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Compt. rend.

THE NON-RELATIVISTIC WAVE FUNCTIONS ASSOCIATED WITH THE EXTENDED PARTICLE

P. Hillon, Jean Pierre Vigier

Published May 1, 1960• 0 citations
OpenAlex

Abstract

The non-relativistic wave mechanics of extended particles is characterized by its invariance under the group of Galilean transformations with rotation which is the direct product of the Galileo group and the R/sub 3/ group of tridimensional rothtions. From the direct product of the irreduci ble representations of these last two groups the nonrelativistic wave functions associated with the extended particle are deduced. (tr-auth)

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The non-relativistic wave mechanics of extended particles is characterized by its invariance under the group of Galilean transformations with rotation which is the direct product of the Galileo group and the R/sub 3/ group of tridimensional rothtions. From the direct product of the irreduci ble representations of these last two groups the nonrelativistic wave functions associated with the extended particle are deduced. (tr-auth)
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The Galilean symmetry of a free particle in one-dimensional space is examined under the scope of non-Abelian quantum kinematics. Within the Hilbert space that carries the regular ray representation of the Galilei group the Schrödinger operator appears as one of the three fundamental invariants of the extended kinematic algebra. By means of a superselection rule the physical Hilbert subspaces of the system are identified, in which a complementary ray representation of the Galilean transformation produces the time-dependent Schrödinger equation, and the Feynman space-time propagator. The quantization approach used in this paper is purely group theoretic and relativistic.
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The representation theory of the Galilean group extended by a one-dimensional group of dilations (which leave invariant the free Schrodinger equation) is studied. The existence of position and time operators is investigated from the standpoint of the imprimitivity theorem, for quantum particles whose states carry a projective representation of the extended group. Position and time operators are shown to exist for some two-component direct sums with non-trivial multipliers and for vector representations with zero helicity.
The irreducible representations of the extended Galilean group are used to derive infinite sets of symmetric and asymmetric second-order differential equations with constant coeffcients. All derived equations are local and their Lagrangians exist. It is shown that the asymmetric equations are Galilean invariant but the symmetric ones are not. By specifying quantum and classical physical settings, the constants in the equations are determined and the fundamental wave equations, including the Schrödinger, Schrödinger-like and new asymmetric equations, are obtained; the derived wave equation is non-fundamental. Formulation of wave theories based on the fundamental and non-fundamental wave equaions is considered, and physical implications of these theories on the wave description are discussed.
A study was made of the general conditions required to set up a non- relativistic functional theory for systems of particles. Representation of each particle of a system by a space-time function uj(P,T). Formal classification for the particles of the system with respect to indiscernability and the exclusion principle for the fermions. Using the general theory of predictions, the arguments of the functional for predictions are fixed, functional configuration- space is introduced, and the general form of the equation for the predictional elements is determined. Conditions incurred by the limiting cases of the usual wave mechanics for systems and of the single particle in the functional theory of particles. Fundamental principle for the equations of motion concerning the particles of a system: each function uj(P,T) describing a particle of the system obeys a non-linear equation which depends upon the functions u/sub k/(P,T) of the other particles of the system. Brief investigation of the limiting case of classical mechanics. (auth)