Normalized Gaussian path integrals

Path Integrals Without Integrals∗

Recently, we have developed an efficient recursive approach for analytically calculating the short-time expansion of the propagator to extremely high orders for a general many-body quantum system. Here we give brief overview of this approach and then demonstrate application of this technique by numerically studying the thermodynamical properties of a rotating ideal Bose gas of Rb atoms in an an...

Gaussian integrals

Path Integrals for Stochastic Neurodynamics Path Integrals for Stochastic Neurodynamics

We present here a method for the study of stochastic neurodynamics in the framework of the "Neural Network Master Equation" proposed by Cowan. We consider a model neural network composed of two{state neurons subject to simple stochastic kinetics. We introduce a method based on a spin choerent state path integral to compute the moment generating function of such a network. A formal construction ...

II.G Gaussian Integrals

It can be reduced to a product of N one dimensional integrals by diagonalizing the matrix K ≡ Ki,j . Since we need only consider symmetric matrices (Ki,j = Kj,i), the eigenvalues are real, and the eigenvectors can be made orthonormal. Let us denote the eigenvectors and eigenvalues of K by q̂ and Kq respectively, i.e. Kq̂ = Kqq̂. The vectors {q̂} form a new coordinate basis in the original N dimensi...

Conditionally Gaussian stochastic integrals

We derive conditional Gaussian type identities of the form E [ exp ( i ∫ T 0 utdBt ) ∣∣∣∣ ∫ T 0 |ut|dt ] = exp ( − 2 ∫ T 0 |ut|dt ) , for Brownian stochastic integrals, under conditions on the process (ut)t∈[0,T ] specified using the Malliavin calculus. This applies in particular to the quadratic Brownian integral ∫ t 0 ABsdBs under the matrix condition A †A2 = 0, using a characterization of Yo...