Computational Physics

468 Computational methods for lattice field theories

Ais a constant, and some special conditions are needed for the boundaries. In a
quantum mechanical description we again use the path integral, which gives us the
probability density for the chain to go from an initial configurationi={φ(ni)}
at timetito a final configurationf={φn(f)}at timetf(note that(i,f)denotes a
complete chain configuration):
∫
[D(t)]e−iS/
(15.4)

where the path integral is over all possible configurations of the fieldin the course
of time, with fixed initial and final configurations.
We now want to formulate this problem in continuum space. To this end we
replace the discrete indexnby a continuous indexx 1 , and we replace the interaction
term occurring in the summand by the continuous derivative:

S=

1

2

∫tf

ti

dt

∫

dx 1

{

mφ( ̇t,x 1 )^2 −A

[

∂φ(t,x 1 )

∂x 1

] 2 }

. (15.5)

The fieldφ(t,x 1 )can be thought of as a sheet whose shape is given as a height
φ(t,x 1 )above the (1+1) dimensional space-time plane. In the path integral, we
must sum over all possible shapes of the sheet, weighted by the factor eiS/
. The
field can be rescaled at will, as it is integrated over in the path integral (this rescaling
results in an overall prefactor), and the time and space units can be defined such as
to give the time derivative a prefactor 1/cwith respect to the spatial derivative (cis
the speed of light), and we obtain

S=

∫

d^2 x

1

2

∂μφ(x)∂μφ(x), (15.6)

where we have usedxto denote the combined space-time coordinatex≡(t,x 1 )≡
(x 0 ,x 1 ). From now on, we putc≡
≡1 and we use the Einstein summation
convention according to which repeated indices are summed over. The partial
space-time derivatives∂μ,∂μare denoted by:

∂μ=

∂

∂xμ

, ∂μ=

∂

∂xμ

. (15.7)

Furthermore we use the Minkowski metric:

aμaμ=a^20 −a^2. (15.8)

The fact that we choosec≡
≡1 leaves only one dimension for distances in
space-time, and masses and energies. The dimension of inverse distance is equal to
the energy dimension, which is in turn equal to the mass dimension.
Using partial integration, we can reformulate the action as

S=−

∫

d^2 x

1

2

φ(x)∂μ∂μφ(x), (15.9)