Computational Physics

15.2 Quantum field theory 471

for arbitraryM, which denotes the number of particles present. It is possible to have
the samekioccurring more than once in this state (with an appropriate normalisation
factor): this means that there is more than one particle with the same momentum.
The state| 0 〉is the vacuum state: it corresponds to having no particles. The lowest
energy above the vacuum energy is that corresponding to a single particle at rest
(k =0): the energy is equal to the mass. InSection 11.2we have seen that for
a statistical field theory the inverse of the lowest excitation energy is equal to the
correlation length:
m≈ 1 /(ξa). (15.23)

However, this holds for a statistical field theory where we do not have complex
weight factors; these can be made real by an analytical continuation of the physics
into imaginary time:t→it(see alsoSection 12.2.4). In that case the (continuous)
action indspace-time dimensions reads

S=

∫

ddx

1

2

[∂μφ∂μφ+m^2 φ^2 ] (15.24)

where now

∂μφ∂μφ=(∇φ)^2 +

(

∂

∂t

φ

) 2

(15.25)

i.e. the Minkowski metric has been replaced by the Euclidean metric. The matrix
elements of the time-evolution operator now read exp(−S)instead of exp(−S/i)
(for
≡1). We have now a means to determine the particle mass: simply by
measuring the correlation length. In the free field theory, the inverse correlation
length is equal to the mass parametermin the Lagrangian, but if we add extra terms
to the Lagrangian (see below) then the inverse correlation length (or the physical
mass) is no longer equal tom.
It might seem that we have been a bit light-hearted in switching to the Euclidean
field theory. Obviously, expectation values of physical quantities can be related for
the Minkowski and Euclidean versions by an analytic continuation. In the numerical
simulations we use the Euclidean metric to extract information concerning the
Hamiltonian. This operator is the same in both metrics – only the time evolution and
hence the Lagrangian change when going from one metric to the other. Euclidean
field theory can therefore be considered merely as a trick to study the spectrum of
a quantum Hamiltonian of a field theory which in reality lives in Minkowski space.
If we add another term to the Lagrangian:

S=

∫

ddx

{

1

2

φ(x)(−∂μ∂μ+m^2 )φ(x)+V[φ(x)]

}

, (15.26)

whereVis not quadratic (in that case it would simply contribute to the mass term),
then interactions between the particles are introduced. Usually one considers

V=gφ^4 (x)/ 2 (15.27)