Integral formulation of the quantum mechanics in the phase space

Abstract

A formulation of quantum mechanics is introduced based on a $2D$-dimensional phase-space wave function $\text{\reflectbox{\text{p}}}\mkern-3mu\text{p}\left(q,p\right)$ which might be computed from the position-space wave function $\psi\left(q\right)$ with a transformation related to the Gabor transformation. The equation of motion for conservative systems can be written in the form of the Schr\"{o}dinger equation with a $4D$-dimensional Hamiltonian with classical terms on the diagonal and complex off-diagonal couplings. The Hamiltonian does not contain any differential operators and the quantization is achieved by replacing $q$ and $p$ with $2D$-dimensional counterparts $\left(q+q'\right)/2$ and $\left(p+p'\right)/2$ and by using a complex-valued factor $e^{i\left(q\cdot p'-q'\cdot p\right)/2}$ in phase-space integrals. Despite the fact that the formulation increases the dimensionality, it might provide a way towards exact multi-dimensional computations as it may be evaluated directly with Monte-Carlo algorithms.