472 Computational methods for lattice field theories
and the Lagrangian describes the simplest interesting field theory for interacting
particles, thescalarφ^4 theory. The name ‘scalar’ denotes thatφ(x)is not a vector.
Vector theories do exist, we shall encounter examples later on. When a potential is
present, the energy is no longer a sum of one-particle energies: the particles interact.
We have mentioned the probability of going from a particular initial state to
another (final) state as an example of the problems studied in field theory. Our
experimental knowledge on particles is based on scattering experiments. This is a
particular example of such a problem: given two particles with certain initial states,
what are the probabilities for different resulting reaction products? That is, which
particles do we have in the end and what are their momenta? In scalar field theory
we have only one type of particle present. As we have seen in the first chapter of this
book, experimental information on scattering processes is usually given in terms of
scattering cross sections. These scattering cross sections can be calculated – they
are related to an object called theS-matrix, which is defined as
Sfi= lim
ti→−∞
tf→∞〈ψf|U(ti,tf)|ψi〉. (15.28)Our initial state is one with a particular set of initial momenta as in(15.22), and sim-
ilarly for the final state;U(ti,tf)is the time-evolution operator,^2 and the statesψi,f
usually contain a well-defined number of free particles with well-defined momenta
(or positions, depending on the representation).
Scattering cross sections are expressed directly in terms of the S-matrix, and the
latter is related to theGreen’s functionsof the theory by the so-called Lehmann–
Symanzik–Zimmermannrelation,whichcanbefoundforexampleinRef. [2].
These Green’s functions depend on a set of positionsx 1 ,...,xnand are given by
G(x 1 ,...,xn)=∫
[Dφ]φ(x 1 )···φ(xn)e−S//∫
[Dφ]e−S/. (15.29)Note thatxiis a space-time vector, the subscripts do not denote space-time com-
ponents. The scattering cross sections are evaluated in the Euclidean metric; the
Minkowskian quantities are obtained by analytical continuation:t→it.
As the initial and final states in a scattering experiment are usually given by
the particle momenta, we need the Fourier transform of the Green’s function,
defined as
G(k 1 ,...,kn)δ(k 1 +···+kn)( 2 π)d=∫
ddx 1 ...ddxneik^1 ·x^1 +···+ikn·xnG(x 1 ,...,xn). (15.30)(^2) More precisely, the time-evolution operator is that for a theory with an interaction switched on at a time
much later thantiand switched off again at a time long beforetf.