We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, $\Delta$, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...

In this paper, we introduce a simple formula for conditional Wiener integrals over $C_0\left(𝔹\right)$, the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on $C_0\left(𝔹\right)$ in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral...

Existence, uniqueness and ergodicity of weak solutions to the equation of stochastic quantization in finite volume is obtained as a simple consequence of the Girsanov theorem.

In analogy with earlier work on the forward-backward case, we consider an explicit construction of the forward-forward double stochastic product integral ${\prod }^{\to \to }(1+dr)$ with generator $dr=\lambda (d{A}^{†}\otimes dA-dA\otimes d{A}^{†})$. The method of construction is to approximate the product integral by a discrete double product ${\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\Gamma \left(R_{m,n}^{(j,k)}\right)=\Gamma \left({\prod }_{(j,k)\in {\mathbb{N}}_m\times \mathbb{N}ₙ}^{\to \to }\left(R_{m,n}^{(j,k)}\right)\right)$ of second quantised rotations $R_{m,n}^{(j,k)}$ in different planes using the embedding of ${ℂ}^m\oplus ℂⁿ$ into L²(ℝ) ⊕ L²(ℝ) in which the standard orthonormal bases of ${ℂ}^m$ and ℂⁿ are mapped to the orthonormal sets consisting of normalised indicator functions of...

We study a certain class of von Neumann algebras generated by selfadjoint elements ${\omega }_i=a_i+a{⁺}_i$, where $a_i,a{⁺}_i$ satisfy the general commutation relations: $a_{i}a{⁺}_j={\sum }_{r,s}{t}_j^i{r}_{s}a{⁺}_r{a}_s+{\delta }_{ij}Id$. We assume that the operator T for which the constants $t_j^i{r}_s$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the number of generators...

We present quantum stochastic calculus in terms of diagrams taking weights in the algebra of observables of some quantum system. In particular, we note the absence of non-time-consecutive Goldstone diagrams. We review recent results in Markovian limits in these terms.