PHY5667 Problem Set #2 (due Tue Sep 7) (1) In quantum mechanics, the operators ˆ x and ˆ p satisﬁes a commutation relation [ˆ x, ˆ p ] = i (in the ~ = 1 units as usual). Then, in the basis spanned by the eigenstates of ˆ x (deﬁned as ˆ x | x i = x | x i ), the representation ˆ p =-i∂/∂x realizes this commutator. Now, consider a QFT for a ﬁeld operator Ψ( ~x ) with a commutation relation [Ψ( ~x ) , Ψ † ( ~ y )] = δ ( ~x-~ y ). In the basis spanned by the eigenstates of Ψ( ~x ) (deﬁned as Ψ( ~x ) | ψ i = ψ ( ~x ) | ψ i ), ﬁnd a representation of the operator Ψ † ( ~x ) that realizes the commutation relation. [Hint: Follow the analogy with quantum mechanics!] (2) Consider a QFT in 1+1 dimensions for a Hermitian ﬁeld Φ and its conjugate momentum Π described by the Hamiltonian density H ( x ) = v 2 2 [Π( x )] 2 + 1 2 [ ∂ x Φ( x )] 2 , with commutation relations [Φ( x ) , Π( y )] = i δ ( x-y ) and [Φ( x ) , Φ( y )] = [Π(
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