Questions: Fock Space and Particle Interpretation

Question 1

In non-relativistic quantum mechanics, the Hilbert space for N identical particles is fixed. Why does quantum field theory require a Hilbert space (Fock space) that includes all particle numbers simultaneously?

• A. Because relativistic particles move at speeds close to c, so more dimensions are needed
• B. Because the relativistic energy-momentum relation E = sqrt(p^2 + m^2) allows particle creation and annihilation — processes that change particle number — so the state space must accommodate every possible particle number
• C. Because quantum fields are classical objects that do not have a fixed particle number
• D. Because the uncertainty principle prevents precise measurement of particle number

Answer: (B)

In relativistic physics, energy can convert to mass and vice versa. A sufficiently energetic photon can create an electron-positron pair, and an electron and positron can annihilate into photons. These processes change the total number of particles. A Hilbert space with a fixed particle number cannot describe such processes. Fock space solves this by being the direct sum of all N-particle spaces: F = H_0 + H_1 + H_2 + ..., where H_0 is the vacuum (zero particles), H_1 is the one-particle space, and so on. Any state in Fock space can be a superposition of states with different particle numbers.

Question 2

The vacuum state |0> in Fock space has zero particles and zero energy. It is the simplest possible state — essentially 'nothing.'

• T. True
• F. False

Answer: False

The vacuum |0> has zero particles by definition: a_p|0> = 0 for all p. After normal ordering, it has zero energy by convention. But it is far from 'nothing.' The vacuum has non-trivial properties: quantum fields fluctuate around zero (vacuum fluctuations), virtual particle-antiparticle pairs constantly appear and disappear, and measurable effects like the Casimir force and the Lamb shift arise from the vacuum structure. The vacuum is the ground state of the field — the state of lowest energy — but it is a dynamically rich quantum state, not empty space.

Question 3

Two identical bosons created by a_p-dagger a_q-dagger|0> are automatically in a symmetrized state because [a_p-dagger, a_q-dagger] = 0. What is the analogous statement for fermions, and why does it enforce the Pauli exclusion principle?

Model answer: For fermions, the creation operators satisfy anticommutation relations {c_p-dagger, c_q-dagger} = 0. This means c_p-dagger c_q-dagger = -c_q-dagger c_p-dagger, so the two-particle state is automatically antisymmetric under exchange of p and q. Setting p = q gives (c_p-dagger)^2 = 0, which means you cannot create two fermions in the same state — the result is the zero vector, not a physical state. The Pauli exclusion principle is not an additional postulate but a direct algebraic consequence of the anticommutation relations.

This is the spin-statistics connection at work in Fock space. Bosonic commutation relations produce symmetric multi-particle states with unlimited occupation per mode. Fermionic anticommutation relations produce antisymmetric states with at most one particle per mode. The choice between commutation and anticommutation is not arbitrary — it is dictated by the spin-statistics theorem.

Question 4

A state |psi> in Fock space can be a superposition of components with different particle numbers, such as alpha|0> + beta|1_p> + gamma|2_{p,q}>. In what physical situation would such a superposition arise?

• A. This never occurs in nature — physical states always have a definite particle number
• B. In the ground state of an interacting field theory, where the true vacuum is a superposition over all particle numbers due to virtual pair creation
• C. Only in theories with massless particles
• D. Only when an external classical source drives the field

Answer: (B)

In a free field theory, energy eigenstates have definite particle numbers. But interactions mix sectors of different particle number. The ground state of an interacting theory (the 'interacting vacuum') is not the Fock vacuum |0> but a complicated superposition involving virtual pairs. Similarly, scattering processes involve transitions between different particle-number sectors. Coherent states (describing laser light, for example) are explicit superpositions over all photon numbers. The ability to describe such superpositions is precisely why Fock space is necessary.