Questions: Klein-Gordon Field (Canonical Quantization)

Question 1

In canonical quantization of the Klein-Gordon field, one imposes [phi(x,t), pi(y,t)] = i delta^3(x-y). A student asks: 'Why is this a delta function rather than a Kronecker delta?' What is the correct explanation?

• A. The delta function is an approximation that becomes a Kronecker delta on a lattice
• B. Because x and y are continuous labels for the infinite degrees of freedom of the field — the commutation relation is the continuum generalization of [q_i, p_j] = i delta_{ij}
• C. The delta function ensures that the commutator is Lorentz invariant
• D. The Kronecker delta only applies to fermionic fields

Answer: (B)

In particle mechanics, [q_i, p_j] = i hbar delta_{ij} involves a Kronecker delta because i and j label discrete degrees of freedom. In field theory, the 'label' on each degree of freedom is the continuous spatial position x. The field phi(x) at each point x is analogous to a separate q_i, and pi(x) is the conjugate momentum at that point. The Dirac delta function delta^3(x-y) is exactly the continuum limit of the Kronecker delta: it says that field operators at different spatial points commute, while the field and its conjugate momentum at the same point have a canonical commutation relation.

Question 2

After quantization, the Klein-Gordon field phi(x) can be expanded as phi(x) = integral [a_p e^{ipx} + a_p-dagger e^{-ipx}] d^3p / (2pi)^3 (2E_p). The operator a_p-dagger creates a particle with momentum p. What happens if you try to define a position-space 'particle creation operator' by Fourier transforming a_p-dagger?

• A. You obtain a well-defined operator that creates a particle localized at a point
• B. The resulting operator is phi-dagger(x) itself, which creates a particle at position x — but the particle cannot be perfectly localized due to the energy-momentum relation, and the state is not a position eigenstate in the non-relativistic sense
• C. The Fourier transform diverges and no such operator exists
• D. The operator creates an antiparticle rather than a particle

Answer: (B)

For a real scalar field, phi(x) itself serves as the operator that creates and destroys particles at position x. For a complex field, phi-dagger(x) creates a particle at x. However, perfect localization is impossible in relativistic quantum field theory: attempting to confine a particle to a region smaller than its Compton wavelength costs enough energy to create particle-antiparticle pairs. The position-space 'creation operator' creates a state that is spread out over roughly a Compton wavelength, not a delta-function-localized state. This is a fundamental departure from non-relativistic quantum mechanics.

Question 3

The vacuum energy of the quantized Klein-Gordon field is the sum of (1/2) hbar omega_p over all momentum modes, which diverges. This infinity is physically meaningful and must be included in all calculations.

• T. True
• F. False

Answer: False

The infinite vacuum energy is the sum of zero-point energies from each mode's harmonic oscillator. In non-gravitational physics, only energy differences are observable, so this infinite constant can be subtracted by normal ordering — redefining the Hamiltonian so that the vacuum has zero energy. Normal ordering places all creation operators to the left of annihilation operators, automatically removing the vacuum energy. This is the first infinity encountered in QFT and the simplest to handle. The situation is more subtle in gravity, where absolute energy density matters — the cosmological constant problem.

Question 4

Explain why canonical quantization of the Klein-Gordon field produces a theory of particles, even though the starting point is a continuous classical field.

Model answer: The classical Klein-Gordon field decomposes into independent Fourier modes, each behaving as a harmonic oscillator with frequency omega_p = sqrt(p^2 + m^2). Canonical quantization promotes each mode to a quantum harmonic oscillator with creation operator a_p-dagger and annihilation operator a_p. The energy spectrum of each mode is discrete: (n_p + 1/2) hbar omega_p, where n_p is a non-negative integer. Interpreting n_p as the number of particles with momentum p, the quantized field naturally describes a system with a variable number of particles — each quantum of excitation of mode p is a particle with momentum p, energy omega_p, and mass m.

This is the conceptual core of quantum field theory: particles are not fundamental objects put in by hand but are quantized excitations of underlying fields. A photon is a quantum of the electromagnetic field, an electron is a quantum of the Dirac field, and a Higgs boson is a quantum of the Higgs field. The field is primary; the particle is derived.