Feynman Path Integral Formulation

5.5 Method of In and Out Vacua 159
[
]
φ(x) =∑ f k (x)a k + ¯f k (x)a † k
, (5.78)
k
in terms of a complete set { f k } of (generally complex) c-number solutions of the
original wave equation,
g μν ∇ μ ∇ ν f k (x) =0 . (5.79)
The f k ’s contain asymptotically only ingoing, and positive frequency components:
they only contain positive frequency particles on past null infinity, here denoted
by I − (we follow here the original notation, where a state of positive frequency is
assumed to have a time dependence e iωt ). The position independent operators a † k and
a k are therefore interpreted as creation and destruction operators for these incoming
particles. Such operators define in the usual way a vacuum, here denoted by |0 − 〉,
which is devoid of a quanta
a k |0 − 〉 = 0 , (5.80)
and from it a corresponding Fock space.
The field operator φ can also be equivalently expanded in a different, but still
complete, set of c-number solutions of the original wave equation. These will now
be denoted by p k and q k , with their complex conjugate counterparts ¯p k and ¯q k .The
p k ’s are chosen to be asymptotically outgoing, positive frequency, solutions of the
wave equation, subject to the condition that they be zero on the horizon: they will
only contain a positive frequency part on the future null horizon I + .Thep k ’s do
not form a complete set, and that is where the q k ’s come in: they represent solutions
which contain no outgoing component, and are zero on the future null horizon I + .
No restriction is needed on the frequency part of the q k ’s.
In this second basis the quantum operator φ has the expansion
φ(x) =∑
k
[
p k (x)b k + ¯p k (x)b † k + q k(x)c k + ¯q k (x)c † k
with p k and q k c-number solutions of the original wave equation,
]
, (5.81)
g μν ∇ μ ∇ ν p k (x) =g μν ∇ μ ∇ ν q k (x) =0 , (5.82)
and the b † k , b k, c † k , c k the corresponding creation and destruction operators for particles
in the corresponding mode.
Since the two sets of c-number solutions both individually form a complete set,
and are equivalent, they should be related to each other by a linear transformation,
p k = ∑
k ′ [
αkk ′ f k ′ + β kk ′ ¯f k ′
]
, (5.83)
with a similar expressions for q k . It is easy to see that the mixing between the f k ’s
and the p k ’s will in general involve complex coefficients due to the mixing of positive
and negative frequencies taking place during the collapse, as a consequence