Skip to content

Ch. 4 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

B-1. Reading Off Values from the Equal-Time Commutation Relations

Using the equal-time commutation relation

\[ [\hat{\phi}(t, \mathbf{x}),\, \hat{\pi}(t, \mathbf{y})] = i\,\delta^{(3)}(\mathbf{x} - \mathbf{y}) \]

find each of the following quantities.

(a) When \(\mathbf{x} = (1, 2, 3)\) and \(\mathbf{y} = (4, 5, 6)\), find the value of \([\hat{\phi}(t, \mathbf{x}),\, \hat{\pi}(t, \mathbf{y})]\).

(b) Find the value of \([\hat{\phi}(t, \mathbf{x}),\, \hat{\pi}(t, \mathbf{y})]\) integrated over all space with respect to \(\mathbf{y}\), namely

\[ \int d^3y\, [\hat{\phi}(t, \mathbf{x}),\, \hat{\pi}(t, \mathbf{y})] \]

(c) Express \([\hat{\pi}(t, \mathbf{x}),\, \hat{\phi}(t, \mathbf{y})]\) in terms of \([\hat{\phi}(t, \mathbf{x}),\, \hat{\pi}(t, \mathbf{y})]\).

Hint

(a) The Dirac delta function is zero when \(\mathbf{x} \neq \mathbf{y}\). (b) Use the fundamental property of the delta function: \(\int d^3y\, f(\mathbf{y})\,\delta^{(3)}(\mathbf{x}-\mathbf{y}) = f(\mathbf{x})\). (c) Recall the antisymmetry of the commutator: \([A, B] = -[B, A]\).

→ View solution

B-2. Calculation of Commutation Relations for Creation and Annihilation Operators

Using the commutation relations

\[ [\hat{a}_{\mathbf{p}},\, \hat{a}_{\mathbf{q}}^\dagger] = \delta^{(3)}(\mathbf{p} - \mathbf{q}), \qquad [\hat{a}_{\mathbf{p}},\, \hat{a}_{\mathbf{q}}] = 0, \qquad [\hat{a}_{\mathbf{p}}^\dagger,\, \hat{a}_{\mathbf{q}}^\dagger] = 0 \]

compute the following.

(a) \([\hat{a}_{\mathbf{p}},\, \hat{a}_{\mathbf{q}}^\dagger \hat{a}_{\mathbf{q}}]\)

(b) \([\hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}},\, \hat{a}_{\mathbf{q}}^\dagger]\)

(c) \([\hat{a}_{\mathbf{p}},\, (\hat{a}_{\mathbf{q}}^\dagger)^2]\)

Hint

Use the commutator identity \([A, BC] = [A, B]C + B[A, C]\) repeatedly. The procedure is the same as computing the commutation relations of the number operator \(\hat{n} = a^\dagger a\) with \(a\) and \(a^\dagger\) in quantum mechanics.

→ View solution

B-3. Action of the Annihilation Operator on the Vacuum

Using the definition of the vacuum state \(\hat{a}_{\mathbf{p}} |0\rangle = 0\) (for all \(\mathbf{p}\)) and the definition of the one-particle state \(|\mathbf{q}\rangle = \hat{a}_{\mathbf{q}}^\dagger |0\rangle\), compute the following.

(a) \(\hat{a}_{\mathbf{p}} |\mathbf{q}\rangle\)

(b) \(\langle \mathbf{p} | \mathbf{q} \rangle\)

(c) \(\hat{a}_{\mathbf{p}} \hat{a}_{\mathbf{q}} |0\rangle\)

Hint

(a) Decompose \(\hat{a}_{\mathbf{p}} \hat{a}_{\mathbf{q}}^\dagger |0\rangle\) using the commutation relation into \(\hat{a}_{\mathbf{q}}^\dagger \hat{a}_{\mathbf{p}} |0\rangle + [\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}^\dagger] |0\rangle\). (b) Use the result from (a), noting that \(\langle 0 | \hat{a}_{\mathbf{p}}\) is \((\hat{a}_{\mathbf{p}}^\dagger |0\rangle)^\dagger\).

→ View solution

B-4. Verifying Orthogonality of the Fourier Transform

Using the orthogonality of the Fourier transform

\[ \int \frac{d^3x}{(2\pi)^3}\, e^{i(\mathbf{p}-\mathbf{q})\cdot\mathbf{x}} = \delta^{(3)}(\mathbf{p} - \mathbf{q}) \]

compute the following.

(a) Express \(\displaystyle \int \frac{d^3x}{(2\pi)^3}\, e^{i(\mathbf{p}+\mathbf{q})\cdot\mathbf{x}}\) in terms of \(\delta^{(3)}\).

(b) Express \(\displaystyle \int \frac{d^3x}{(2\pi)^3}\, e^{i\mathbf{p}\cdot\mathbf{x}}\, e^{-i\mathbf{q}\cdot\mathbf{x}}\, e^{i\mathbf{k}\cdot\mathbf{x}}\) in terms of \(\delta^{(3)}\).

(c) Compute \(\displaystyle \int d^3x\, \int \frac{d^3p}{(2\pi)^3}\, e^{i\mathbf{p}\cdot(\mathbf{x}-\mathbf{y})} f(\mathbf{x})\), where \(f(\mathbf{x})\) is an arbitrary smooth function.

Hint

(a) Regard the exponent as \(e^{i(\mathbf{p}-(-\mathbf{q}))\cdot\mathbf{x}}\). (b) Combine the exponents using the exponent rule. (c) Consider the order of the \(\mathbf{x}\) integration and the \(\mathbf{p}\) integration, then use the definition of the delta function.

→ View solution

B-5. Explicit Calculation of \(\omega_{\mathbf{p}}\)

Using the energy-momentum relation

\[ \omega_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2 + m^2} \]

find \(\omega_{\mathbf{p}}\) in the following cases (in natural units \(\hbar = c = 1\)).

(a) \(\mathbf{p} = \mathbf{0}\) (rest mode)

(b) \(|\mathbf{p}| = m\)

(c) \(|\mathbf{p}| \gg m\) (ultra-relativistic limit). Approximate \(\omega_{\mathbf{p}}\) to leading order in \(|\mathbf{p}|\).

(d) \(m = 0\) (massless case)

Hint

(a) Simply substitute \(|\mathbf{p}| = 0\). (c) Write \(\sqrt{p^2 + m^2} = |p|\sqrt{1 + m^2/p^2}\) and perform a Taylor expansion for \(m/|p| \ll 1\).

→ View solution

B-6. Verifying Hermiticity of the Field Operator

Show that the mode expansion of the real scalar field

\[ \hat{\phi}(\mathbf{x}) = \int \frac{d^3p}{(2\pi)^{3/2}} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \left(\hat{a}_{\mathbf{p}}\,e^{i\mathbf{p}\cdot\mathbf{x}} + \hat{a}_{\mathbf{p}}^\dagger\, e^{-i\mathbf{p}\cdot\mathbf{x}}\right) \]

satisfies \(\hat{\phi}^\dagger(\mathbf{x}) = \hat{\phi}(\mathbf{x})\) by computing its Hermitian conjugate \(\hat{\phi}^\dagger(\mathbf{x})\).

Hint

Taking the Hermitian conjugate gives \(\hat{a}_{\mathbf{p}} \to \hat{a}_{\mathbf{p}}^\dagger\) and \(e^{i\mathbf{p}\cdot\mathbf{x}} \to e^{-i\mathbf{p}\cdot\mathbf{x}}\) (since \(\mathbf{x}\) is real). In the resulting expression, perform the change of integration variable \(\mathbf{p} \to -\mathbf{p}\) and use \(\omega_{\mathbf{p}} = \omega_{-\mathbf{p}}\).

→ View solution

B-7. Derivation of the Mode Expansion of the Conjugate Momentum Density

Given the mode expansion of the field (in the form including time dependence)

\[ \hat{\phi}(t, \mathbf{x}) = \int \frac{d^3p}{(2\pi)^{3/2}} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \left(\hat{a}_{\mathbf{p}}\,e^{i(\mathbf{p}\cdot\mathbf{x} - \omega_{\mathbf{p}} t)} + \hat{a}_{\mathbf{p}}^\dagger\, e^{-i(\mathbf{p}\cdot\mathbf{x} - \omega_{\mathbf{p}} t)}\right) \]

compute the conjugate momentum density \(\hat{\pi}(t, \mathbf{x}) = \partial_0 \hat{\phi}(t, \mathbf{x})\) and derive Eq. (4.12).

Hint

Carry out the time derivative using \(\partial_0 e^{-i\omega_{\mathbf{p}} t} = -i\omega_{\mathbf{p}}\, e^{-i\omega_{\mathbf{p}} t}\) and \(\partial_0 e^{+i\omega_{\mathbf{p}} t} = +i\omega_{\mathbf{p}}\, e^{+i\omega_{\mathbf{p}} t}\). Compare with the expression at \(t = 0\).

→ View solution

B-8. Verification of the Discrete→Continuous Correspondence

When placed in a box of volume \(V = L^3\), momentum is discretized as \(\mathbf{p} = \frac{2\pi}{L}\mathbf{n}\) (\(\mathbf{n} \in \mathbb{Z}^3\)). Verify the following correspondences.

(a) Show that in the continuous limit \(L \to \infty\), \(\displaystyle\sum_{\mathbf{n}} \to \int \frac{V\, d^3p}{(2\pi)^3}\).

(b) Starting from the discrete commutation relation \([\hat{a}_{\mathbf{n}}, \hat{a}_{\mathbf{m}}^\dagger] = \delta_{\mathbf{n}, \mathbf{m}}\), what scaling relationship between \(\hat{a}_{\mathbf{p}}\) and \(\hat{a}_{\mathbf{n}}\) is required so that the continuous limit reproduces \([\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}^\dagger] = \delta^{(3)}(\mathbf{p} - \mathbf{q})\)?

Hint

(a) Regard the discrete sum as a Riemann sum. The spacing between adjacent momenta is \(\Delta p = 2\pi/L\). (b) Use the relation \(\delta_{\mathbf{n},\mathbf{m}} \to \frac{(2\pi)^3}{V}\delta^{(3)}(\mathbf{p}-\mathbf{q})\) to derive a relationship such as \(\hat{a}_{\mathbf{p}} = \sqrt{V/(2\pi)^3}\, \hat{a}_{\mathbf{n}}\).

→ View solution

Medium

M-1. Derivation of Creation and Annihilation Operator Commutation Relations from Equal-Time Commutation Relations

Substitute the field mode expansions (4.11) and (4.12) into the equal-time commutation relation \([\hat{\phi}(t, \mathbf{x}), \hat{\phi}(t, \mathbf{y})] = 0\), and show that the commutation relations for the creation and annihilation operators

\[ [\hat{a}_{\mathbf{p}},\, \hat{a}_{\mathbf{q}}] = 0, \qquad [\hat{a}_{\mathbf{p}}^\dagger,\, \hat{a}_{\mathbf{q}}^\dagger] = 0 \]

are required.

Specifically, derive what conditions are imposed on \([\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}]\) in order for \([\hat{\phi}(\mathbf{x}), \hat{\phi}(\mathbf{y})] = 0\) to hold.

Hint

Expanding \([\hat{\phi}(\mathbf{x}), \hat{\phi}(\mathbf{y})]\) yields terms containing \([\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}]\) and \([\hat{a}_{\mathbf{p}}^\dagger, \hat{a}_{\mathbf{q}}^\dagger]\). Use the orthogonality of Fourier transforms to extract the condition that the commutators vanish for arbitrary \(\mathbf{p}, \mathbf{q}\). The relation \([\hat{a}_{\mathbf{p}}^\dagger, \hat{a}_{\mathbf{q}}^\dagger] = ([\hat{a}_{\mathbf{q}}, \hat{a}_{\mathbf{p}}])^\dagger\) can also be used.

→ View solution

M-2. Expression of the Hamiltonian in Terms of Creation and Annihilation Operators

Substitute the mode expansions (4.11), (4.12) into the Klein-Gordon field Hamiltonian

\[ H = \int d^3x\, \mathcal{H} = \int d^3x\, \left[\frac{1}{2}\hat{\pi}^2 + \frac{1}{2}(\nabla\hat{\phi})^2 + \frac{1}{2}m^2\hat{\phi}^2\right] \]

and rewrite it in the following form:

\[ H = \int \frac{d^3p}{(2\pi)^3}\, \omega_{\mathbf{p}} \left(\hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}} + \frac{1}{2}\,\delta^{(3)}(\mathbf{0})\right) \]

At each step, clearly state where the orthogonality of the Fourier transform (4.15) and the commutation relations (4.13) are used.

Hint

Write out the three terms \(\hat{\pi}^2\), \((\nabla\hat{\phi})^2\), and \(m^2\hat{\phi}^2\) separately using the mode expansion. When \(\nabla\) acts on \(e^{i\mathbf{p}\cdot\mathbf{x}}\), it produces a factor of \(i\mathbf{p}\). Using the orthogonality of the Fourier transform in the \(\mathbf{x}\) integration yields \(\delta^{(3)}(\mathbf{p} \pm \mathbf{q})\), and the terms separate into those of the type \(\hat{a}_{\mathbf{p}} \hat{a}_{-\mathbf{p}}\) and those of the type \(\hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}}\). Verify that the former cancels when the contributions from all three terms are combined.

→ View solution

M-3. Zero-Point Energy and Normal Ordering

(a) Explain what physical problem the \(\frac{1}{2}\delta^{(3)}(\mathbf{0})\) term in the result of Standard Problem S2 causes. In particular, calculate the vacuum energy \(\langle 0 | H | 0 \rangle\) and show that it diverges.

(b) Define normal ordering \(:\!\hat{O}\!:\) as "the operation of rearranging all creation operators to the left of all annihilation operators." Show that for the normal-ordered Hamiltonian

\[ :\!H\!: \;= \int \frac{d^3p}{(2\pi)^3}\, \omega_{\mathbf{p}}\, \hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}} \]

we have \(\langle 0 | :\!H\!: | 0 \rangle = 0\).

(c) Normal ordering removes the absolute value of the zero-point energy, but differences in zero-point energy are physically observable. Briefly explain this claim in the context of the Casimir effect.

Hint

(a) Using \(\hat{a}_{\mathbf{p}} |0\rangle = 0\), we have \(\langle 0 | \hat{a}_{\mathbf{p}}^\dagger \hat{a}_{\mathbf{p}} | 0 \rangle = 0\), but the \(\frac{1}{2}\delta^{(3)}(\mathbf{0})\) term remains. \(\delta^{(3)}(\mathbf{0})\) corresponds to the box volume \(V/(2\pi)^3\), and furthermore the integral over \(\omega_{\mathbf{p}}\) is ultraviolet divergent. (c) The presence or absence of boundary conditions changes the allowed modes, and the difference in zero-point energy manifests as a finite force.

→ View solution

M-4. Quantization of the Complex Scalar Field and Particles/Antiparticles

Consider the Lagrangian density of a complex scalar field:

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

(a) Treating \(\phi\) and \(\phi^*\) as independent fields, find the conjugate momentum densities \(\pi_\phi\) and \(\pi_{\phi^*}\) for each.

(b) The mode expansion of the complex scalar field is written using two types of creation and annihilation operators \(\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{p}}^\dagger\) and \(\hat{b}_{\mathbf{p}}, \hat{b}_{\mathbf{p}}^\dagger\) as

\[ \hat{\phi}(\mathbf{x}) = \int \frac{d^3p}{(2\pi)^{3/2}} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \left(\hat{a}_{\mathbf{p}}\,e^{i\mathbf{p}\cdot\mathbf{x}} + \hat{b}_{\mathbf{p}}^\dagger\, e^{-i\mathbf{p}\cdot\mathbf{x}}\right) \]

Express \(\hat{\phi}^\dagger(\mathbf{x})\) in terms of \(\hat{a}_{\mathbf{p}}^\dagger\) and \(\hat{b}_{\mathbf{p}}\).

(c) Explain the physical difference between particles created by \(\hat{a}_{\mathbf{p}}^\dagger\) and particles created by \(\hat{b}_{\mathbf{p}}^\dagger\), from the perspective of the sign of the Noether charge corresponding to the \(U(1)\) symmetry \(\phi \to e^{i\alpha}\phi\) studied in Ch. 3.

Hint

(a) Compute \(\pi_\phi = \partial\mathcal{L}/\partial(\partial_0 \phi)\). Differentiate treating \(\phi\) and \(\phi^*\) as independent variables. (b) Simply take the Hermitian conjugate of \(\hat{\phi}\). Note that unlike the real scalar field, \(\hat{\phi}^\dagger \neq \hat{\phi}\). (c) When rewriting the Noether charge \(Q = i\int d^3x\, (\phi^* \dot{\phi} - \dot{\phi}^* \phi)\) in terms of creation and annihilation operators, the terms \(\hat{a}^\dagger \hat{a}\) and \(\hat{b}^\dagger \hat{b}\) acquire opposite signs.

→ View solution

Advanced

A-1. Quantitative Calculation of the 1-Dimensional Casimir Effect

Consider a massless (\(m = 0\)) scalar field in one spatial dimension, with perfectly reflecting walls (Dirichlet boundary conditions \(\hat{\phi}(0) = \hat{\phi}(L) = 0\)) placed at \(x = 0\) and \(x = L\).

(a) Show that the allowed momentum modes between the walls are \(p_n = n\pi/L\) (\(n = 1, 2, 3, \ldots\)).

(b) The zero-point energy between the walls can be formally written as

\[ E(L) = \frac{1}{2}\sum_{n=1}^{\infty} \omega_n = \frac{\pi}{2L}\sum_{n=1}^{\infty} n \]

Replace this divergent sum with a finite value using zeta function regularization. Specifically, use the analytic continuation of the Riemann zeta function \(\zeta(s) = \sum_{n=1}^{\infty} n^{-s}\) to \(s = -1\), namely \(\zeta(-1) = -1/12\).

(c) From the obtained Casimir energy, find the force \(F = -dE/dL\) acting between the walls, and determine whether this force is attractive or repulsive.

(d) (Advanced discussion) In this calculation, the ultraviolet divergence was handled by regularization. From the perspective of "the difference in zero-point energy with and without boundary conditions," discuss why it is physically justified to discard the divergent part and extract only the finite part.

Hint

(a) From the Dirichlet boundary conditions, only modes of the form \(\sin(p_n x)\) are allowed. (b) Replace \(\sum n\) with \(\sum n^{-s}\) and substitute \(s = -1\). (c) Differentiate \(E(L) = -\pi/(24L)\) with respect to \(L\). (d) Consider the difference between the zero-point energy without walls (continuous modes) and with walls (discrete modes), and argue that the divergent parts cancel. You may verify this explicitly by introducing an exponential cutoff \(e^{-\epsilon n}\).

→ View solution

A-2. Lorentz Invariance of Field Commutation Relations and Causality

In canonical quantization, time is treated specially by imposing equal-time commutation relations, but the resulting theory must ultimately be Lorentz invariant. In this problem, we examine the causal structure of field commutation relations.

(a) Show that when the spacetime interval between two points \(x - y\) is spacelike (\((x - y)^2 < 0\)), an appropriate Lorentz transformation can make \(x^0 = y^0\) (simultaneous).

(b) Using the equal-time commutation relation for a real scalar field \([\hat{\phi}(t, \mathbf{x}), \hat{\phi}(t, \mathbf{y})] = 0\), argue that for two spacelike-separated points

\[ [\hat{\phi}(x), \hat{\phi}(y)] = 0 \qquad \text{for } (x-y)^2 < 0 \]

holds. Explain why this is called the microcausality condition.

(c) Show that if one attempts to quantize the Klein-Gordon field with Fermi statistics (anticommutation relations) instead of Bose statistics (commutation relations), microcausality is violated.

(Hint: Compute the anticommutator \(\{\hat{\phi}(\mathbf{x}), \hat{\phi}(\mathbf{y})\}\) and verify that it does not vanish for spacelike-separated points.)

This result illustrates one aspect of the spin-statistics theorem—fields with integer spin obey Bose statistics, while fields with half-integer spin obey Fermi statistics.

Hint

(a) For a spacelike interval, \(|\Delta \mathbf{x}| > |\Delta t|\), so a Lorentz transformation with boost velocity \(v = \Delta t / |\Delta \mathbf{x}| < 1\) can make the events simultaneous. See also the discussion of Lorentz transformations in Ch. 2. (b) If the interval is spacelike, there exists an inertial frame in which the events are simultaneous, where the commutation relation vanishes. The commutation relation of a Lorentz-invariant scalar field is independent of the coordinate system. (c) When using anticommutation relations, contributions from \([\hat{a}_{\mathbf{p}}, \hat{a}_{\mathbf{q}}^\dagger]_+\) remain in \(\{\hat{\phi}(\mathbf{x}), \hat{\phi}(\mathbf{y})\}\), and it does not vanish even for spacelike separation.

→ View solution