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.