Skip to content

Ch. 3 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

B-1. Euler-Lagrange Equation from the Lagrangian Density (Klein-Gordon Field)

The Lagrangian density for a real scalar field \(\phi(x)\) is given by

\[ \mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2 \]

Using the Euler-Lagrange equation (3.7), explicitly compute each of the following steps.

(a) Find \(\dfrac{\partial \mathcal{L}}{\partial \phi}\).

(b) Find \(\dfrac{\partial \mathcal{L}}{\partial(\partial_\mu \phi)}\).

(c) Substitute the results of (a) and (b) into the Euler-Lagrange equation to derive the Klein-Gordon equation.

Hint

In (b), rewrite \(\mathcal{L}\) as \(\frac{1}{2}\eta^{\alpha\beta}\partial_\alpha\phi\,\partial_\beta\phi - \frac{1}{2}m^2\phi^2\) and take the partial derivative with respect to \(\partial_\mu\phi\). Kronecker deltas \(\delta^\mu_\alpha\) will appear, and the metric tensor plays the role of raising the index.

→ View solution

B-2. Expanding the Indices of the Kinetic Term

Explicitly expand the kinetic term \(\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi\) of the Lagrangian density into temporal and spatial components using the Minkowski metric \(\eta^{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)\), and show that

\[ \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi = \frac{1}{2}\dot{\phi}^2 - \frac{1}{2}(\nabla\phi)^2 \]

where \(\dot{\phi} = \partial_0\phi\) and \(\nabla\phi = (\partial_1\phi, \partial_2\phi, \partial_3\phi)\).

Hint

Use \(\partial^\mu\phi = \eta^{\mu\nu}\partial_\nu\phi\) to confirm that \(\partial^0\phi = \partial_0\phi\) and \(\partial^i\phi = -\partial_i\phi\) (\(i=1,2,3\)), then take the sum over \(\mu\).

→ View solution

B-3. Dispersion Relation for a Massless Field

From the Lagrangian density \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi\), substitute the plane wave solution

\[ \phi(x) = A\,e^{-ik_\mu x^\mu} = A\,e^{-iEt + i\mathbf{p}\cdot\mathbf{x}} \]

into the resulting equation of motion, and derive the dispersion relation \(E^2 = |\mathbf{p}|^2\). Here, \(k^\mu = (E, \mathbf{p})\).

Hint

Use \(\partial_0\phi = -iE\,\phi\) and \(\partial_i\phi = ip_i\,\phi\) to compute \(\partial_\mu\partial^\mu\phi\).

→ View solution

B-4. Lagrangian with Interaction Term

Given that the Lagrangian density of a real scalar field is

\[ \mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2 - \frac{\lambda}{4!}\phi^4 \]

(where \(\lambda\) is the coupling constant), derive the Euler-Lagrange equation and show that

\[ (\partial_\mu\partial^\mu + m^2)\phi + \frac{\lambda}{3!}\phi^3 = 0 \]
Hint

Since the \(\phi^4\) term does not contain \(\partial_\mu\phi\), it does not contribute to \(\dfrac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\). Use \(\dfrac{\partial}{\partial\phi}\left(\frac{\lambda}{4!}\phi^4\right) = \frac{\lambda}{3!}\phi^3\).

→ View solution

B-5. Component Expression of the d'Alembertian

Write out the d'Alembertian \(\Box \equiv \partial_\mu\partial^\mu\) explicitly in terms of time and spatial derivatives. Furthermore, write the Klein-Gordon equation \((\Box + m^2)\phi = 0\) in \((t, x, y, z)\) components.

Hint

Use \(\partial_\mu\partial^\mu = \eta^{\mu\nu}\partial_\mu\partial_\nu\) with \(\eta^{00} = +1\) and \(\eta^{ii} = -1\).

→ View solution

B-6. Component Expansion of the Continuity Equation

Suppose a conserved current \(j^\mu = (j^0, j^1, j^2, j^3)\) satisfies \(\partial_\mu j^\mu = 0\).

(a) Separate this equation into temporal and spatial components and write it in the form of the continuity equation.

(b) Using Gauss's theorem, show that the conserved charge \(Q = \int d^3x\,j^0\) is time-independent. Assume that \(\mathbf{j}\) falls off sufficiently rapidly at spatial infinity.

Hint

(a) Expand as \(\partial_\mu j^\mu = \partial_0 j^0 + \partial_i j^i = \frac{\partial j^0}{\partial t} + \nabla\cdot\mathbf{j}\). (b) Compute \(\frac{dQ}{dt}\) and use the result from (a) together with Gauss's theorem \(\int d^3x\,\nabla\cdot\mathbf{j} = \oint_{\partial V} \mathbf{j}\cdot d\mathbf{S}\).

→ View solution

B-7. Partial Derivatives of the Complex Scalar Field

We treat the complex scalar field \(\phi(x)\) and its complex conjugate \(\phi^*(x)\) as independent variables. Given the Lagrangian density

\[ \mathcal{L} = \partial_\mu\phi^*\,\partial^\mu\phi - m^2\phi^*\phi \]

compute the following.

(a) \(\dfrac{\partial \mathcal{L}}{\partial \phi^*}\)

(b) \(\dfrac{\partial \mathcal{L}}{\partial(\partial_\mu \phi^*)}\)

(c) Write down the Euler-Lagrange equation for \(\phi^*\).

Hint

Treat \(\phi\) and \(\phi^*\) as independent variables. When differentiating with respect to \(\phi^*\), regard \(\phi\) as a constant. In (b), write \(\partial_\mu\phi^*\,\partial^\mu\phi = \eta^{\alpha\beta}\partial_\alpha\phi^*\,\partial_\beta\phi\) and then differentiate with respect to \(\partial_\mu\phi^*\).

→ View solution

B-8. Construction of Noether Current (Application of the Formula)

According to Noether's theorem, when the Lagrangian density is invariant (\(\delta\mathcal{L} = 0\)) under an infinitesimal transformation \(\phi \to \phi + \delta\phi\) of the field \(\phi\), the conserved current is given by

\[ j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\delta\phi \]

For the Klein-Gordon field \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2\), consider an infinitesimal constant shift \(\delta\phi = \epsilon\) (where \(\epsilon\) is an infinitesimal constant).

(a) Compute \(\delta\mathcal{L}\) under this transformation and verify that this is not a symmetry when \(m \neq 0\).

(b) Verify that this transformation is a symmetry when \(m = 0\), and write down the corresponding conserved current \(j^\mu\).

Hint

Since \(\delta\phi = \epsilon\) is a constant, \(\partial_\mu(\delta\phi) = 0\). Therefore \(\delta(\partial_\mu\phi) = 0\). The variation \(\delta\mathcal{L}\) reduces to \(\frac{\partial\mathcal{L}}{\partial\phi}\,\delta\phi\) only.

→ View solution

Medium

M-1. Derivation of Conjugate Momentum and Hamiltonian Density

For the Lagrangian density of a real scalar field \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2\), perform the following.

(a) Define the momentum density \(\pi(x)\) conjugate to the field \(\phi\) by

\[ \pi(x) \equiv \frac{\partial\mathcal{L}}{\partial\dot{\phi}} \]

and find its explicit expression.

(b) Construct the Hamiltonian density \(\mathcal{H}\) via the Legendre transformation

\[ \mathcal{H} = \pi\dot{\phi} - \mathcal{L} \]

and write it in terms of \(\pi\), \(\phi\), and \(\nabla\phi\).

(c) Explain the physical meaning of each term in the resulting \(\mathcal{H}\) by analogy with the particle mechanics Hamiltonian \(H = T + V\).

Hint

(a) Note that \(\dot{\phi} = \partial_0\phi\), and expand \(\mathcal{L}\) into the form of Eq. (3.9) before differentiating with respect to \(\dot{\phi}\). (b) Use \(\dot{\phi} = \pi\) to eliminate \(\dot{\phi}\) from \(\mathcal{L}\).

→ View solution

M-2. Internal Symmetry and Noether Current of the Complex Scalar Field

The Lagrangian density of the complex scalar field

\[ \mathcal{L} = \partial_\mu\phi^*\,\partial^\mu\phi - m^2\phi^*\phi \]

is invariant under the global \(U(1)\) transformation

\[ \phi(x) \to e^{i\alpha}\phi(x), \qquad \phi^*(x) \to e^{-i\alpha}\phi^*(x) \]

where \(\alpha\) is a real constant.

(a) Using the infinitesimal transformation (\(\alpha \ll 1\)) with \(\delta\phi = i\alpha\phi\) and \(\delta\phi^* = -i\alpha\phi^*\), explicitly verify that \(\delta\mathcal{L} = 0\).

(b) Using Noether's theorem, derive the conserved current \(j^\mu\).

(c) Write down the conserved charge \(Q = \int d^3x\,j^0\) and qualitatively explain why it corresponds to "particle number \(-\) antiparticle number."

Hint

(b) For the complex scalar field, the Noether current takes the form \(j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi + \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi^*)}\delta\phi^*\). Extract the coefficient of \(\alpha\).

→ View solution

M-3. Spacetime Translational Invariance and the Energy-Momentum Tensor

Suppose the Lagrangian density \(\mathcal{L}(\phi, \partial_\mu\phi)\) is invariant under spacetime translations \(x^\mu \to x^\mu + a^\mu\) (where \(a^\mu\) is an infinitesimal constant vector). Under this transformation, the field changes as \(\delta\phi = -a^\nu\partial_\nu\phi\).

(a) Show that the change in the Lagrangian density itself can be written in the form of a total derivative: \(\delta\mathcal{L} = -a^\nu\partial_\nu\mathcal{L} = \partial_\mu(-a^\nu\delta^\mu{}_\nu\mathcal{L})\).

(b) Using the generalization of Noether's theorem (when \(\delta\mathcal{L} = \partial_\mu K^\mu\), the conserved current is \(j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\delta\phi - K^\mu\)), derive the canonical energy-momentum tensor

\[ T^{\mu}{}_\nu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\partial_\nu\phi - \delta^\mu{}_\nu\,\mathcal{L} \]

(c) For the Klein-Gordon field with Lagrangian density \(\mathcal{L} = \frac{1}{2}\partial_\alpha\phi\,\partial^\alpha\phi - \frac{1}{2}m^2\phi^2\), compute \(T^{00}\) and verify that it coincides with the Hamiltonian density \(\mathcal{H}\).

Hint

(a) Since \(\mathcal{L}\) does not depend explicitly on \(x^\mu\), \(\delta\mathcal{L}\) arises only through the variation of the fields. On the other hand, the total change of \(\mathcal{L}\) viewed as a function of \(x\) equals \(a^\nu\partial_\nu\mathcal{L}\). (b) Set \(K^\mu = -a^\nu\delta^\mu{}_\nu\mathcal{L}\), and since \(a^\nu\) is arbitrary, a conserved current is obtained for each \(\nu\).

→ View solution

M-4. Euler-Lagrange Equations for the Dirac Field

The Lagrangian density for the Dirac field \(\psi(x)\) (a 4-component spinor) is given by

\[ \mathcal{L}_{\mathrm{Dirac}} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi \]

where \(\bar{\psi} = \psi^\dagger\gamma^0\) is the Dirac conjugate and \(\gamma^\mu\) are the gamma matrices.

(a) Treating \(\bar{\psi}\) as an independent variable, derive the Euler-Lagrange equation with respect to \(\bar{\psi}\) and obtain the Dirac equation

\[ (i\gamma^\mu\partial_\mu - m)\psi = 0 \]

(b) Derive the Euler-Lagrange equation with respect to \(\psi\) and obtain the adjoint Dirac equation for \(\bar{\psi}\)

\[ \bar{\psi}(i\overleftarrow{\partial}_\mu\gamma^\mu + m) = 0 \]

where \(\overleftarrow{\partial}_\mu\) denotes a derivative acting to the left.

Hint

(a) View \(\mathcal{L}\) as a function of \(\bar{\psi}\) and \(\partial_\mu\bar{\psi}\). Noting that there are no terms containing \(\partial_\mu\bar{\psi}\), the Euler-Lagrange equation reduces to \(\frac{\partial\mathcal{L}}{\partial\bar{\psi}} = 0\). (b) When differentiating with respect to \(\psi\), integration by parts is required.

→ View solution

Advanced

A-1. Generalization of Noether's Theorem

Consider the case where, under the field transformation \(\phi \to \phi + \delta\phi\), the Lagrangian density is not invariant but changes by a total derivative (4-dimensional divergence) expressed using some 4-vector \(K^\mu\):

\[ \delta\mathcal{L} = \partial_\mu K^\mu \]

(a) Show that the variation \(\delta S\) of the action \(S = \int d^4x\,\mathcal{L}\) vanishes in this case as well (under appropriate boundary conditions), and confirm that the equations of motion remain unchanged.

(b) Prove that the modified Noether current

\[ j^\mu = \frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}\,\delta\phi - K^\mu \]

satisfies \(\partial_\mu j^\mu = 0\) when the equations of motion hold.

(c) As an application, consider a Lorentz boost transformation (in the \(x\) direction with infinitesimal rapidity \(\delta\omega\)) for the real scalar field \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - V(\phi)\):

\[ \delta\phi = -\delta\omega\,(t\,\partial_x\phi + x\,\partial_t\phi) \]

Construct the corresponding conserved current. Discuss the physical meaning of the conserved charge.

Hint

(a) \(\delta S = \int d^4x\,\partial_\mu K^\mu\) becomes a surface integral over the boundary by Gauss's theorem, and vanishes due to the boundary conditions. (b) Organize \(\delta\mathcal{L}\) following the same procedure as the derivation of the Euler-Lagrange equations, and compare with \(\partial_\mu K^\mu\). (c) Under a Lorentz boost, \(\delta x^0 = -\delta\omega\,x^1\) and \(\delta x^1 = -\delta\omega\,x^0\), and one uses the scalar field transformation rule \(\delta\phi = -\delta x^\nu\partial_\nu\phi\). The conserved charge corresponds to the boost generator (a quantity related to center-of-mass motion).

→ View solution

A-2. Lagrangian of the Maxwell Field and Gauge Invariance

The Lagrangian density of the electromagnetic field is given by

\[ \mathcal{L}_{\mathrm{Maxwell}} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} \]

where the field strength tensor is \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu\), and \(A_\mu(x)\) is the 4-vector potential.

(a) Derive the Euler-Lagrange equations for \(A_\nu\) and obtain the vacuum Maxwell equations (without sources)

\[ \partial_\mu F^{\mu\nu} = 0 \]

Explicitly show the computation of \(\dfrac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)}\) in the derivation.

(b) Show that \(F_{\mu\nu}\) is invariant under the gauge transformation \(A_\mu(x) \to A_\mu(x) + \partial_\mu\Lambda(x)\) (where \(\Lambda(x)\) is an arbitrary scalar function), and thereby confirm that \(\mathcal{L}_{\mathrm{Maxwell}}\) is also gauge invariant.

(c) Discuss what happens when one attempts to directly apply Noether's theorem to the gauge transformation. In particular, explain how the fact that the gauge transformation parameter \(\Lambda(x)\) is a function of spacetime (a local transformation) relates to Noether's theorem, which assumes a global symmetry, and mention the existence of Noether's second theorem.

Hint

(a) Expand \(F_{\mu\nu}F^{\mu\nu} = (\partial_\mu A_\nu - \partial_\nu A_\mu)(\partial^\mu A^\nu - \partial^\nu A^\mu)\) and differentiate with respect to \(\partial_\mu A_\nu\). Using the antisymmetry \(F_{\mu\nu} = -F_{\nu\mu}\), one obtains \(\frac{\partial\mathcal{L}}{\partial(\partial_\mu A_\nu)} = -F^{\mu\nu}\). (b) Use \(\partial_\mu(\partial_\nu\Lambda) - \partial_\nu(\partial_\mu\Lambda) = 0\) (commutativity of partial derivatives). (c) The Noether current associated with the local parameter \(\Lambda(x)\) is identically conserved (it becomes an identity that holds without using the equations of motion). This is the content of Noether's second theorem, and it is related to the redundant degrees of freedom in gauge theories.

→ View solution