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

Presymmetry of Classical Relativistic Fields

Y. Avishai, H. Ekstein

Published Feb 15, 1973• 25 citations
Publisher

Abstract

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 $\mathcal{O}$ of observation procedures, it induces automorphisms, not of $\mathcal{O}$ but of a point subalgebra ${\mathcal{O}}_{x}\ensuremath{\subset}\mathcal{O}$ that is associated to a single point $x$. 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 $\mathcal{O}$ in which local subsets associated to a single point are proper (although unbounded) subalgebras, whose automorphisms are discussed.

DOI
10.1103/physrevd.7.983

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1 online resource (PDF, page 403-432)

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The physical act of accelerating observation instruments has an obvious counterpart in Newtonian physics: It is the transformation induced by (x\ensuremath→,t)\ensuremath→(x\ensuremath→+12a\ensuremath→t2,t)(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}, t)\ensuremath{\rightarrow}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}}+\frac{1}{2}\stackrel{\ensuremath{\rightarrow}}{\mathrm{a}}{t}^{2}, t)(x\ensuremath→,t)\ensuremath→(x\ensuremath→+21​a\ensuremath→t2,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 Ox\ensuremath⊂O{\mathcal{O}}_{x}\ensuremath{\subset}\mathcal{O}Ox​\ensuremath⊂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⋃(x∣x0=0)Oxc{\ensuremath{\bigcup}}_{(x|{x}_{0}=0)}{\mathcal{O}}_{\mathrm{xc}}\ensuremath⋃(x∣x0​=0)​Oxc​ 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.
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 laws of physics were not handed down from above. Neither are they rules somehow built into the structure of the universe. They are ingredients of the models that physicists invent to describe observations. Rather than being restrictions on the behavior of matter, the laws of physics are restrictions on the behavior of physicists. If the models of physics are to describe observations based on an objective reality, then those models cannot depend on the point of view of the observer. This suggests a principle of point-of-view invariance that is equivalent to the principle of covariance when applied to space-time. As Noether showed, space-time symmetries lead to the principles of energy, linear momentum, and angular momentum conservation--essentially all of classical mechanics. It also leads to Lorentz invariance and special relativity. When generalized to the abstract space of functions such as the quantum state vector, point-of-view invariance is identified with gauge invariance. Quantum mechanics is then just the mathematics of gauge transformations with no additional assumptions needed to obtain its rules, including the superposition and uncertainty principles. The conservation and quantization of electric charge follow from global gauge invariance. The electromagnetic force is introduced to preserve local gauge invariance. Although not discussed here, the other forces in the standard model of elementary particles are also fields introduced to preserve local gauge invariance. Gravity can also be viewed as such a field. Thus practically all of fundamental physics as we know it follows directly from the single principle of point-of-view invariance.
The description of relativistic effects requires a preliminary definition of events localised in space-time while the clocks used for time definition and the fields used in synchronisation or localisation procedures are necessarily quantum systems. We outline an algebraic framework where basic requirements of quantum theory and relativity are consistently dealt with. This approach which may be termed as a `quantum relativity' is built on the algebra of symmetries. It contains the definition of quantum observables associated with Einstein localisation in space-time as well as the evaluation of their commutation relations including the energy-time relation. These commutators also describe the relativistic shifts undergone under frame transformations, not only for inertial frames but also for uniformly accelerated frames. The quantum redshift laws differ from their classical counterparts while still obeying universal metric properties. The paper presents this approach in terms closely connected to the seminal Einstein conception of localisation in space-time and refers to already published papers for more technical developments.
Even in the presence of external fields, space-time symmetry implies nontrivial relations between observables at one time, i.e., kinematical relations. Symmetry operations at one time---translations, rotations, and (for Galilei symmetry) velocity shifts---can be performed on observation-producing and on state-producing instruments, regardless of the existence of an external field. Furthermore, it is possible to give an operational definition of every initial state intrinsically, i.e., regardless of the external field. The precise statement of this empirical fact explains, for example, why a particle in an external field has integral or half-integral eigenvalues of the spin, why a Hamiltonian exists even in the presence of a time-dependent external field, and why (for Galilei symmetry) the canonical commutation relations are still valid, although the full space-time symmetry from which these results can be derived has been destroyed. It is pointed out that the rigorous validity of kinematical relations, in spite of strong breaking of the underlying space-time symmetry, is analogous to the rigorous validity of equal-time current commutation rules, in spite of the breaking of the underlying U(3)U(3)U(3) symmetry.
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 $H{P}_{\mathrm{ac}}=\frac{{|{\mathrm{P}}^{\ensuremath{'}}|}^{2}}{2\ensuremath{\mu}}$. 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 $\ensuremath{\mu}$ 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 ${r}^{\ensuremath{-}\ensuremath{\epsilon}}$, for any $\ensuremath{\epsilon}>0$, provided that the potential is repulsive at large separations between the particles.