How to Quantize Phases and Moduli!

A typical classical interference pattern of two waves with intensities I_1, I_2 and relative phase phi = phi_2-phi_1 may be characterized by the 3 observables p = sqrt{I_1 I_2}, p cos\phi and -p sin\phi. They are, e.g. the starting point for the semi-classical operational approach by Noh, Fougeres and Mandel (NFM) to the old and notorious phase problem in quantum optics. Following a recent group theoretical quantization of the symplectic space S = {(phi in R mod 2pi, p > 0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper applies those results to that controversial problem of quantizing moduli and phases of complex numbers: The Poisson brackets of the classical observables p cos\phi, -p sin\phi and p > 0 form the Lie algebra of the group SO(1,2). The corresponding self-adjoint generators K_1, K_2 and K_3 of that group may be obtained from its irreducible unitary representations. For the positive discrete series the modulus operator K_3 has the spectrum {k+n, n = 0, 1,2,...; k > 0}. Self-adjoint operators for cos phi and sin phi can be defined as ((1/K_3)K_1 + K_1/K_3)/2 and -((1/K_3)K_2 + K_2/K_3)/2 which have the theoretically desired properties for k > or = 0.5. The approach advocated here solves, e.g. the modulus-phase quantization problem for the harmonic oscillator and appears to provide a full quantum theoretical basis for the NFM-formalism.Comment: 27 pages, Latex; Refs. added, additional remarks on the harmonic oscillato

Wigner functions for angle and orbital angular momentum: Operators and dynamics

Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces S^1 x R, i.e. for the canonical pair angle and angular momentum, was presented (arXiv:1601.02520), main properties of those functions derived, discussed and their usefulness illustrated by examples. The present paper is a continuation which compares properties of the new Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions on planar ones in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase space functions is discussed. The star product formalism is shown to be completely implementable. In addition basic dynamical laws for the new Wigner and Moyal functions are derived as generalized quantum Liouville and energy equations. They are very similar to those of the planar case, but also show characteristic differences.Comment: 14 pages, continuation of paper Phys. Rev. A 94, 062113 (2016

Canonical Quantum Statistics of Schwarzschild Black Holes and Ising Droplet Nucleation

Recently is was shown that the imaginary part of the canonical partition function of Schwarzschild black holes with an energy spectrum E_n = \sigma \sqrt{n} E_P, n= 1,2, ..., has properties which - naively interpreted - leads to the expected unusual thermodynamical properties of such black holes (Hawking temperature, Bekenstein-Hawking entropy etc). The present paper interprets the same imaginary part in the framework of droplet nucleation theory in which the rate of transition from a metastable state to a stable one is proportional to the imaginary part of the canonical partition function. The conclusions concerning the emerging thermodynamics of black holes are essentially the same as before. The partition function for black holes with the above spectrum was calculated exactly recently. It is the same as that of the primitive Ising droplet model for nucleation in 1st-order phase transitions in 2 dimensions. Thus one might learn about the quantum statistics of black holes by studying that Ising model, the exact complex free energy of which is presented here for negative magnetic fields, too.Comment: 17 pages, LateX ; a brief note added indicating the generalization of a key result from 3+1 to d+1 dimensions; paper to appear in Physics Letters

Exact Partition Functions for the Primitive Droplet Nucleation Model in 2 and 3 Dimensions

The grand canonical partition functions for primitive droplet nucleation models with an excess energy epsilon_n = - mu n + sigma n^{1-eta}, eta = 1/d, for droplets of n constituents in d dimensions are calculated exacly in closed form in the cases d=2 and 3 for all (complex) mu by exploiting the fact that the partition functions obey simple PDE.Comment: 10 pages, Latex; References and a cross-check of the main result - unchanged - added, misprints correcte

Symmetric States in Quantum Geometry

Symmetric states are defined in the kinematical sector of loop quantum gravity and applied to spherical symmetry and homogeneity. Consequences for the physics of black holes and cosmology are discussed.Comment: 9 pages, talk at the Ninth Marcel Grossmann Meeting, Rome, July 2-8, 200

Quantum Symmetry Reduction for Diffeomorphism Invariant Theories of Connections

Given a symmetry group acting on a principal fibre bundle, symmetric states of the quantum theory of a diffeomorphism invariant theory of connections on this fibre bundle are defined. These symmetric states, equipped with a scalar product derived from the Ashtekar-Lewandowski measure for loop quantum gravity, form a Hilbert space of their own. Restriction to this Hilbert space yields a quantum symmetry reduction procedure in the framework of spin network states the structure of which is analyzed in detail. Three illustrating examples are discussed: Reduction of 3+1 to 2+1 dimensional quantum gravity, spherically symmetric electromagnetism and spherically symmetric gravity