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Ch. 1 Problems

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Basic

Medium

Advanced

Basic

B-1. Substituting a plane wave solution into the Klein-Gordon equation to derive the dispersion relation

\[ \frac{\partial^2 \phi}{\partial t^2} - \nabla^2 \phi + m^2 \phi = 0 \]

Substitute the plane wave solution \(\phi(\mathbf{x}, t) = A\, e^{i(\mathbf{p} \cdot \mathbf{x} - Et)}\) into the above equation and derive the dispersion relation \(E^2 = |\mathbf{p}|^2 + m^2\). Show the calculation of each derivative explicitly.

Hint

Computing \(\partial^2 \phi / \partial t^2\) gives \((-iE)^2 \phi = -E^2 \phi\). For \(\nabla^2 \phi\), similarly consider the square of \((i\mathbf{p})\).

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B-2. Calculation of Probability Density for Negative Energy Solutions

\[ \rho = \frac{i}{2m}\left(\phi^* \frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t}\right) \]

Substitute the negative energy solution \(\phi = A\, e^{i(\mathbf{p} \cdot \mathbf{x} + |E|t)}\) (where \(E = -|E|\), \(|E| = \sqrt{|\mathbf{p}|^2 + m^2}\)) into this expression, and express \(\rho\) in terms of \(|A|^2\), \(|E|\), and \(m\). Confirm that \(\rho < 0\).

Hint

The time derivative of the plane wave with \(E = -|E|\) substituted gives \(\partial\phi/\partial t = i|E|\,\phi\). Follow the same procedure as the derivation of Eq. (1.10), paying attention to the sign of \(E\).

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B-3. Anticommutation Relations from the Dirac Equation Condition (1.12)

Using

\[ \{\alpha^i, \alpha^j\} = 2\delta^{ij}, \qquad \{\alpha^i, \beta\} = 0, \qquad \beta^2 = 1 \]

show the following:

(a) \((\alpha^1)^2 = 1\)

(b) \(\alpha^1 \beta = -\beta \alpha^1\)

(c) The eigenvalues of \(\alpha^i\) are only \(\pm 1\) (Hint: use \((\alpha^i)^2 = 1\))

Hint

(a) Set \(i = j\) in \(\{\alpha^i, \alpha^j\} = 2\delta^{ij}\). (c) If \(A^2 = 1\) and \(A\mathbf{v} = \lambda \mathbf{v}\), then \(A^2 \mathbf{v} = \lambda^2 \mathbf{v} = \mathbf{v}\), so \(\lambda^2 = 1\).

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B-4. Proof of the Continuity Equation Identity

\[ \phi^* \frac{\partial^2 \phi}{\partial t^2} - \phi \frac{\partial^2 \phi^*}{\partial t^2} = \frac{\partial}{\partial t}\left(\phi^* \frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t}\right) \]
Hint

Differentiate the right-hand side with respect to time and expand. Use identities such as \(\frac{\partial}{\partial t}(\phi^* \frac{\partial \phi}{\partial t}) = \frac{\partial \phi^*}{\partial t}\frac{\partial \phi}{\partial t} + \phi^* \frac{\partial^2 \phi}{\partial t^2}\), and find the terms that cancel.

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B-5. Calculation of the Compton Wavelength

  • \(\hbar = 1.055 \times 10^{-34}\) J·s
  • \(m_e = 9.109 \times 10^{-31}\) kg
  • \(c = 2.998 \times 10^{8}\) m/s

Furthermore, calculate the Compton wavelength of the proton (\(m_p = 1.673 \times 10^{-27}\) kg) and compare it with the electron case.

Hint

Simply substitute the numerical values into \(\lambda_C = \hbar/(mc)\). The Compton wavelength of the proton should be approximately \(m_e/m_p \approx 1/1836\) times that of the electron.

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B-6. Mass Dimension in Natural Units

(a) Time \(t\)

(b) Length \(x\)

(c) Mass \(m\)

(d) The field \(\phi(\mathbf{x}, t)\) of the Klein-Gordon equation (in 3+1 dimensions, using the fact that the action \(S = \int d^4x\,\mathcal{L}\) with Lagrangian density \(\mathcal{L} = \frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi - \frac{1}{2}m^2\phi^2\) is dimensionless)

Hint

In \(\hbar = c = 1\), we have \([E] = [p] = [m] = 1\) (mass dimension 1) and \([x] = [t] = -1\) (mass dimension \(-1\)). Since the action \(S\) has \([\hbar] = 0\) (dimensionless), determine \([\mathcal{L}]\) from \([d^4x] + [\mathcal{L}] = 0\), and then determine \([\phi]\).

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B-7. Time Scale of Pair Creation from the Uncertainty Relation

(a) Estimate the maximum time \(\Delta t\) during which an electron-positron pair (rest energy \(2m_e c^2 \approx 1.022\) MeV) can be virtually created.

(b) Calculate the distance light travels during that time and compare it with the Compton wavelength.

Hint

(a) \(\Delta t \sim \hbar / \Delta E = \hbar / (2m_e c^2)\). (b) \(c \cdot \Delta t = \hbar/(2m_e c) = \lambda_C / 2\).

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B-8. d'Alembert Operator and the Covariant Form of the Klein-Gordon Equation

Using

\[ \Box \equiv \frac{\partial^2}{\partial t^2} - \nabla^2 = \partial_\mu \partial^\mu \]

verify that equation (1.1) can be written as \((\Box + m^2)\phi = 0\). Show this by expanding \(\partial_\mu \partial^\mu\) in components using the metric tensor \(\eta^{\mu\nu} = \mathrm{diag}(+1, -1, -1, -1)\).

Hint

Since \(\partial_\mu = (\partial/\partial t, \partial/\partial x^1, \partial/\partial x^2, \partial/\partial x^3)\) and \(\partial^\mu = \eta^{\mu\nu}\partial_\nu = (\partial/\partial t, -\partial/\partial x^1, -\partial/\partial x^2, -\partial/\partial x^3)\), we have \(\partial_\mu \partial^\mu = (\partial/\partial t)^2 - (\nabla)^2\).

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Medium

M-1. Lorentz Covariance of the Probability Current Density for the Klein-Gordon Equation

Show that \(\rho\) and \(\mathbf{j}\) defined in equations (1.6) and (1.7) of the text can be written in a unified form as the 4-current density

\[ j^\mu = \frac{i}{2m}\left(\phi^* \partial^\mu \phi - \phi\, \partial^\mu \phi^*\right) \]

Specifically:

(a) Verify that \(j^0 = \rho\) (equation (1.6)).

(b) Verify that \(j^k\) (\(k = 1, 2, 3\)) agrees with each component of \(\mathbf{j}\) in equation (1.7) (pay attention to the sign convention of the metric).

(c) Show that the continuity equation (1.5) can be written as \(\partial_\mu j^\mu = 0\), and discuss why this is a Lorentz scalar condition.

Hint

Using the metric \(\eta^{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)\), we have \(\partial^0 = \partial/\partial t\) but \(\partial^k = -\partial/\partial x^k\). In (b), be careful with the sign handling. For (c), state that \(\partial_\mu j^\mu = 0\) does not change its form under Lorentz transformations (it is a scalar equation).

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M-2. Non-negativity of Probability Density in the Dirac Equation

Starting from the Dirac equation (1.11) and its Hermitian conjugate, derive that the probability density \(\rho = \psi^\dagger \psi\) and the probability current density \(\mathbf{j} = \psi^\dagger \boldsymbol{\alpha} \psi\) satisfy the continuity equation

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 \]

Make each step of the derivation explicit.

Hint

From equation (1.11), \(i\partial_t \psi = H_D \psi\) (where \(H_D = -i\boldsymbol{\alpha}\cdot\nabla + \beta m\)). The Hermitian conjugate is \(-i\partial_t \psi^\dagger = \psi^\dagger H_D^\dagger\). Use the fact that \(H_D\) is Hermitian (i.e., \(\alpha^i\) and \(\beta\) are Hermitian matrices) to compute \(\partial_t(\psi^\dagger\psi)\).

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M-3. Compton Wavelength and Position Localization

Consider a thought experiment in which a scalar particle of mass \(m\) is confined to a one-dimensional box of width \(L\).

(a) Using the uncertainty principle, estimate the momentum uncertainty of the particle in the box as \(\Delta p \sim \hbar/L\), and find the corresponding relativistic energy uncertainty \(\Delta E\) (discuss both the \(\Delta p \gg mc\) limit and the \(\Delta p \ll mc\) limit).

(b) From the condition \(\Delta E \geq 2mc^2\), express the critical value of the box width \(L\) in terms of the Compton wavelength \(\lambda_C = \hbar/(mc)\).

(c) Explain how this result is consistent with the discussion in the main text that "the single-particle picture breaks down at distance scales shorter than the Compton wavelength."

Hint

(a) The relativistic energy is \(E = \sqrt{(pc)^2 + (mc^2)^2}\), but in the case \(\Delta p \gg mc\), we have \(\Delta E \approx c\,\Delta p\). (b) Solving \(c \cdot \hbar / L \geq 2mc^2\) gives \(L \lesssim \lambda_C / 2\).

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M-4. General Solution of the Klein-Gordon Field and Positive/Negative Frequency Modes

The general solution of the Klein-Gordon equation is written as a Fourier expansion:

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

where \(\omega_{\mathbf{p}} = \sqrt{|\mathbf{p}|^2 + m^2} > 0\).

(a) Verify by direct substitution that this \(\phi\) satisfies the Klein-Gordon equation.

(b) If \(\phi\) is a real scalar field (\(\phi = \phi^*\)), what relation must hold between \(a(\mathbf{p})\) and \(b(\mathbf{p})\)?

(c) If \(\phi\) is a complex scalar field, state that \(a(\mathbf{p})\) and \(b(\mathbf{p})\) are independent, and discuss the connection to the argument in the main text that the latter corresponds to "antiparticles."

Hint

(a) Acting with \(\Box + m^2\) on each Fourier mode yields \((-\omega_{\mathbf{p}}^2 + |\mathbf{p}|^2 + m^2) = 0\). (b) Compare the condition \(\phi^* = \phi\) for each Fourier component.

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Advanced

A-1. Causality and the Klein-Gordon Propagator

For the free Klein-Gordon field, define the retarded Green's function (propagator) \(G_R(x - y)\) by

\[ (\Box_x + m^2)\, G_R(x - y) = -\delta^4(x - y) \]

(a) Find the Fourier representation of \(G_R\). Make the pole prescription (\(i\varepsilon\) prescription) explicit, and show how the retarded boundary condition (\(G_R = 0\) for \(x^0 < y^0\)) is realized through a discussion of the integration contour in the complex \(p^0\) plane.

(b) Show (or argue) that \(G_R(x - y) = 0\) for spacelike separations \((x - y)^2 < 0\), and explain why this is a manifestation of causality.

(c) Based on this result, discuss how one can mathematically justify the statement in the text that "forces are transmitted at or below the speed of light through the exchange of virtual particles."

Hint

(a) Under Fourier transformation, \((\Box + m^2) \to (-p^2 + m^2)\), giving \(\tilde{G}_R(p) = 1/(p^2 - m^2)\). The retarded condition places both poles below the real axis (\(p^0 \to p^0 + i\varepsilon\)). (b) For spacelike separations, the sign of \(x^0 - y^0\) can be reversed by a Lorentz transformation; use the \(\theta(x^0 - y^0)\) factor in \(G_R\).

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A-2. Equivalence of the Dirac Sea and Quantum Field Theory

In the main text, it was pointed out that the Dirac sea picture cannot be applied to bosons. Here we explore this issue further.

(a) Show that in the Dirac sea picture, the energy of the vacuum state formally diverges as \(E_{\text{vac}} = -\sum_{\mathbf{p}, s} \omega_{\mathbf{p}}\) (all negative energy states occupied), where the sum runs over all momenta \(\mathbf{p}\) and spins \(s\).

(b) In the framework of quantum field theory (second quantization), antiparticles are described not as "holes in the sea" but as "positive-energy excitations created by antiparticle creation operators." Show that these two pictures give the same results for observable physical quantities (energy differences, charge differences) for one specific physical process (e.g., electron-positron pair creation).

(c) Clearly explain from the perspective of the Pauli exclusion principle why the Dirac sea picture cannot be constructed for scalar fields (spin 0, Bose statistics). Furthermore, describe how this problem is avoided in quantum field theory.

Hint

(a) Since all negative energy states \(E = -\omega_{\mathbf{p}}\) are occupied, the energy is \(\sum_{\mathbf{p},s}(-\omega_{\mathbf{p}})\). (b) Compare the energy cost \(+\omega_{\mathbf{p}}\) of creating a "hole" in the Dirac sea with the energy \(+\omega_{\mathbf{p}}\) of creating an antiparticle in quantum field theory. (c) Since bosons can occupy the same state without limit, it is impossible to "fill all states." In quantum field theory, the choice of commutation relations (bosons) or anticommutation relations (fermions) automatically incorporates the statistics.

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