ps07soln - PHYS651 Problem Set 7 Solutions 1 a We need ψ x...

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Unformatted text preview: PHYS651: Problem Set 7 Solutions 1 . ( a ) We need: ψ ( x ) = Z d 3 p (2 π ) 3 1 p 2 E ~ p X s a s ~ p u s ( p ) e- ip · x + b s † ~ p v s ( p ) e ip · x ψ † ( y ) = Z d 3 q (2 π ) 3 1 p 2 E ~ q X r a r † ~ q u r † ( q ) e- iq · y + b r ~ q v r † ( q ) e iq · x and the following commutation relation and spin sums, { a r ~ p , a s † ~ q } = { b r ~ p , b s † ~ q } = (2 π ) 3 δ (3) ( ~ p- ~ q ) δ r s (1) X s u s u s † = X s u s ¯ u s γ = ( γ · p + m ) γ X s u s a u s † b = ( γ · pγ + mγ ) ab (2) X s v s a v s † b = ( γ · pγ- mγ ) ab . (3) Most of those relations are in P&S, or are simple generalization of what is in P&S. { ψ a ( x ) , ψ † b ( y ) } = Z d 3 pd 3 q (2 π ) 6 1 2 p E ~ p E ~ q X r,s h u s a ( p ) u r † b ( q ) e- i ( p · x- q · y ) { a s ~ p , a r † ~ q } + v s a ( p ) v r † b ( q ) e i ( p · x- q · y ) { b s † ~ p , b r ~ q } i using (1) and the fact that commutation relation are always evaluated at equal time ( x = y ). = Z d 3 p (2 π ) 3 1 E ~ p X s h u s a ( p ) u s † b ( p ) e i~ p · ( ~x- ~ y ) + v s a ( p ) v s † b ( p ) e- i~ p ( ~x- ~ y ) i = Z d 3 p (2 π ) 3 1 E ~ p e i~ p ( ~x- ~ y ) X s h u s a ( p ) u s † b ( p ) + v s a ( q ) v s † b ( q ) i Where q = ( p ,- ~ p ) (this is only a change of variable to get the same exponential for both terms). Now using the spins sums rules (2, 3). X s h u s a ( p ) u s † b ( p ) + v s a ( q ) v s † b ( q ) i = ( γ · pγ + mγ + γ · qγ- mγ ) ab = 2 E ~ p δ ab Hence, { ψ a ( x ) , ψ † b ( y ) } = Z d 3 p (2 π ) 3 e- i~ p · ( ~x- ~ y ) δ...
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ps07soln - PHYS651 Problem Set 7 Solutions 1 a We need ψ x...

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