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hw3 - Homework Set No 3 Physics 880.08 Deadline Monday 1 In...

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Homework Set No. 3, Physics 880.08 Deadline – Monday, November 22, 2010 1. In class we quantized free real scalar field theory with the Lagrangian density L = 1 2 μ ϕ∂ μ ϕ - m 2 2 ϕ 2 . The field operator was shown to be ϕ ( x ) = integraldisplay d 3 k (2 π ) 3 2 E k bracketleftBig ˆ a vector k e i k · x + ˆ a vector k e i k · x bracketrightBig , (1) and the canonical momentum operator was given by π = ˙ ϕ . Above k · x = E k t - vector k · vectorx . (a) (10 pts) Show that canonical commutation relations [ ϕ ( vectorx,t ) ( vectorx ,t )] = ( vectorx - vectorx ) [ ϕ ( vectorx,t ) ( vectorx ,t )] = 0 [ π ( vectorx,t ) ( vectorx ,t )] = 0 (2) require that the particle creation and annihilation operators obey the following commutation relations bracketleftBig ˆ a vector k , ˆ a vector k bracketrightBig = (2 π ) 3 2 E k δ 3 ( vector k - vector k ) , bracketleftbig ˆ a vector k , ˆ a vector k bracketrightbig = bracketleftBig ˆ a vector k , ˆ a vector k bracketrightBig = 0 . (3) (Note that simply showing that Eq. (3) makes the field
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