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Appendix B Problems

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Basic

Medium

Advanced

Basic

B-1. Antisymmetry of Infinitesimal Lorentz Transformations

Substitute the infinitesimal Lorentz transformation \(\Lambda^\mu{}_\nu = \delta^\mu{}_\nu + \omega^\mu{}_\nu\) into the metric preservation condition

\[ \Lambda^\mu{}_\alpha\,\Lambda^\nu{}_\beta\,\eta^{\alpha\beta} = \eta^{\mu\nu} \]

and, keeping terms only to first order in \(\omega\), derive \(\omega_{\mu\nu} + \omega_{\nu\mu} = 0\) (antisymmetry).

Hint

Substitute \(\Lambda^\mu{}_\alpha = \delta^\mu{}_\alpha + \omega^\mu{}_\alpha\) and expand, discarding the second-order term in \(\omega\): \(\omega^\mu{}_\alpha \omega^\nu{}_\beta \eta^{\alpha\beta}\). Use identities such as \(\delta^\mu{}_\alpha \eta^{\alpha\beta} = \eta^{\mu\beta}\), and lower the index using \(\omega^{\mu\nu} = \omega^\mu{}_\alpha \eta^{\alpha\nu}\) to simplify.

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B-2. Verifying Matrix Elements of Generators

Using the definition from equation (B.10)

\[ (M^{[\rho\sigma]})^{\mu}{}_{\nu} = \eta^{\rho\mu}\,\delta^{\sigma}{}_{\nu} - \eta^{\sigma\mu}\,\delta^{\rho}{}_{\nu} \]

write out all components of the \(4 \times 4\) matrix for \(M^{[23]}\) (the generator of rotations in the \(yz\) plane). Use the metric \(\eta^{\mu\nu} = \mathrm{diag}(-1, +1, +1, +1)\).

Hint

Set \(\rho = 2, \sigma = 3\), and compute \((M^{[23]})^\mu{}_\nu = \eta^{2\mu}\delta^3{}_\nu - \eta^{3\mu}\delta^2{}_\nu\) for each combination of \(\mu, \nu = 0, 1, 2, 3\). Note that \(\eta^{2\mu}\) equals \(+1\) only when \(\mu = 2\) and is \(0\) otherwise.

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B-3. Matrix Representation of Boost Generator \(K^2 = M^{[02]}\)

Compute the \(4 \times 4\) matrix representation of \(K^2 = M^{[02]}\) from equation (B.10), and confirm that the infinitesimal boost transformation in the \(y\) direction

\[ \Lambda^\mu{}_\nu = \delta^\mu{}_\nu + \phi\,(M^{[02]})^\mu{}_\nu \]

gives \(t' \approx t - \phi\, y\), \(y' \approx y - \phi\, t\) (with \(x, z\) unchanged).

Hint

Compute \((M^{[02]})^\mu{}_\nu = \eta^{0\mu}\delta^2{}_\nu - \eta^{2\mu}\delta^0{}_\nu\). Note that \(\eta^{00} = -1\). Substitute the resulting matrix into \(x'^\mu = (\delta^\mu{}_\nu + \phi\,(M^{[02]})^\mu{}_\nu)\,x^\nu\) and read off each component.

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B-4. Direct Calculation of Commutation Relations for Rotation Generators

Using the \(4 \times 4\) matrix representation (B.11), directly calculate the commutator \([J^3, J^1]\) of \(J^3 = M^{[12]}\) and \(J^1 = M^{[23]}\) (obtained in D2) as a matrix product, and verify that \([J^3, J^1] = iJ^2\) holds. Here, also obtain the matrix for \(J^2 = M^{[31]}\) from equation (B.10).

Hint

Calculate the \(4 \times 4\) matrix product \(J^3 \cdot J^1 - J^1 \cdot J^3\) component by component. Since there are few non-zero components, it is efficient to focus only on the non-zero columns. \(M^{[31]}\) is obtained from equation (B.10) by setting \(\rho = 3, \sigma = 1\) (alternatively, one can set \(\rho = 1, \sigma = 3\) and use the relation \(M^{[13]} = -M^{[31]}\)).

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B-5. Recovering Rotation Generators Using the Levi-Civita Symbol

Using the definition \(J^i = \frac{1}{2}\varepsilon^{ijk}M^{[jk]}\), verify that \(J^1\), \(J^2\), \(J^3\) are equal to \(M^{[23]}\), \(M^{[31]}\), \(M^{[12]}\) respectively (expand carefully paying attention to the summation convention).

Hint

For \(J^1 = \frac{1}{2}\varepsilon^{1jk}M^{[jk]}\), the only cases where \(\varepsilon^{1jk} \neq 0\) are \((j,k) = (2,3)\) and \((3,2)\). Since \(M^{[jk]}\) is defined for \(j < k\), use \(M^{[32]} = -M^{[23]}\), and note that \(\varepsilon^{132} = -1\).

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B-6. Expressing \(\mathbf{J}\) and \(\mathbf{K}\) in terms of \(\mathbf{J}_+\) and \(\mathbf{J}_-\)

Starting from the definitions in Eq. (B.18),

\[ \mathbf{J}_+ = \frac{\mathbf{J} + i\mathbf{K}}{2}, \qquad \mathbf{J}_- = \frac{\mathbf{J} - i\mathbf{K}}{2} \]

solve inversely to express \(\mathbf{J}\) and \(\mathbf{K}\) in terms of \(\mathbf{J}_+\) and \(\mathbf{J}_-\).

Hint

This is simply linear algebra involving adding or subtracting the two equations. Compute \(\mathbf{J}_+ + \mathbf{J}_-\) and \(\mathbf{J}_+ - \mathbf{J}_-\).

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B-7. Calculating Dimensions of Representations

For each of the following Lorentz group representations \((j_+, j_-)\), calculate the dimension of the representation space \((2j_+ + 1)(2j_- + 1)\):

(a) \((1, 0)\)　　(b) \((1, 1)\)　　(c) \((3/2, 0)\)　　(d) \((1/2, 1)\)

Hint

The dimension of the spin \(j\) representation of \(\mathrm{SU}(2)\) is \(2j + 1\). Simply calculate for each \(j_+\) and \(j_-\), then take the product.

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B-8. Verification of \([K^1, K^2] = -iJ^3\)

Using the \(4 \times 4\) matrix representation, directly compute \([K^1, K^2]\) (i.e., \([M^{[01]}, M^{[02]}]\)) and verify that the result equals \(-iJ^3 = -iM^{[12]}\).

Hint

\(M^{[01]}\) is given by Eq. (B.12). \(M^{[02]}\) is the one obtained in D3. Compute the matrix product \(M^{[01]}M^{[02]} - M^{[02]}M^{[01]}\). Note that the generators \(M^{[\rho\sigma]}\) in the text are not Hermitian, so be careful with the handling of factors of \(i\). Be aware of the difference between the convention where Lorentz transformations are written as \(\Lambda = \exp(i\omega_{\rho\sigma}M^{\rho\sigma}/2)\) and the convention where they are written as \(\Lambda = \exp(\omega_{\rho\sigma}\mathcal{J}^{\rho\sigma}/2)\). The text adopts the form of Eq. (B.14), \(\Lambda = \exp(i\boldsymbol{\theta}\cdot\mathbf{J} + i\boldsymbol{\phi}\cdot\mathbf{K})\), so the commutation relations include factors of \(i\).

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Medium

M-1. Complete Derivation of \([J^i_+, J^j_-] = 0\)

Using the definitions in equation (B.18), expand \([J^i_+, J^j_-]\) and substitute the Lorentz algebra commutation relations (B.15)–(B.17) to show that all terms cancel and the result is zero. Show each intermediate term explicitly.

Hint

Compute \([J^i_+, J^j_-] = \frac{1}{4}([J^i, J^j] - i[J^i, K^j] + i[K^i, J^j] + [K^i, K^j])\). For the third term \([K^i, J^j]\), reverse the sign of \([J^j, K^i]\) and use (B.16). Note that \(\varepsilon^{jik} = -\varepsilon^{ijk}\).

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M-2. Explicit Form of Boost Generators in the \((1/2, 0)\) Representation

In the left-handed Weyl spinor representation \((j_+, j_-) = (1/2, 0)\), \(\mathbf{J}_+\) is represented by the spin-\(1/2\) representation (\(\boldsymbol{\sigma}/2\), where \(\boldsymbol{\sigma}\) are the Pauli matrices), and \(\mathbf{J}_- = 0\).

(a) Using \(\mathbf{J} = \mathbf{J}_+ + \mathbf{J}_-\) and \(\mathbf{K} = -i(\mathbf{J}_+ - \mathbf{J}_-)\), express the rotation generators \(\mathbf{J}\) and boost generators \(\mathbf{K}\) in this representation in terms of the Pauli matrices.

(b) Compute the \(2 \times 2\) matrix \(\Lambda_L = \exp(i\phi K^3)\) for a boost in the \(z\)-direction with rapidity \(\phi\), and express the result in terms of hyperbolic functions.

(c) Verify that the obtained \(\Lambda_L\) is not unitary, and explain the physical meaning of this fact.

Hint

(a) Substitute \(\mathbf{J}_+ = \boldsymbol{\sigma}/2\) and \(\mathbf{J}_- = 0\) into \(\mathbf{K} = -i(\mathbf{J}_+ - \mathbf{J}_-)\). (b) Compute \(e^{i\phi K^3} = e^{i\phi \cdot (-i\sigma^3/2)} = e^{\phi\sigma^3/2}\) using the diagonal nature of \(\sigma^3\). (c) The exponential of a Hermitian matrix is not unitary. Relate this to the fact that boosts are non-compact transformations.

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M-3. Correspondence Between the \((1/2, 1/2)\) Representation and 4-Vectors

The \((1/2, 1/2)\) representation can be identified with the space of \(2 \times 2\) Hermitian matrices. For an arbitrary 4-vector \(V^\mu\), define

\[ \tilde{V} = V^\mu \sigma_\mu = \begin{pmatrix} V^0 + V^3 & V^1 - iV^2 \\ V^1 + iV^2 & V^0 - V^3 \end{pmatrix} \]

where \(\sigma_\mu = (\mathbf{1}, \boldsymbol{\sigma})\).

(a) Show that \(\det \tilde{V} = -(V^\mu V_\mu)\) (sign reversal of the Minkowski norm).

(b) Explain how Lorentz transformations are realized in the form \(\tilde{V} \to M\,\tilde{V}\,M^\dagger\) (\(M \in \mathrm{SL}(2, \mathbb{C})\)), and verify that \(\det \tilde{V}\) is invariant.

Hint

(a) Directly compute the determinant of the \(2 \times 2\) matrix. (b) Use \(\det(M\tilde{V}M^\dagger) = |\det M|^2 \det \tilde{V}\) together with the \(\mathrm{SL}(2,\mathbb{C})\) condition \(\det M = 1\).

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M-4. Quantitative Proof of Why Spin \(1/3\) is Forbidden

In an irreducible representation of \(\mathrm{SU}(2)\), derive that \(J^2 = j(j+1)\mathbf{1}\) and the eigenvalues of \(J_3\) are \(m = -j, -j+1, \ldots, j-1, j\), using the properties of the raising and lowering operators \(J_\pm = J_1 \pm iJ_2\):

\[ J_\pm |j, m\rangle = \sqrt{j(j+1) - m(m \pm 1)}\,|j, m \pm 1\rangle \]

and the conditions \(J_+ |j, j\rangle = 0\), \(J_- |j, -j\rangle = 0\). In particular, show that \(2j\) must be a non-negative integer.

Hint

From \(J_+|j,j\rangle = 0\), verify that \(j(j+1) - j(j+1) = 0\). From the requirement that the norm of the state obtained by applying \(J_-\) a total of \(n\) times must be non-negative, it follows that \(j - n \geq -j\), i.e., \(n \leq 2j\) is required. Since \(n\) is a non-negative integer, \(2j\) must also be a non-negative integer.

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Advanced

A-1. Covering Group \(\mathrm{SL}(2, \mathbb{C})\) of the Lorentz Group and \(2\pi\) Rotation of Spinors

\(\mathrm{SL}(2, \mathbb{C})\) is the universal covering group of the Lorentz group \(SO^+(1,3)\), and there exists a 2-to-1 homomorphism \(\mathrm{SL}(2, \mathbb{C}) \to SO^+(1,3)\).

(a) For the rotation matrix \(U(\theta) = \exp(i\theta J^3)\) (with \(J^3 = \sigma^3/2\)) of the left-handed Weyl spinor obtained in S2, show that the sign of the spinor is reversed when \(\theta = 2\pi\) (i.e., \(U(2\pi) = -\mathbf{1}\)).

(b) On the other hand, in the 4-vector representation (the \((1/2, 1/2)\) representation from S3), since \(\tilde{V} \to M\tilde{V}M^\dagger\), verify that \(\tilde{V}\) is invariant when \(M = -\mathbf{1}\), and explicitly explain the 2-to-1 correspondence \(\pm M \to \Lambda\).

(c) Discuss how this result corresponds to the physical fact that "a spin-\(1/2\) particle acquires a phase of \(-1\) under a \(360°\) rotation," including its connection to quantum mechanics (the angular momentum discussion in Chapter N).

Hint

(a) Compute \(e^{i\cdot 2\pi \cdot \sigma^3/2} = e^{i\pi\sigma^3}\). Since the eigenvalues of \(\sigma^3\) are \(\pm 1\), rather than using \(e^{i\pi\sigma^3} = \cos\pi\,\mathbf{1} + i\sin\pi\,\sigma^3\), directly use \(e^{i\pi} = -1\) by treating it as a diagonal matrix. (b) Verify that \((-\mathbf{1})\tilde{V}(-\mathbf{1})^\dagger = \tilde{V}\). (c) Discuss the relationship between projective representations and the universal covering group.

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A-2. \((1, 0) \oplus (0, 1)\) Representation and the Electromagnetic Field Tensor

The electromagnetic field tensor \(F^{\mu\nu}\) is an antisymmetric rank-2 tensor with 6 independent components.

(a) Show that the self-dual part \(F^+_{\mu\nu} = \frac{1}{2}(F_{\mu\nu} + \frac{i}{2}\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma})\) and the anti-self-dual part \(F^-_{\mu\nu} = \frac{1}{2}(F_{\mu\nu} - \frac{i}{2}\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma})\) of \(F^{\mu\nu}\) belong to the \((1, 0)\) representation and the \((0, 1)\) representation of the Lorentz group, respectively, by examining the transformation properties of \(\mathbf{E} \pm i\mathbf{B}\).

(b) Confirm by dimension counting that \(F^{\mu\nu}\) as a whole forms the \((1, 0) \oplus (0, 1)\) representation, and explain how this decomposition is consistent with the condition that \(F^+\) and \(F^-\) are complex conjugates of each other under real Lorentz transformations.

(c) Using this result, interpret the symmetry of Maxwell's equations in vacuum (electromagnetic duality) in the language of Lorentz representation theory.

Hint

(a) Compute how the combinations \(\mathbf{F}_\pm = \mathbf{E} \pm i\mathbf{B}\) transform under boosts. Show that \(\mathbf{J}_+\) acts only on \(\mathbf{F}_+\) and \(\mathbf{J}_-\) acts only on \(\mathbf{F}_-\). (b) \((1,0)\) is 3-dimensional and \((0,1)\) is also 3-dimensional, giving a total of 6 dimensions. The reality condition \((F^+)^* = F^-\) yields the real 6 components. (c) Connect the fact that the transformation \(\mathbf{F}_+ \to e^{i\alpha}\mathbf{F}_+\) preserves the source-free Maxwell equations with the correspondence between phase rotations on the \((1,0)\) representation space and duality rotations.

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