Fock Space Representations

The Fock space can be depicted in several ways. The occupation-number representation is arguably the easiest, most natural way to think about it, but there are unitary representations, utilizing the Wigner theorem by which physically observable quantities are invariant under (anti-)unitary representations on the transformation group G. At this point, the representation of G is most important. With (anti-)unitary transformation groups and an ONB consisting of occupation-number states, fixing the elements of G, then the transformed elements of the ONB also form an ONB themselves. The transformed particle operators are

and they behave just the way they're expected to, due to the (anti-)unitarity of the transformation. The space thus retains its structure. Note that the transformation is an operator that is unique down to a phase. A family of linear maps whose kernels intersect to the span of some vector of the space, can be combined linearly with invertible constants are constructed such that the dimension of the intersection of its linear combinations equals 1.

The one-particle Hilbert space specifically is interesting to quantum physics. Its unitary representation retains the vacuum state, involves time-reversal and charge conjugation symmetry. The observables can either be transformed actively or passively. The active transformation features the Ansatz:

The passive transformation doesn't transform the observable itself, but rather the the particle operators.

Given an automorphic Hamiltonian on a Hilbert space and a group of transformations G with (anti-)unitary representation D: G → GL(V), so that H is symmetric under the representation D of G on V, then [D(a), H] = 0 for all a ∈ G. For G = SO(3) and the Hamiltonian of a particle in a geometric potential, the representation

with the angular momentum J, then D is unitary. The resulting eigenenergies is dependent on the quantum numbers nlm, but only actually differ in the index n. This means that it has several degrees of accidental degeneracy in the indices l and m. These degeneracies are "symmetry-induced". For linear Hamiltonians with commuting unitary representations, written as an outer sum

with the multiplicity n as the number of equivalent irreducible representations, then

Then there exists a basis of each invariant subspace V which is independent of H, so that in that basis,

The eigenvalues of H are labeled by the (α, i) of the symmetry group.

The product representation is reducible by choosing the symmetry-adapted ONB consisting of eigenstates of the spin momentum operator and the completeness relation.