hw2 - with anti-symmetric structure constants f abc . (a)...

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Homework Set No. 2, Physics 880.02 Deadline – Thursday, February 5, 2009 1. If a photon was a massive particle of mass m its Lagrangian density (in the absence of sources) would have been given by the so-called Proca Lagrangian L Proca = - 1 4 F μν F μν + m 2 2 A μ A μ (1) where A μ is the 4-vector photon Feld. (a) (5 pts) ±ind the equations of motion for the Proca Lagrangian (known as the Proca equations). (b) (5 pts) Take a 4-divergence of the Proca equations obtained in (a) to show that if m 6 = 0 Proca equations require Lorenz gauge condition μ A μ = 0 to always be valid. (Hence Proca Lagrangian in Eq. (1) is not gauge-invariant!) Rewrite Proca equations imposing Lorenz gauge condition. 2. Consider generators of some Lie group obeying Lie algebra commutation relations [ X a , X b ] = i f abc X c (2)
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Unformatted text preview: with anti-symmetric structure constants f abc . (a) (5 pts) Prove the Jacobi identity [ X a , [ X b , X c ]] + [ X b , [ X c , X a ]] + [ X c , [ X a , X b ]] = 0 by expanding out the commutators. (b) (5 pts) Use the commutation relation (2) for X a ’s in the Jacobi identity to show that f bcd f ade + f abd f cde + f cad f bde = 0 , which is also often referred to as the Jacobi identity. 3. (5 pts) Using Gell-Mann matrices (and their commutators) Fnd the structure constants f 156 and f 678 of the group SU (3). 4. Using Young tableaux method decompose the following product representations of the group SU (3) into sums of irreducible representations: (a) (5 pts) 8 ⊗ 8 , (b) (5 pts) 8 ⊗ 3 . 1...
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This note was uploaded on 07/28/2011 for the course PHYSICS 880.02 taught by Professor Kovchegov during the Winter '09 term at Ohio State.

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