3.4 Reduced density matrix and entropy
The density matrix
defined by tracing over
-modes is given by the Euclidean path
integral over field configurations on the complete instanton
with a cut along the axis
where the field
in the path integral takes the values
and
below and above
the cut respectively. The trace
is obtained by equating the fields across the cut and doing the
unrestricted Euclidean path integral on the complete Euclidean instanton
. Analogously,
is given
by the path integral over field configurations defined on the n-fold cover
of the complete instanton.
This space is described by the metric (50) where angular coordinate
is periodic with period
. It has a conical singularity on the surface
so that in a small vicinity of
the
total space
is a direct product of
and a two-dimensional cone
with angle deficit
. Due to the abelian isometry generated by the Killing vector
this construction can be
analytically continued to arbitrary (non-integer)
. So that one can define a partition function
by the path integral over field configurations over
, the
-fold cover of the instanton
. For a
bosonic field described by the field operator
one has that
. Defining the effective
action as
, the entanglement entropy is still given by formula (16), i.e., by
differentiating the effective action with respect to the angle deficit. Clearly, only the term linear in
contributes to the entropy. Thus, the problem reduces to the calculation of this term in the
effective action.