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)kin terms of a complete set { f k } of (generally complex) c-number solutions of theoriginal 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 denotedby I − (we follow here the original notation, where a state of positive frequency isassumed to have a time dependence e iωt ). The position independent operators a † k anda k are therefore interpreted as creation and destruction operators for these incomingparticles. Such operators define in the usual way a vacuum, here denoted by |0 − 〉,which is devoid of a quantaa 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 stillcomplete, set of c-number solutions of the original wave equation. These will nowbe denoted by p k and q k , with their complex conjugate counterparts ¯p k and ¯q k .Thep k ’s are chosen to be asymptotically outgoing, positive frequency, solutions of thewave equation, subject to the condition that they be zero on the horizon: they willonly contain a positive frequency part on the future null horizon I + .Thep k ’s donot form a complete set, and that is where the q k ’s come in: they represent solutionswhich 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 expansionφ(x) =∑k[p k (x)b k + ¯p k (x)b † k + q k(x)c k + ¯q k (x)c † kwith 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 particlesin 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 ’sand the p k ’s will in general involve complex coefficients due to the mixing of positiveand negative frequencies taking place during the collapse, as a consequence