Chapter 37 Quiz: From Quantum Mechanics to Quantum Field Theory

Instructions: This quiz covers the core concepts from Chapter 37. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. The primary motivation for quantum field theory over quantum mechanics is:

(a) Quantum mechanics gives wrong predictions for hydrogen energy levels (b) QM cannot describe particle creation/annihilation, has relativistic pathologies, and cannot explain vacuum phenomena (c) QM is mathematically inconsistent (d) QFT is simpler and easier to compute with

Q2. In the canonical quantization of the free scalar field, the classical field amplitude $a_\mathbf{k}$ is promoted to:

(a) A wavefunction $\psi(\mathbf{k})$ (b) An annihilation operator $\hat{a}_\mathbf{k}$ satisfying $[\hat{a}_\mathbf{k}, \hat{a}^\dagger_{\mathbf{k}'}] = (2\pi)^3\delta^{(3)}(\mathbf{k} - \mathbf{k}')$ (c) A classical probability distribution $P(\mathbf{k})$ (d) A spinor field $\psi_\alpha(\mathbf{k})$

Q3. A "particle" in quantum field theory is best described as:

(a) A tiny, indivisible ball of matter (b) A quantized excitation of an underlying quantum field (c) A mathematical abstraction with no physical reality (d) A special solution of the Schrodinger equation

Q4. The vacuum state $|0\rangle$ in QFT is:

(a) Empty space with absolutely nothing in it (b) The lowest energy state of the quantum field, with no particles but non-trivial quantum fluctuations (c) A state with exactly one particle of each type (d) Undefined in quantum field theory

Q5. Internal lines in Feynman diagrams represent:

(a) Real particles that are detected in the experiment (b) Virtual particles — mathematical terms in the perturbation series (c) Classical electromagnetic waves (d) Measurement apparatus

Q6. The coupling constant of QED is:

(a) $G_N = 6.67 \times 10^{-11}$ (Newton's constant) (b) $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137$ (c) $\lambda = 0.13$ (Higgs self-coupling) (d) $g_s \approx 1.2$ (strong coupling)

Q7. The spin-statistics theorem in QFT states that:

(a) Particles with integer spin have higher energy than those with half-integer spin (b) Particles with integer spin must be bosons, and particles with half-integer spin must be fermions (c) Spin is quantized in units of $\hbar/2$ (d) Particles with higher spin are heavier

Q8. The Higgs mechanism explains:

(a) Why gravity is so weak compared to other forces (b) How the $W^\pm$ and $Z^0$ gauge bosons acquire mass despite gauge invariance (c) Why protons are heavier than electrons (d) The origin of dark matter

Q9. The "cosmological constant problem" refers to:

(a) The difficulty of measuring the age of the universe (b) The discrepancy of $\sim 10^{120}$ between QFT's estimate of vacuum energy and the observed dark energy (c) The fact that the speed of light is constant (d) The non-zero curvature of spacetime

Q10. Renormalization in QFT is:

(a) A mathematical trick that hides the theory's inconsistencies (b) A systematic procedure for absorbing infinities into redefinitions of physical parameters, reflecting the scale-dependence of physical quantities (c) The process of setting all coupling constants to zero (d) A method for solving the Schrodinger equation exactly

True/False with Justification (4 questions)

Q11. TRUE or FALSE: The Klein-Gordon equation, when interpreted as a single-particle wave equation, gives a valid positive-definite probability density.

Justification:

Q12. TRUE or FALSE: In the Standard Model, all gauge bosons are massless.

Justification:

Q13. TRUE or FALSE: A free quantum field is mathematically equivalent to an infinite collection of independent quantum harmonic oscillators.

Justification:

Q14. TRUE or FALSE: Quantum field theory has been reconciled with general relativity into a consistent theory of quantum gravity.

Justification:

Short Answer (4 questions)

Q15. The QED prediction for the electron's anomalous magnetic moment agrees with experiment to better than one part in $10^{12}$. Explain in 2-3 sentences why this is considered the most impressive prediction in all of physics, and what type of calculations (involving which mathematical objects) are required to achieve this precision.

Q16. Explain the conceptual difference between the vacuum in quantum mechanics (truly empty) and the vacuum in quantum field theory (not empty). Give one experimental consequence of the QFT vacuum being non-trivial.

Q17. In the Standard Model, the gauge group is SU(3) $\times$ SU(2) $\times$ U(1). Briefly describe what each factor corresponds to physically and name the gauge bosons associated with each.

Q18. Section 37.6 outlines the content of a graduate QFT course sequence. Identify three specific topics from this textbook (with chapter numbers) that serve as direct prerequisites for QFT, and explain briefly how each connects.

Applied Scenarios (2 questions)

Q19. Vacuum Energy and the Casimir Effect

The Casimir force between two parallel conducting plates of area $A$ separated by distance $L$ is $F = -\frac{\pi^2\hbar c}{240 L^4}A$.

(a) Calculate the force for $A = 1\,\text{cm}^2$ and $L = 100\,\text{nm}$. (4 points)

(b) Is this force attractive or repulsive? Why? (2 points)

(c) A typical atomic force microscope (AFM) can measure forces down to $\sim 10^{-12}$ N. At what plate separation $L$ does the Casimir force for a $1\,\text{cm}^2$ plate equal $10^{-12}$ N? (4 points)

(d) The Casimir effect is often cited as evidence for vacuum fluctuations. However, some physicists argue it can be explained without invoking vacuum energy (using van der Waals forces between the plate atoms). Does this controversy affect the validity of QFT? Explain briefly. (5 points)

Q20. Particle Content of the Standard Model

(a) How many fundamental fermions are there in the Standard Model? List them by category (quarks and leptons), organized by generation. (4 points)

(b) How many gauge bosons are there? Name each and the force it mediates. (4 points)

(c) The Higgs boson was discovered at the LHC in 2012 with a mass of approximately $125$ GeV. Using $E = mc^2$, calculate this mass in kg. Compare to the proton mass. How many times heavier is the Higgs boson? (4 points)

(d) The top quark mass is $\sim 173$ GeV, making it heavier than the Higgs boson. Why is a fundamental particle so heavy? What determines fermion masses in the Standard Model? (3 points)

Answer Key

Multiple Choice

  1. (b) — QM is internally consistent and spectacularly successful within its domain. The need for QFT arises from particle creation/annihilation, relativistic pathologies (negative probabilities in KG, Dirac sea), vacuum effects (Casimir, Lamb shift), and the need to derive (not assume) spin-statistics.
  2. (b) — Canonical quantization promotes classical amplitudes to operators with canonical commutation relations. This is the same procedure used for the QHO (Chapter 8) and second quantization (Chapter 34).
  3. (b) — In QFT, fields are fundamental and particles are their quantized excitations. An electron is a quantum of the Dirac field, a photon is a quantum of the electromagnetic field.
  4. (b) — The vacuum is not empty but is the ground state of the quantum field, with zero particle number but non-zero field fluctuations. The Casimir effect and Lamb shift are measurable consequences.
  5. (b) — Internal lines are virtual particles — they do not satisfy the on-shell condition and are not directly observed. They are terms in the perturbative expansion of the scattering amplitude.
  6. (b) — The fine-structure constant $\alpha \approx 1/137$ governs electromagnetic interactions. Its smallness ensures rapid convergence of the perturbation series in QED.
  7. (b) — The spin-statistics theorem proves that consistent relativistic QFT requires integer-spin fields to be quantized with commutators (bosons) and half-integer-spin fields with anticommutators (fermions).
  8. (b) — Gauge invariance forbids mass terms for gauge bosons. The Higgs mechanism (spontaneous symmetry breaking) circumvents this: the Higgs field's non-zero vacuum expectation value gives mass to $W^\pm$ and $Z^0$ while preserving gauge invariance at a deeper level.
  9. (b) — QFT predicts vacuum energy density $\sim M_P^4$, while the observed dark energy density is $\sim (10^{-3}\,\text{eV})^4$. The discrepancy of $\sim 10^{120}$ is the worst prediction in the history of physics.
  10. (b) — Renormalization is a rigorous procedure, not a trick. It reflects the physical fact that coupling constants, masses, and field strengths depend on the energy scale at which they are probed. The Wilsonian picture makes this manifestly physical.

True/False

  1. FALSE. The Klein-Gordon "probability density" $\rho \propto \phi^*\partial_t\phi - \phi\partial_t\phi^*$ can be negative, making it invalid as a probability density. This was one of the pathologies that motivated reinterpreting the KG equation as a field equation (where $\rho$ becomes a charge density, which can be negative).
  2. FALSE. The photon and the 8 gluons are massless, but the $W^\pm$ ($\sim 80$ GeV) and $Z^0$ ($\sim 91$ GeV) are massive. Their mass arises from the Higgs mechanism (spontaneous electroweak symmetry breaking).
  3. TRUE. Each mode $\mathbf{k}$ of a free quantum field is an independent QHO with frequency $\omega_\mathbf{k} = \sqrt{|\mathbf{k}|^2 + m^2}$. The field's Fock space is the tensor product of the Fock spaces of these oscillators.
  4. FALSE. Quantum gravity remains the deepest unsolved problem in theoretical physics. General relativity is not a renormalizable QFT, and all attempts to quantize gravity (string theory, loop quantum gravity, etc.) remain incomplete.

Short Answer (Key Points)

  1. The electron $g-2$ prediction requires computing Feynman diagrams up to five loops (over 12,000 diagrams at fifth order), involving intricate multi-dimensional integrals. The calculation incorporates contributions from QED, the weak force, the strong force, and hadronic effects. The agreement to $10^{-12}$ precision between a calculation involving thousands of terms and an independent experiment is unmatched by any other prediction in science — it is the ultimate stress test of QFT.

  2. In QM, the vacuum is the state with no particles and no properties — truly empty. In QFT, the vacuum $|0\rangle$ has zero particle number but non-trivial field fluctuations: each mode of the quantum field has zero-point fluctuations $\langle 0|\hat{\phi}^2|0\rangle \neq 0$. The Casimir effect — the measurable force between uncharged conducting plates — is a direct consequence of these vacuum fluctuations modifying the mode structure of the electromagnetic field.

  3. SU(3) is the gauge group of the strong force (quantum chromodynamics, QCD); its 8 gauge bosons are the gluons, which mediate the strong interaction between quarks. SU(2) is the gauge group of the weak isospin force; its 3 gauge bosons are $W^+$, $W^-$, and $Z^0$ (after electroweak symmetry breaking), mediating weak interactions (e.g., beta decay). U(1) is the gauge group of weak hypercharge; after electroweak symmetry breaking, it combines with the neutral SU(2) component to produce the photon (electromagnetism) and the $Z^0$.

  4. (1) Chapter 4 (QHO): The creation/annihilation operator algebra of the QHO is the building block of QFT — every free field is a collection of QHOs. (2) Chapter 34 (Second Quantization): The Fock space construction and the promotion of field amplitudes to operators is exactly the canonical quantization procedure of QFT. (3) Chapter 31 (Path Integrals): The path integral formulation generalizes directly to QFT by integrating over field configurations instead of particle trajectories, providing the most powerful method for deriving Feynman rules and proving renormalizability.

Applied Scenarios

  1. (a) $F = \frac{\pi^2 (1.055 \times 10^{-34})(3 \times 10^8)}{240 (10^{-7})^4}(10^{-4}) = \frac{\pi^2 \times 3.165 \times 10^{-26}}{240 \times 10^{-28}} \times 10^{-4} = \frac{3.124 \times 10^{-25}}{2.4 \times 10^{-26}} \times 10^{-4} = 13.0 \times 10^{-4} = 1.3 \times 10^{-3}$ N. (b) Attractive. Between the plates, only modes with $\lambda_n = 2L/n$ fit; outside, all wavelengths contribute. The vacuum energy density is lower between the plates, creating a pressure pushing them together. (c) $F = 10^{-12}$ N: $L^4 = \frac{\pi^2\hbar c A}{240 \times 10^{-12}} = \frac{9.87 \times 3.165 \times 10^{-26} \times 10^{-4}}{2.4 \times 10^{-10}} = 1.30 \times 10^{-20}$. $L = (1.30 \times 10^{-20})^{1/4} = 1.07 \times 10^{-5}$ m $= 10.7\,\mu$m. This is experimentally accessible with AFM. (d) The Casimir effect can indeed be derived from the retarded van der Waals interaction between plate atoms (Lifshitz theory), without explicit reference to vacuum energy. This does not affect QFT's validity — the two descriptions are mathematically equivalent, just using different languages. The deeper point is that QFT predicts the correct Casimir force regardless of which physical picture one prefers.

  2. (a) 12 fundamental fermions (plus 12 antiparticles = 24 total). Gen I: up, down quarks; electron, $\nu_e$. Gen II: charm, strange quarks; muon, $\nu_\mu$. Gen III: top, bottom quarks; tau, $\nu_\tau$. Each quark comes in 3 colors. (b) 12 gauge bosons: photon ($\gamma$, EM), $W^+$, $W^-$, $Z^0$ (weak), 8 gluons (strong). Plus the Higgs boson (scalar, not a gauge boson). (c) $m_H = 125\,\text{GeV}/c^2 = 125 \times 10^9 \times 1.602 \times 10^{-19} / (3 \times 10^8)^2 = 2.23 \times 10^{-25}$ kg. Proton mass $= 1.673 \times 10^{-27}$ kg. Ratio: $m_H/m_p \approx 133$. The Higgs is about 133 times heavier than the proton. (d) Fermion masses in the Standard Model are determined by their Yukawa coupling to the Higgs field: $m_f = y_f v/\sqrt{2}$, where $y_f$ is the Yukawa coupling constant and $v \approx 246$ GeV. The top quark is heavy because its Yukawa coupling is large ($y_t \approx 1$). The Standard Model does not explain why the Yukawa couplings have the values they do — this is one of its unexplained parameters.