John V. Corbett

The pauli problem, state reconstruction and quantum-real numbers

The Pauli problem is the uniqueness component of a prototypical quantum state reconstruction problem. We show that when the operators have continuous spectrum a commixing of unique and nonunique vector states can occur. A realistic state reconstruction problem is posed and solved using the quantum real number interpretation instead of the statistical interpretation.

About SIC POVMs and discrete Wigner distributions

A set of d2 vectors in a Hilbert space of dimension d is called equiangular if each pair of vectors encloses the same angle. The projection operators onto these vectors define a POVM which is distinguished by its high degree of symmetry. Measures of this kind are called symmetric informationally complete, or SIC POVMs for short, and could be applied for quantum state tomography. Despite its simple geometrical description, the problem of constructing SIC POVMs or even proving their existence seems to be very hard. It is our purpose to introduce two applications of discrete Wigner functions to the analysis of the problem at hand. First, we will present a method for identifying symmetries of SIC POVMs under Clifford operations. This constitutes an alternative approach to a structure described before by Zauner and Appleby. Further, a simple and geometrically motivated construction for an SIC POVM in dimensions two and three is given (which, unfortunately, allows no generalization). Even though no new structures are found, we hope that the re-formulation of the problem may prove useful for future inquiries.

Are wave functions uniquely determined by their position and momentum distributions

Abstract The problem of determining a square integrable function from both itsmodulus and the modulus of its Fourier transform is studied. It is shownthat for a large class of real functions the function is uniquely determinedfrom this data. We also construct fundamental subsets of functions that arenot uniquely determined. In quantum mechanical language, bound states areuniquely determined by their position and momentum distributions but, ingeneral, scattering states are not. 1. Introduction The Pauli problemIn a footnote to hi Handbuchs der Physik article on the general principles of wavemechanics [4], Pauli raised the question of whether the ijs(q) wav wae functios nuniquely determined by the probability densities | ^{q)f and |^(p)| 2 in configurationand momentum space. In this article we show that there are fundamental subsetswhere the answer is no and there is a large and interesting class of functions (q) eL 2 (R) for which the answer is yes.The question was motivated by the fact that if the wave function was uniquelydetermined by its position and momentum probability densities then we have, atleast in principle, a method of determining the wave function from experiment.Nevertheless, we should point out that the mathematical problem is of interest inother contexts, such as control theory and crystallography. We will commentfurther on the physical meaning o thif s problem and our results in the final section.The mathematical problem is that of determining the phase of a complex-valuedsquare integrable function given both the amplitude of the function and the

The geometry of state space

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is that of flag manifolds.

A sheaf model for intuitionistic quantum mechanics

We analyze the passing of a single particle through a double-slit apparatus. This is done using a sheaf model of intuitionistic logic. We develop an algebra of slits and study the non-classical behaviour of quantum slits. A specific interference formula is obtained by averaging. This formula extends the usual one.

Quantum mechanical measurement of non-orthogonal states and a test of non-locality

Abstract We propose a model of measurement of a non-hermitan observable whose states are described by non-orthogonal vectors. This model is used to analyse a proposal for testing non-locality in quantum mechanics using a system whose physically observable states are not mutually orthogonal.

Convergence of the Born Series

Three types of Born series which can be associated with a transition amplitude are discussed, and the criteria for the convergence of the different series are compared. With regard to the divergence of the Born series, the ordering, matrix‐element series diverges implies vector series diverges implies operator series diverges, is obtained for the natural vector and operator Born series that can be abstracted from the expression for the transition amplitude. The conclusion that the divergence of the operator Born series does not ensure that the Born series of physical matrix‐elements divergences is applied to an example of three‐body rearrangement scattering.

Entanglement and the quantum spatial continuum

The non‐locality of entangled systems provides more evidence that the spatial continuum of quantum particles is not classical. We assume that physical quantities take Dedekind real numbers in a topos for their numerical values. This means that the quantum spatial continuum is isomorphic to RD(EJ(M))3, where RD(EJ(M)) the sheaf of Dedekind real numbers in the topos Shv(EJ(M) of sheaves on the state space of the quantum system. In such a continuum, a single particle can have a quantum trajectory which passes through two classically separated slits and two particles in an entangled condition stay close to each other in their quantum space and hence Einstein locality is retained.

Particles and simple scattering theory

A necessary and sufficient condition for a pair of subspaces to be the in and out scattering subspaces for a simple scattering system is obtained. It involves the existence of certain representations of the Galilean presymmetry group on these subspaces. The physical interpretation is that, in scattering, particles remain as particles even in the presence of the interaction.

A Topos Theory Foundation for Quantum Mechanics

The theory of quantum mechanics is examined using non-standard real numbers, called quantum real numbers (qr-numbers), that are constructed from standard Hilbert space entities. Our goal is to resolve some of the paradoxical features of the standard theory by providing the physical attributes of quantum systems with numerical values that are Dedekind real numbers in the topos of sheaves on the state space of the quantum system. The measured standard real number values of a physical attribute are then obtained as constant qr-number approximations to variable qr-numbers. Considered as attributes, the spatial locations of massive quantum particles form non-classical spatial continua in which a single particle can have a quantum trajectory which passes through two classically separated slits and the two particles in the Bohm-Bell experiment stay close to each other in quantum space so that Einstein locality is retained.