JHEP10(2017)092
2.1 Particles in the vacuum
Consider the free massless scalar field. The field operator
ˆ
φ can be expanded over the basis
of Minkowski plane waves f
p
as
ˆ
φ =
Z
dp (f
p
ˆa
p
+ f
∗
p
ˆa
†
p
),
thus associating the mode operator ˆa
p
with the Minkowski plane wave f
p
. The Minkowski
vacuum |0i
M
is defined as the state for which
ˆa
p
|0i
M
= 0 for all p. (2.1)
Alternatively, the field could be expanded over a basis {g
k
},
ˆ
φ =
Z
dk (g
k
ˆ
b
k
+ g
∗
k
ˆ
b
†
k
).
Here k need not be momentum, but should be thought of as an index that labels the basis
modes. Thus the operator
ˆ
b
k
is associated with the mode g
k
. This operator can be written
in terms of the mode operators ˆa
p
as
ˆ
b
k
=
Z
dp
α
∗
kp
ˆa
p
− β
∗
kp
ˆa
†
p
, (2.2)
where α
kp
and β
kp
are the so-called Bogolyubov coefficients. These coefficients can be
computed by
α
kp
= (g
k
, f
p
), β
kp
= −(g
k
, f
∗
p
), (2.3)
where
∗
is complex conjugation, and the inner product (·, ·) is given by the Klein-Gordon
inner product. In Minkowski space it reads
(g, f) = −i
Z
dx [ g ∂
t
f
∗
− (∂
t
g)f
∗
] . (2.4)
Now by using (2.1) and (2.2), we see that the expectation value of the number operator
of the new modes equals
h
ˆ
b
†
k
ˆ
b
k
i
M
=
Z
dp |β
kp
|
2
. (2.5)
So, even though the overall state is the vacuum with respect to the modes f
p
, the modes
g
k
are in an excited state. As will become clear in due course, the modes g
k
will separately
often be in a mixed state due to entanglement with other modes. Thus they will have
some non-zero von Neumann entropy. Since the overall state |0i
M
is pure, this entropy is
solely due to entanglement, and we will therefore refer to this quantity as the entanglement
entropy of the mode.
In this paper, the overall state is always the Minkowski vacuum, so we will henceforth
drop the subscript M .
– 5 –
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