Questions: Path Integral Quantization

Question 1

In the path integral for a scalar field, the partition function is Z = integral D[phi] e^{iS[phi]}, where D[phi] denotes integration over all field configurations. What does 'all field configurations' mean concretely?

• A. Integration over all possible values of phi at every spacetime point — an infinite-dimensional integral, one ordinary integral for each point in spacetime
• B. A sum over all possible particle trajectories
• C. Integration over the Fourier coefficients of the field
• D. Both A and C are correct descriptions — they are related by a change of variables from position space to momentum space, and both represent the same infinite-dimensional integral

Answer: (D)

The functional integral D[phi] is an integral over all functions phi(x). This can be thought of as an independent integral over the value of phi at each spacetime point (continuum formulation) or equivalently as an integral over all Fourier modes of the field (momentum space formulation). On a lattice, it becomes a finite (but very high) dimensional integral — one variable per lattice site. The two descriptions are related by a unitary change of basis and give the same physics.

Question 2

In the path integral, the classical solution (the field configuration that extremizes S) dominates when S >> hbar. The quantum corrections come from fluctuations around the classical solution.

• T. True
• F. False

Answer: True

When S is large compared to hbar (the classical limit), the phase e^{iS/hbar} oscillates rapidly for configurations far from the classical solution, and their contributions cancel. Only configurations near the stationary point (where delta S = 0, the classical equation of motion) contribute coherently. This is the stationary-phase approximation. Expanding S to second order around the classical solution gives the one-loop correction (a Gaussian integral over fluctuations). Higher orders in the expansion give higher-loop corrections. The path integral therefore naturally organizes the perturbative expansion: tree level = classical, one loop = leading quantum correction, etc.

Question 3

The path integral for gauge theories requires a gauge-fixing procedure (Faddeev-Popov). Without gauge fixing, the path integral gives an infinite answer.

• T. True
• F. False

Answer: True

In a gauge theory, distinct field configurations related by gauge transformations describe the same physics. The naive path integral integrates over all configurations, including the infinite volume of gauge-equivalent copies (the gauge orbit). This overcounting gives infinity. The Faddeev-Popov procedure inserts a gauge-fixing condition (like Lorenz gauge) and a corresponding determinant (which can be expressed using ghost fields) to integrate over each physical configuration exactly once. The resulting gauge-fixed path integral gives finite, well-defined answers.

Question 4

Explain the advantages of path integral quantization over canonical quantization for gauge theories, and identify one situation where canonical quantization is more natural.

Model answer: Path integral advantages: (1) Manifest Lorentz covariance — there is no need to single out a time direction, unlike canonical quantization where equal-time commutation relations break manifest covariance. (2) Gauge theories are handled systematically via the Faddeev-Popov procedure, which is most naturally formulated in the path integral. (3) Non-perturbative effects like instantons (field configurations that tunnel between different vacuum sectors) are naturally included as saddle points of the Euclidean path integral. (4) Lattice field theory directly discretizes the path integral for numerical computation. Canonical quantization is more natural for: understanding the Hilbert space structure, the particle interpretation of the theory, and for systems where Hamiltonian methods are needed (e.g., bound state problems, real-time evolution).

In practice, most modern QFT calculations use the path integral for deriving Feynman rules and computing scattering amplitudes, while canonical quantization provides the conceptual framework (Fock space, particle states, S-matrix). The two approaches are equivalent but have complementary strengths.