The problem of localization of relativistic particle in quantum physics originates from the fact that position variable is non-Hermitian in nature. Kalney1 in a series of papers, has tried to construct a position operator for a massive as well as massless particle in a unified way incorporating Lorentz invariance. His observation of non-Hermitian and non-normal nature of position operator suggests that it gives rise to the possibility of getting more information from complex eigenvalues. Kalney and Toledo2 have considered that a relativistic particle may not be a point particle and position be extended to certain region. Non-Hermitian nature of position variable is suitable for description of such extended body as each coordinate is associated with a pair of variables.
The problem of localization of relativistic particle in quantum physics originates from the fact that position variable is non-Hermitian in nature. Kalney1 in a series of papers, has tried to construct a position operator for a massive as well as massless particle in a unified way incorporating Lorentz invariance. His observation of non-Hermitian and non-normal nature of position operator suggests that it gives rise to the possibility of getting more information from complex eigenvalues. Kalney and Toledo2 have considered that a relativistic particle may not be a point particle and position be extended to certain region. Non-Hermitian nature of position variable is suitable for description of such extended body as each coordinate is associated with a pair of variables.
The purpose of this paper is to find to what extent the notion of position in the relativistic quantum mechanics of free elementary systems of nonzero mass is defined (or precluded) by the requirement that the descriptions of a localized state as seen from two observers in different frames of reference should be physically consistent. Neither the existence of a time component of the position operator nor the compatibility of the space components of position is assumed. Hermiticity with respect to the quantum-relativistic scalar product in state-vector space is neither assumed nor rejected. There are three possibilities: (i) The position is represented as usual by a Hermitian operator; this is the point-type position. (ii) Non-Hermitian operators are accepted, and the k component of position can represent segments of the k axis with length of the order of m1; when the operator is not only non-Hermitian, but also non-normal, this is the extended-type position. (iii) Non-Hermitian operators are accepted, but the segments are reduced to points; this is the limiting case of the extended-type position. Case (ii) is not studied in this paper. General conditions are found for any spin as regards cases (i) and (iii), and their consequences are obtained for spin 0 and spin \textonehalf{}. The results are the following: There is no position operator in case (i), but in case (iii) the position operator is unique for spin 0 and unique up to a parameter for spin \textonehalf{}. For both spins, in spite of the lack of relativistic Hermiticity, the eigenvalues of position are real, and the velocity is the expected one. The eigenstates of one component of the position are found. The components of the position are compatible with each other for spin 0, and incompatible for spin \textonehalf{}. In the former case, the simultaneous eigenstates of all components are Philip's class-I localized states. Related work is discussed.
The present series of papers is devoted to the localization problem in relativistic quantum mechanics. The consequences of imposing the condition that the descriptions of a localized state seen from two different frames of reference should be physically consistent were investigated in Paper II. Only instantaneous localization was studied. Considering elementary systems of nonzero mass and spin 0 and \textonehalf{}, the following results were found. (a) As regards the localization problem, one cannot avoid accepting at least one of the following strong departures from the usual ideas: (i) Position has no meaning; (ii) it violates the physical equivalence of inertial frames; (iii) it is the only quantum variable which cannot be represented by an operator; (iv) it is non-Hermitian; (v) some unusual interaction effects do not disappear when the interaction is switched off. (b) If a component of position is to be precisely measurable as a point, then for spin 0 there is only one possible position operator; for spin \textonehalf{} the operator is unique up to a parameter; in both cases the operators are non-Hermitian with respect to the relativistically invariant scalar product, but in spite of this the eigenvalues of all components are real. The components of position are compatible with each other for spin 0 and incompatible for spin \textonehalf{}. The communication relations of position with linear momentum are the standard ones. The velocity operator, which is Hermitian, has the expected form. Several authors have stated that the localization problem can have a solution for mass greater than zero, but for the zero-mass case a solution does not exist. In this paper the localization problem is studied for zero-mass elementary systems by only imposing, as in Paper II, the above requirement of physical consistency. We consider systems with spin 0, \textonehalf{}, and 1. We prove that the consequences are essentially the same as those obtained for systems of mass greater than zero. The parameter is no longer arbitrary.
In parts II and III of the present series of papers, the localization problem was studied for the relativistic nonzero-mass (spin 0 and \textonehalf{}) and zero-mass (spin 0, \textonehalf{}, and 1) cases. With a set of postulates of which the basic one was the imposition of Lorentz invariance of localization (which is a self-consistency requirement), it was possible to define the position operator uniquely (except for a constant for spin \textonehalf{}, nonzero mass). However, the position operator has unusual properties, e.g., non-Hermiticity. This does not preclude its having physical meaning, but is such a strange result that in the present paper we test the validity of the postulates in the nonrelativistic case (spin 0 and \textonehalf{}) where the position operator is well known. We use the same set of postulates, except that here we impose the Galilean invariance of localization. The result of the test is positive; the position operator found is the usual one. Moreover, with our set of postulates, we restate the notion of localization in a more physical basis, giving a precise definition of it without imposing avoidable requirements.
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 thesis is about the peculiar probabilistic structure of quantum mechanics. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables. A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. A method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability is proposed and applied to a series of model to reobtain the quantum behaviour od a particle. In particular, using ordinary probability theory, the kinematics of a point-like particle jumping at random over a discrete random space is studied. After the removal of the random space from the model, the position and velocity of the particle do not commute, when represented as operators on the same Hilbert space. Using a suitable random process for the space one is able to recover non-relativistic quantum mechanics in a suitable limit.
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