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Appendix D Problems

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Table of Contents

Basic

Medium

Basic

B-1. Additive Group of Integers

Verify that integer addition \((\mathbb{Z}, +)\) satisfies the four group axioms.

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B-2. Multiplicative Group of Positive Real Numbers

Verify that the set of all positive real numbers with multiplication \((\mathbb{R}^+, \times)\) forms a group. What are the identity element and the inverse element?

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B-3. Multiplication of Integers Does Not Form a Group

Show that \((\mathbb{Z}, \times)\), the set of all integers \(\mathbb{Z}\) equipped with ordinary multiplication \(\times\) as the operation, does not form a group, by checking the group axioms (closure, associativity, identity element, inverse element) one by one.

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B-4. Group Conditions of \(U(1)\)

Verify that the elements \(e^{i\theta}\) of \(U(1)\) satisfy the group axioms.

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B-5. 3-Dimensional \(z\)-Axis Rotation

The rotation matrix about the \(z\)-axis in 3-dimensional space is given by

\[ R_z(\theta) = \begin{pmatrix}\cos\theta & -\sin\theta & 0\\\sin\theta & \cos\theta & 0\\0 & 0 & 1\end{pmatrix} \]

Show that \(R_z(\theta_1) R_z(\theta_2) = R_z(\theta_1 + \theta_2)\) holds by explicitly computing the matrix product.

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B-6. Commutation Relation of Pauli Matrices \([\sigma_1, \sigma_2]\)

For the Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\), verify that \([\sigma_1, \sigma_2] = 2i\sigma_3\) by explicitly computing the matrix products.

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B-7. Commutation Relations of Pauli Matrices (All Pairs)

Compute \([\sigma_2, \sigma_3]\) and \([\sigma_3, \sigma_1]\) as well, and verify that \([\sigma_i, \sigma_j] = 2i\varepsilon_{ijk}\sigma_k\) holds for all combinations.

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B-8. Antisymmetry of the Commutator

Show from the definition that \([A, B] = -[B, A]\) (antisymmetry of the commutator).

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B-9. Eigenvalues and Eigenvectors of Spin 1/2

The state of a spin \(1/2\) particle is represented by a two-component vector \(|\psi\rangle = \begin{pmatrix}\alpha\\\beta\end{pmatrix}\). Find the eigenvalues and eigenvectors of \(\sigma_3\), and confirm that they correspond to "spin up" and "spin down."

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Medium

M-1. Exponential Representation of \(SU(2)\) Elements

Explicitly compute the \(2 \times 2\) matrix \(U = e^{i\theta\sigma_3/2}\) (using \(e^{iA} = I + iA + (iA)^2/2! + \cdots\); since \(\sigma_3\) is a diagonal matrix, this is straightforward). Verify that \(U^\dagger U = I\) and \(\det U = 1\).

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M-2. Jacobi Identity (Verification with Pauli Matrices)

Using the Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\), verify by explicit calculation that the Jacobi identity

\[ [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0 \]

holds for the case \(A = \sigma_1,\; B = \sigma_2,\; C = \sigma_3\).

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M-3. Parameter Count of \(SU(N)\)

Show that the number of parameters of \(SU(N)\) is \(N^2 - 1\) by the argument "subtract 1 from the \(N^2\) parameters of an \(N \times N\) unitary matrix due to the condition \(\det = 1\)." Verify this for the cases \(N = 2, 3\).

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