QuantumPhysics.dvi

15.3 Imaginary time path Integrals

If we formally replacet→−iτand considerτreal, then we obtain theimaginary timeorEuclidean
formulation of quantum mechanics. Often, one also refers tothis procedure asanalytic continuation
to imaginary timebecause the substitutiont→ −iτcan be regarded as an analytic continuation.
The evolution operator becomes

UE(τ) =e−τH/ ̄h (15.24)

and is formally related to the customary evolution operatorbyUE(τ) =U(iτ). Notice, however,
thatU(t) andUE(τ) are very different objects: most importantly,UE(τ) is NOT a unitary operator
for realτ.

It is possible to obtain a path integral representation for the matrix elements ofUE(τ), just as
we obtained one forU(t). Instead of working out this path integral representationfrom scratch,
we may simply obtain it by analytic continuation,t→−iτ,

〈qb|UE(τb−τa)|qa〉=

∫

Dqexp

{

−

1

̄h

∫τb

τa

dτ LE(q,q ̇)

}

(15.25)

subject to the boundary conditions,

q(τa) = qa

q(τb) = qb (15.26)

Here, the prefactor− 1 / ̄h, and theEuclidean LagrangianLE are obtained as follows. In view of
t→−iτ, we have,
∫
dt→−i

∫

dτ (15.27)

and the Euclidean Lagrangian becomes,

L

(

q(t),

dq(t)

dt

)

=−LE

(

q(τ),

dq(τ)

dτ

)

(15.28)

Strictly speaking, we should also put a subscriptEon the basic variablesq, so thatq(t) =qE(τ),
but we shall omit this to save some notation. For the simplestLagrangians, we have,

L =

1

2

m

(

dq

dt

) 2

−V(q)

LE =

1

2

m

(

dq

dτ

) 2

+V(q) (15.29)

Note that, at least for these simplest cases, the argument ofthe exponential in the path integral is
now real and generally leads to a damped fall-off for largeq(τ). This renders the Euclidean path
integral well-defined even to the standards of mathematicalrigor!