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Basic

B-1. Dimensional Analysis in Natural Units

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Problem:

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 \(d^4x = dt\,d^3x\))

Strategy: With \(\hbar = c = 1\), unify the dimensions of all physical quantities as \([\text{mass}]^n\). \([\hbar] = [E][t] = 1 \Rightarrow [t] = [E]^{-1}\), \(c = [\ell]/[t] = 1 \Rightarrow [\ell] = [t]\). The reference is \([E] = [\text{mass}]\).

Quantity Mass dimension \(n\) Reason
(a) \(E\) \(+1\) \([E] = [\text{mass}]\) (by definition)
(b) \(\ell\) \(-1\) \([\ell] = [t] = [E]^{-1}\)
(c) \(t\) \(-1\) From \([\hbar] = [E][t] = 1\)
(d) \(p\) \(+1\) \([p] = [E]/c = [E] \cdot [\text{mass}]^0 = [\text{mass}]\)
(e) \(S\) \(0\) \([S] = [\hbar] = 1\), so dimensionless

Check: \([d^4x] = [t][\ell]^3 = [\text{mass}]^{-4}\), so \([\mathcal{L}] = [\text{mass}]^4\) is required (used in A2).

B-2. Inner Product of 4-Vectors

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Problem:

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.

\(\eta_{\mu\nu} = \mathrm{diag}(+1,-1,-1,-1)\) (mostly minus).

(a) \(p_\mu = \eta_{\mu\nu}p^\nu\):

\(p_0 = +E, \quad p_1 = -p_x, \quad p_2 = -p_y, \quad p_3 = -p_z\)

(b) Invariant:

\(p^\mu p_\mu = \eta_{\mu\nu}p^\mu p^\nu = (p^0)^2 - |\mathbf{p}|^2 = E^2 - |\mathbf{p}|^2\)

(c) Restoring conventional units:

In natural units, \(p^\mu p_\mu = m^2\). Restoring dimensions with \([E] \to [E]/c\) (aligning components by dividing by c) and \([m] \to mc^2\):

\(\frac{E^2}{c^2} - |\mathbf{p}|^2 = m^2 c^2 \quad\Rightarrow\quad E^2 = |\mathbf{p}|^2 c^2 + m^2 c^4\)

This is the standard relativistic energy-momentum relation. \(\boxed{E^2 = |\mathbf{p}|^2 c^2 + m^2 c^4}\)

Consistency check: At rest \(\mathbf{p} = 0\), we get \(E = mc^2\). For \(m = 0\) (photon), we get \(E = |\mathbf{p}|c\). Both limits are consistent with known results.

B-3. Threshold for Particle Production

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Problem:

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).

(a) Invariant mass \(\sqrt{s}\):

\(p_1^\mu = (E, \mathbf{p}_1), \quad p_2^\mu = (M, \mathbf{0})\) (target at rest)

\(s = (p_1 + p_2)^\mu(p_1 + p_2)_\mu = (E + M)^2 - |\mathbf{p}_1|^2\)

Substituting \(|\mathbf{p}_1|^2 = E^2 - m^2\):

\(s = (E + M)^2 - (E^2 - m^2) = E^2 + 2EM + M^2 - E^2 + m^2 = m^2 + M^2 + 2EM\)

\(\boxed{\sqrt{s} = \sqrt{m^2 + M^2 + 2EM}}\)

(b) Threshold for Higgs production:

For \(p + p \to p + p + H\), the threshold is \(\sqrt{s} = 2m_p + m_H\). Substituting \(m = M = m_p\) and \(E = m_p + T\):

\((2m_p + m_H)^2 = 2m_p^2 + 2(m_p + T)m_p = 2m_p^2 + 2m_p^2 + 2m_p T = 4m_p^2 + 2m_p T\)

\(T = \frac{(2m_p + m_H)^2 - 4m_p^2}{2m_p} = \frac{4m_p^2 + 4m_p m_H + m_H^2 - 4m_p^2}{2m_p} = 2m_H + \frac{m_H^2}{2m_p}\)

Substituting numerical values (\(m_p = 0.938\ \mathrm{GeV}\), \(m_H = 125\ \mathrm{GeV}\)):

\(T_{\mathrm{thr}} = 2(125) + \frac{125^2}{2 \times 0.938} = 250 + \frac{15625}{1.876} \approx 250 + 8329 \approx 8579\ \mathrm{GeV}\)

\(\boxed{T_{\mathrm{thr}} \approx 8.58\ \mathrm{TeV}}\)

Verification: This quantitatively confirms the phenomenon that the threshold energy in asymmetric collisions becomes orders of magnitude larger compared to symmetric collisions (center-of-mass frame). Since the LHC uses 13 TeV center-of-mass colliding beams in a symmetric collision configuration, the same physics is accessible at far lower energy than the 8.58 TeV fixed-target equivalent required for Higgs (125 GeV) production. This is the advantage of colliding beam experiments.

B-4. Matrix Operations for Lorentz Boosts

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Problem:

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).

(a) Calculation of \(\gamma\):

\(\beta = 3/5, \quad \beta^2 = 9/25, \quad 1 - \beta^2 = 16/25\)

\(\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{16/25}} = \frac{5}{4}\)

\(\boxed{\gamma = 5/4}\)

(b) Four-momentum after the boost:

\(\gamma = 5/4, \quad \gamma\beta = (5/4)(3/5) = 3/4\)

Acting on \(p^\mu = (5m, 3m, 0, 0)\):

\(p'^0 = \gamma p^0 - \gamma\beta p^1 = (5/4)(5m) - (3/4)(3m) = 25m/4 - 9m/4 = 16m/4 = 4m\)

\(p'^1 = -\gamma\beta p^0 + \gamma p^1 = -(3/4)(5m) + (5/4)(3m) = -15m/4 + 15m/4 = 0\)

\(p'^2 = 0, \quad p'^3 = 0\)

\(\boxed{p'^\mu = (4m, 0, 0, 0)}\)

(c) Conservation of the Lorentz invariant:

\(p^\mu p_\mu = (5m)^2 - (3m)^2 - 0 - 0 = 25m^2 - 9m^2 = 16m^2\)

\(p'^\mu p'_\mu = (4m)^2 - 0 - 0 - 0 = 16m^2\)

Both agree: \(\boxed{p^\mu p_\mu = p'^\mu p'_\mu = 16m^2}\).

Physical interpretation: \(p'^\mu = (4m, 0, 0, 0)\) is the four-momentum in the rest frame. This means the boost is a transformation to the particle's rest frame (the particle's velocity \(v = p^1/p^0 \cdot c = 3c/5\) equals the boost velocity). The rest mass is \(4m\), and \(m_{\mathrm{rest}}^2 = 16m^2\) is consistent with the invariant.

B-5. Kinematics of Electron-Positron Pair Production

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Problem:

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 \(M \to \infty\)).

(a) Invariant mass:

Photon \(k^\mu = (E_\gamma, E_\gamma, 0, 0)\) (massless), nucleus \(P_N^\mu = (M, \mathbf{0})\).

\(s = (k + P_N)^\mu(k + P_N)_\mu = (E_\gamma + M)^2 - E_\gamma^2\)

Expanding: \(E_\gamma^2 + 2E_\gamma M + M^2 - E_\gamma^2 = 2E_\gamma M + M^2\)

\(\boxed{\sqrt{s} = \sqrt{M^2 + 2E_\gamma M}}\)

(b) Threshold condition \(\sqrt{s} = M + 2m_e\):

\(M^2 + 2E_\gamma M = (M + 2m_e)^2 = M^2 + 4Mm_e + 4m_e^2\)

\(E_\gamma^{\min} = \frac{4Mm_e + 4m_e^2}{2M} = 2m_e + \frac{2m_e^2}{M}\)

Approximation for \(M \gg m_e\): \(\boxed{E_\gamma^{\min} \approx 2m_e}\)

(c) Numerical value:

In the limit \(M \to \infty\), \(E_\gamma^{\min} = 2m_e = 2 \times 0.511 = 1.022\ \mathrm{MeV}\)

\(\boxed{E_\gamma^{\min} \approx 1.022\ \mathrm{MeV}}\)

Physical interpretation: The nucleus exists solely as a "recoil partner" to satisfy momentum conservation. When its mass is sufficiently large, it carries away virtually no energy, so pair production occurs when the photon energy equals exactly the rest energy of the electron-positron pair, \(2m_e c^2\). In vacuum (without a recoil partner), pair production is forbidden by the requirement of momentum conservation.

B-6. Index Contraction Practice

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Problem:

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).

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

\(\eta^{\mu\nu}\eta_{\mu\nu} = \delta^\mu{}_\mu = 4\) (the dimension of spacetime)

\(\boxed{\eta^{\mu\nu}\eta_{\mu\nu} = 4}\)

(b) \(\partial_\mu x^\mu\):

\(\partial_\mu x^\mu = \partial_\mu x^\nu \cdot \delta^\mu_\nu = \delta^\mu_\mu = 4\)

Alternatively, writing it out explicitly: \(\partial_0 x^0 + \partial_1 x^1 + \partial_2 x^2 + \partial_3 x^3 = 1 + 1 + 1 + 1 = 4\)

\(\boxed{\partial_\mu x^\mu = 4}\)

(c) The d'Alembert operator \(\Box\phi\):

\(\eta^{\mu\nu}\partial_\mu\partial_\nu = \eta^{00}\partial_0^2 + \eta^{11}\partial_1^2 + \eta^{22}\partial_2^2 + \eta^{33}\partial_3^2 = \partial_0^2 - \partial_1^2 - \partial_2^2 - \partial_3^2\)

\(\boxed{\Box\phi = \frac{\partial^2\phi}{\partial(x^0)^2} - \frac{\partial^2\phi}{\partial(x^1)^2} - \frac{\partial^2\phi}{\partial(x^2)^2} - \frac{\partial^2\phi}{\partial(x^3)^2}}\)

Or \(\Box = \partial_t^2 - \nabla^2\) (in natural units \(c = 1\), \(x^0 = t\)).

Verification: The Klein-Gordon equation \((\Box + m^2)\phi = 0\) for a plane wave \(\phi = e^{-ip \cdot x}\) reduces to \((-p^\mu p_\mu + m^2) = 0\), i.e., \(p^\mu p_\mu = m^2\) (the mass-shell condition).

B-7. Sense of Scale

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Problem:

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 \(\sqrt{s} = 13\;\mathrm{TeV}\)

Scale Value eV equivalent
(a) \(m_e c^2\) \(0.511\ \mathrm{MeV}\) \(5.11 \times 10^5\ \mathrm{eV}\)
(b) \(m_H c^2\) \(125\ \mathrm{GeV}\) \(1.25 \times 10^{11}\ \mathrm{eV}\)
(c) CMB photon \(k_B T \approx 8.617 \times 10^{-5} \times 2.725\) \(\approx 2.35 \times 10^{-4}\ \mathrm{eV}\)
(d) LHC \(13\ \mathrm{TeV}\) \(1.3 \times 10^{13}\ \mathrm{eV}\)

In descending order:

\(\boxed{\text{LHC}\ (10^{13}) > m_H\ (10^{11}) > m_e\ (10^{5.7}) > \text{CMB}\ (10^{-3.6})}\)

Span: With the highest at \(10^{13}\) eV and the lowest at \(10^{-4}\) eV, the difference is approximately 17 orders of magnitude. This still falls short of the "40 orders of magnitude" mentioned in the text. The 40 orders of magnitude difference can be reached by including, for example, the Planck scale (\(10^{28}\) eV, see S4) and the neutron decay constant (energy resolution on the order of \(10^{-25}\) eV).

Medium

M-1. Uncertainty Principle and Particle Number Change

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Problem:

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.

(a) Critical length \(\Delta x_c\):

From a relativistic perspective, the momentum uncertainty is \(\Delta p \gtrsim \hbar/\Delta x\). The corresponding energy uncertainty is \(\Delta E \gtrsim c\Delta p \gtrsim \hbar c/\Delta x\) (ultra-relativistic regime).

The condition for the energy to exceed \(mc^2\):

\(\frac{\hbar c}{\Delta x} \gtrsim mc^2 \quad\Rightarrow\quad \Delta x \lesssim \frac{\hbar}{mc}\)

The right-hand side is precisely the Compton wavelength \(\lambda_C = \hbar/(mc)\):

\(\boxed{\Delta x_c = \lambda_C = \frac{\hbar}{mc}}\)

(b) Inevitability of particle number change:

When \(\Delta x < \lambda_C\), we have \(\Delta E > mc^2\), and within the range of energy uncertainty, virtual particle-antiparticle pairs of mass \(m\) can be created (energy fluctuations of order \(E = mc^2\) are permitted). This is "vacuum fluctuation"—the dynamical fluctuation of the vacuum as the ground state of a field.

In single-particle quantum mechanics (the Schrödinger equation), the number of particles is treated as a conserved constant, but in the regime \(\Delta x < \lambda_C\), this assumption breaks down, and a framework capable of describing the creation and annihilation of new particle pairs is required → quantum field theory.

(c) Compton wavelength of the electron:

\(\lambda_C = \frac{\hbar c}{m_e c^2} = \frac{197\ \mathrm{MeV}\cdot\mathrm{fm}}{0.511\ \mathrm{MeV}} \approx 386\ \mathrm{fm} = 3.86 \times 10^{-13}\ \mathrm{m}\)

\(\boxed{\lambda_C \approx 386\ \mathrm{fm}}\)

Comparison and conclusion:

  • Atomic scale (\(\sim 10^5\ \mathrm{fm} = 0.1\ \mathrm{nm}\)): Since \(\Delta x \gg \lambda_C\), particle number fluctuations are negligible → can be described by ordinary quantum mechanics
  • Nuclear scale (\(\sim\) a few fm): Since \(\Delta x \lesssim \lambda_C\), particle number fluctuations are essential → quantum field theory is required

This boundary represents the physical threshold at which the transition from quantum mechanics to quantum field theory becomes inevitable.

Sanity check: \(\lambda_C\) can also be understood as "the length scale at which the electron and a photon carry the same energy." The fact that the Compton scattering formula \(\Delta\lambda = \lambda_C(1 - \cos\theta)\) directly uses \(\lambda_C\) shows that this is a quantity with direct experimental relevance.

M-2. String Vibrations and "Particles"

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Problem:

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.

(a) d'Alembert equation:

\(\mu\partial_t^2\phi = T\partial_x^2\phi \quad\Rightarrow\quad \partial_t^2\phi - \frac{T}{\mu}\partial_x^2\phi = 0\)

Setting \(v = \sqrt{T/\mu}\):

\(\partial_t^2\phi - v^2\partial_x^2\phi = 0\)

Using the 1+1 dimensional d'Alembert operator \(\Box = \partial_t^2 - v^2\partial_x^2\) (with \(c\) replaced by \(v\)):

\(\boxed{\Box\phi = 0}\)

(b) Solution for a string fixed at both ends:

From the boundary conditions \(\phi(0,t) = \phi(L,t) = 0\), the spatial part takes the form \(\sin(n\pi x/L)\):

\(\phi(x,t) = \sum_{n=1}^\infty q_n(t)\sin\!\left(\frac{n\pi x}{L}\right)\)

Substituting into \(\Box\phi = 0\), each mode is a harmonic oscillator:

\(\ddot{q}_n(t) + \omega_n^2 q_n(t) = 0, \qquad \omega_n = \frac{n\pi v}{L}\)

\(\boxed{\omega_n = \frac{n\pi v}{L}, \quad n = 1, 2, 3, \ldots}\)

(c) Analogy as a prototype of quantum field theory:

Each vibrational mode \(n\) of the string is an independent one-dimensional harmonic oscillator. Upon quantizing the harmonic oscillator, the energy levels of the \(n\)-th mode are \(E_n = \hbar\omega_n(N_n + 1/2)\), where \(N_n\) is "the number of excitation quanta present in that mode."

In quantum field theory, the field \(\phi(x,t)\) is viewed as the "string," and vibrational modes \(n\) are placed in one-to-one correspondence with "momentum \(\mathbf{k}\)":

String Quantum Field Theory
Entire string \(\phi(x,t)\) Scalar field \(\phi(x)\)
Vibrational mode \(n\) Momentum mode \(\mathbf{k}\)
Mode frequency \(\omega_n = n\pi v/L\) $\omega_\mathbf{k} = \sqrt{
Mode quantum number \(N_n\) Particle number \(N_\mathbf{k}\)
Mode energy \(\hbar\omega_n(N_n + 1/2)\) Mode energy \(\omega_\mathbf{k}(N_\mathbf{k} + 1/2)\)

Connection to the main text: Lina's explanation that "vibrational modes of a field are particles" corresponds precisely to the fact that each vibrational mode of the string is a harmonic oscillator, and its excitation quanta (\(N_n\)) correspond to particles. In string theory (The Quest for Quantum Gravity), this is boldly extended so that these vibrational modes correspond to "various particle species (particles with different masses and spins)."

M-3. Falsifiability and Precision Agreement

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Problem:

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.

(a) Numerical value of the lowest-order term:

\(\frac{\alpha}{2\pi} = \frac{1}{2\pi \times 137.036} = \frac{1}{861.02} \approx 1.1614 \times 10^{-3}\)

\(\boxed{\alpha/(2\pi) \approx 0.001161}\) (4 significant figures)

Agreement with the experimental value \(a_e \approx 0.001160\):

The lowest-order term \(0.001161\) and the experimental value \(0.001160\) agree up to \(1.16 \times 10^{-3}\). This lowest-order term alone, which Schwinger (1948) computed by hand, reproduces more than 99.9% of the experimental value.

(b) Significance of computing higher-order terms (from the perspective of falsifiability):

Although the lowest-order term already accounts for the bulk of the experimental value, this does not establish that "QED is correct." As experimental measurement precision improves, contributions from higher-order terms (\(\alpha^2\), \(\alpha^3\), …) appear at deeper decimal places. If the predictions from higher-order terms deviate from the experimental value, this constitutes evidence that the QED model is missing some unknown physical effect (for example, loop contributions from undiscovered particles), necessitating modification or extension of the model. Conversely, if agreement continues to all orders, the reliability of QED is guaranteed to that level of precision. "Can be calculated → can be predicted → can be tested → the model's reliability either increases or the model is rejected"—it is precisely this falsifiability that makes QED a science rather than a mere philosophical assertion.

(c) Response to the hypothetical scenario (disagreement at the 15th digit):

This should be interpreted not as "it is wrong" but as "the limits of its applicability have become visible." Just as Newtonian mechanics remains perfectly valid at low speeds and macroscopic scales even after the advent of relativity and quantum mechanics, QED remains valid within the range of measured digits. Even if disagreement appears at the 15th digit, this does not constitute a wholesale rejection of QED but rather means that "effects from a deeper theory (for example, the Standard Model, or beyond that a grand unified theory, or further still quantum gravity) have begun to manifest at that level of precision." The philosophy of science stance presented in the text is that "models are always hypotheses with a domain of validity, and as precision improves, steps toward more fundamental theories become visible"—disagreement at the 15th digit is precisely part of that process.

M-4. Dimensional Analysis of Planck Scales

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Problem:

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."

(a) Derivation of the Planck mass:

Let \(M_P = G^a \hbar^b c^d\) and determine the exponents from the dimension \([M_P] = \mathrm{kg}\).

Dimension table:

\([G] = \mathrm{m^3\,kg^{-1}\,s^{-2}}, \quad [\hbar] = \mathrm{kg\,m^2\,s^{-1}}, \quad [c] = \mathrm{m\,s^{-1}}\)

\([G^a \hbar^b c^d] = \mathrm{m}^{3a+2b+d}\,\mathrm{kg}^{-a+b}\,\mathrm{s}^{-2a-b-d}\)

For the dimension to be \(\mathrm{kg}\) only: \(3a + 2b + d = 0\), \(-a + b = 1\), \(-2a - b - d = 0\)

From the 2nd equation: \(b = 1 + a\). Substituting into the 1st equation: \(3a + 2(1 + a) + d = 0 \Rightarrow d = -5a - 2\). Substituting into the 3rd equation: \(-2a - (1 + a) - (-5a - 2) = 0 \Rightarrow 2a + 1 = 0 \Rightarrow a = -1/2\). Therefore \(b = 1/2\), \(d = 1/2\).

\(\boxed{M_P = \sqrt{\frac{\hbar c}{G}}}\)

(b) Planck length and Planck time:

Following the same procedure (or deriving directly from dimensional relations):

\(\boxed{\ell_P = \sqrt{\frac{\hbar G}{c^3}}, \qquad t_P = \sqrt{\frac{\hbar G}{c^5}}}\)

Relations: \(\ell_P = c \cdot t_P\), \(\ell_P = \hbar/(M_P c)\) (the Compton wavelength of the Planck mass), \(t_P = \ell_P/c\).

(c) Numerical values:

  • \(M_P = \sqrt{\frac{1.055 \times 10^{-34} \times 3.0 \times 10^8}{6.674 \times 10^{-11}}} = \sqrt{\frac{3.165 \times 10^{-26}}{6.674 \times 10^{-11}}} = \sqrt{4.74 \times 10^{-16}} \approx 2.18 \times 10^{-8}\ \mathrm{kg}\)

  • \(\ell_P = \sqrt{\frac{1.055 \times 10^{-34} \times 6.674 \times 10^{-11}}{(3.0 \times 10^8)^3}} = \sqrt{\frac{7.04 \times 10^{-45}}{2.7 \times 10^{25}}} = \sqrt{2.61 \times 10^{-70}} \approx 1.62 \times 10^{-35}\ \mathrm{m}\)

  • \(t_P = \ell_P/c = \frac{1.62 \times 10^{-35}}{3.0 \times 10^8} \approx 5.39 \times 10^{-44}\ \mathrm{s}\)

\(\boxed{M_P \approx 2.18 \times 10^{-8}\ \mathrm{kg}, \quad \ell_P \approx 1.62 \times 10^{-35}\ \mathrm{m}, \quad t_P \approx 5.39 \times 10^{-44}\ \mathrm{s}}\)

(d) Conversion to GeV and comparison with the LHC:

\(M_P c^2 = 2.18 \times 10^{-8} \times (3.0 \times 10^8)^2 = 2.18 \times 10^{-8} \times 9.0 \times 10^{16} = 1.96 \times 10^9\ \mathrm{J}\)

\(\frac{1.96 \times 10^9}{1.602 \times 10^{-10}} \approx 1.22 \times 10^{19}\ \mathrm{eV} = 1.22 \times 10^{10}\ \mathrm{GeV}\)

\(\boxed{M_P c^2 \approx 1.22 \times 10^{19}\ \mathrm{GeV} = 1.22 \times 10^{16}\ \mathrm{TeV}}\)

Ratio to the LHC (13 TeV):

\(\frac{M_P c^2}{\sqrt{s}_{\mathrm{LHC}}} = \frac{1.22 \times 10^{16}}{13} \approx 10^{15}\)

Discussion:

The Planck scale is approximately \(10^{15}\) times higher than the LHC. When gravity is naively incorporated into quantum field theory, the gravitational coupling strength grows with energy and the perturbative expansion breaks down near the Planck scale. There is no prospect of directly reaching this energy regime experimentally, so we must rely on the internal consistency of the theory itself to probe this frontier.

Advanced

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

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Problem:

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.

(a) Treatment in quantum mechanics (postulate):

In quantum mechanics, the wave function of identical particles (e.g., two electrons) must be symmetric for bosons and antisymmetric for fermions, based on the requirement that "exchanging the positions of particles 1 and 2 yields a physically indistinguishable state." This is imposed by hand at the level of wave function construction as the symmetrization postulate. That is, in quantum mechanics, when constructing a two-particle wave function, one is axiomatically required to symmetrize/antisymmetrize as \(|\psi_1\rangle \otimes |\psi_2\rangle + \epsilon |\psi_2\rangle \otimes |\psi_1\rangle\) (\(\epsilon = +1\) for bosons, \(\epsilon = -1\) for fermions).

(b) Automatic derivation of boson indistinguishability:

From the commutation relation \([\hat{a}^\dagger_{\mathbf{p}_1}, \hat{a}^\dagger_{\mathbf{p}_2}] = \hat{a}^\dagger_{\mathbf{p}_1}\hat{a}^\dagger_{\mathbf{p}_2} - \hat{a}^\dagger_{\mathbf{p}_2}\hat{a}^\dagger_{\mathbf{p}_1} = 0\):

\(\hat{a}^\dagger_{\mathbf{p}_1}\hat{a}^\dagger_{\mathbf{p}_2} = \hat{a}^\dagger_{\mathbf{p}_2}\hat{a}^\dagger_{\mathbf{p}_1}\)

Acting on the vacuum:

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

\(\boxed{|\mathbf{p}_1, \mathbf{p}_2\rangle = |\mathbf{p}_2, \mathbf{p}_1\rangle}\)

(c) Antisymmetry of fermions and the Pauli exclusion principle:

From the anticommutation relation \(\{\hat{b}^\dagger_{\mathbf{p}_1,s_1}, \hat{b}^\dagger_{\mathbf{p}_2,s_2}\} = \hat{b}^\dagger_{\mathbf{p}_1,s_1}\hat{b}^\dagger_{\mathbf{p}_2,s_2} + \hat{b}^\dagger_{\mathbf{p}_2,s_2}\hat{b}^\dagger_{\mathbf{p}_1,s_1} = 0\):

\(\hat{b}^\dagger_{\mathbf{p}_1,s_1}\hat{b}^\dagger_{\mathbf{p}_2,s_2} = -\hat{b}^\dagger_{\mathbf{p}_2,s_2}\hat{b}^\dagger_{\mathbf{p}_1,s_1}\)

Therefore, the two-fermion state:

\(|\mathbf{p}_1 s_1, \mathbf{p}_2 s_2\rangle = \hat{b}^\dagger_{\mathbf{p}_1,s_1}\hat{b}^\dagger_{\mathbf{p}_2,s_2}|0\rangle = -\hat{b}^\dagger_{\mathbf{p}_2,s_2}\hat{b}^\dagger_{\mathbf{p}_1,s_1}|0\rangle = -|\mathbf{p}_2 s_2, \mathbf{p}_1 s_1\rangle\)

Antisymmetric under exchange. Furthermore, when \(\mathbf{p}_1 = \mathbf{p}_2\), \(s_1 = s_2\), from the anticommutation relation \(\{A, A\} = 2A^2 = 0 \Rightarrow A^2 = 0\):

\((\hat{b}^\dagger_{\mathbf{p},s})^2|0\rangle = 0\)

\(\boxed{\text{Two fermions cannot occupy the same quantum state (Pauli exclusion principle)}}\)

(d) Shift in worldview (approximately 300 characters):

In quantum mechanics, symmetrization/antisymmetrization was required to be "imposed by hand" when constructing two-particle wave functions. This is the symmetrization postulate, and a postulate is a provisional requirement that "agrees with experiment when assumed." In quantum field theory, particles are not independent entities but quantum excitations of fields, and excitation quanta created from the same mode of the same field are inherently indistinguishable from the moment of their creation. Simply by incorporating the algebraic structure of commutation relations (bosons) / anticommutation relations (fermions) of creation operators into the framework of field quantization, symmetry, antisymmetry, and the Pauli exclusion principle are automatically derived as consequences. This shift—where a postulate is demoted to a theorem—quantitatively demonstrates the change in worldview from "particles are the fundamental entities" to "fields are the fundamental entities." Furthermore, the spin-statistics theorem determines which fields are quantized with commutation relations and which with anticommutation relations, based on the Lorentz transformation properties of the field.

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

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Problem:

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."

(a) Dimension of the QED coupling constant \(e\):

Each factor in \(\mathcal{L}_{\mathrm{int}} = -e\,\bar{\psi}\gamma^\mu\psi\,A_\mu\):

  • \([\bar{\psi}\psi] = [\psi]^2 = [\text{mass}]^{3}\)
  • \([\gamma^\mu] = [\text{mass}]^0\) (dimensionless constant matrices)
  • \([A_\mu] = [\text{mass}]^1\)

Therefore \([\bar{\psi}\gamma^\mu\psi A_\mu] = [\text{mass}]^{3+0+1} = [\text{mass}]^4\). Since \([\mathcal{L}] = [\text{mass}]^4\):

\([e] = [\text{mass}]^{4-4} = [\text{mass}]^0\)

\(\boxed{[e] = 0 \quad(\text{dimensionless})}\)

(b) Dimensions of the gravitational constant \(G\) and \(\kappa\):

In the Einstein-Hilbert action \(S_{\mathrm{EH}} = \frac{1}{16\pi G}\int d^4x\sqrt{-g}\,R\), we have \([d^4x] = [\text{mass}]^{-4}\), \([R] = [\text{mass}]^2\), \([\sqrt{-g}] = [\text{mass}]^0\), \([S] = 0\):

\(0 = [G]^{-1} \cdot [\text{mass}]^{-4} \cdot [\text{mass}]^2 \quad\Rightarrow\quad [G]^{-1} = [\text{mass}]^{2}\)

\(\boxed{[G] = [\text{mass}]^{-2}}\)

\(\kappa = \sqrt{32\pi G}\):

\(\boxed{[\kappa] = [G]^{1/2} = [\text{mass}]^{-1}}\)

(c) Renormalizability classification:

Coupling constant Mass dimension \(\delta\) Classification
\(e\) (QED) \(0\) Renormalizable
\(G\) (gravity) \(-2\) Non-renormalizable
\(\kappa\) (linearized gravitational coupling) \(-1\) Non-renormalizable

(d) Why quantizing gravity requires a framework beyond quantum field theory (brief explanation):

In a non-renormalizable theory (\(\delta < 0\)), new types of divergences (with increasingly higher momentum powers) appear at each loop order, requiring infinitely many counterterms to absorb them. Since these cannot be controlled by a finite number of parameters, the theory loses its predictive power. The fundamental reason why \([G] = [\text{mass}]^{-2}\) in quantum gravity traces back to the fact that gravity describes "the geometry of spacetime itself," so higher-dimensional quantities (higher derivatives of the Ricci tensor) combine with the coupling constant to proliferate divergences. The idea of string theory—where Rina explained in the main text that "point particles are replaced by strings with finite extent"—structurally alleviates these UV divergences. The spatial extent of strings (\(\sim \ell_s\)) softens interactions at high momenta (\(\mathbf{k} \gg 1/\ell_s\)), circumventing the barrier of non-renormalizability inherent to quantum field theory. This is the core reason why quantizing gravity requires a framework that goes beyond quantum field theory.