Algebras of distributions suitable for phase-space quantum mechanics. I

Abstract

Phasespace approach to relativistic quantum mechanics. I. Coherentstate representation for massive scalar particlesThe twisted product off unctions on R2N is extended to a *-algebra of tempered distributions that contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.

Harmonic oscillator in twisted Moyal plane: Eigenvalue problem and relevant properties

“…Proof: The results are immediate by induction, performing similar analysis as in [8] to construct the right states f R m0 such thatā ⋆ f R m0 = θ(m + 1)f R m+1,0 . Similarly, the study of the ho left states provides the following result.…”

A Note on Q-algebras and Quantization

“…We define an idempotent element p = 2 n exp(− |y| 2 ) in (O C (T x X), * ), where | · | denotes the riemannian metric. It is easy to check that p * p = p , and according to [7], Equation (27), p * ȳ i = 0 for 1 ≤ i ≤ n. Theorem 4, [7] proves that p * O C is a subspace of S(R 2n ).…”

“…We find that the proposed algebra is hard to work with in the case of noncommutative tori. Gracia-Bondia and Várilly [7] studied the largest possible * -subalgebra of tempered distributions on R 2n where the "twisted product"(convolution product) is defined. This * -subalgebra is invariant under Fourier transform and is very "big", and contains rapidly decreasing smooth functions, distributions of compact support and all polynomials, etc.…”

Non-commutative Renormalization

“…A representation of the algebra A Θ is given by some set of functions on R d equipped with a non-commutative product: the GroenwaldMoyal product. What follows is based on [88].…”

“…In this basis, the Moyal product becomes a simple matrix product. Each field is then represented by an infinite matrix [88,39,95].…”