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Ch. 2 Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Raising and Lowering Indices

Under the sign convention \(\eta_{\mu\nu} = \text{diag}(+1, -1, -1, -1)\), for the 4-vector \(V^\mu = (E, p_x, p_y, p_z) = (5, 1, -2, 3)\), write down all the covariant components \(V_\mu\).

Hint

Compute \(V_\mu = \eta_{\mu\nu} V^\nu\) for each component. Since \(\eta_{\mu\nu}\) is a diagonal matrix, we have \(V_0 = \eta_{00} V^0\), \(V_1 = \eta_{11} V^1\), ... and so on.

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B-2. Inner Product of 4-Vectors

For the 4-vectors \(A^\mu = (4, 1, 0, -1)\) and \(B^\mu = (2, 3, 1, 0)\), calculate the Lorentz-invariant inner product \(A^\mu B_\mu\).

Hint

Use \(A^\mu B_\mu = A^0 B^0 - A^1 B^1 - A^2 B^2 - A^3 B^3\) (see Eq. (2.4) in the text). You may either first find \(B_\mu\) and then contract, or directly apply this expansion.

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B-3. Expanding the Einstein Summation Convention

Expand the following expression explicitly by writing out all terms for \(\mu, \nu = 0, 1, 2, 3\) according to the Einstein summation convention (using the fact that \(\eta_{\mu\nu}\) is a diagonal matrix, keep only the non-zero terms).

\[ \eta_{\mu\nu}\, A^\mu\, B^\nu \]
Hint

\(\eta_{\mu\nu}\) is non-zero only when \(\mu = \nu\). Therefore, all terms with \(\mu \neq \nu\) vanish. Write out the remaining 4 terms.

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B-4. Application of Lorentz Boost

Using the Lorentz boost \(t' = \gamma(t - vx),\, x' = \gamma(x - vt)\) to an inertial frame moving with velocity \(v = 3/5\) (in natural units) in the \(x\) direction, find the transformed coordinates \((t', x')\) of the spacetime point \((t, x) = (5, 3)\).

Hint

First compute the Lorentz factor \(\gamma = 1/\sqrt{1 - v^2}\). Since \(v = 3/5\), we have \(v^2 = 9/25\), \(1 - v^2 = 16/25\). Then substitute into \(t' = \gamma(t - vx)\), \(x' = \gamma(x - vt)\).

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B-5. Calculation of Rapidity

Find the rapidity \(\beta\) corresponding to velocity \(v = 4/5\) (in natural units). Also, verify the values of \(\cosh\beta\) and \(\sinh\beta\).

Hint

From \(v = \tanh\beta\), we get \(\beta = \text{arctanh}(v) = \frac{1}{2}\ln\frac{1+v}{1-v}\). The relations \(\cosh\beta = \gamma\) and \(\sinh\beta = \gamma v\) can also be used.

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B-6. Practice with Index Contraction

Using the Kronecker delta \(\delta^\mu{}_\nu\) (equal to 1 when \(\mu = \nu\), and 0 otherwise), compute the following contractions.

\[ \delta^\mu{}_\nu\, A^\nu = \text{?} \]
\[ \eta_{\mu\nu}\, \eta^{\nu\rho} = \text{?} \]
Hint

First expression: \(\delta^\mu{}_\nu\) plays the role of "replacing an index." Second expression: \(\eta_{\mu\nu}\) and \(\eta^{\nu\rho}\) are inverse matrices of each other. What does the matrix product \((\eta)(\eta^{-1})\) give?

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B-7. Hyperbolic Function Representation of the Boost Matrix

For the \(x\)-direction boost matrix using rapidity \(\beta\)

\[ \Lambda^\mu{}_\nu = \begin{pmatrix} \cosh\beta & -\sinh\beta & 0 & 0 \\ -\sinh\beta & \cosh\beta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

compute the result \(x'^\mu = \Lambda^\mu{}_\nu\, x^\nu\) of boosting the 4-vector \(x^\mu = (3, 1, 0, 0)\) with \(\beta = \ln 2\).

Hint

When \(\beta = \ln 2\), use \(\cosh\beta = \frac{e^\beta + e^{-\beta}}{2} = \frac{2 + 1/2}{2} = \frac{5}{4}\), \(\sinh\beta = \frac{e^\beta - e^{-\beta}}{2} = \frac{2 - 1/2}{2} = \frac{3}{4}\).

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B-8. Dimensional Analysis in Natural Units

In natural units (\(c = 1\), \(\hbar = 1\)), express the dimensions of the following physical quantities as "powers of mass" (\([\text{mass}]^n\)).

(a) Length　　(b) Time　　(c) Energy　　(d) Momentum　　(e) Force

Hint

From \(c = 1\), we have \([\text{length}] = [\text{time}]\). From \(\hbar = 1\), we have \([\text{energy}] \times [\text{time}] = [\text{dimensionless}]\). Therefore \([\text{time}] = [\text{energy}]^{-1} = [\text{mass}]^{-1}\). Force = Energy / Length.

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Medium

M-1. Derivation of the Condition on the Lorentz Transformation Matrix

Show that the condition for the Lorentz transformation \(x'^\mu = \Lambda^\mu{}_\nu\, x^\nu\) to preserve the invariant interval \(\eta_{\mu\nu}\, x^\mu\, x^\nu\) is

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

Furthermore, confirm that the matrix form of this condition is \(\Lambda^T \eta \Lambda = \eta\), and derive from this that \(\det\Lambda = \pm 1\).

Hint

Substitute \(x'^\mu = \Lambda^\mu{}_\alpha\, x^\alpha\) into \(\eta_{\mu\nu}\, x'^\mu\, x'^\nu = \eta_{\alpha\beta}\, x^\alpha\, x^\beta\). Since this must hold for arbitrary \(x^\alpha\), compare the coefficients of \(x^\alpha x^\beta\). For the determinant, compute the determinant of both sides of \(\det(\Lambda^T \eta \Lambda) = \det\eta\).

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M-2. Additivity of Rapidity

Consider two successive boosts in the \(x\) direction. First boost with rapidity \(\beta_1\), then boost with rapidity \(\beta_2\). Show by computing the product of the boost matrices that the rapidity of the composite transformation is \(\beta_1 + \beta_2\). Furthermore, derive from this the velocity addition rule

\[ v = \frac{v_1 + v_2}{1 + v_1 v_2} \]
Hint

Compute the product of the two boost matrices and use the addition theorems for hyperbolic functions: \(\cosh(\beta_1 + \beta_2) = \cosh\beta_1\cosh\beta_2 + \sinh\beta_1\sinh\beta_2\) and \(\sinh(\beta_1 + \beta_2) = \sinh\beta_1\cosh\beta_2 + \cosh\beta_1\sinh\beta_2\). The velocity is given by \(v = \tanh\beta\), and apply the addition theorem for \(\tanh\).

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M-3. Four-Momentum and the Mass-Shell Condition

Define the four-momentum of a relativistic particle as \(p^\mu = (E, \mathbf{p})\).

(a) For a particle of mass \(m\), compute the Lorentz invariant \(p^\mu p_\mu\) and show that the mass-shell condition (on-shell condition)

\[ p^\mu p_\mu = m^2 \]

is equivalent to \(E^2 = \mathbf{p}^2 + m^2\) (in natural units).

(b) State what the mass-shell condition becomes for a massless particle (photon), and derive the relationship between energy and momentum.

(c) Write down how the four-momentum \(p^\mu\) transforms under a Lorentz boost (in the \(x\)-direction with velocity \(v\)), and by boosting from the rest frame \(\mathbf{p} = 0\), derive \(E = \gamma m\) and \(p_x = \gamma m v\).

Hint

(a) Use \(p^\mu p_\mu = \eta_{\mu\nu} p^\mu p^\nu = (p^0)^2 - (p^1)^2 - (p^2)^2 - (p^3)^2 = E^2 - |\mathbf{p}|^2\). (b) Substitute \(m = 0\). (c) Since the four-momentum is a four-vector, it obeys the same Lorentz transformation law as coordinates.

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M-4. Group Structure of Lorentz Transformations

Show that the set of all Lorentz transformations forms a group by verifying the following four conditions.

(i) Closure: The composition of two Lorentz transformations is also a Lorentz transformation. (ii) Associativity: \((\Lambda_1 \Lambda_2)\Lambda_3 = \Lambda_1(\Lambda_2 \Lambda_3)\). (iii) Existence of identity: The identity transformation \(\Lambda^\mu{}_\nu = \delta^\mu{}_\nu\) satisfies the Lorentz transformation condition. (iv) Existence of inverse: For any Lorentz transformation \(\Lambda\), the inverse \(\Lambda^{-1}\) exists and is also a Lorentz transformation.

Furthermore, state that the subgroup satisfying \(\det\Lambda = +1\) and \(\Lambda^0{}_0 \geq 1\) is called the proper orthochronous Lorentz group \(SO^+(1,3)\), and explain that it consists only of transformations (rotations and boosts) that are continuously connected to the identity.

Hint

(i) From \(\Lambda_1^T \eta \Lambda_1 = \eta\) and \(\Lambda_2^T \eta \Lambda_2 = \eta\), show that \((\Lambda_1\Lambda_2)^T \eta (\Lambda_1\Lambda_2) = \eta\). (iv) From \(\det\Lambda = \pm 1 \neq 0\), the inverse matrix exists. Also show that the inverse matrix satisfies the Lorentz condition. The cases \(\det\Lambda = -1\) or \(\Lambda^0{}_0 \leq -1\) correspond to spatial inversion (parity) or time reversal.

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Advanced

A-1. Transformation Rules for Contravariant and Covariant Tensors, and Application to the Electromagnetic Field Tensor

(a) Explain how the Lorentz transformation rule for a rank-2 contravariant tensor \(T^{\mu\nu}\),

\[ T'^{\mu\nu} = \Lambda^\mu{}_\alpha\, \Lambda^\nu{}_\beta\, T^{\alpha\beta} \]

follows as a natural extension of the four-vector transformation rule \(V'^\mu = \Lambda^\mu{}_\nu V^\nu\).

(b) The electromagnetic field tensor \(F^{\mu\nu}\) is an antisymmetric tensor whose components are given by

\[ F^{\mu\nu} = \begin{pmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & -B_z & B_y \\ E_y & B_z & 0 & -B_x \\ E_z & -B_y & B_x & 0 \end{pmatrix} \]

(in natural units). By explicitly computing \(F'^{\mu\nu} = \Lambda^\mu{}_\alpha\, \Lambda^\nu{}_\beta\, F^{\alpha\beta}\), determine how the electric and magnetic fields mix under a boost in the \(x\)-direction with velocity \(v\). In particular, derive \(E_y' = \gamma(E_y - vB_z)\) and \(B_z' = \gamma(B_z - vE_y)\).

(c) Express the Lorentz invariant \(F^{\mu\nu}F_{\mu\nu}\) in terms of the electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\). Discuss the physical meaning of this invariant.

Hint

(a) The transformation rule for a general rank-2 tensor is derived from the transformation of the tensor product \(A^\mu B^\nu\). (b) It is not necessary to compute the full \(4 \times 4\) matrix product; one only needs to evaluate specific values of \(\mu, \nu\) (for example, \(\mu=0, \nu=2\) yields \(E_y'\)). (c) Lower the indices using \(F_{\mu\nu} = \eta_{\mu\alpha}\eta_{\nu\beta}F^{\alpha\beta}\) and then contract. The result is \(2(\mathbf{B}^2 - \mathbf{E}^2)\).

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A-2. Generators of the Lorentz Group and the Lie Algebra

Write an infinitesimal Lorentz transformation as \(\Lambda^\mu{}_\nu = \delta^\mu{}_\nu + \omega^\mu{}_\nu\) (where \(\omega^\mu{}_\nu\) is an infinitesimal parameter).

(a) From the Lorentz condition \(\eta_{\mu\nu}\, \Lambda^\mu{}_\alpha\, \Lambda^\nu{}_\beta = \eta_{\alpha\beta}\), show that \(\omega_{\mu\nu} \equiv \eta_{\mu\alpha}\omega^\alpha{}_\nu\) is antisymmetric (\(\omega_{\mu\nu} = -\omega_{\nu\mu}\)). How many independent parameters are there? State which physical transformation (rotation or boost) each corresponds to.

(b) Introduce the generators \(M^{\mu\nu}\) of the Lorentz group, and state that a finite Lorentz transformation can be written as

\[ \Lambda = \exp\left(-\frac{i}{2}\omega_{\mu\nu} M^{\mu\nu}\right) \]

Verify that the 4-vector representation (4×4 matrix) of \(M^{\mu\nu}\)

\[ (M^{\mu\nu})^\alpha{}_\beta = i(\eta^{\mu\alpha}\delta^\nu{}_\beta - \eta^{\nu\alpha}\delta^\mu{}_\beta) \]

is correct by considering the case of a boost in the \(x\)-direction (\(\omega_{01} = -\omega_{10} = \beta\), all others zero).

(c) Verify the Lie algebra commutation relation satisfied by the generators

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

for a specific set of components (for example \([M^{01}, M^{02}]\)) using the 4-vector representation from (b). Explain that this algebraic structure serves as the starting point for classifying spin (scalar fields, vector fields, spinor fields) in quantum field theory.

Hint

(a) Substitute \(\Lambda^\mu{}_\nu = \delta^\mu{}_\nu + \omega^\mu{}_\nu\) into the Lorentz condition and neglect terms of second order and higher in \(\omega\). The number of independent components of a \(4\times 4\) antisymmetric matrix is \(4 \times 3/2 = 6\). (b) When only \(\omega_{01}\) is nonzero, \(\Lambda = I - \frac{i}{2}(\omega_{01}M^{01} + \omega_{10}M^{10}) = I - i\omega_{01}M^{01}\). Check whether this reproduces the infinitesimal boost \(\Lambda^0{}_1 = -\beta\), etc. (c) Compute the matrix products directly. In quantum field theory, different representations of the Lorentz group (scalar: trivial representation, vector: 4-dimensional representation, spinor: 2-dimensional representation) correspond to fields of different spin.

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