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I |ψ > = 0 for any constraint χi . 26) and as ψ˙ µ = 0 in the Heisenberg picture, then γ µ is a constant and satisfies the algebra, {γ µ , γ ν } = 2g µν ({ , } → anticommutator). 3. QUANTIZATION OF THE SPINNING PARTICLE. 35 This is the massless Dirac equation. G. 43) 0 −1 36 CHAPTER 2. GRASSMANN VARIABLES Eddington showed that all representations of αi , β in 4-d are unitarily equivalent. 45) Aside: 1. This can be generalized to n dimensions. 2n/2 (n=even) dimension of (α, β) is 2(n−1)/2 (n=odd) 2.

T α∗ T −1 = −α 3. , time). Under the rotation subgroup SO(3), these are spinors with s = 1/2. Oct. 13/99 60 CHAPTER 2. GRASSMANN VARIABLES Chapter 3 Bargmann-Wigner Equations The Bargman-Wigner fields are fields of higher spin (s = 1, 3/2, 2, 5/2, . ) that are built up out of the spin (1/2) fields. e. ψα → spin 1/2 ψαβ = ψβα → spin 1 etc. 2) The wave functions satisfy 2s equations: 0 = 0 = .. 7)  0 e 1 62 CHAPTER 3. BARGMANN-WIGNER EQUATIONS (1) (1) δ1α1 δ1α2 . . α2s (2) δ2α1 δ1α2 δ1α3 . . δ1α2s + δ1α1 δ2α2 δ1α3 .

Spin 1/2 ψαβ = ψβα → spin 1 etc. 2) The wave functions satisfy 2s equations: 0 = 0 = .. 7)  0 e 1 62 CHAPTER 3. BARGMANN-WIGNER EQUATIONS (1) (1) δ1α1 δ1α2 . . α2s (2) δ2α1 δ1α2 δ1α3 . . δ1α2s + δ1α1 δ2α2 δ1α3 . . δ1α2s + ... + δ1α1 δ1α2 δ1α3 . . α2s (3) δ2α1 δ2α2 δ1α3 . . δ1α2s + 2s 2 symmetrized terms. ) (2s+1) (2s+1) δ2α1 δ2α2 δ2α3 . . α2s Similarly, there are (2s + 1) negative energy solutions. ,α2s β2s = (σz )α1 β1 δα2 β2 δα3 β3 . . δα2s β2s + δα1 β1 (σz )α2 β2 δα3 β3 . . δα2s β2s ..

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