## Path Integrals in Field Theory

For a scalar field in a finite volume V = [0, L]3, the Hamiltonian is given in the Fourier representation by

[ see Equation (2.6)

] H =

( 1 2 |π˜ ( k)|2 + 12 ( k2 + m2)|ϕ˜( k)|2 + V(ϕ˜) + ϕ˜( k) J˜ (−k, t)

) . (7.1)

As ϕ(x) is real we have ϕ˜∗( k) = ϕ˜(−k). It is customary to write the quadratic term in the fields (the mass term) explicitly, such that the potential V(ϕ) only contains the interaction terms. If we like, we could split the Fourier modes in their real and imaginary components

[ the cos( x · k) and sin( x · k)

modes ] . Or even simpler is to use Dirichlet boundary conditions, i.e., ϕ(x) = 0

at the boundaries of the volume, such that the Fourier modes are given by∏ j sin(πnj xj/L) (with nj > 0), with real coefficients. In either case, for V(ϕ) =

0 the Hamiltonian simply describes an infinite set of decoupled harmonic oscillators, which can be truncated to a finite set by introducing a so-called momentum cutoff |k| ≤ . In this case we know how to write the path integral, even in the presence of interactions. The introduction of a cutoff is called a regularisation. The field theory is called renormalisable if the limit → ∞ can be defined in a suitable way, often by varying the parameters in a suitable way with the cutoff. The class of renormalisable field theory is relatively small. For a finite momentum cutoff, the path integral is nothing but a simple generalisation of the one we defined for quantum mechanics in n dimensions, or in the absence of interactions

Z = lim N→∞

∏ k

(2π it)−N/2 ∫ N−1∏

∏ k

dϕ˜ j ( k) exp [

it N−1∑ j=0

∑ k

|ϕ˜ j+1( k) − ϕ˜ j ( k)|2 2t2

− 12 ( k 2 + m2)|ϕ˜ j ( k)|2 − ϕ˜ j ( k) J˜ (−k, jt)

]

≡ ∫ Dϕ˜( k, t) exp

( i ∫ T

∑ k

{ 1 2 | ˙˜ϕ( k, t)|2 − 12 ( k2 + m2)|ϕ˜( k, t)|2

− ϕ˜( k, t) J˜ (−k, t)}dt )

. (7.2)

A in

One of course identifies ϕ˜ j ( k) = ϕ˜( k, t = jt) and performing the Fourier transformation once more, one can write∫ T

0 dt

∑ k

{ 1 2 | ˙˜ϕ( k, t)|2 − 12 ( k

2 + m2)|ϕ˜( k, t)|2 − ϕ˜( k, t) J˜ (−k, t)}

= ∫ T

0 dt

∫ V

d3 x {

( ∂tϕ( x, t)

)2 − 12 (∂iϕ( x, t))2 − 12 m2ϕ2( x, t) − ϕ( x, t) J ( x, t)}

= ∫

V×[0,T] d4x

{ 1 2 ∂µϕ(x)∂

µϕ(x) − 12 m2ϕ2(x) − ϕ(x) J (x) } . (7.3)

The last expression is manifestly Lorentz invariant apart from the dependence on the boundary conditions on the fields (which should disappear once we take L and T to infinity). This will allow us to perform perturbation theory in a Lorentz covariant way. (Things are somewhat subtle as any finite choice of the momentum cutoff does break the Lorentz invariance, and there are some theories where this is not restored when removing the cutoff, i.e., taking the limit → ∞.) This achieves a substantial simplification over Hamiltonian perturbation theory. It is now also trivial to reintroduce the interactions by adding the potential term to the Lagrange density, and we find in yet another shorthand notation for the measure the following expression for the path integral (implicitly assuming that the boundary values ϕ( x, 0) and ϕ( x, T) are fixed, prescribed functions)

Z = ∫ Dϕ(x) exp

( i ∫

V×[0,T] d4x

{ 1 2 ∂µϕ(x)∂

µϕ(x) − 12 m2ϕ2(x)

− V(ϕ) − ϕ(x) J (x)}). (7.4) In principle a path integral should be independent of the discretisation used

in order to define it. For the Euclidean path integral, one particular way that is used quite often is the lattice discretisation, where instead of a momentum cutoff one makes not only time but also space discrete. This means that the field now lives on a lattice and its argument takes the values ja where j ∈ ZZ4 and a is the so-called lattice spacing, which in the end should be taken to zero. By suitably restricting the components of j , with appropriate boundary conditions on the fields, one keeps space and time finite, V = a3 M3 and T = a N. This leads to an integral of the form

(2πa )−NM 3/2

∫ ∏ j

dϕ j exp

( −a4

∑ j

∑ µ

(ϕ j+eµ − ϕ j )2 a2

+ 12 m2ϕ2j + V(ϕ j ) + J jϕ j })

(7.5)

where eµ is a unit vector in the µ direction, and ϕ j is identified with ϕ(a j). In a sense, the momentum cutoff is similar to the space-time cutoff 1/a . The lattice formulation is very suitable for numerically evaluating the path integral, whereas the momentum cutoff is suitable for performing perturbation theory around the quadratic approximation of the action. For the latter we will compute, using the path integral, the same quantity as was calculated in Chapter 5, using Hamiltonian perturbation theory.