HW6 - x L described by the action S = Z d t Z L d x 1 2 (...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY5667 Problem Set #6 (due Tue Oct 5) (1) In the lectures as well as in the previous HW, we discussed the rotation and boost generators J i and K i , which generate the transformations of the form φ 0 (0) = e i θ i J i +i η i K i φ (0). Similarly, for spacetime translations, find an explicit expression for the generator T μ such that φ ( x + a ) = e i a μ T μ φ ( x ), where a μ is a constant 4-vector. (2) Prove that, under a Lorentz boost ± x 0 0 x 0 1 ² = ± cosh η sinh η sinh η cosh η ²± x 0 x 1 ² , x 0 2 , 3 = x 2 , 3 , arbitrary 4-momenta k and q satisfy k 0 0 δ ( ~ k 0 - ~ q 0 ) = k 0 δ ( ~ k - ~ q ) . (3) Consider a QFT with the Hamiltonian density H = 1 2 Π 2 + 1 2 ( ~ φ ) · ( ~ φ ) + m 2 2 φ 2 , where φ and Π satisfy (in the Heisenberg picture) [ φ ( t,~x ) , Π( t,~ y )] = i δ ( ~x - ~ y ) , [ φ ( t,~x ) ( t,~ y )] = [Π( t,~x ) , Π( t,~ y )] = 0 . Show that the above H and commutation relations together imply Π = 0 φ as well as ( μ μ + m 2 ) φ = 0 . (4) Consider a (classical) theory for a real field φ in 1+1 dimensions on an interval 0
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: x L described by the action S = Z d t Z L d x 1 2 ( )( )-m 2 2 2- 3 3- 4 4 + Z d t M 2 [ ( t,L )] 2 , where m , , , and M are constants. By demanding that S = 0 for an arbitrary variation ( t,x ) (with the only restriction being ( t,x ) 0 for t ), derive (a) the boundary condition for at each boundary x = 0 and L , (b) the equation of motion for in the bulk 0 < x < L , and (c) general solutions of the equation of motion satisfying the boundary conditions, assum-ing m = = = 0 for simplicity but still with M 6 = 0. 1...
View Full Document

Ask a homework question - tutors are online