An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 3.1

The Dirac Field — M. Peskin, D. Schroeder (Perseus, 1995)

Physics — QFT · lorentz-group-algebra

Problem

Lorentz group. Recall from Eq. (3.17) the Lorentz commutation relations, [Jμν,Jρσ]=i(gνρJμσ−gμρJνσ−gνσJμρ+gμσJνρ).[J^{\mu\nu}, J^{\rho\sigma}] = i(g^{\nu\rho}J^{\mu\sigma} - g^{\mu\rho}J^{\nu\sigma} - g^{\nu\sigma}J^{\mu\rho} + g^{\mu\sigma}J^{\nu\rho}). (a) Define the generators of rotations and boosts as Li=12ϵijkJjk,Ki=J0i,L^i = \frac{1}{2}\epsilon^{ijk}J^{jk}, \qquad K^i = J^{0i}, where i,j,k=1,2,3i, j, k = 1, 2, 3. An infinitesimal Lorentz transformation can then be written Φ→(1−iθ⋅L−iβ⋅K)Φ.\Phi \to (1 - i\boldsymbol{\theta}\cdot\mathbf{L} - i\boldsymbol{\beta}\cdot\mathbf{K})\Phi. Write the commutation relations of these vector operators explicitly. (For example, [Li,Lj]=iϵijkLk[L^i, L^j] = i\epsilon^{ijk}L^k.) Show the combinations J+=12(L+iK)andJ−=12(L−iK)\mathbf{J}_+ = \frac{1}{2}(\mathbf{L} + i\mathbf{K}) \quad \text{and} \quad \mathbf{J}_- = \frac{1}{2}(\mathbf{L} - i\mathbf{K}) commute with one another and separately satisfy the commutation relations of angular momentum. (b) The finite-dimensional representations of the rotation group correspond precisely to the allowed values for angular momentum: integers or half-integers. The result of part (a) implies that all finite-dimensional representations of the Lorentz group correspond to pairs of integers or half integers, (j+,j−)(j_+, j_-), corresponding to pairs of representations of the rotation group. Using the fact that J=σ/2\mathbf{J} = \boldsymbol{\sigma}/2 in the spin-1/2 representation of angular momentum, write explicitly the transformation laws of the 2-component objects transforming according to the (12,0)(\frac{1}{2}, 0) and (0,12)(0, \frac{1}{2}) representations of the Lorentz group. Show that these correspond precisely to the transformations of ψL\psi_L and ψR\psi_R given in (3.37). (c) The identity σT=−σ2σσ2\boldsymbol{\sigma}^T = -\sigma^2\boldsymbol{\sigma}\sigma^2 allows us to rewrite the ψL\psi_L transformation in the unitarily equivalent form ψ′→ψ′(1+iθ⋅σ2+β⋅σ2),\psi' \to \psi'(1 + i\boldsymbol{\theta}\cdot\frac{\boldsymbol{\sigma}}{2} + \boldsymbol{\beta}\cdot\frac{\boldsymbol{\sigma}}{2}), where ψ′=ψLTσ2\psi' = \psi_L^T\sigma^2. Using this law, we can represent the object that transforms as (12,12)(\frac{1}{2}, \frac{1}{2}) as a 2×22 \times 2 matrix that has the ψR\psi_R transformation law on the left and, simultaneously, the transposed ψL\psi_L transformation on the right. Parametrize this matrix as (V0+V3V1−iV2V1+iV2V0−V3).\begin{pmatrix} V^0 + V^3 & V^1 - iV^2 \\ V^1 + iV^2 & V^0 - V^3 \end{pmatrix}. Show that the object VμV^\mu transforms as a 4-vector.

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