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Ch. 6 Problems

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Basic

Medium

Advanced

Basic

B-1. Antisymmetry of the Field Strength Tensor and Verification of Components

Using the definition \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\), calculate the following.

(a) Write \(F_{01}\) in terms of partial derivatives of \(A_0, A_1\).

(b) Write \(F_{10}\) in the same manner, and verify that \(F_{01} = -F_{10}\).

(c) Write \(F_{12}\) in terms of partial derivatives of \(A_1, A_2\), and by comparing with the matrix in Eq. (6.4), confirm that this corresponds to \(-B_z\).

Hint

Directly substitute the indices into the definition \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\). Use \(\partial_0 = \partial/\partial t\), \(\partial_1 = \partial/\partial x\), etc., and compare with \(E_x = -\partial_0 A_1 - \partial_1 A_0\) (pay attention to sign conventions) and \(B_z = \partial_1 A_2 - \partial_2 A_1\).

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B-2. Expansion of the Lagrangian

Express the Lagrangian density \(\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\) in terms of the electric field \(\mathbf{E}\) and the magnetic field \(\mathbf{B}\). That is, show that

\[ \mathcal{L} = \frac{1}{2}(\mathbf{E}^2 - \mathbf{B}^2) \]

by using the components from Eq. (6.4) and contracting all indices explicitly.

Hint

Compute \(F_{\mu\nu}F^{\mu\nu} = \eta^{\mu\alpha}\eta^{\nu\beta}F_{\mu\nu}F_{\alpha\beta}\). Due to antisymmetry, the only independent components are those with \((\mu,\nu)\) satisfying \(\mu < \nu\). It is clearest to separate the calculation into the \(F_{0i}F^{0i}\) part and the \(F_{ij}F^{ij}\) part. Note the metric \(\eta^{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)\).

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B-3. Invariance of Electric and Magnetic Fields under Gauge Transformations

Under the gauge transformation \(A_\mu \to A_\mu + \partial_\mu \lambda\), use the definitions in Eq. (6.3)

\[ \mathbf{E} = -\nabla A_0 - \frac{\partial \mathbf{A}}{\partial t}, \qquad \mathbf{B} = \nabla \times \mathbf{A} \]

to show by direct calculation that \(\mathbf{E}\) and \(\mathbf{B}\) are each invariant.

Hint

Substitute \(A_0 \to A_0 + \partial_0 \lambda\) and \(\mathbf{A} \to \mathbf{A} + \nabla\lambda\). For \(\mathbf{E}\), use the fact that \(-\nabla(\partial_0 \lambda) - \partial_t(\nabla \lambda)\) vanishes. For \(\mathbf{B}\), use \(\nabla \times (\nabla \lambda) = 0\) (the curl of the gradient of any scalar field is zero).

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B-4. Gauge Transformation Law of the Covariant Derivative

Substitute the transformations \(\psi \to e^{i\alpha(x)}\psi\), \(A_\mu \to A_\mu - \frac{1}{q}\partial_\mu \alpha\) into the covariant derivative \(D_\mu = \partial_\mu + iqA_\mu\), and verify that

\[ D_\mu \psi \to e^{i\alpha(x)} D_\mu \psi \]

holds by reproducing the calculation from the text with your own hands. Write out each intermediate term without omitting any steps.

Hint

Expand \(D_\mu' \psi' = (\partial_\mu + iq A_\mu - i\partial_\mu\alpha)(e^{i\alpha}\psi)\). Apply the product rule to \(\partial_\mu(e^{i\alpha}\psi)\), and confirm that the term \(e^{i\alpha}(i\partial_\mu\alpha)\psi\) cancels with \(-i(\partial_\mu\alpha)e^{i\alpha}\psi\).

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B-5. Calculation of Conjugate Momenta

For the Lagrangian \(\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\), calculate the conjugate momenta \(\pi^\mu = \partial\mathcal{L}/\partial(\partial_0 A_\mu)\) for each component \(\mu = 0, 1, 2, 3\), and verify the following.

(a) \(\pi^0 = 0\)

(b) \(\pi^i = F^{0i} = E^i\) (\(i = 1, 2, 3\))

Hint

Using the result from D2, separating \(\mathcal{L}\) into terms that contain \(\partial_0 A_\mu\) and terms that do not gives a clearer picture. The key to \(\pi^0 = 0\) is that \(F_{0i} = \partial_0 A_i - \partial_i A_0\) does not contain \(\partial_0 A_0\).

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B-6. Transversality Condition for Polarization Vectors

When the wave vector \(\mathbf{k} = k(0, \sin\theta, \cos\theta)\) lies in the \(yz\) plane, explicitly write down two independent polarization vectors \(\boldsymbol{\epsilon}(\mathbf{k}, 1)\), \(\boldsymbol{\epsilon}(\mathbf{k}, 2)\) that satisfy the Coulomb gauge transversality condition \(\mathbf{k} \cdot \boldsymbol{\epsilon}(\mathbf{k}, \lambda) = 0\). Ensure that they also satisfy the orthonormality condition \(\boldsymbol{\epsilon}(\mathbf{k}, \lambda) \cdot \boldsymbol{\epsilon}(\mathbf{k}, \lambda') = \delta_{\lambda\lambda'}\).

Hint

Find two directions perpendicular to \(\mathbf{k}\). One is the \(x\) direction \((1, 0, 0)\), which is obviously orthogonal to \(\mathbf{k}\). The other is taken as the direction within the \(yz\) plane that is orthogonal to \(\mathbf{k}\). Computing the cross product of \(\mathbf{k}\) and the first polarization vector is efficient.

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B-7. Verification of the Bianchi Identity

Verify the Bianchi identity

\[ \partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0 \]

for the case \((\lambda, \mu, \nu) = (0, 1, 2)\) by substituting \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\) into each term, writing them out explicitly, and confirming that the whole expression vanishes.

Hint

Substituting \(F_{12} = \partial_1 A_2 - \partial_2 A_1\), etc., yields 6 terms. Use the commutativity of partial derivatives \(\partial_\mu\partial_\nu = \partial_\nu\partial_\mu\) to show that the terms cancel in pairs.

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B-8. Dispersion Relation of the Photon

Substitute the plane wave solution \(A_i \propto e^{-i(\omega t - \mathbf{k}\cdot\mathbf{x})}\) into the wave equation for a massless field \(\Box A_i = 0\), and derive the dispersion relation \(\omega = |\mathbf{k}|\). Compare this with the dispersion relation \(\omega = \sqrt{|\mathbf{k}|^2 + m^2}\) for a Klein-Gordon field of mass \(m\), and discuss the correspondence with the fact that the photon has zero mass.

Hint

When \(\Box = \partial_t^2 - \nabla^2\) acts on the plane wave, it yields \((-\omega^2 + |\mathbf{k}|^2)\). Set this equal to zero.

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Medium

M-1. Deriving Maxwell's Equations from the Euler-Lagrange Equation

Starting from the Lagrangian density \(\mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\), apply the Euler-Lagrange equation for the field \(A_\nu\)

\[ \partial_\mu \frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)} - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 \]

to derive the equation of motion \(\partial_\mu F^{\mu\nu} = 0\). Furthermore, write out the cases \(\nu = 0\) and \(\nu = i\) in terms of 3-dimensional vectors, and show that they correspond to Gauss's law \(\nabla \cdot \mathbf{E} = 0\) and the Ampère-Maxwell law \(\partial_t \mathbf{E} = \nabla \times \mathbf{B}\), respectively.

Hint

Since \(\mathcal{L}\) does not depend on \(A_\nu\) itself but only on \(\partial_\mu A_\nu\), we have \(\partial\mathcal{L}/\partial A_\nu = 0\). When computing \(\partial\mathcal{L}/\partial(\partial_\mu A_\nu)\), differentiate \(F_{\alpha\beta}F^{\alpha\beta}\) with respect to \(\partial_\mu A_\nu\). Use the fact that \(F_{\alpha\beta}\) is linear in \(\partial_\mu A_\nu\) and simplify using antisymmetry. For \(\nu = 0\), use \(F^{i0} = E^i\), and for \(\nu = i\), rewrite the components \(F^{0i}\) and \(F^{ji}\) in terms of the electric and magnetic fields.

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M-2. Counting Degrees of Freedom in Coulomb Gauge

Following the steps below, demonstrate that the electromagnetic field has 2 physical degrees of freedom.

(a) \(A_\mu\) has 4 components. Explain how the constraint \(\pi^0 = 0\) excludes \(A_0\) from the independent dynamical variables.

(b) Explain how the Coulomb gauge condition \(\nabla \cdot \mathbf{A} = 0\) removes one further degree of freedom from the remaining 3 components, by expressing it in Fourier space as \(\mathbf{k} \cdot \tilde{\mathbf{A}}(\mathbf{k}) = 0\).

(c) In vacuum, show that combining Gauss's law \(\nabla \cdot \mathbf{E} = 0\) with the Coulomb gauge condition leads to \(A_0 = 0\) (boundary condition: \(A_0 \to 0\) at infinity).

(d) Summarize the above and state that the remaining physical degrees of freedom correspond to two transverse polarizations.

Hint

(a) \(\pi^0 = 0\) means that canonical commutation relations cannot be established between \(A_0\) and \(\pi^0\). (b) For a 3-dimensional vector \(\tilde{\mathbf{A}}\), the condition \(\mathbf{k} \cdot \tilde{\mathbf{A}} = 0\) is one scalar constraint. (c) Use the fact that in Coulomb gauge, \(\nabla \cdot \mathbf{E} = -\nabla^2 A_0 - \partial_t(\nabla \cdot \mathbf{A}) = -\nabla^2 A_0 = 0\).

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M-3. Completeness Relation for Polarization Vectors

In the Coulomb gauge, the two polarization vectors \(\boldsymbol{\epsilon}(\mathbf{k}, \lambda)\) (\(\lambda = 1, 2\)) satisfy the orthonormality condition

\[ \boldsymbol{\epsilon}(\mathbf{k}, \lambda) \cdot \boldsymbol{\epsilon}(\mathbf{k}, \lambda') = \delta_{\lambda\lambda'} \]

and the transversality condition \(\mathbf{k} \cdot \boldsymbol{\epsilon}(\mathbf{k}, \lambda) = 0\).

(a) Show that the completeness relation

\[ \sum_{\lambda=1}^{2} \epsilon_i(\mathbf{k}, \lambda)\, \epsilon_j(\mathbf{k}, \lambda) = \delta_{ij} - \frac{k_i k_j}{|\mathbf{k}|^2} \]

holds.

(b) Verify that the right-hand side \(\delta_{ij} - k_i k_j / |\mathbf{k}|^2\) is a transverse projector that removes the projection onto the \(\mathbf{k}\) direction, by contracting with \(k_j\).

Hint

(a) \(\hat{\mathbf{k}} = \mathbf{k}/|\mathbf{k}|\) together with \(\boldsymbol{\epsilon}(\mathbf{k}, 1)\) and \(\boldsymbol{\epsilon}(\mathbf{k}, 2)\) form an orthonormal basis of 3-dimensional space. Use the completeness relation for the 3-dimensional identity matrix: \(\delta_{ij} = \hat{k}_i \hat{k}_j + \sum_\lambda \epsilon_i \epsilon_j\). (b) Compute \((\delta_{ij} - k_i k_j/|\mathbf{k}|^2) k_j\).

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M-4. Canonical Commutation Relations in Coulomb Gauge and the Transverse Delta Function

In the Coulomb gauge, starting from the Fourier expansion (6.20) of \(\mathbf{A}\) and the commutation relations of creation and annihilation operators

\[ [a(\mathbf{k}, \lambda),\, a^\dagger(\mathbf{k}', \lambda')] = (2\pi)^3 \delta^3(\mathbf{k} - \mathbf{k}') \delta_{\lambda\lambda'} \]

derive the equal-time commutation relation

\[ [A_i(\mathbf{x}, t),\, \pi_j(\mathbf{y}, t)] = i\delta_{ij}^{\perp}(\mathbf{x} - \mathbf{y}) \]

Here \(\pi_j = E_j = \dot{A}_j\) (Coulomb gauge), and \(\delta_{ij}^{\perp}\) is the transverse delta function

\[ \delta_{ij}^{\perp}(\mathbf{x} - \mathbf{y}) = \int \frac{d^3k}{(2\pi)^3} \left(\delta_{ij} - \frac{k_i k_j}{|\mathbf{k}|^2}\right) e^{i\mathbf{k}\cdot(\mathbf{x} - \mathbf{y})} \]

Explain the physical reason why the transverse delta function appears instead of the usual \(i\delta_{ij}\delta^3(\mathbf{x}-\mathbf{y})\).

Hint

Substitute the Fourier expansions of \(A_i\) and \(\dot{A}_j\), and use the \([a, a^\dagger]\) commutation relations. The sum over polarization vectors is handled using the completeness relation from S3. For the physical reason, explain that since the Coulomb gauge condition \(\nabla \cdot \mathbf{A} = 0\) eliminates the longitudinal component of \(A_i\), the transverse projection is also reflected in the commutation relations.

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Advanced

A-1. Comparison with the Proca Field (Massive Vector Field)

The Lagrangian for a vector field with mass \(m\) (the Proca Lagrangian) is given by

\[ \mathcal{L}_{\text{Proca}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{2}m^2 A_\mu A^\mu \]

(a) Derive the Euler-Lagrange equations and show that the equation of motion is

\[ \partial_\mu F^{\mu\nu} + m^2 A^\nu = 0 \]

(b) Act with \(\partial_\nu\) on both sides of the above equation of motion and show that \(\partial_\nu A^\nu = 0\) (the Lorenz condition) follows automatically as a consequence of the equation of motion when \(m \neq 0\).

(c) Using this result, argue by counting degrees of freedom that the massive vector field has 3 physical degrees of freedom (2 transverse polarizations + 1 longitudinal polarization).

(d) Discuss qualitatively how the longitudinal polarization disappears and gauge symmetry is restored in the \(m \to 0\) limit. State your expectations for how this discussion relates to the Higgs mechanism, which will be studied from Ch. 17 onward.

Hint

(a) Note that \(\partial\mathcal{L}/\partial A_\nu = m^2 A^\nu\). (b) \(\partial_\nu \partial_\mu F^{\mu\nu}\) vanishes due to the antisymmetry of \(F^{\mu\nu}\). (c) \(A_\mu\) has 4 components, and \(\partial_\nu A^\nu = 0\) provides 1 constraint, giving \(4 - 1 = 3\). Since there is no gauge symmetry, no additional gauge fixing is needed. (d) Since the mass term breaks gauge symmetry, the symmetry is restored as \(m \to 0\), and the gauge degree of freedom increases by one, reducing the physical degrees of freedom from \(3 \to 2\). The Higgs mechanism corresponds to the reverse process (a gauge field acquires mass, and the degrees of freedom increase from \(2 \to 3\)).

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A-2. Lagrangian with Gauge-Fixing Term and the Photon Propagator

To implement the Lorenz gauge condition \(\partial_\mu A^\mu = 0\), we consider the Lagrangian with a gauge-fixing term added:

\[ \mathcal{L}_{\text{gf}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} - \frac{1}{2\xi}(\partial_\mu A^\mu)^2 \]

where \(\xi\) is the gauge parameter.

(a) Derive the Euler-Lagrange equation of motion for \(A_\nu\) from this Lagrangian, and show that

\[ \left[\eta^{\mu\nu}\Box - \left(1 - \frac{1}{\xi}\right)\partial^\mu \partial^\nu\right] A_\nu = 0 \]

(b) Fourier transform to momentum space via \(A_\nu(x) = \int \frac{d^4k}{(2\pi)^4}\, \tilde{A}_\nu(k)\, e^{-ikx}\), and rewrite the above equation as an algebraic equation for \(\tilde{A}_\nu\).

(c) Determine the photon propagator (Feynman propagator) \(D_F^{\mu\nu}(k)\) as the inverse of the above differential operator, and show that

\[ D_F^{\mu\nu}(k) = \frac{-1}{k^2 + i\epsilon}\left[\eta^{\mu\nu} - (1 - \xi)\frac{k^\mu k^\nu}{k^2}\right] \]

In particular, verify that for \(\xi = 1\) (Feynman gauge) the propagator takes its simplest form \(D_F^{\mu\nu} = -\eta^{\mu\nu}/(k^2 + i\epsilon)\).

(d) Write down the propagator in the case \(\xi = 0\) (Landau gauge), and verify that \(k_\mu D_F^{\mu\nu}(k) = 0\) holds. Explain why this is the momentum-space expression of the Lorenz condition \(\partial_\mu A^\mu = 0\).

Hint

(a) When applying the Euler-Lagrange equation to \(\mathcal{L}\) with respect to \(A_\nu\), the contribution from \(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\) is the same as in S1. Compute the contribution from the gauge-fixing term \(-\frac{1}{2\xi}(\partial_\mu A^\mu)^2\). (b) Replace \(\partial_\mu \to -ik_\mu\). (c) To find the inverse, assume \(D_F^{\mu\nu}\) is a linear combination of the two independent tensor structures \(\eta^{\mu\nu}\) and \(k^\mu k^\nu / k^2\), and determine the coefficients from the condition that the product with the differential operator equals \(\eta^\mu{}_\nu\). (d) Substitute \(\xi = 0\) and contract with \(k_\mu\).

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