From Classical Mechanics to Quantum Field Theory

44 From Classical Mechanics to Quantum Field Theory. A Tutorial

To calculateZ, it is possible to proceed as in the previous section by taking into
account the following two minor differences^22 :

• in the expression we havee−βHinstead ofe−
  ıtH
  ;


• boundary conditions are nowx(t)=x(0) = x, on which we have to
  integrate.

Leaving all details to the reader, we write here only the final result:

Z= lim
→ 0

∫

x 0 =xM

(M

∏

n=1

d^3 xn

)(

m

2 π

) 3 M 2

exp

{

−^



∑∞

n=1

[

(xn−xn− 1 )^2

2

+V(xn)

]}

,

(1.216)

which can be written, formally, as:

Z=

∫

x(0)=x(β)

[Dx(τ)] exp

{

−

1



∫β

0

dτ′H[x(τ′)]

}

. (1.217)

We remark also that the same result could have been obtained by performing a
“Wick-rotation”, i.e. the analytic continuation from the real variabletto the
imaginary oneτ =ıt^23. The measure that appears in (1.217) is the same one
used in the context of stochastic processes, i.e. Wiener measure. In this case it is
possible to give a rigorous mathematical treatment to make sense of the continuous
version of Feynman path-integral[ 20 ].

1.3.2.3 Path integral with coherent states

In this section, we will work out the path integral formulation of QM using coherent
states. In the following, we will concentrate on one single degree of freedom,
either bosonic or fermionic, but the construction can be generalized to many body
systems and quantum field theory, giving one of the most exploited technique in
this research area.
Either in the bosonic or in fermionic case, let us denote with|φi〉and〈φf|the
initial and final coherent states. Then we write the propagator in the following

(^22) See sect. 3.2.1 of the third part of this volume for an analogue definition in the Euclidean
approach to QFT. 23
Indeed this would have changed the expressions of the derivatives appearing in the actionS
according to:dx/dt=ıdx/dτ,(dx/dt)^2 =−(dx/dτ)^2.