Heat semigroups on Weyl algebra

We study the algebra of semigroups Laplacians on Weyl algebra. consider first-order partial differential operators $\nabla^\pm_i$ forming Lie $[\nabla^\pm_j,\nabla^\pm_k]= i\mathcal{R}^\pm_{jk}$ and $[\nabla^+_j,\nabla^-_k] =i\frac{1}{2}(\mathcal{R}^+_{jk}+\mathcal{R}^-_{jk})$ with some anti-symmetric matrices $\mathcal{R}^\pm_{ij}$ define corresponding $\Delta_\pm=g_\pm^{ij}\nabla^\pm_i\nabla^\pm_j$ positive $g_\pm^{ij}$. show that heat $\exp(t\Delta_\pm)$ can be represented as a Gaussian average $\exp\left<\xi,\nabla^\pm\right>$ use these representations to compute product semigroups, $\exp(t\Delta_+)\exp(s\Delta_-)$ kernel.