Og. Smolyanov et Mo. Smolyanova, TRANSFORMATIONS OF FEYNMAN-INTEGRALS UNDER NONLINEAR TRANSFORMATIONS OF THE PHASE-SPACE, Theoretical and mathematical physics, 100(1), 1994, pp. 803-810
The Feynman measure is defined as a linear continuous functional on a
test-function space (introduced in the paper). The functional is given
by means of its Fourier transform. Not only a positive-definite corre
lation operator but also one without fixed sign is considered (the lat
ter case corresponds to the so-called symplectic, or Hamiltonian, Feyn
man measure). The Feynman integral is the value of the Feynman measure
on a function (ire the test-function space). The effect on the Feynma
n measure of nonlinear transformations of the phase space in the form
of shifts along vector fields or along integral curves of vector field
s is described. Analogs of the well-known Cameron--Martin, Girsanov-Ma
ruyama, and Ramer formulas in the theory of Gaussian measures are obta
ined. The results of the paper can be regarded as,formulas for a chang
e of variable in Feynman integrals.