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hw1 - Homework Set No 1 Physics 880.02 Deadline Thursday 1...

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Homework Set No. 1, Physics 880.02 Deadline – Thursday, January 22, 2009 1. Consider a free (real) scalar theory with the Lagrangian density L = 1 2 μ ϕ∂ μ ϕ - m 2 2 ϕ 2 . Define the Hamiltonian by H ( t ) = Z d 3 x [ π ( ~x, t ) ˙ ϕ ( ~x, t ) - L ] . a. (3 pts) Show that for classical field configurations d dt H ( t ) = 0 . b. (2 pts) Write H ( t ) in terms of π and ϕ with no ˙ ϕ ’s appearing. c. (5 pts) Now imagine that the field is quantized. Use canonical quantization commu- tators [ ϕ ( ~x, t ) , π ( ~x 0 , t )] = ( ~x - ~x 0 ) (with all other commutators being zero) to show that H ( t ) (now an operator) generates time translations, i.e., show that i ∂ 0 ϕ = [ ϕ, H ( t )] i ∂ 0 π = [ π, H ( t )] . 2. The same as in problem 1, but now for Dirac field: start with a theory with Lagrangian density L = ¯ ψ ( μ μ - m ) ψ. a. (3 pts) Construct a Hamiltonian H ( t ) and show that for classical field configurations d dt H ( t ) = 0 . b. (2 pts) Write H ( t ) in terms of π and ψ . c. (5 pts) For quantized field ψ use the anti-commutation relations n ψ α ( ~x, t ) , ψ β ( ~x 0 , t ) o = δ α β δ ( ~x - ~x 0 ) to show that i ∂ 0 ψ

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hw1 - Homework Set No 1 Physics 880.02 Deadline Thursday 1...

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