Abstract
The Weyl correspondence and the related Wigner formalism lie at the core of traditional quantum mechanics. We discuss here an alternative quantization scheme, the idea of which goes back to Born and Jordan, and which has recently been revived in another context, namely time–frequency analysis. We show in particular that the uncertainty principle does not enjoy full symplectic covariance properties in the Born and Jordan scheme, as opposed to what happens in the Weyl quantization.
Highlights
The problem of “quantization” of an “observable” harks back to the early days of quantum theory; mathematically speaking, and to use a modern language, it is the problem of assigning to a symbol a pseudo-differential operator in a way which is consistent with certain requirements
We show that in particular the uncertainty principle does not enjoy full symplectic covariance properties in the Born and Jordan scheme, as opposed to what happens in the Weyl quantization
The first is widely used in the theory of partial differential equations and in time-frequency analysis, the second is the traditional quantization used in quantum mechanics
Summary
The problem of “quantization” of an “observable” harks back to the early days of quantum theory; mathematically speaking, and to use a modern language, it is the problem of assigning to a symbol a pseudo-differential operator in a way which is consistent with certain requirements (symmetries under a group of transformations, positivity, etc.). The first is widely used in the theory of partial differential equations and in time-frequency analysis (mainly for numerical reasons), the second is the traditional quantization used in quantum mechanics Both are particular cases of Shubin’s pseudo-differential calculus, where one can associate to a given symbol a an infinite family (Aτ )τ of pseudo-differential operators parametrized by a real number τ, the cases τ. Φ = a, Wigτ (ψ, φ) and this observation has recently been used by researchers in time-frequency analysis to obtain more realistic phase-space distributions (more about this in the discussion at the end of the paper) They went one step further by introducing a new distribution by averaging Wigτ for the values of τ in the interval [0, 1]. We denote by S(R2n) the Schwartz space of rapidly decreasing smooth functions and by S′(R2n) its dual (the tempered distributions)
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