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HW3 - ψ ·· ψ t,~x ψ t,~x a † ~ k a † ~ k | i =...

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PHY5667 Problem Set #3 (due Tue Sep 14) (1) Consider a nonrelativistic QFT for a fermion ψ . Prove the following identity: h 0 | ˆ T ψ ( t 1 , ~x 1 ) ψ ( t 2 , ~x 2 ) ψ ( t 3 , ~x 3 ) ψ ( t 4 , ~x 4 ) | 0 i = P ( t 1 , ~x 1 ; t 4 , ~x 4 ) P ( t 2 , ~x 2 ; t 3 , ~x 3 ) - P ( t 1 , ~x 1 ; t 3 , ~x 3 ) P ( t 2 , ~x 2 ; t 4 , ~x 4 ) , where the function P is the propagator for ψ , i.e., P ( t, ~x ; t 0 , ~x 0 ) = Z d ω 2 π d 3 ~ k (2 π ) 3 i ω - k 2 2 m + i ε e i ~ k · ( ~x - ~x 0 ) - i ω ( t - t 0 ) (2) Again, consider a nonrelativistic QFT for a fermion ψ . Reduce the time-ordered prod- uct h 0 | ˆ T ψ ( t 1 , ~x 1 ) ψ ( t 2 , ~x 2 ) ψ ( t 3 , ~x 3 ) ψ ( t 4 , ~x 4 ) ψ ( t 5 , ~x 5 ) ψ ( t 6 , ~x 6 ) | 0 i to a form containing only propagators, as in question (1). Be sure to simplify your expression as much as possible.
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Unformatted text preview: ψ : ··· ψ ( t,~x ) ψ ( t ,~x ) a † ( ~ k ) a † ( ~ k ) | i = ··· ³ 1 (2 π ) 3 / 2 e i ~ k · ~x-i k 2 2 m t · 1 (2 π ) 3 / 2 e i ~ k · ~x-i k 2 2 m t ± 1 (2 π ) 3 / 2 e i ~ k · ~x-i k 2 2 m t · 1 (2 π ) 3 / 2 e i ~ k · ~x-i k 2 2 m t ´ | i , where + or-corresponds to the case where ψ is bosonic or fermionic, respectively. 1...
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