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Locality in non-relativistic and relativistic quantum mechanics

D. R. E. Timpson

Published Jan 1, 1987• 0 citations
OpenAlex

Abstract

This thesis investigates the description of localised states and observables in quantum mechanics.Motivated by the fact that actual experimental apparatus is of only finite size, a method of localising any observable ("selfadjoint operator on Hilbert space) is introduced, which is applicable to both bounded and unbounded operators.This method makes use of objects called localising isometries, which also prove useful in modifying the evolution of non-relativistic particles, so that they remain well localised at all times: this modification does not jeopardise the well tested predictions of standard quantum mechanics.It is mathematically more convenient when describing systems of well separated identical particles.Pairs of correlated particles, such as those considered in EPR-type thought experiments, undergo chronological disordering at large times, and the analysis of this phenomenon is used to argue physically in favour of asymptotic algebras, which do not contain two-particle observables at infinite times.The second part of the thesis investigates the description of the position of a spinless relativistic particle.Existing Lorentz covariant currents are non-local, i.e. they describe particles which are not strictly localised in a bounded region of space, and these currents are modified so as to describe strictly localised particles, which evolve causally.vi

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This thesis investigates the description of localised states and observables in quantum mechanics.Motivated by the fact that actual experimental apparatus is of only finite size, a method of localising any observable ("selfadjoint operator on Hilbert space) is introduced, which is applicable to both bounded and unbounded operators.This method makes use of objects called localising isometries, which also prove useful in modifying the evolution of non-relativistic particles, so that they remain well localised at all times: this modification does not jeopardise the well tested predictions of standard quantum mechanics.It is mathematically more convenient when describing systems of well separated identical particles.Pairs of correlated particles, such as those considered in EPR-type thought experiments, undergo chronological disordering at large times, and the analysis of this phenomenon is used to argue physically in favour of asymptotic algebras, which do not contain two-particle observables at infinite times.The second part of the thesis investigates the description of the position of a spinless relativistic particle.Existing Lorentz covariant currents are non-local, i.e. they describe particles which are not strictly localised in a bounded region of space, and these currents are modified so as to describe strictly localised particles, which evolve causally.vi
The concept of position is a basic one in our construction of a model of thenphysical world: yet the theory of localisation in quantum mechanics is fraughtnwith difficulties, especially in the relativistic case. A strictly localised state forna nonrelativistic particle cannot be described by an ordinary function. In thenrelativistic case the problems are manifold. For a free electron, using the Diracnposition operator x, we cannot construct positive-energy states which localisenthe particle within a compact region. Furthermore it has been suggested that anconsistent one-particle description of localisation on a scale small compared tonthe Compton wavelength of the particle is ruled out because pair-productionnwould take place. These matters are dicussed at greater length in Chaptern1, where several proposed localisation schemes (that is, sets of definitions ofnlocalised states and position operators) are reviewed: none of these is deemednwholly satisfactory. Introductory remarks are also made in this chapter aboutnthe remarkable phenomenon of 'probability backflow', which is discussed further below, and which is relevant to our discussion of localisation.In Chapter 2 we show that while strict localisation of the free Dirac particle is not describable within the usual mathematical formalism, in terms of Dirac's position operator x it is possible to describe sequences of positive-energy states whose spread Dx = ((x -- x0)2) about a point X0 approaches zero.In Chapter 3 the component states of these sequences are used as the basis of a full causally well-behaved solution to the localisation problem for the free electron, with natural space-time transformation properties. It is shown that although x is not an observable in the usual sense, the four-vector density (p(x, t), j(x, t)/c) is observable, and can be localised arbitrarily precisely about any point in space, at any instant of time, using only positive-energy states. A suitable spin operator can be diagonalised at the same time.n n
This thesis explores the possibility of an algebraic formulation of non-relativistic quantum theory in which certain paradoxes associated with non-locality may be resolved. It is shown that the localisation of a free quantum mechanical wave function at large time coincides approximately with the localisation of an ensemble of classical particles having the same momentum range. This result is used to give a formal definition of spatially separating states and spatially separating particles. We then study certain C*-algebras on which expectation values converge in an infinite time limit. By considering such algebras which contain local observables it is possible to introduce states at infinity as limits of states described by wave functions. In such a state at infinity there is zero probability of a position measurement finding the system in any bounded region in configuration space. It is shown that a C*-algebra exists on which any coherent superposition of spatially separating states will converge in an infinite time limit to a mixture of disjoint states. This allows us to obtain an asymptotic resolution of de Broglie's paradox and the Einstein, Podolsy and Rosen paradox. These results are obtained for the simplest types of quantum systems i.e. a one particle system without spin having configuration space IRⁿ and a system consisting of two such particles which may be distinguished from each other.
Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, with the introduction of Time and Space operators; and with (iii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a "chronon". [This chapter is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk]. PACS numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 03.65.Yz; 11.10.St; 11.10.-z; 11.90.+t; 02.00.00; 03.00.00; 24.10.Ht; 03.65.Yz; 21.60.-u; 11.10.Ef; 03.65.Fd; 02.40.Dr; 98.80.Jk. Keywords: time operator, space-time operator, non-selfadjoint operators, non-hermitian operators, bilinear operators, time operator for discrete energy spectra, time-energy uncertainty relations, Klein-Gordon equation, chronon, quantum dissipation, decoherence, nuclear optical model, cosmology, projective relativity.
It has been claimed that quantum mechanics is a non-local theory, i.e., that the predictions of quantum mechanics are at odds with a conception of the world where all the interactions are mediated by local mechanisms. The cited reason is the violation of the Bell inequality in cases where measurements are performed on quantum systems of particles that have interacted in the past. Quantum mechanics predicts that the state of the particles in such a system are correlated, a result obtained from considering a thought-experiment often known as EPR. In this thesis, I will argue that the violation of the Bell inequality in EPR forces us to either accept that causes sometimes act both backwards and forwards in time, or give up some widely-held assumptions about causal relations or the role of causal explanations. First, I will present three principles regarding causality that I take be widely accepted. I will then present the background for the claim the quantum mechanics is a non-local theory, and discuss the implications of non-local causal influences in light of the Special Theory of Relativity. One argument against a causal explanation of the EPR-correlations will be discussed, and different models for the causal structure of EPR will be assessed. I will argue that a retrocausal model – models that include instances of retrocausation, where the direction of causation is the opposite of the direction of time – is the only viable option if we want to uphold the three principles presented in chapter 1. I will discuss the plausibility of rejecting one of these principles, and argue that at least one of the can be restricted in a way that still accounts for our commitment to it. Finally, I will assess the non-local models against the retrocausal models, and argue that a retrocausal model offers interpretational advantages over the non-local alternatives.
A type of mechanics will be presented that possesses some distinctive properties. On the one hand, its physical description & rules of operation are readily comprehensible & intuitively clear. On the other, it fully satisfies all observable predictions of non-relativistic quantum mechanics. Within it, particles exist at points in space, follow continuous, piecewise differentiable paths, and their linear momentum is equal to their mass times their velocity along their path. Yet the probabilities for position and momentum, conditioned on the state of the particle's environment, follow the rules of quantum theory. Indeed, all observable consequences of quantum theory are satisfied; particles can be entangled, have intrinsic spin, this spin is not local to the particle, particle identity can effect probabilities, and so forth. All the rules of quantum mechanics are obeyed, and all arise in a straightforward fashion. After this is established, connections will be drawn out between this type of mechanics and other types of quantum worlds; those that obey Bohmian mechanics, stochastic mechanics, the many worlds interpretation, and physical collapse. In the final section, a relativistic version of the mechanics will be presented.