Questions: Feynman Diagrams and Perturbative Expansion

Question 1

A virtual photon appears as an internal line in a Feynman diagram for electron-electron scattering. Which statement about this virtual photon is correct?

• A. It is a real photon that briefly exists during the interaction and then annihilates
• B. It is a field excitation that may carry any energy, including values that violate E² = p²c² + m²c⁴, because it is never directly observed
• C. It must satisfy the photon dispersion relation E = pc, because all photons are massless
• D. It represents the average over all possible photon trajectories between the two electrons

Answer: (B)

Virtual particles are 'off-shell': they are internal lines in Feynman diagrams representing intermediate field excitations that need not satisfy the on-shell relation E² = p²c² + m²c⁴. This is possible because they are never directly measured — only the external lines (real, observable particles) are required to be on-shell. Energy and momentum are conserved at every vertex, but the propagator for an internal line allows all values of p², which is why integrating over unmeasured loop momenta can give divergent results. Options A and D mischaracterize what internal lines mean; option C incorrectly imposes the on-shell condition.

Question 2

In the perturbative expansion of a quantum field theory amplitude, what does a 'tree-level' diagram represent?

• A. A diagram drawn on paper (as opposed to a computer-generated diagram)
• B. The leading-order term in the expansion — a diagram with no closed loops, corresponding to the lowest power of the coupling constant
• C. A diagram that only involves photons, not electrons or other fermions
• D. An approximation valid only for strong coupling constants where higher-order terms are negligible

Answer: (B)

Tree-level diagrams are those with no closed loops. In the perturbative expansion in powers of the coupling constant λ, the tree-level diagrams give the lowest-order (leading) contributions. Each additional loop adds a factor of λ (or λ²), so loop diagrams are higher-order corrections. Tree-level amplitudes correspond to the classical field theory limit and are finite; loop diagrams introduce integrals over unconstrained momenta that can diverge. The term 'tree' refers to the topological structure (graph with no cycles), not to the physical medium or the content of the diagram.

Question 3

Virtual particles in Feynman diagrams are field excitations that need not satisfy the energy-momentum relation required of real, observable particles.

• T. True
• F. False

Answer: True

True. Internal lines in Feynman diagrams represent propagators — mathematical factors encoding the quantum amplitude for a field excitation to carry a given four-momentum. These excitations are 'off-shell': the four-momentum squared p² need not equal m². Because they are never directly measured (only external, observable particles enter detectors), there is no physical requirement that they be on-shell. This off-shell freedom is what makes the integration over internal momenta in loop diagrams possible — and is also why those integrals diverge at large momenta.

Question 4

A Feynman diagram represents the literal spacetime path taken by particles during a quantum interaction.

• T. True
• F. False

Answer: False

False. Feynman diagrams are bookkeeping devices for terms in a perturbative expansion of the path integral — they encode specific mathematical contributions to a quantum amplitude, not literal particle trajectories. In quantum mechanics, there is no single definite path; the path integral sums over all possible histories. Each diagram represents one term in a power-series expansion of that sum. Internal lines (virtual particles) do not correspond to real particles with definite trajectories; they represent the quantum superposition of all possible intermediary field configurations at that order in the expansion.

Question 5

Why do loop diagrams in quantum field theory produce divergent integrals, and what does renormalization do to handle them?

Model answer: Loop diagrams require integrating over all momenta flowing around the loop — there is no kinematic constraint fixing the loop momentum, because energy-momentum conservation at vertices still leaves one free integration variable per loop. As the loop momentum goes to infinity (the ultraviolet limit), the integrand typically does not decay fast enough, producing a divergent result. Renormalization absorbs these infinities into redefinitions of physical parameters (mass, charge, field strength), which are then measured experimentally rather than computed from the bare theory. After renormalization, predictions for observable quantities are finite and in remarkable agreement with experiment.

The key insight is that the bare parameters in the Lagrangian are not the measurable physical parameters — renormalization establishes the relationship between them by requiring that loop-corrected quantities equal observed values. For QED, this procedure yields predictions (like the electron's anomalous magnetic moment) accurate to 12 significant figures, making it one of the most precisely tested theories in physics.