A general formula for the calculation of Gaussian path-integrals in two and three euclidean dimensions

摘要

The method which was used in a preceding article for the analytical evaluation of Gaussian path-integrals in one euclidean dimension is generalized. As before, the final results are formulae directly applicable to all Gaussian path-integrals, this time in two and in three euclidean dimensions, respectively. The classical action is almost entirely integrated, and the proportionality factor F(tb,ta) in front of the exponential part is expressed in terms of a number of time-dependent functions which one encounters in the description of the motion along the classical path. The quadratic Lagrange function is kept as general as possible, e.g., involving twenty-eight terms in the three-dimensional case. The well-known time-discretization procedure for path-integrals has been avoided. Instead, one of the main steps in the theoretical development consists in applying the convolution property of quantum-mechanical Green's functions, i.e., in three euclidean dimensions, K(rb,tb;ra,ta)=∫∫∫−∞+∞K(rb,tb;r,t)K(r,t;ra,ta)dr. This equality leads to a non-linear algebraic relation between F(tb,ta), F(tb,t) and F(t,ta). The solution of this equation yields the F-factor which appears in the propagator represented by the Gaussian path-integral. At two locations, the use of some remarkable determinantal identity, previously unknown to the author, has been indispensable in order to attain the desired final result. In the Appendix, the “n-dimensional” generalization of these identities is formulated.