In this thesis we consider the problem of calculating the best possible bounds on integrals of quasidensity functions over an arbitrary subregion of phase space. The setting is the phase space theory of quantum mechanics for systems with one linear degree of freedom, although extensions to N degrees of freedom are also discussed. In particular, it is the problem of bounding the integral of arbitrary Wigner functions over a specified subregion S of the phase plane that is studied here. In solving this problem a class of quantum observables is introduced which can be considered as quantisations of the region, and these observables are referred to as region operators. It can then be shown that the integral of a Wigner function over S, which we refer to as a quasiprobability integral (qpi) is bounded by the infimum and supremum of the spectrum of the corresponding region operator.
This thesis begins with some introductory comments in Chapter 1 followed by a review of the relevant aspects of the phase space theory of quantum mechanics, which appears in Chapter 2. This background is then used to develop the theory of quasiprobability integrals in Chapter 3. Two realisations of the spectral problem for region operators are considered, and we show that they lead to a Fredholm integral equation of the second kind or, alternatively, an infinite-dimensional matrix eigenvalue problem on the basis of states formed from the eigenstates of the simple harmonic oscillator. Extensions of these results to quantum systems with N-degrees of freedom, and to the problem of bounding integrals of smoothed S-parameterised quasidensity functions over S are also presented.
In Chapter 4 through 6, these results are then applied to a number of specific classes of subregions, for which the
best-possible bounds on qpis of Wigner functions are calculated by using a combination of analytical and computational techniques. In Chapter 4, computational techniques are used to calculate the bounds on qpis over subregions of the phase plane enclosed by regular polygons and elliptical sectors and also the non-compact family of subregions bounded by infinite wedges.
It is demonstrated that the symmetries of a subregion can be used to simplify the corresponding region operator and, in Chapter 5, it is shown that in some cases this leads to an exact solution of the spectral problem for the associated region operator. These 'special' subregions are related to conic sections and the exact solutions arise because these subregions are invariant under subgroups of the group of linear canonical transformations of the phase plane. For the particular case in which the subregion is formed by the interior of an ellipse, the bounds on qpis can be determined exactly for
all values of the area.
Examples of 2 N-dimensional subregions are also considered, and in Chapter 6 exact results for the bounds on qpis over the interior of a 2 N-dimensional hypersphere are derived. Extensions of the results for Wigner functions to s-parameterised quasidensities are also discussed for circular discs and subregions bounded by pairs of parallel lines in the phase plane.
Some further generalisations are discussed in the conclusion: in particular to quantum spin systems and to the problem of bounding integrals of quasidensity functions over curves in the phase-plane. The possibility of connecting the bounds on different kinds of regions or of similar regions on differing spaces via a process of contraction is also considered.