The Path Integral in Euclidean Spaces

Emanuele Curotto

Hyegsoon An, Ang M

In this paper we present the concept of time continuation to develop the path integral quantization given in Ref, [1—2] into a time continuation theory, which can deal with the Minkowski and the Euclidean formulation of path integral quantization in an unified description, and give the general effective Lagrangian and the convergence factor for functional integral. Especially, the convergence factors in the Feynman and the Lee-Yang cases in the Minkowski space are proved to be the positive definite Hamiltonian. But in the Euclidean space, it is the effective Lagrangian which is equivalent to the Hamiltonian, no damping factor is needed, so mathematically the Euclidean formulation has some advantages. It is pointed out that when Hamiltonian involves a high order of momentum P, there appears singular term in the effective Lagrangian.

Belal Ehsan Baaquie

An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.

H. Dekker