On Hansen’s Version of Spectral Theory and the Moyal Product

The story of the Moyal product begins in 1927 with Weyl’s introduction of the unitary operators associated with position and momentum in the Schrödinger representation, and more particularly with the collection of unitary operators W (a, b) = exp i(aP + bQ) (a, b ∈ R) [27, 28]. If f ∈ L(E) is an integrable function on phase space E = R, the operator W (f) = ∫ f(x)W (x) dx can be defined. In [18, 19] von Neumann gave an explicit formula for the product f ◦ g defined by a W (f ◦ g) = W (f)W (g) for f, g ∈ L(E). This product f ◦ g is now known as the twisted convolution of f and g. Then the formula Δ[f ] = 1 2π (Ff) defines the Weyl quantization of the function f (here F denotes the Fourier transform). Associated with Weyl quantization is the Moyal product f g = 1 2πFf ◦ Fg, so that Δ[f g] = Δ[f ]Δ[g]. This formalism was further developed by Wigner in 1932, [29], who introduced what is known as the Wigner transform. Inter alia, this transform permits a more rigorous handling of the formalisms of Weyl quantization. The