We give a brief review of geometric quantisation up to and including the Blattner-Kostant-Sternberg kernal. In general this leads to symmetric operators that are not essentially self-adjoint so motivating a study of Hermitian operators as observables in a generalised quantum mechanics. We show that a generalised squaring axiom can reproduce the results of Blattner-Kostant-Sternberg quantisation. We also show that quantisation with respect to polarisations with compact leaves gives results that conflict with the nonlocal nature of quantum mechanics. We develop a front form quantum mechanics of a free scalar particle using geometric quantisation. The front and instant forms are related via unitary maps derived from the pairing which intertwines quantisations with respect to the forms. The front form position operator has a maximally symmetric component so we are compelled to work within the framework of a generalised quantum mechanics; the result in there being no Hegerfeldt type instantaneous spreading of initially localised wavefunctions in the front form. Finally we show that this model can be lifted to a many particle free field theory.
We give a brief review of geometric quantisation up to and including the Blattner-Kostant-Sternberg kernal. In general this leads to symmetric operators that are not essentially self-adjoint so motivating a study of Hermitian operators as observables in a generalised quantum mechanics. We show that a generalised squaring axiom can reproduce the results of Blattner-Kostant-Sternberg quantisation. We also show that quantisation with respect to polarisations with compact leaves gives results that conflict with the nonlocal nature of quantum mechanics. We develop a front form quantum mechanics of a free scalar particle using geometric quantisation. The front and instant forms are related via unitary maps derived from the pairing which intertwines quantisations with respect to the forms. The front form position operator has a maximally symmetric component so we are compelled to work within the framework of a generalised quantum mechanics; the result in there being no Hegerfeldt type instantaneous spreading of initially localised wavefunctions in the front form. Finally we show that this model can be lifted to a many particle free field theory.
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
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.
Mackey's scheme (1963) for the quantisation of classical momenta generating complete vector fields (complete momenta) is introduced, the differential operators corresponding to these momenta are introduced and discussed, and an isomorphism is shown to exist between the subclass of first-order self-adjoint differential operators, whose symmetric restrictions are essentially self-adjoint, and the complete classical momenta. Difficulties in the quantisation of incomplete momenta are discussed, and a critique given. Finally, in an attempt to relate the concept of completeness to measurability, concepts of classical and quantum global measurability are introduced, and shown to require completeness. These results afford strong physical insight into the nature of complete momenta, and lead us to suggest a quantisability condition based upon global measurability.
A formulation of quantum mechanics with additive and multiplicative (q-)difference operators instead of differential operators is studied from first principles. Borel-quantisation on smooth configuration spaces is used as guiding quantisation method. After a short discussion this method is translated step-by-step to a framework based on difference operators. To restrict the resulting plethora of possible quantisations additional assumptions motivated by simplicity and plausibility are required. Multiplicative difference operators and the corresponding q-Borel kinematics are given on the circle and its N-point discretisation; the connection to q-deformations of the Witt algebra is discussed. For a "natural" choice of the q-kinematics a corresponding q-difference evolution equation is obtained. This study shows general difficulties for a generalisation of a physical theory from a known one to a "new" framework.
We present a mathematical formalism of non-Hermitian quantum mechanics, following the Dirac-von Neumann formalism of quantum mechanics. In this formalism, the state postulate is the same as in the Dirac-von Neumann formalism, but the observable postulate should be changed to include para-Hermitian operators (spectral operators of scalar type with real spectrum) representing observable, as such both the measurement postulate and the evolution postulate must be modified accordingly. This is based on a Stone type theorem as proved here that the dynamics of non-Hermitian quantum systems is governed by para-unitary time evolution. The Born formula on the expectation of an observable at a certain state is given in the non-Hermitian setting, which is proved to be equal to the usual Born rule for every Hermitian observable, but for a non-Hermitian one it may depend on measurement via the choice of a metric operator associated with the non-Hermitian observable under measurement. Our formalism is nether Hamiltonian-dependent nor basis-dependent, but can recover both PT-symmetric and biorthogonal quantum mechanics, and it reduces to the Dirac-von Neumann formalism of quantum mechanics in the Hermitian setting. As application, we study observable-geometric phases for non-Hermitian quantum systems.
Abstract In this chapter we present the mathematical framework of describing wave functions as vectors in a Hilbert space and observables as Hermitian operators on such a vector space. We describe the mathematical formulation of these ideas for finite and infinite-dimensional cases via the Dirac notation, and discuss the example of the Hilbert space of photon polarization. Important classes of linear operators on Hilbert spaces are discussed, including projectors, Hermitian and self-adjoint operators, as well as unitary operators. As an important centre piece for the mathematical language of quantum mechanics we present the spectral theorem. We discuss the subtleties of this framework in infinite dimensions, giving a brief overview of the distributional aspects of quantum mechanics and the conceptually important problem of finding self-adjoint extensions via the Weyl-von Neumann theorem