ABSTRACT
We list some practically useful Wiener path integrals which can be explicitly calculated and expressed in terms of elementary functions. Recall that dWx(τ ) is the Wiener path-integral measure
dWx(τ ) def≡ exp
{ − 1
4D
dτ x˙2(τ ) } t∏ τ=t0
dx(τ )√ 4πD dτ
and the basic Wiener formula is the integral with this measure over the set of all continuous but nondifferentiable functions:∫
C{x0,t0;x,t} dWx(τ ) = 1√4πD(t − t0) exp
{ − (x − x0)
4D(t − t0)
} . (II.1)
The generating formula for the rest of the path integrals which we list below is the path integral∫ C{x0,t0;x,t}
dWx(τ ) exp { − ∫ t
t0 dτ [k2x2(τ )− η(τ)x(τ )]
}
= [
k 2π √
D sinh(2k √
D(t − t0)) ] 1
{ −k (x
2 + x20) cosh(2k √
D(t − t0))− 2x0x 2 √
D sinh(2k √
D(t − t0))
}
× exp {√
D ∫ t
t0 dτ ′ η(τ)η(τ ′) sinh(2k
√ D(t − τ )) sinh(2k√D(τ ′ − t0)) k sinh(2k
√ D(t − t0))
} × exp
dτ η(τ ) x0 sinh(2k
√ D(t − τ ))+ x sinh(2k√D(τ − t0))
sinh(2k √
D(t − t0)) }
(II.2)
for the driven oscillator (an oscillator in a field of an external force) which was obtained in chapter 1 (see (1.2.262)). As we have learned, this path integral is, actually, the generating (characteristic) functional. Therefore, all other path integrals presented in this appendix can be derived from it via functional differentiation or as particular cases for specific forms of the external force. For the reader’s convenience, we present them in explicit form. The corresponding Feynman path integral can be reduced to the Wiener one by the transition to the Euclidean time, as has been explained in section 2.1. • Transition to the limit k → 0 and an appropriate functional differentiation of (II.2) give the following
sequence of path integrals:
dWx(τ ) exp {∫ t
t0 dτ η(τ )x(τ )
} = 1√
4πD(t − t0) exp { − (x − x0)
4D(t − t0)
}
× exp {
2D ∫ t
t0 dτ ′ η(τ)η(τ ′) (t − τ )(τ
′ − t0) t − t0
+ ∫ t
t0 dτ η(τ )
(t − τ )x0 + (τ − t0)x t − t0
} (II.3)∫
C{x0,t0;x,t} dWx(τ ) xn(s) = 1√4πD(t − t0) exp
{ − (x − x0)
4D(t − t0)
}
n! k!(n − 2k)!
