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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Galilean Symmetry, Measurement, and Scattering as an Isomorphism between Two Subalgebras of Observables

J. V. Corbett

Published Jun 15, 1970• 12 citations
Publisher

Abstract

A model of quantum-mechanical scattering is given in terms of the Abelian subalgebras of observables which are associated with the initial preparation and final measurement procedures. The scattering is described by an operator, the $S$ matrix, which defines an isomorphism between these two subalgebras. By using the concept of presymmetry for Galilean-invariant systems, these algebras are shown to be generated by the generalized momenta \ensuremath{'}P which are related to the dynamical situation by $H{P}_{\mathrm{ac}}=\frac{{|{\mathrm{P}}^{\ensuremath{'}}|}^{2}}{2\ensuremath{\mu}}$. Here $H$ is the total Hamiltonian effective at the time of measurement, ${P}_{\mathrm{ac}}$ is the projection onto its absolutely continuous spectrum, and $\ensuremath{\mu}$ is the mass of the particle. The $S$ matrix which is defined in this way gives the same formal expressions for the differential cross sections as the usual one, and it is automatically unitary. Furthermore, since the differential cross section is defined as a relative probability in terms of the results of the two measurement procedures, it can be finite even when the classical cross section is infinite. Scattering can therefore be defined in this model for spherically symmetric potentials which decrease as slowly as ${r}^{\ensuremath{-}\ensuremath{\epsilon}}$, for any $\ensuremath{\epsilon}>0$, provided that the potential is repulsive at large separations between the particles.

DOI
10.1103/physrevd.1.3331

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A model of quantum-mechanical scattering is given in terms of the Abelian subalgebras of observables which are associated with the initial preparation and final measurement procedures. The scattering is described by an operator, the SSS matrix, which defines an isomorphism between these two subalgebras. By using the concept of presymmetry for Galilean-invariant systems, these algebras are shown to be generated by the generalized momenta \ensuremath{'}P which are related to the dynamical situation by HPac=∣P\ensuremath′∣22\ensuremathμH{P}_{\mathrm{ac}}=\frac{{|{\mathrm{P}}^{\ensuremath{'}}|}^{2}}{2\ensuremath{\mu}}HPac​=2\ensuremathμ∣P\ensuremath′∣2​. Here HHH is the total Hamiltonian effective at the time of measurement, Pac{P}_{\mathrm{ac}}Pac​ is the projection onto its absolutely continuous spectrum, and \ensuremathμ\ensuremath{\mu}\ensuremathμ is the mass of the particle. The SSS matrix which is defined in this way gives the same formal expressions for the differential cross sections as the usual one, and it is automatically unitary. Furthermore, since the differential cross section is defined as a relative probability in terms of the results of the two measurement procedures, it can be finite even when the classical cross section is infinite. Scattering can therefore be defined in this model for spherically symmetric potentials which decrease as slowly as r\ensuremath−\ensuremathϵ{r}^{\ensuremath{-}\ensuremath{\epsilon}}r\ensuremath−\ensuremathϵ, for any \ensuremathϵ>0\ensuremath{\epsilon}>0\ensuremathϵ>0, provided that the potential is repulsive at large separations between the particles.
The association between laboratory procedures and self-adjoint operators (observables), usually implicit and allusive, is made explicit by the introduction of two collections of nonmathematical objects; the observation procedures and the state-preparing procedures. By a process of idealization and extrapolation from empirical facts, these two collections are turned into mathematical sets ("hardware spaces") with the structure of a star algebra O\mathcal{O}O and a convex linear set S\mathcal{S}S, respectively. There is a many-one mapping from O\mathcal{O}O into the space U\mathfrak{U}U of observables (the operators on Hilbert space), and physical laws or solutions of equations of motion are embodied in the mapping $\ensuremath{\Phi}:\mathcal{O}\ensuremath{\rightarrow}\mathfrak{U}$. Certain material motions of observation instruments and state-preparing instruments induce automorphisms of the hardware spaces but in general, not automorphisms of U\mathfrak{U}U. An extension of space-time invariance theory to accelerations [$\mathbf{x}\ensuremath{\rightarrow}\mathbf{x}+\ensuremath{\Sigma}{2}^{\ensuremath{\infty}}{(n!)}^{\ensuremath{-}1}{\mathbf{r}}_{n}{t}^{n}$], to external symmetry-breaking fields, and to subsystems under the influence of other subsystems becomes possible. The main results of this paper are a derivation of Newton's second law and of the gravitational equivalence principle for nonrelativistic quantum mechanics from invariance and causality principles.
The thesis is divided in two parts. In the first part of the thesis, I present an updated version of the theory of covariant quantum mechanics and of the associated classical background. This theory has originally been proposed by Jadczyk and Modugno and further developed in cooperation with other people. It provides a geometric model for quantum mechanics of a scalar and spin particle in a curved spacetime with absolute time equipped with a given classical gravitational and electromagnetic field. Starting from minimal axioms, this model yields in a covariant way the Schrödinger equation by means of a global Lagrangian formalism and the quantum operators associated with classical quantisable functions by means of classification of distinguished quantum vector fields. In the second part of the thesis, I investigate systematically the infinitesimal symmetries of the classical and quantum geometric structures. This part deals with the main original contributions of the thesis. Some results fit well-known facts of other mathematical models of quantum mechanics, but are derived by innovative procedures. Other results are completely new. The most important original contributions can be summarized by the following two statements: The quantum symmetries constitute Lie algebras, which are naturally isomorphic to subalgebras of the new classical Lie algebra of quantisable functions. Let me stress that this algebra is not a Poisson subalgebra. On the other hand, by means of the quantum Lagrangian formalism, we can associate a conserved quantum current with any element of a distinguished classical subalgebra. These results might be the prequantum source of the probabilistic interpretation of quantum mechanics.
The physical act of accelerating observation instruments has an obvious counterpart in Newtonian physics: It is the transformation induced by $(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}, t)\ensuremath{\rightarrow}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}+\frac{1}{2}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}{t}^{2}, t)$. In special relativity, a theoretical counterpart for physical acceleration has been introduced only for a small subset of measurement procedures, such as clocks and yardsticks, but not for such instruments as accelerometers. We propose a general theoretical counterpart for physical accelerations in Minkowski space. In the algebra O\mathcal{O}O of observation procedures, it induces automorphisms, not of O\mathcal{O}O but of a point subalgebra ${\mathcal{O}}_{x}\ensuremath{\subset}\mathcal{O}$ that is associated to a single point xxx. From the postulates of presymmetry, we deduce a generalized version of Newton's second law; an acceleration-invariant subset ${\ensuremath{\bigcup}}_{(x|{x}_{0}=0)}{\mathcal{O}}_{\mathrm{xc}}$ of instant observation procedures provides complete predictive power. The main technical contribution of the paper is the introduction of a topological algebra O\mathcal{O}O in which local subsets associated to a single point are proper (although unbounded) subalgebras, whose automorphisms are discussed.
A unified description of classical and quantal scattering is presented. This description is unified both in the mathematical language and in the conceptual pictures that are used. Statistical states and observables, represented in phase space, are emphasized while both the time dependent trajectory and stationary state beam pictures are used. Semiclassical-type approximation schemes to the generalized differential cross section are presented for the single and double potential cases. Connection is made with the Born and distorted wave Born approximations, for both quantum and classical mechanics. A similar semiclassical-type approximation scheme for the double potential internal state kinetic cross section is presented. Connection is also made with the 'constant acceleration approximation' of Oppenheim and Bloom. Zwanzig's projection operator method is used to express the kinetic cross sections in terms of memory effects for the interacting observables.
The scattering matrix is used to relate particle states before and after undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory. When quantum theory got developed, the classical descriptions for scattering were no longer sufficient and other methods began to develop. One of them being the S-matrix. In this thesis we present assumptions made in order to define the S-matrix, as well as its different properties and some one-dimensional examples. The definition of the S-matrix relates to the probability amplitude for different outcomes in a scattering experiment and the elements of the matrix are called scattering amplitudes. The S-matrix is unitary and if the scattering potential V (x) is real, it is also time-reversal invariant. In one dimension it is showcased how particles can transmit through or reflect off of a potential as well as how the symmetry of the potential effects the S-matrix. Lastly we give some examples of the S-matrix in one dimension and we end with a brief outlook of how it is defined and applied in quantum field theory where special relativity is taken into account.