Fundamental phase space formula for the similitude group.

In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, Sim(2). This formula takes the form ∫ a(Y)P(Y)d(Y) = 〈Â〉, where Y is the phase space variable and 〈Â〉 is a linear operator on Hilbert space representing a quantum dynamical observable. 〈Â〉 is the quantum expected value of the observable in a state of the system. The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators. The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution. The formula is intimately related to the quantization-dequantization problem a(Y) ⟷ Â which assigns a quantum operator to the classical phase space function a(Y). [ABSTRACT FROM AUTHOR]