Computational Physics

15.3 Interacting fields and renormalisation 475

theory is renormalisable is quite restrictive. For example, only theφ^4 potential has
this property; changing theφ^4 into aφ^6 destroys the renormalisability of the theory.
In computational physics we usually formulate the theory on a lattice. We then
choose values for the bare mass and coupling constant and calculate various phys-
ical quantities in units of the lattice constanta(or its inverse). Comparison with
experiment then tells us what the actual value of the lattice constant is. Therefore
the procedure is somehow the reverse of that followed in ordinary renormalisation,
although both are intimately related. In ordinary renormalisation theory we find the
bare coupling constant and mass as a function of the cut-off from a comparison with
experiment. In computational field theory we find the lattice constant as a function
of the bare coupling constant from comparison with experimental data.
Let us consider an example. We take the Euclideanφ^4 action in dimensiond=4:

S=

1

2

∫

d^4 x{[∂μφ(x)][∂μφ(x)]+m^2 φ^2 (x)+gφ^4 (x)} (15.36)

and discretise this on the lattice, with a uniform lattice constanta. Lattice points are
denoted by the four-indexn=(n 0 ,n 1 ,n 2 ,n 3 ). A lattice pointncorresponds to the
physical pointx=(an 0 ,an 1 ,an 2 ,an 3 )=an. The discretised lattice action reads

SLattice=

1

2

∑

n

a^4







∑^3

μ= 0

[

φ(n+eμ)−φ(n)

a

] 2

+m^2 φ^2 n+gφ^4 (n)







. (15.37)

We rescale theφ-field, the mass and the coupling constant according to

φ(n)→φ(n)/a; m→m/a and g→g, (15.38)

to make the lattice action independent of the lattice constanta:

SLattice=

1

2

∑

n







∑^3

μ= 0

[φ(n+eμ)−φ(n)]^2 +m^2 φn^2 +gφ^4 (n)







. (15.39)

Now we perform a Monte Carlo or another type of simulation for particular values
ofmandg. We can then ‘measure’ the correlation length in the simulation. This
should be the inverse of the experimental mass, measured in units of the lattice
constanta. Suppose we know this mass from experiment, then we can infer what
the lattice constant is in real physical units.
Life is, however, not as simple as the procedure we have sketched suggests. The
problem is that in the generic case, the correlation length is quite small in units
of lattice constants. However, a lattice discretisation is only allowed if the lattice
constant ismuch smallerthan the typical length scale of the physical problem.
Therefore, the correlation length should be an order of magnitude larger than the
lattice constant. Only close to a critical point does the correlation length assume