Daniel A. Dubin

Mathematical aspects of Weyl quantization and phase

Part 1 Fundamentals: some remarks on classical mechanics the bounded model the smooth model representations of the CCR probability in quantum mechanics dynamical systems Weyl quantization. Part Quantization and phase: quantization in polar co-ordinates phase operators the laser model Weyl dequantization the moyal product ordered quantization asymptotics measurements.

The Weyl Quantization of Phase Angle

We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.

On representing observables in quantum mechanics

There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact poses a serious question on the physical meaning of such a self-adjoint operator and its associated operator measures.

Thermal states of the vector meson model in two dimensions

Abstract We compute the KMS states for the vector meson model in two-dimensional space-time. We also consider their limit to those of the Thirring model.

The pointwise product in Weyl quantization

We study the ⊙-product of Bracken [1], which is the Weyl quantized version of the pointwise product of functions in phase space. We prove that it is not compatible with the algebras of finite rank and Hilbert–Schmidt operators. By solving the linearization problem for the special Hermite functions, we are able to express the ⊙-product in terms of the component operators, mediated by the linearization coefficients. This is applied to finite rank operators and their matrices, and operators whose symbols are radial and angular distributions.

The meaning of phase in the Dicke laser model

Abstract We find that of three phase operator models, only the Weyl quantization of the classical phase angle relates directly to the statistical mechanics treatment of a basic laser model.