Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 10.2

Spinors — Schwartz M.D. z-lib.org

Physics — QFT · su2-representations

Problem

In this problem you will construct the finite-dimensional irreducible representations of SU(2)\mathrm{SU}(2). By definition, such a representation is a set of three n×nn \times n matrices τ1\tau_1, τ2\tau_2 and τ3\tau_3 satisfying the algebra of the Pauli matrices [τi,τj]=iεijkτk[\tau_i, \tau_j] = i\varepsilon_{ijk}\tau_k. It is also helpful to define the linear combinations τ±=τ1±iτ2\tau^\pm = \tau^1 \pm i\tau^2. (a) In any such representation we can diagonalize τ3\tau_3. Its eigenvectors are nn complex vectors VjV_j with τ3Vj=λjVj\tau_3 V_j = \lambda_j V_j. Show that τ+Vj\tau^+ V_j and τ−Vj\tau^- V_j either vanish or are eigenstates of τ3\tau_3 with eigenvalues λj+1\lambda_j + 1 and λj−1\lambda_j - 1 respectively. (b) Prove that exactly one of the eigenstates Vmax⁡V_{\max} of τ3\tau_3 must satisfy τ+Vmax⁡=0\tau^+ V_{\max} = 0. The eigenvalue λmax⁡=j\lambda_{\max} = j of Vmax⁡V_{\max} is known as the spin. Similarly, there will be an eigenvector Vmin⁡V_{\min} of τ3\tau_3 with τ−Vmin⁡=0\tau^- V_{\min} = 0. (c) Since there are a finite number of eigenvectors, Vmin⁡=(τ−)NVmax⁡V_{\min} = (\tau^-)^N V_{\max} for some integer NN. Prove that N=2JN = 2J so that n=2J+1n = 2J+1. (d) Construct explicitly the five-dimensional representation of SU(2)\mathrm{SU}(2).

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