORENTZ GROUP IN TERMS OF AN ANGULAR MOMENTUM BASIS: DERIVATION FOR THE CASES OF NONZERO MASS AND ZERO MASS, DISCRETE SPIN,

In a previous paper the authors showed how the infinitesimal generators of the proper, orthochronous, inhomogeneous Lorentz group acted in a basis in which the square of the angular momentum, the z component of the angular momentum, the helicity and the energy were diagonal for the irreducible representations which correspond to the cases of nonzero and zero mass, discrete spin. In that paper no derivation of the results were given. It was possible, however, to verify them directly. In the present paper we carry out the derivation.

The kinematics of the angular momentum of a particle

SummaryThe way a particle changes its angular momentum under «homogeneous Lorentz transformations is well-known classically. The object of the present paper and a later one is to consider the problem quantuni-mechanically for particles of any mass and any spin. In the present paper we shall consider in detail the case where a particle has a definite angular momentum in one frame of reference and we shall calculate the probability distribution of angular momentum in a frame translated with respect to the original frame. Tn a later paper we shall treat the case where the two frames of reference are moving with respect to one another. The basic mathematical tool is the form for the infinitesimal generators of the inhomogeneous Lorentz group devised by Lomont and Moses in which the Hamiltonian, square of the angular momentum,z-component of angular momentum, and helicity are diagonal. The present paper and the projected one are important in multiple scattering problems, for it is possible using the results to take into account, to a certain degree at least, the effect of the selection rules. These rules are almost always ignored in multiple scattering problems. For example, it is shown that when the density of a gas is sufficiently low, radiative cooling goes on at a much faster rate when selection rules are taken into account than when they are ignored.RiassuntoII modo in cui una particella camlbia il suo momento angolare in seguito a trasformazioni di Lorentz inomogenee è elassicamente ben noto. L’oggetto del presente lavoro, e di uno successive, è di considerare il problema dal punto di vista della meccanica quantistica per particelle di massa e spin qualsiasi. Nel présente lavoro esamineremo in dettaglio il caso in cui una particella ha un momento angolare definite in un sistema di riferimento e calcoleremo la probability di distribuzione del momento angolare in un sistema di riferimento traslato rispetto al sistema originale. In un successivo lavoro tratteremo il caso in cui i due sistemi di riferimento si muovono l’uno rispetto all’altro. Lo strumento matematico fondamentale è la forma dei generatori infinitesimali del gruppo inomogeneo di Lorentz ideati da Lomont e Moses, in cui l’hamiltoniana, il quadrato del momento angolare, la componentez del momento angolare, e Félicità sono diagonali. Il présente lavoro e quello progettato sono importanti nei problemi di scattering multiplo, in quanto è possibile, utilizzandone i risultati, di tener conto, almeno fino a un certo limite, dell’effetto delle regole di selezione. Nei problemi di scattering multiplo tali regole sono quasi sempre ignorate. Si dimostra, ad esempio, che quando la densità di un gas è sufficientemente bassa., il raflreddamento radiativo procede molto più rapidamente se si tien conto delle regole di selezione di quanto non avvenga se si trascurano.

The assignment of wave functions to energy densities and probability densities

SummaryIn many problems involving wave propagation, the squares of the absolute values of the wave functions rather than the wave functions themselves are the physically observable quantities. These quantities are interpreted as energy densities in the case of electromagnetic radiation, for example, and as probability densities in quantum mechanics. The objective of the present paper is to give a theorem involving Fourier integrals which indicates a set of physical measurements of energy densities and probability densities in one-dimensional problems which lead to an essentially unique determination of the wave functions. For use in quantum mechanics, the measurements which are required can be expressed as the mean values of certain operators constructed from the position and momentum operators. These will also be given.RiassuntoIn molti problemi che interessant) la propagazione delle onde, le grandezze flsicamente osservabili sono i quadrati dei valori assoluti delle funzioni d‘onda piuttosto che le funzioni d‘onda stesse. Queste grandezze vengono interpretate corne densità d‘energia nel oaso della radiazione elettromagnetica, per esempio, e corne densità di probabilità nella meccanica quantistiea. Lo scopo del presente lavoro è di dare un teorema che implica integrali di Fourier, il quale indica un gruppo di misure fisiche delle densità di energia e delle densità di probabilità in problemi unidimensionali che porta ad una determinazione essenzialmente unica delle funzioni d‘onda. Per servire in meccanica quantistiea, le misure richieste possono essere espresse come valori medi di certi operatori costruiti dagli operatori di posizione e di impulso. Si danno anche questi ultimi.

Transformation from a Linear Momentum to an Angular Momentum Basis for Relativistic Particles of Nonzero Mass and Any Spin

The infinitesimal generators of the inhomogeneous Lorentz group have been given in a basis in which the components of the linear‐momentum operators are diagonal and in another basis in which the square of the angular momentum is diagonal for all unitary irreducible ray representations of the group. In a previous paper we showed how the two bases were related for representations corresponding to zero mass and any finite spin. In the present paper we show how the two bases are related for representations corresponding to nonzero mass and any spin. Thus this paper and the preceding one enable us to expand relativistic plane waves into relativistic spherical waves and vice‐versa for particles of any spin and any mass.In the previous paper we used the relation between the linear‐ and angular‐momentum bases to integrate the infinitesimal generators in the angular‐momentum basis and thereby obtain closed expressions which show how the angular momentum of particles of zero mass and any finite spin transform under...

Generalizations of the Jost Functions

The Jost functions have proved valuable in the study of the analytic properties of the scattering phase for the radial Schrodinger equation. In the present paper we shall present an alternative definition of the Jost functions, prove the equivalence of the new definition to the usual one, and generalize the new definition to the one‐dimensional Schrodinger equation (− ∞ <x< ∞), the three‐dimensional nonseparated Schrodinger equation, and the three‐dimensional nonseparated Dirac equation. It is hoped that these generalizations lead to a better understanding of the analytic properties of the scattering operator for these and related dynamical systems. The generalized Jost functions are shown to be operators in the variables which label the degeneracy of the continuous spectrum of the Hamiltonians which are considered.

Equivalence and Antiequivalence of Irreducible Sets of Operators. I. Finite Dimensional Spaces

A fundamental problem which arises in determining whether two quantum mechanical systems are essentially identical is whether a unitary or antiunitary transformation exists which maps one set of dynamical variables into another. Since an elementary dynamical system is specified by giving an irreducible set of dynamical variables, we are led to investigate the following problem: Given two irreducible sets of operators with a one‐to‐one correspondence between them, find the algebraic properties of the two sets which make it possible to infer the existence of a unitary or antiunitary operator relating them. A series of theorems is obtained from such considerations for finite dimensional spaces. It is shown that if the second set of operators contains some of the algebraic properties of the first set, the two sets are related by a similarity transformation. By altering the requirements, this transformation is a unitary transformation. Indications are also given to show how the theorems can be extended to Hilb...

EXPONENTIAL REPRESENTATION OF COMPLEX LORENTZ MATRICES

SummaryIt is shown that every proper, complex Lorentz matrixL can be expressed in the formL=Σ exp [AG], whereA is an antisymmetric matrix,G is the metric matrix, and Σ=±l. It is then shown that not every proper, complex Lorentz matrixL can be expressed in the formL=exp [AG], withA andG as above.RiassuntoSi dimostra che ogni matrice di Lorentz propria e oomplessa,L, puÒ essere espressa nella formaL = Σ exp [AG], doveA è una matrice antisimmetrica,G è la matrice metrica, e Σ=±1. Si dimostra poi che non tutte le matrici di Lorentz proprie e complesse,L, si possono esprimere nella formaL = exp [AG], conA eG come sopra.