Identical particles and Fock space

For two identical particles in ordinary three-dimensional space, bosonic states are symmetric and fermionic states antisymmetric under exchange of the complete single-particle labels. If $|a\rangle$ and $|b\rangle$ are distinct orthonormal orbitals, the normalized combinations are $(|a\rangle|b\rangle\pm|b\rangle|a\rangle)/\sqrt2$. The signs refer to exchange, not to particles carrying ordinary positive or negative probabilities. Spin and spatial labels must both be included in the complete state.

Antisymmetrizing two identical copies of the same complete single-particle state gives zero. This is Pauli exclusion. It does not forbid two electrons from occupying the same spatial orbital with opposite spin because those are different complete states. In nonrelativistic many-particle mechanics we impose the appropriate statistics for the species. The connection between integer spin and bosonic statistics, and half-integer spin and fermionic statistics, needs assumptions of relativistic quantum field theory; it is not derived merely by permuting a two-particle wavefunction.

Choose a one-particle mode basis and write a many-particle state using occupations $|n_1,n_2,\ldots\rangle$. Bosonic mode occupations can be any nonnegative integer; each fermionic mode has occupation zero or one. For noninteracting particles with one-particle energies $\epsilon_j$, the fixed-number energy is the sum $\sum_j\epsilon_jn_j$, with any separately specified vacuum convention. Interactions generally make occupation-basis states mix and require additional Hamiltonian terms.

Bosonic creation and annihilation operators change a mode's occupation with square-root factors. Fermionic operators obey anticommutation relations instead, giving exclusion and ordering-dependent signs across different modes. The sign convention depends on a chosen order for the modes, while measurable predictions must be consistent with that convention. “Identical” therefore does not mean all particles have the same state; it means the species provides no physical particle labels that distinguish otherwise exchanged descriptions.

Fock space is the direct sum of the zero-particle, one-particle, two-particle, and higher appropriately symmetrized or antisymmetrized sectors. A fixed-number problem lives in one sector. Including all sectors makes a variable particle number representable, but does not force it to change: a number-conserving Hamiltonian keeps an initial state inside its fixed-number sector. The vacuum means zero occupation in the chosen mode description, not necessarily zero energy or the absence of every kind of physical field.

For one mechanical oscillator, the number ladder originally counted excitations of one coordinate. In a quantized free field, independent modes behave like oscillators and their excitation numbers acquire a particle interpretation appropriate to the setup. This needs a field Hamiltonian, commutation rules, and a choice of modes; the oscillator analogy alone does not establish relativistic locality or an interacting theory.