# DECIDE: A Deterministic Mixed Quantum-Classical Dynamics Approach
## 2 Wigner-Weyl Transform  
Before deriving the DECIDE equations of motion, we present an overview of the Wigner-Weyl transform and its properties. The Wigner-Weyl transform, which was first introduced by [PERSON] in 1932 [44], is a starting point for deriving mixed quantum-classical dynamics methods. [PERSON]'s original goal was to find correction terms that would bridge quantum and classical mechanics [44; 45]. The transform can be understood in terms of probability functions. For a wave function expressed in position space, \(\psi(x)\), the probability density is \(|\psi(x)|^{2}\). The wave function can also be expressed in momentum space as  
\[\phi(p)=\frac{1}{\sqrt{h}}\int e^{-ixp/h}\psi(x)\ dx\equiv\langle p|\psi\rangle, \tag{1}\]  
and its associated probability density is \(|\phi(p)|^{2}\) (\(h=h/2\pi\) and \(h\) is [PERSON]'s constant). Each probability density above depends on either \(x\) or \(p\). The Wigner-Weyl transform provides a way to represent the probability distribution in terms of both position \((x)\) and momentum \((p)\). By definition, the Wigner-Weyl transform of an operator \(\hat{A}\) is \[A_{W}(x,p) = \int e^{-ipy/\hbar}\psi^{*}(x+\frac{y}{2})\hat{A}\psi(x-\frac{y}{2}) \;dy, \tag{2}\] \[\equiv \int e^{-ipy/\hbar}\Big{\langle}x+\frac{y}{2}|\hat{A}|x-\frac{y}{2} \Big{\rangle}\;dy,\]  
where \(x\) and \(p\) are vectors containing the positions and momenta of all DOF in the system and \(y\) is an arbitrary integration variable. Using the above definition, one can obtain several properties of this transform. First, the identity operator \(\hat{1}\) is given by  
\[1_{W} = \int e^{-ipy/\hbar}\Big{\langle}x+\frac{y}{2}|\hat{1}|x-\frac{y}{ 2}\Big{\rangle}\;dy \tag{3}\] \[= \int e^{-ipy/\hbar}\delta(x+\frac{y}{2}-x+\frac{y}{2})\;dy=1.\]  
Second, for an operator \(\hat{A}\) that is only a function of the position operator \(\hat{x}\), the Wigner-Weyl transform is  
\[\int e^{-ipy/\hbar}\Big{\langle}x+\frac{y}{2}|\hat{A}|x-\frac{y}{2}\Big{\rangle} \;dy=\int e^{-ipy/\hbar}A(x-\frac{y}{2})\delta(y)\;dy=A(x). \tag{4}\]  
Similarly, for an operator \(\hat{A}\) that is only a function of \(\hat{p}\), its Wigner-Weyl transform is \(A(p)\). Third, the trace of the product of two operators \(\hat{A}\) and \(\hat{B}\) is  
\[\text{Tr}(\hat{A}\hat{B})=\frac{1}{\hbar}\int\int A_{W}(x,p)B_{W}(x,p)\;dxdp. \tag{5}\]  
The Wigner function \(W(x,p)\), which corresponds to the probability density of \(x\) and \(p\), is defined in terms of the Wigner transform of the density operator \(\hat{p}=|\psi\rangle\langle\psi|\) as  
\[\frac{\rho_{W}(x,p)}{h}=\frac{1}{h}\int\int e^{-ipyh}|x+\frac{y}{2}\rangle \langle x-\frac{y}{2}|\;dxdy\equiv W(x,p). \tag{6}\]  
For example, the Wigner functions for the ground and first-excited states of the quantum harmonic oscillator are shown in Figure 1.  
Figure 1: Wigner functions for the **(a)** ground state, \(W_{0}(x,p)=2\exp(-p^{2}-x^{2})\), and **(b)** excited state, \(W_{1}(x,p)=2(-1+2p^{2}+2x^{2})\exp(-p^{2}-x^{2})\), of the quantum harmonic oscillator whose dimensionless Hamiltonian is \(\hat{H}=-\frac{1}{2}\frac{d^{2}}{dx^{2}}+\frac{1}{2}x^{2}\). The expressions for the Wigner functions are taken from Ref. [45]. In the Wigner-Weyl representation, the expectation value of an operator \(\hat{A}\) is given by  
\[A_{W}(x,p) = \int e^{-ipy/\hbar}\psi^{*}(x+\frac{y}{2})\hat{A}\psi(x-\frac{y}{2}) \;dy, \tag{2}\]0  
where \(W(x,p)\) is normalized, i.e.,  
\[A_{W}(x,p) = \int e^{-ipy/\hbar}\psi^{*}(x+\frac{y}{2})\hat{A}\psi(x-\frac{y}{2}) \;dy, \tag{2}\]1  
The Wigner-Weyl transform of the product of two operators is [46]  
\[A_{W}(x,p) = \int e^{-ipy/\hbar}\psi^{*}(x+\frac{y}{2})\hat{A}\psi(x-\frac{y}{2}) \;dy, \tag{2}\]2  
In the above equation, \(*\) denotes the Moyal star product [47; 48] and \(\Lambda\) is the Poisson bracket operator,  
\[A_{W}(x,p) = \int e^{-ipy/\hbar}\psi^{*}(x+\frac{y}{2})\hat{A}\psi(x-\frac{y}{2}) \;dy, \tag{2}\]3  
where an arrow denotes the direction of the operation. (The derivation of Equation (9) may be found in Ref. [46].)