### 2.6 The Euclidean path integral representation and the replica method

A technical method very useful for the calculation of the entanglement entropy in a field theory is the the replica trick, see [33]. Here we illustrate this method for a field theory described by a second-order Laplace-type operator. One considers a quantum field in a -dimensional spacetime and chooses the Cartesian coordinates , where is Euclidean time, such that the surface is defined by the condition and are the coordinates on . In the subspace it will be convenient to choose the polar coordinate system and , where the angular coordinate varies between and . We note that if the field theory in question is relativistic, then the field operator is invariant under the shifts , where is an arbitrary constant.

One first defines the vacuum state of the quantum field in question by the path integral over a half of the total Euclidean spacetime defined as such that the quantum field satisfies the fixed boundary condition on the boundary of the half-space,

where is the action of the field. The surface in our case is a plane and the Cartesian coordinate is orthogonal to . The co-dimension 2 surface defined by the conditions and naturally separates the hypersurface into two parts: and . These are the two sub-regions and discussed in Section 2.1.

The boundary data is also separated into and . By tracing over one defines a reduced density matrix

where the path integral goes over fields defined on the whole Euclidean spacetime except a cut . In the path integral the field takes the boundary value above the cut and below the cut. The trace of the -th power of the density matrix (14) is then given by the Euclidean path integral over fields defined on an -sheeted covering of the cut spacetime. In the polar coordinates the cut corresponds to values . When one passes across the cut from one sheet to another, the fields are glued analytically. Geometrically this -fold space is a flat cone with angle deficit at the surface . Thus, we have
where is the Euclidean path integral over the -fold cover of the Euclidean space, i.e., over the cone . Assuming that in Eq. (15) one can analytically continue to non-integer values of , one observes that
where is the renormalized matrix density. Introduce the effective action , where is the partition function of the field system in question on a Euclidean space with conical singularity at the surface . In the polar coordinates the conical space is defined by making the coordinate periodic with period , where is very small. The invariance under the abelian isometry helps to construct without any problem the correlation functions with the required periodicity starting from the -periodic correlation functions. The analytic continuation of to different from 1 in the relativistic case is naturally provided by the path integral over the conical space . The entropy is then calculated by the replica trick
One of the advantages of this method is that we do not need to care about the normalization of the reduced density matrix and can deal with a matrix, which is not properly normalized.