From Classical Mechanics to Quantum Field Theory

40 From Classical Mechanics to Quantum Field Theory. A Tutorial

which gives the probability of finding the particle at pointxat timet,given
that it was at pointx′at time 0 (we are supposingt>0 and invariance under
time-translations).

1.3.2.1 Path integral in the space of coordinates

Let us start by considering a system with a Hamiltonian of the form:

H=T+V, T=

p^2

2 m

,V=V(x) (1.192)

describing a particle with massmmoving in a potentialV(x).
In the following, we will use the so-called Trotter formula: for any two self-
adjoint operatorsA, Bon some Hilbert spaceH, one can trivially write:

eıt(A+B)=[eı(A+B)]M,t≡M. (1.193)

Then, using the fact that

eı(A+B)=eıAeıB+O( 2 ) (1.194)

we get:

eı(A+B)= lim→ 0

[

eıAeıB

]M

, (1.195)

where the limit means →0andM →∞so thatM =tis kept constant.
This formula holds in the operator-norm sense ifHis finite-dimensional. In the
infinite-dimensional case, the limit has to be understood in the strong sense and
applied to vectors in the appropriate domains.
For the Hamiltonian (1.192), supposingT+V to be self-adjoint on a dense
domainD(T)∩D(V), we can re-write the kernel in the following form:

〈x|e−

ıtH

|x′〉= lim→ 0 〈x|

[

e−i

T

e−ı

V]M

|x′〉

= lim→ 0

∫

dx 1 ...

∫

dxM− 1 〈x|e−ı

T

e−ı

V

|xM− 1 〉

×〈xM− 1 |e−ı

T

e−ı

V

xM− 2 〉···〈x 1 |e−ı

T

e−ı

V

|x′〉

= lim→ 0

∫

dx 1 ...

∫

dxM− 1

∏M

n=1

〈xn|e−i

T

e−ı

V

|xn− 1 〉. (1.196)

To obtain this expression, in the second line we have insertedMresolutions of the
identities in coordinate space (I=

∫

dxj|xj〉〈xj|) and defined:xM=x,x 0 =x′.