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Physical Review

Presymmetry. II

H. Ekstein

Published Aug 25, 1969• 45 citations
Publisher

Abstract

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 $\mathcal{O}$ and a convex linear set $\mathcal{S}$, respectively. There is a many-one mapping from $\mathcal{O}$ into the space $\mathfrak{U}$ 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 $\mathfrak{U}$. 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.

DOI
10.1103/physrev.184.1315

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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.
Abstract Hilbert space may be regarded as a convenient standard approximation by interpolation and extrapolation to a unitary space, in general decomposable into a sum of semisimple spaces. In the limit we expect a particle theory expanded in powers of h according to the number of interacting particles. In the classical limit h tending to zero, phase space, with equations of motion reducible to Hamiltonian form, replaces Hilbert space. The modification of states by observing them is taken care of by considering probability distributions of petty ensembles. If the equations for any single observer can be made autonomous by replacing empirical time by universal time with an arrow we have a causal system. We then obtain the relation between probability and negentropy required for the second law of thermodynamics. An approximate Newtonian theory provides proximate particles with internal and external variables and admits the Poincaré group. For the internal variables we have approximately the Breit interaction. For the external variables we have equivalence for observers of the homogeneous Lorentz group of relativity. We introduce grand ensembles of ultimate particles, and nebulae as proximate particles in the large. We assume the Einstein principle of equivalence for the ten parameter set of observers suggested by relativity and suppose the second law of thermodynamics holds for each observer. The Einstein law of gravitation follows in classical theory to the order of the reciprocal of the large constant, in general with positive natural curvature as well as that corresponding to mass. Replacing particle interactions by fields we include them in classical theory.
Linear algebra is the natural mathematical language of quantum mechanics. For this reason, the present chapter starts with a review of Hilbert spaces for vectors and operators. We introduce vector and Hilbert spaces along with inner products and metrics. The Dirac notation is developed for the Euclidean vector spaces as a starting point for the concepts of complete orthonormal sets of vectors, closure, dual vector spaces, and adjoint operators. The Dirac delta function in various forms and the principal part is introduced in Appendix 5 as an essential tool. The chapter turns to the main use of Dirac notation for function spaces; the concepts of norm, inner product, and closure are discussed. Fourier, Cosine, and Sine series are discussed as examples of expansions in complete orthonormal sets of functions. Although Hilbert spaces are interesting mathematical objects with important physical applications, the study of linear algebra remains incomplete without a study of linear operators (i.e., linear transformations). In fact, the set of linear transformations itself forms a vector space and therefore has a basis set. The basis set for the operator is linked with the basis sets for the spaces that it operates between. The linear operator can be discussed as an abstract operator or through an isomorphism as a matrix or as a generalized expansion in operator space. A Hermitian (a.k.a., self-adjoint) operator produces a basis set within a Hilbert space. The basis set comes from the eigenvector equation for the particular operator. The fact that a Hermitian operator produces a complete set (of orthornomal vectors) has special importance for quantum mechanics. Observables such as energy or momentum correspond to Hermitian operators. Complete sets make it possible to represent every possible result of a measurement of the observable by an object (vector) in the theory. The Hermitian operators have real eigenvalues which represent the results of the measurement.