Prologue Problems¶

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Table of Contents

Basic

B-1. Dimensional Analysis in Natural Units
B-2. Inner Product of 4-Vectors
B-3. Threshold for Particle Production
B-4. Matrix Operations for Lorentz Boosts
B-5. Kinematics of Electron-Positron Pair Production
B-6. Index Contraction Practice
B-7. Sense of Scale

Medium

M-1. Uncertainty Principle and Particle Number Change
M-2. String Vibrations and "Particles"
M-3. Falsifiability and Precision Agreement
M-4. Dimensional Analysis of Planck Scales

Advanced

A-1. Indistinguishability of Identical Particles (A Consequence in Quantum Field Theory)
A-2. Dimensional Analysis of the Gravitational Coupling Constant and Non-Renormalizability

Basic¶

B-1. Dimensional Analysis in Natural Units¶

In quantum field theory, natural units $\hbar = c = 1$ are frequently used. In this unit system, the dimensions of all physical quantities can be expressed as "powers of mass" $[\text{mass}]^n$. Find the mass dimension $n$ for each of the following physical quantities.

(a) Energy $E$

(b) Length $\ell$

(c) Time $t$

(d) Momentum $p$

(e) Action $S = \int d^4x\,\mathcal{L}$ (where $d^4x = dt\,d^3x$)

Hint

Setting $\hbar = c = 1$, we have $[E] = [\text{mass}]$, and from $[\hbar] = [E][t] = 1$ we can determine $[t]$. From $c = [\ell]/[t] = 1$ we can also determine $[\ell]$. Since the action $S$ has the units of $\hbar$, $[S] = [\hbar] = ?$.

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B-2. Inner Product of 4-Vectors¶

Let the Minkowski metric be $\eta_{\mu\nu} = \mathrm{diag}(+1, -1, -1, -1)$ ("mostly minus" convention). For the 4-momentum $p^\mu = (E,\, p_x,\, p_y,\, p_z)$, calculate the following.

(a) Write down each component of $p_\mu = \eta_{\mu\nu}\,p^\nu$.

(b) Express the invariant $p^\mu p_\mu$ in terms of $E$ and $|\mathbf{p}|$.

(c) Verify that the on-shell condition $p^\mu p_\mu = m^2$ (in natural units) corresponds to $E^2 = |\mathbf{p}|^2 c^2 + m^2 c^4$ in conventional units.

Hint

Since $p_\mu = \eta_{\mu\nu}p^\nu$, we have $p_0 = +E$, $p_i = -p^i$. The inner product is $p^\mu p_\mu = E^2 - |\mathbf{p}|^2$. To restore $c = 1$ from natural units to conventional units, track the dimensions with $E \to E/c$, $m \to mc$.

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B-3. Threshold for Particle Production¶

In natural units ($c = 1$), consider the case where an incident particle (mass $m$, kinetic energy $T$) collides with a stationary target particle (mass $M$) to produce particles with total mass $m_1 + m_2 + \cdots + m_n$.

(a) Writing the total energy of the incident particle as $E = m + T$, express the invariant mass of the center-of-mass system $\sqrt{s}$ in terms of $E$, $m$, and $M$, where

\[ s = (p_1 + p_2)^\mu (p_1 + p_2)_\mu \]

(b) Consider the minimal reaction $p + p \to p + p + H$ for producing a Higgs boson (mass $m_H \approx 125\;\mathrm{GeV}$) by directing a proton beam at a stationary proton target (mass $m_p \approx 0.938\;\mathrm{GeV}$). From the threshold condition $\sqrt{s} = 2m_p + m_H$, find the minimum kinetic energy $T_{\mathrm{thr}}$ required for the incident proton (give the numerical value in GeV).

Hint

Since the target is at rest, $p_2^\mu = (M, \mathbf{0})$. Expand $s = (E + M)^2 - |\mathbf{p}_1|^2$ and use $E^2 - |\mathbf{p}_1|^2 = m^2$. At threshold, all particles are produced at rest in the center-of-mass frame, so $\sqrt{s} = \sum m_{\text{final}}$.

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B-4. Matrix Operations for Lorentz Boosts¶

The transformation matrix for a Lorentz boost in the $x$ direction is given by

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

where $\beta = v/c$ and $\gamma = 1/\sqrt{1-\beta^2}$.

(a) Calculate $\gamma$ when $\beta = 3/5$.

(b) Apply this boost to the 4-momentum $p^\mu = (5m,\, 3m,\, 0,\, 0)$ and find $p'^\mu = \Lambda^\mu{}_\nu\, p^\nu$.

(c) Verify by direct calculation that $p'^\mu p'_\mu = p^\mu p_\mu$ holds (conservation of the Lorentz invariant).

Hint

If $\beta = 3/5$, then $\beta^2 = 9/25$, $1 - \beta^2 = 16/25$, $\gamma = 5/4$. Compute the matrix-vector product component by component.

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B-5. Kinematics of Electron-Positron Pair Production¶

Consider the reaction $\gamma + N \to N + e^- + e^+$ in which a photon ($m_\gamma = 0$) produces an electron-positron pair ($e^-e^+$) in the vicinity of a stationary nucleus (mass $M \gg m_e$).

(a) Given the photon energy $E_\gamma$, express the invariant mass $\sqrt{s}$ of this system in terms of $E_\gamma$ and $M$.

(b) From the threshold condition for pair production $\sqrt{s} = M + 2m_e$, find the minimum photon energy $E_\gamma^{\min}$ required. Simplify your answer using the approximation $M \gg m_e$.

(c) Taking $m_e = 0.511\;\mathrm{MeV}$, find the numerical value of $E_\gamma^{\min}$ in MeV (in the limit $M \to \infty$).

Hint

The photon 4-momentum is $k^\mu = (E_\gamma, E_\gamma, 0, 0)$ (since the mass is zero, $|\mathbf{k}| = E_\gamma$). Expand $s = (k + P_N)^\mu(k + P_N)_\mu$. For $M \gg m_e$, we have $E_\gamma^{\min} \approx 2m_e + 2m_e^2/M \approx 2m_e$.

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B-6. Index Contraction Practice¶

In 4-dimensional Minkowski spacetime, perform the following index contractions.

(a) $\eta^{\mu\nu}\eta_{\mu\nu}$

(b) $\partial_\mu x^\mu$ (where $x^\mu = (x^0, x^1, x^2, x^3)$)

(c) Write out $\eta^{\mu\nu}\partial_\mu\partial_\nu \phi \equiv \Box\phi$ explicitly in terms of partial derivatives with respect to $(x^0, x^1, x^2, x^3)$ ($\Box$ is the d'Alembert operator).

Hint

(a) $\eta^{\mu\nu}\eta_{\mu\nu} = \delta^\mu{}_\mu = $ dimension of spacetime. (b) Use $\partial_\mu x^\nu = \delta^\nu_\mu$. (c) $\eta^{00} = +1$, $\eta^{ii} = -1$.

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B-7. Sense of Scale¶

In the main text, it was stated that "quantum field theory can describe phenomena spanning more than 40 orders of magnitude in scale, from the magnetic moment of a single electron to the structure of the entire universe." Estimate the following energy scales in units of eV and arrange them in descending order.

(a) The rest energy of the electron $m_e c^2$

(b) The rest energy of the Higgs boson $m_H c^2 \approx 125\;\mathrm{GeV}$

(c) The typical energy of a CMB (Cosmic Microwave Background) photon (estimated from the temperature $T \approx 2.725\;\mathrm{K}$ using $E \sim k_B T$, where $k_B \approx 8.617 \times 10^{-5}\;\mathrm{eV/K}$)

(d) The center-of-mass collision energy at the LHC $\sqrt{s} = 13\;\mathrm{TeV}$

Hint

$1\;\mathrm{GeV} = 10^9\;\mathrm{eV}$, $1\;\mathrm{TeV} = 10^{12}\;\mathrm{eV}$, $m_e c^2 \approx 0.511\;\mathrm{MeV} = 5.11 \times 10^5\;\mathrm{eV}$. The CMB is on the order of $\sim 10^{-4}\;\mathrm{eV}$.

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Medium¶

M-1. Uncertainty Principle and Particle Number Change¶

Combining the energy-time uncertainty relation learned in quantum mechanics

\[ \Delta E \cdot \Delta t \gtrsim \hbar \]

with the mass-energy equivalence $E = mc^2$ from special relativity, discuss the following.

(a) When attempting to localize a particle of mass $m$ within a region of size $\Delta x$ or smaller, derive the condition under which the kinetic energy of the particle exceeds $mc^2$, starting from the momentum uncertainty $\Delta p \gtrsim \hbar / \Delta x$. Express this critical length scale $\Delta x_c$ in terms of $m$, $\hbar$, and $c$ (explicitly stating its relation to the Compton wavelength $\lambda_C$).

(b) Discuss how in the region $\Delta x < \lambda_C$, the energy uncertainty exceeds $mc^2$, and therefore new particle-antiparticle pairs can be created via $E = mc^2$. Explain why this implies the breakdown of "single-particle quantum mechanics with a fixed number of particles."

(c) Calculate the numerical value of the electron's Compton wavelength $\lambda_C = \hbar/(m_e c)$ (in units of fm), and verify that it is of the same order as the size of atomic nuclei ($\sim$ a few fm). Discuss why changes in particle number can be ignored in atomic physics ($\sim 0.1\;\mathrm{nm}$ scale) but cannot be ignored in nuclear and particle physics.

Hint

(a) From $\Delta p \sim \hbar/\Delta x$, the condition for the kinetic energy $\sim (\Delta p)^2/(2m)$ to exceed $mc^2$, or more directly using the relativistic relation $c\Delta p \sim mc^2$. (b) Discuss how the energy for virtual pair creation falls within the range of the uncertainty. (c) $\hbar c \approx 197\;\mathrm{MeV \cdot fm}$, $m_e c^2 \approx 0.511\;\mathrm{MeV}$.

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M-2. String Vibrations and "Particles"¶

The transverse vibration $\phi(x, t)$ of an infinitely long string (linear mass density $\mu$, tension $T$) obeys the wave equation

\[ \mu\,\frac{\partial^2 \phi}{\partial t^2} = T\,\frac{\partial^2 \phi}{\partial x^2} \]

(a) Letting $v = \sqrt{T/\mu}$, rewrite this equation in the form $\Box\phi = 0$ (the d'Alembert equation in 1+1 dimensions).

(b) Write down the general solution for a string of length $L$ (with both ends fixed) as a Fourier series, and find the frequency $\omega_n$ of each mode $n$.

(c) Recalling the quantization of the harmonic oscillator from quantum mechanics, the energy of each mode $n$ is

\[ E_n = \hbar\omega_n\left(N_n + \frac{1}{2}\right), \quad N_n = 0, 1, 2, \ldots \]

If we interpret $N_n$ as "the number of particles present in mode $n$," this is precisely the prototype of quantum field theory. Using this analogy, restate Lina's explanation from the main text that "vibrational modes of a field are particles" in the language of string vibrations.

Hint

(a) $\partial_t^2 \phi - v^2 \partial_x^2 \phi = 0$. (b) Setting $\phi(x,t) = \sum_n q_n(t)\sin(n\pi x/L)$, we get $\omega_n = n\pi v/L$. (c) "The entire string" corresponds to the field, and "the energy quanta of each vibrational mode" correspond to particles.

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M-3. Falsifiability and Precision Agreement¶

In the main text, Lina stated that "quantum field theory is not truth but a model," and cited the example of Newton's gravitational model being forced into revision by the perihelion precession of Mercury.

(a) The QED theoretical value of the electron's anomalous magnetic moment $a_e = (g-2)/2$ is currently calculated as a power series expansion in $\alpha$ (the fine-structure constant):

\[ a_e = \frac{\alpha}{2\pi} + c_2\left(\frac{\alpha}{\pi}\right)^2 + c_3\left(\frac{\alpha}{\pi}\right)^3 + \cdots \]

Calculate the numerical value of the lowest-order term $\alpha/(2\pi)$ to 4 significant figures, using $\alpha \approx 1/137.036$.

(b) This value is compared with $a_e \approx 0.00116$. After confirming that even the lowest-order term alone explains the bulk of the experimental value, discuss in approximately 200 words why "computing higher-order terms and comparing them with experiment" is important from the standpoint of scientific methodology, specifically from the perspective of falsifiability.

(c) Suppose that in the future, the theoretical and experimental values of $a_e$ were found to disagree at the 15th decimal place. Does this mean that quantum field theory (QED) is "wrong"? Discuss based on the philosophy of science stance presented in the main text.

Hint

(a) $\alpha/(2\pi) = 1/(2\pi \times 137.036)$. (b) Higher-order terms are sensitive to the effects of new physics (unknown particles, etc.). Agreement increases confidence in the model, while disagreement necessitates modification of the model (or discovery of new physics). (c) This should be interpreted as "the limits of the model's domain of applicability have been found," and it does not mean the model is entirely "wrong." This is analogous to how Newtonian mechanics remains valid at everyday scales.

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M-4. Dimensional Analysis of Planck Scales¶

Derive the Planck mass $M_P$, Planck length $\ell_P$, and Planck time $t_P$ by dimensional analysis from the gravitational constant $G$, Planck's constant $\hbar$, and the speed of light $c$.

(a) Show that $M_P = \sqrt{\hbar c / G}$ by dimensional analysis of $[G]$, $[\hbar]$, and $[c]$.

(b) Express $\ell_P$ and $t_P$ in terms of $G$, $\hbar$, and $c$.

(c) Using $G \approx 6.674 \times 10^{-11}\;\mathrm{m^3\,kg^{-1}\,s^{-2}}$, $\hbar \approx 1.055 \times 10^{-34}\;\mathrm{J \cdot s}$, and $c \approx 3.0 \times 10^8\;\mathrm{m/s}$, compute the numerical values of $M_P$, $\ell_P$, and $t_P$ in SI units.

(d) Convert $M_P c^2$ to GeV and compare it with the LHC center-of-mass energy of $13\;\mathrm{TeV}$. In approximately 100 characters, explain the relationship between the enormity of this scale and the statement in the text that "attempts to incorporate gravity into quantum field theory lead to a breakdown."

Hint

$[G] = \mathrm{m^3\,kg^{-1}\,s^{-2}}$, $[\hbar] = \mathrm{kg\,m^2\,s^{-1}}$, $[c] = \mathrm{m\,s^{-1}}$. Set $M_P = G^a \hbar^b c^d$ and determine $a, b, d$ from the condition $[M_P] = \mathrm{kg}$. Use $1\;\mathrm{GeV} \approx 1.602 \times 10^{-10}\;\mathrm{J}$.

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Advanced¶

A-1. Indistinguishability of Identical Particles (A Consequence in Quantum Field Theory)¶

In the main text, Lina stated that "all electrons are perfectly identical because they are the same type of vibration of the same field." In quantum mechanics, the indistinguishability of identical particles is imposed as a postulate, but in quantum field theory it is derived as a theorem. Through the following arguments, retrace this paradigm shift in worldview.

(a) Recall quantum mechanics (the content learned in Chapters 16–17) and briefly describe how the requirement that the wave function of two identical bosons must be symmetric under exchange was justified within the framework of quantum mechanics (point out that it was imposed as a postulate).

(b) In quantum field theory, using the creation operators $\hat{a}^\dagger_{\mathbf{p}}$ appearing in the Fourier expansion of the scalar field $\hat{\phi}(x)$, a two-particle state is defined as

\[ |\mathbf{p}_1, \mathbf{p}_2\rangle = \hat{a}^\dagger_{\mathbf{p}_1}\hat{a}^\dagger_{\mathbf{p}_2}|0\rangle \]

Show that from the bosonic commutation relation $[\hat{a}^\dagger_{\mathbf{p}_1}, \hat{a}^\dagger_{\mathbf{p}_2}] = 0$, the equality $|\mathbf{p}_1, \mathbf{p}_2\rangle = |\mathbf{p}_2, \mathbf{p}_1\rangle$ follows automatically.

(c) Similarly, from the anticommutation relation $\{\hat{b}^\dagger_{\mathbf{p}_1,s_1}, \hat{b}^\dagger_{\mathbf{p}_2,s_2}\} = 0$ for the fermionic creation operators $\hat{b}^\dagger_{\mathbf{p},s}$ of the Dirac field, show that the two-fermion state is antisymmetric under exchange, and that when $\mathbf{p}_1 = \mathbf{p}_2$, $s_1 = s_2$, the state vanishes (the Pauli exclusion principle).

(d) Based on the above, discuss in approximately 300 characters (or a short paragraph) the claim that "the indistinguishability of identical particles and the spin-statistics relation are not postulates but consequences in quantum field theory," touching on the difference in worldview between quantum mechanics and quantum field theory.

Hint

(b) $\hat{a}^\dagger_{\mathbf{p}_1}\hat{a}^\dagger_{\mathbf{p}_2} = \hat{a}^\dagger_{\mathbf{p}_2}\hat{a}^\dagger_{\mathbf{p}_1} + [\hat{a}^\dagger_{\mathbf{p}_1}, \hat{a}^\dagger_{\mathbf{p}_2}]$. Since the commutator is zero, the left-hand side equals the first term on the right-hand side. (c) From the anticommutation relation $\{A, B\} = AB + BA = 0$, we get $AB = -BA$. When $A = B$, $A^2 = -A^2$ implies $A^2 = 0$. (d) Emphasize that what was imposed in an ad hoc manner as the "symmetrization postulate" in quantum mechanics emerges naturally from the structure of field quantization (commutation vs. anticommutation relations).

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A-2. Dimensional Analysis of the Gravitational Coupling Constant and Non-Renormalizability¶

In the main text, it was stated that "when one tries to incorporate gravity into quantum field theory, calculations diverge to infinity and become uncontrollable." The core of this problem lies in the fact that the gravitational coupling constant has a negative mass dimension. Through the following discussion, detect the "scent" of non-renormalizability from dimensional analysis alone.

(a) The coupling constant of QED (the electric charge $e$) is dimensionless in natural units. To confirm this, determine the mass dimension of each factor in the QED interaction Lagrangian density

\[ \mathcal{L}_{\mathrm{int}} = -e\,\bar{\psi}\gamma^\mu\psi\, A_\mu \]

and show that $[e] = [\text{mass}]^0$. Assume that in 4-dimensional spacetime, $[\mathcal{L}] = [\text{mass}]^4$, the fermion field $[\psi] = [\text{mass}]^{3/2}$, and the gauge field $[A_\mu] = [\text{mass}]^1$.

(b) The Einstein-Hilbert action of general relativity (see general relativity Ch. 6) is

\[ S_{\mathrm{EH}} = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R \]

In natural units, taking $[S] = [\text{mass}]^0$, $[d^4x] = [\text{mass}]^{-4}$, and the Ricci scalar $[R] = [\text{mass}]^2$, determine the mass dimension of $[G]$. Furthermore, find the mass dimension of the gravitational coupling constant $\kappa = \sqrt{32\pi G}$.

(c) As a general result in quantum field theory, when the coupling constant $g$ has mass dimension $[g] = [\text{mass}]^\delta$: - $\delta > 0$: super-renormalizable - $\delta = 0$: renormalizable - $\delta < 0$: non-renormalizable

(details will be studied in Ch. 16). Classify QED and gravity according to this scheme.

(d) In non-renormalizable theories, new types of divergences appear with each increasing loop order, and they cannot be absorbed by a finite number of parameters. Combining this fact with Lina's explanation in the main text that string theory alleviates divergences by replacing "point particles" with "strings of finite extent," discuss in approximately 300 characters "why the quantization of gravity requires a framework beyond quantum field theory."

Hint

(a) $[\bar{\psi}\gamma^\mu\psi\, A_\mu] = [3/2 + 3/2 + 1] = [\text{mass}]^4$, so for the dimension of $\mathcal{L}_{\mathrm{int}}$ to match, we need $[e] = 0$. (b) From $[G^{-1}] \cdot [\text{mass}]^{-4} \cdot [\text{mass}]^2 = [\text{mass}]^0$, we get $[G^{-1}] = [\text{mass}]^2$, i.e., $[G] = [\text{mass}]^{-2}$. (c) $[\kappa] = [G]^{1/2} = [\text{mass}]^{-1}$. (d) $\delta < 0$ means the coupling grows stronger at high energies, and new types of divergences appear at each order of the perturbative expansion. They cannot be controlled with a finite number of counterterms.

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