Chapter 37 Exercises: From Quantum Mechanics to Quantum Field Theory

Part A: Conceptual Questions (*)

These questions test your understanding of the core ideas. No calculations required.

A.1 List four specific physical phenomena that quantum mechanics (as developed in Chapters 1–36) cannot describe. For each, explain briefly why QFT is needed.

A.2 What does it mean to say "particles are excitations of fields"? Give a precise statement, and then give an analogy (while noting where the analogy breaks down).

A.3 A friend says: "Virtual particles are just real particles that exist for a very short time, allowed by the energy-time uncertainty principle." Critique this statement carefully. What is right about it, and what is misleading?

A.4 In quantum mechanics, the Pauli exclusion principle is an additional postulate (or a property of the wavefunction under particle exchange). In quantum field theory, it is a theorem. What assumptions go into proving the spin-statistics theorem, and why does the proof require relativity?

A.5 Explain why the vacuum energy of a quantum field is infinite in naive computation, and describe two different attitudes one might take toward this infinity: (a) the practical approach (normal ordering) and (b) the profound problem (cosmological constant).

A.6 Why is the coupling constant $\alpha \approx 1/137$ important for the success of QED perturbation theory? What would happen to QED predictions if $\alpha$ were of order 1?

A.7 The Standard Model describes three of the four fundamental forces. Which force is not included, and why is it so difficult to incorporate into the QFT framework?

A.8 In Section 37.6, we listed several open questions that QFT does not answer. Choose one and explain in your own words why it is a deep problem, not just a detail to be filled in.

Part B: Applied Problems (**)

These problems require direct application of the chapter's key equations.

B.1: Klein-Gordon Dispersion Relation

The dispersion relation for a free massive scalar field is $\omega(\mathbf{k}) = \sqrt{|\mathbf{k}|^2 + m^2}$ (in natural units $\hbar = c = 1$).

(a) Restore factors of $\hbar$ and $c$: write $\omega(\mathbf{k})$ with explicit constants.

(b) In the non-relativistic limit ($|\mathbf{k}| \ll mc/\hbar$), expand $\omega$ to second order in $|\mathbf{k}|$. Show that $\hbar\omega \approx mc^2 + \frac{\hbar^2|\mathbf{k}|^2}{2m}$. Identify the rest energy and the non-relativistic kinetic energy.

(c) In the ultra-relativistic limit ($|\mathbf{k}| \gg mc/\hbar$), show that $\omega \approx c|\mathbf{k}|$ — the dispersion relation of a massless particle (photon).

(d) Plot $\omega(k)$ vs. $k$ for $m = 0$, $m = 1$, and $m = 3$ (in natural units). What is the minimum frequency (gap) for a massive field?

B.2: Counting Degrees of Freedom

(a) A real scalar field $\phi$ has one degree of freedom per spacetime point. A complex scalar field $\Phi = (\phi_1 + i\phi_2)/\sqrt{2}$ has two. Why does a complex field describe a particle AND its antiparticle, while a real field describes a particle that is its own antiparticle?

(b) The Dirac field $\psi$ is a 4-component spinor. After accounting for the equation of motion (which halves the degrees of freedom), how many independent degrees of freedom does it have? What do they correspond to physically?

(c) The electromagnetic field $A_\mu$ has 4 components, but gauge invariance removes one, and the equation of motion removes another. How many physical degrees of freedom remain? What do they correspond to?

(d) Fill in the following table for the Standard Model:

B.3: Vacuum Energy with a Cutoff

The vacuum energy density of a scalar field in a box of volume $V = L^3$ is:

$$\frac{E_0}{V} = \frac{1}{2}\int_0^{\Lambda} \frac{4\pi k^2\, dk}{(2\pi)^3}\,\sqrt{k^2 + m^2}$$

where $\Lambda$ is an ultraviolet cutoff (the maximum momentum we trust the theory to).

(a) Evaluate this integral in the massless case ($m = 0$). Show that $E_0/V = \Lambda^4/(16\pi^2)$.

(b) For a cutoff at the Planck scale, $\Lambda = M_P c/\hbar$ where $M_P = \sqrt{\hbar c/G} \approx 1.22 \times 10^{19}$ GeV$/c^2$, compute the vacuum energy density in units of GeV$^4$ (with $\hbar = c = 1$).

(c) The observed dark energy density is approximately $\rho_\Lambda \approx (2.3 \times 10^{-3}\,\text{eV})^4$. What is the ratio of the QFT prediction to the observed value? (This is the "cosmological constant problem.")

(d) If instead we use a cutoff at the QCD scale ($\Lambda \sim 200$ MeV), what vacuum energy density do we get? Is it closer to the observed value?

B.4: The Casimir Effect

Two perfectly conducting parallel plates of area $A$ are separated by a distance $L$. The Casimir force between them (for the electromagnetic field) is:

$$F = -\frac{\pi^2\hbar c}{240 L^4} A$$

(a) Calculate the Casimir force for plates of area $A = 1\,\text{cm}^2$ separated by $L = 1\,\mu\text{m}$.

(b) Calculate the corresponding pressure (force per unit area). Compare to atmospheric pressure ($\sim 10^5$ Pa).

(c) The Casimir force scales as $L^{-4}$. At what plate separation does the Casimir pressure equal atmospheric pressure? Is this experimentally accessible?

(d) Explain qualitatively why the Casimir force is attractive. (Hint: between the plates, only modes with wavelength $\lambda_n = 2L/n$ can exist. Outside, all modes exist.)

B.5: Feynman Diagram Counting

In $\phi^4$ theory, each vertex connects four lines.

(a) Draw all distinct Feynman diagrams for $2 \to 2$ scattering (two particles in, two particles out) at tree level (order $\lambda^1$). How many are there?

(b) At one-loop level (order $\lambda^2$), how many topologically distinct diagrams contribute to $2 \to 2$ scattering? (There are three: $s$-channel, $t$-channel, and $u$-channel loops.)

(c) At tree level, the process $2 \to 4$ (two particles in, four out) first appears. What order in $\lambda$ is this? Draw one contributing diagram.

(d) The number of Feynman diagrams grows factorially with the order of perturbation theory. At $n$-th order in $\phi^4$ theory, the number of diagrams grows roughly as $(2n-1)!! = 1 \times 3 \times 5 \times \cdots \times (2n-1)$. How many diagrams are there at order $\lambda^5$?

Part C: Computational Problems (-*)

These problems involve quantitative analysis that benefits from numerical computation.

C.1: Mode Expansion and Vacuum Fluctuations (**)

Using the code from code/example-01-qft-preview.py:

(a) Plot the dispersion relation $\omega(k) = \sqrt{k^2 + m^2}$ for $m = 0, 0.5, 1.0, 2.0$ in natural units.

(b) For a 1D scalar field in a box of length $L = 10$ (natural units), with modes $k_n = 2\pi n/L$, compute and plot the cumulative vacuum energy $E_0(N) = \frac{1}{2}\sum_{n=1}^{N}\omega_n$ as a function of the cutoff $N$ (number of modes included). Show that it diverges.

(c) Compute the Casimir energy for a 1D scalar field between two "walls" separated by distance $a$ (modes $k_n = n\pi/a$) using zeta function regularization: $E(a) = \frac{1}{2}\sum_{n=1}^{\infty}\omega_n \to -\frac{\pi}{24a}$ for the massless case. Numerically verify this by computing the difference between the energy inside and outside the walls with a high cutoff.

C.2: Scalar Field Propagator (***)

(a) Compute the 1+1D scalar field propagator numerically:

$$D_F(x, t) = \int_{-\infty}^{\infty} \frac{dk}{2\pi} \frac{e^{ikx - i\omega_k t}}{2\omega_k}$$

for $m = 1$ and plot $|D_F(x, 0)|$ as a function of $x$. Verify that it decays exponentially for $|x| \gg 1/m$.

(b) Plot $|D_F(0, t)|$ as a function of $t$. How does the propagator behave inside vs. outside the light cone ($|x| > |t|$ vs. $|x| < |t|$)?

(c) For the massless case ($m = 0$), compute the propagator and show it is non-zero only on the light cone ($|x| = |t|$, up to distributional subtleties).

C.3: Running Coupling Constant (***)

The QED coupling constant "runs" (changes with energy scale $Q$) according to:

$$\alpha(Q) = \frac{\alpha(m_e)}{1 - \frac{\alpha(m_e)}{3\pi}\ln\left(\frac{Q^2}{m_e^2}\right)}$$

where $\alpha(m_e) = 1/137.036$ is the value at the electron mass scale.

(a) Plot $\alpha(Q)$ for $Q$ from $m_e = 0.511$ MeV to $M_Z = 91.2$ GeV.

(b) At $Q = M_Z$, the measured value is $\alpha(M_Z) \approx 1/128$. Does the one-loop formula reproduce this?

(c) At what energy scale does the one-loop formula predict $\alpha \to \infty$ (the "Landau pole")? Is this energy scale physically meaningful?

(d) For QCD, the coupling runs in the opposite direction: $\alpha_s$ decreases at high energy ("asymptotic freedom"). Plot $\alpha_s(Q) = \frac{12\pi}{(33 - 2n_f)\ln(Q^2/\Lambda_{QCD}^2)}$ with $n_f = 6$ flavors and $\Lambda_{QCD} = 200$ MeV. At what energy does $\alpha_s \approx 1$ (where perturbation theory breaks down)?

Part D: Synthesis Problems (***)

These problems require integrating multiple concepts and thinking beyond direct formulas.

D.1: From QHO to QFT

Trace the harmonic oscillator through this textbook and into QFT:

(a) In Chapter 4, the QHO had creation and annihilation operators $\hat{a}^\dagger, \hat{a}$ with $[\hat{a}, \hat{a}^\dagger] = 1$. Write the QHO Hamiltonian and its energy eigenvalues.

(b) In Chapter 8, the QHO operators were expressed in Dirac notation. The states $|n\rangle = (\hat{a}^\dagger)^n|0\rangle/\sqrt{n!}$ form a Fock space. How does this connect to the multi-particle Fock space of QFT?

(c) In Chapter 34, we introduced creation operators $\hat{a}^\dagger_\mathbf{k}$ for each mode $\mathbf{k}$. How is the second-quantized field related to infinitely many copies of the QHO?

(d) In this chapter, we quantized the scalar field by the same procedure. What is the energy of the vacuum in terms of QHO zero-point energies? Why is this sum infinite, whereas the QHO zero-point energy $\frac{1}{2}\hbar\omega$ is finite?

(e) Argue that every free bosonic QFT is, at its core, a (possibly infinite) collection of quantum harmonic oscillators. What changes when interactions are added?

D.2: Relativistic Invariance and Causality

(a) The equal-time commutator of the free scalar field vanishes for spacelike separations: $[\hat{\phi}(\mathbf{x}, t), \hat{\phi}(\mathbf{y}, t)] = 0$ when $\mathbf{x} \neq \mathbf{y}$. Why is this essential for causality?

(b) The Feynman propagator $D_F(x - y)$ is non-zero for spacelike separations. Does this violate causality? Explain carefully. (Hint: the propagator is not directly observable. What is observable are commutators of field operators.)

(c) For a fermionic (Dirac) field, the commutator is replaced by an anticommutator: $\{\hat{\psi}(\mathbf{x}, t), \hat{\psi}^\dagger(\mathbf{y}, t)\} = \delta^{(3)}(\mathbf{x} - \mathbf{y})$. Show that using commutators (instead of anticommutators) for a spin-1/2 field would violate causality — this is part of the spin-statistics theorem.

D.3: The Standard Model in One Page

Create a comprehensive summary of the Standard Model:

(a) List all fundamental fermions (12 quarks + 12 leptons, counting antiparticles) organized by generation.

(b) List all gauge bosons and the force each mediates.

(c) State the gauge group and identify which fermions are charged under which gauge factors.

(d) Explain in 3–4 sentences how the Higgs mechanism gives mass to the $W$ and $Z$ bosons.

(e) List three phenomena that the Standard Model cannot explain.

D.4: Path Integral vs. Canonical Quantization

(a) In Chapter 31, you learned the path integral formulation of QM: $\langle x_f|e^{-iHt/\hbar}|x_i\rangle = \int \mathcal{D}[x(t)]\, e^{iS[x]/\hbar}$. How does this generalize to QFT? (Replace $x(t)$ with $\phi(x, t)$; the integral is now over all field configurations.)

(b) The partition function of QFT is $Z = \int \mathcal{D}[\phi]\, e^{iS[\phi]}$. All physical quantities can be extracted from $Z$ by taking functional derivatives. Why is this approach often more powerful than canonical quantization?

(c) The path integral makes symmetries manifest. Explain why gauge invariance, Lorentz invariance, and the spin-statistics connection are easier to see in the path integral approach than in canonical quantization.

Part E: Research and Exploration (****)

These problems go beyond the chapter's direct content and require independent investigation.

E.1: The Lamb Shift

Research the measurement and theoretical calculation of the Lamb shift.

(a) What is the Lamb shift, and why does the Dirac equation (without QFT corrections) predict that the $2S_{1/2}$ and $2P_{1/2}$ levels are degenerate?

(b) Which Feynman diagrams contribute to the Lamb shift at lowest order? (Look up "electron self-energy" and "vacuum polarization.")

(c) Willis Lamb measured the shift in 1947 using microwave spectroscopy. Describe the experimental setup.

(d) Hans Bethe computed the first theoretical estimate of the Lamb shift on a train ride from Shelter Island to Schenectady. What approximations did he make, and how close was his result to the measured value?

E.2: The Discovery of the Higgs Boson

Research the discovery of the Higgs boson at the LHC in 2012.

(a) What decay channels were used to identify the Higgs boson? Why are $H \to \gamma\gamma$ and $H \to ZZ^* \to 4\ell$ particularly clean?

(b) What is the significance level of the discovery, and what does "5 sigma" mean statistically?

(c) How does the measured Higgs mass ($m_H \approx 125$ GeV) compare to theoretical expectations? Why was this particular mass value both expected and surprising?

(d) The Higgs mechanism was proposed independently by several groups in 1964. Why did it take nearly 50 years to confirm experimentally?

E.3: Asymptotic Freedom and the Strong Force

Research asymptotic freedom in QCD — the discovery for which Gross, Politzer, and Wilczek received the 2004 Nobel Prize.

(a) What does "asymptotic freedom" mean physically? How does it differ from the running of the QED coupling?

(b) Why does asymptotic freedom explain both the success of the quark model (quarks behave as free particles at high energies) and confinement (quarks cannot be isolated at low energies)?

(c) What is the role of gluon self-interactions in producing asymptotic freedom? Why don't photons produce the same effect in QED?

Part M: Mixed Practice

These problems mix concepts from Chapter 37 with material from earlier chapters.

M.1: From Chapter 4, the QHO ground state energy is $E_0 = \frac{1}{2}\hbar\omega$. (a) A free scalar field in a box of size $L$ has modes with frequencies $\omega_n = \sqrt{(n\pi/L)^2 + m^2}$ (1D, natural units). Write the vacuum energy as a sum over modes. (b) Show that the vacuum energy per unit length diverges as $\sum_n \omega_n/2$. (c) If we impose a momentum cutoff $n_{\max} = \Lambda L/\pi$, the vacuum energy is finite. How does it scale with $\Lambda$?

M.2: From Chapter 22 (scattering theory), the cross-section $\sigma$ is related to the scattering amplitude $f(\theta)$ by $d\sigma/d\Omega = |f(\theta)|^2$. In QFT, $f$ is computed from Feynman diagrams. (a) For $\phi^4$ theory at tree level, $\mathcal{M} = -\lambda$ is independent of the scattering angle. What does this say about the angular distribution? (b) At one-loop, the amplitude becomes angle-dependent. Explain qualitatively why loop corrections introduce angular dependence.

M.3: From Chapter 15, identical bosons have symmetric wavefunctions and identical fermions have antisymmetric wavefunctions. (a) In the QFT framework, show that $[\hat{a}^\dagger_{\mathbf{k}_1}, \hat{a}^\dagger_{\mathbf{k}_2}] = 0$ automatically gives $|\mathbf{k}_1, \mathbf{k}_2\rangle = |\mathbf{k}_2, \mathbf{k}_1\rangle$ (bosons). (b) For a Dirac field with $\{\hat{b}^\dagger_{\mathbf{k}_1}, \hat{b}^\dagger_{\mathbf{k}_2}\} = 0$, show that $|\mathbf{k}_1, \mathbf{k}_2\rangle = -|\mathbf{k}_2, \mathbf{k}_1\rangle$ (fermions). (c) What happens if you try to create two identical fermions: $(\hat{b}^\dagger_\mathbf{k})^2|0\rangle = ?$

M.4: From Chapter 34 (second quantization), the number operator is $\hat{N} = \sum_\mathbf{k} \hat{a}^\dagger_\mathbf{k}\hat{a}_\mathbf{k}$. (a) Show that $[\hat{N}, \hat{a}^\dagger_\mathbf{k}] = \hat{a}^\dagger_\mathbf{k}$ and $[\hat{N}, \hat{a}_\mathbf{k}] = -\hat{a}_\mathbf{k}$. (b) In a free field theory, $[\hat{H}, \hat{N}] = 0$. What conservation law does this express? (c) In an interacting theory like $\phi^4$, does $[\hat{H}, \hat{N}] = 0$ still hold? Explain why or why not.