1549380323-Statistical Mechanics Theory and Molecular Simulation

Functional integrals 463

limit,x 1 ,...,xP+1becomes the complete set of points needed to specify a continuous
functionx(s) satisfyingx(0) =x,x(t) =x′, with the identification

xk=x(s= (k−1)ǫ). (12.4.2)

Moreover, in the limitǫ→0, the quantity (xk+1−xk)/ǫbecomes

lim

ǫ→ 0

(

xk+1−xk

ǫ

)

=

dx

ds

. (12.4.3)

Finally, in the limitǫ→0, the argument of the exponential

ǫ

∑P

k=1

[

m

2

(

xk+1−xk

ǫ

) 2

−

1

2

(U(xk+1) +U(xk))

]

is just a Riemann sum representation of an integral. Thus, we can write

lim

ǫ→ 0

ǫ

∑P

k=1

[

m

2

(

xk+1−xk

ǫ

) 2

−

(

U(xk+1) +U(xk)

2

)]

=

∫t

0

ds

[

1

2

mx ̇^2 (s)−U(x(s))

]

, (12.4.4)

where ̇x(s) = dx/ds. Interestingly, we see that the integrand of eqn. (12.4.4) is the
classical LagrangianL(x,x ̇) = (m/2) ̇x^2 −U(x) for the system (see Section 1.4). In
eqn. (12.4.4), the integral of the Lagrangian is taken along the pathx(s), and this
integral is just the action from Section 1.8:

A[x] =

∫t

0

ds

[

1

2

mx ̇^2 (s)−U(x(s))

]

. (12.4.5)

Thus, the weight factor for a given pathx(s) that begins atxand ends atx′in time
tis just the complex exponential exp(iA[x]/ ̄h).
We turn next to the integration measure dx 2 ···dxP. As noted previously, the
pointsx 1 ,...,xP+1comprise all of the points of the functionx(s) in the limitP→∞,
withx 1 =xandxP+1 =x′. Thus, the integration overx 2 ,...,xP constitutes an
integration over all possible functionsx(s) that satisfy the endpoint conditionsx(0) =
x,x(t) =x′. In other words, asP→∞, integrating overx 2 ,...,xPvaries all points of
the functionx(s), which is equivalent to varying the function, itself keepingx(0) and
x(t) fixed atxandx′, respectively. This type of integration is referred to asfunctional
integration. Symbolically, it is written as follows:

lim

P→∞,ǫ→ 0

( m

2 πiǫ ̄h

)P/ 2

dx 2 ···dxP≡Dx(s). (12.4.6)

Thus, the functional integral representation of the real-time propagator is

U(x,x′;t) =

∫x(t)=x′

x(0)=x

Dx(s) exp

{

i

̄h

∫t

0

ds

[

1

2

mx ̇^2 (s)−U(x(s))

]}