Question

Suppose we have two operators O1 and O2 each with transforms W1:=W[O1] and W2:=W[O2] . Is there a way to express W[O1O2]

in terms of W1 and W2? Meaning, is there some way to decompose W[O1O2]=f(W1,W2)? In particular, is there a way to express
Tr[O1O2rho]=12π∫[infinity]−[infinity]dq∫[infinity]−[infinity]dp W[O1O2](q,p) W[rho](q,p) .in terms of an integral in terms of W1 and W2? (ie. something like ⋯=∬dqdp W1W2W[rho]?)

Answer 1

Quantum mechanics does not generally allow for an expression of W[O1O2] in simple terms of W1 and W2 due to the non-commutative nature of operator multiplication. The Moyal product or star product formalism is often used instead.

Step-by-step explanation:

The question relates to the field of quantum mechanics, more specifically to the Wigner-Weyl transform of operators and how they relate to each other when operators are composed. Looking for an expression for the Wigner function W[O1O2] in terms of W1 and W2, where W1 and W2 are the Wigner transforms of operators O1 and O2, respectively, is not a trivial task. In general, the product of two operators does not translate into a simple product of their Wigner functions due to the non-commutative nature of operator multiplication in quantum mechanics. Instead, one often considers the Moyal product or uses the star product formalism to express the phase space equivalent of the operator product. The trace formula you've mentioned involves an integral over phase space and generally cannot be simplified into a simple integral involving only W1, W2, and W[rho] because of the aforementioned complexities in operator algebra.