Franklin E. Schroeck

Quantum Mechanics on Phase Space

Preface. I: Basic quantum theory and the necessity for its revision. I.1. Classical mechanics of particles and fluids. I.2. Structure of a physical model: state, property (observable), measurement. I.3. Quantum mechanics of a (non-relativistic spinless) particle. I.4. On the connection between classical mechanics and quantum mechanics. I.5. Mathematical appendix. II: Basic experiments suggest generalizing quantum mechanics. II.1. Quantum mechanical descriptions of an experiment. II.2. Capture on a screen, in a bubble chamber, gel, cloud chamber. II.3. The Stern-Gerlach experiment. II.4. Crossed polarizers. II.5. Single slit experiments and inapplicability of Heisenberg uncertainty relations. II.6. Spontaneous decay, Breit-Wigner (Cauchy) distributions, and the inapplicability of Heisenberg type uncertainty relations. II.7. Interferometers. II.8. Imaging processes and signal analysis. II.9. Sensory perception and neuroscience. II.10. Five other subjects and their implications. II.11. Mathematical appendix. III: Construction of quantum mechanics on phase space. III.1. Group representation theory. III.2. The Heisenberg group (Weyl algebra) and the Affine group. III.3. Representations of the Galilei group. III.4. Representations of the Poincare group. III.5. Remarks on the de Sitter group. IV: Consequences of formulating quantum mechanics on phase space. IV.1. The quantum/classical connection. IV.2. Quantum field theory. IV.3. Spring cleaningin the house of quantum mechanics. IV.4. Reprise: Expanding the realm of application of quantum mechanics. IV.5. A discrete (lattice) quantum universe, and computability. V: Foundational aspects. V.1. Relation to generalized quantum logic. V.2. P.O.V.M.s arising on operational manuals. V.3. Relation to quantum mechanical measurement theory. V.4. Philosophical and other foundational aspects. References. Index.

On the reality of spin and helicity

The possibilities of a realistic interpretation of quantum mechanics are investigated by means of a statistical analysis of experiments performed on the simplest type of quantum systems carrying spin or helicity. To this end, fundamental experiments, some new, for measuring polarization are reviewed and (re)analyzed. Theunsharp reality of spin is essential in the interpretation of some of these experiments and represents a natural motivation for recent generalizations of quantum mechanics to a theory incorporating effect-valued measures as unsharp observables and generalized systems of imprimitivity.

Coexistence of observables

Recent results in the theory of integration of complex-valued functions with respect to a positive operator-valued measure are used to generalize the usual notion of coexistent observables. This leads to a connection between effects as observables and the quantization scheme of stochastic quantum mechanics. It also leads to a new viewpoint for the concept of a “classical apparatus” for quantum measurement which does not require a classical mechanical treatment of the apparatus from the outset.

Stochastic quantum mechanics viewed from the language of manuals

The language of manuals may be used to discuss inference in measurement in a general experimental context. Specializing to the context of the frame manual for Hilbert space, this inference leads to state dominance of the inferred state from partial measurements; this in turn, by Sakais theorem, determines observables which are described by positive operator-valued measures. Symmetries are then introduced, showing that systems of covariance, rather than systems of imprimitivity, are natural objects to study in quantum mechanics. Experiments measuring different polarization components simultaneously are reexamined in this language. Finally, implications of the Naimark extension theorem for the manual approach are investigated.

Quantum mechanics on phase space and teleportation

The formalism of quantum mechanics on phase space is used to describe the standard protocol of quantum teleportation with continuous variables in order to partially investigate the interplay between this formalism and quantum information. Instead of the Wigner quasi-probability distributions used in the standard protocol, we use positive definite true probability densities which account for unsharp measurements through a proper wave function representing a non-ideal quantum measuring device. This is based on a result of Schroeck and may be taken on any relativistic or nonrelativistic phase space. The obtained formula is similar to a known formula in quantum optics, but contains the effect of the measuring device. It has been applied in three cases. In the first case, the two measuring devices, corresponding to the two entangled parts shared by Alice and Bob, are not entangled and described by two identical Gaussian wave functions with respect to the Heisenberg group. They lead to a probability density identical to the

Perspectives: Quantum Mechanics on Phase Space

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Bloch's paradox does not appear in quantum mechanics on phase space

Q function which is analyzed and compared with the Wigner formalism. A new expression of the teleportation fidelity