Conjuncted: Unitary Representation of the Poincaré Group is a Fock Representation


The Fock space story is not completely abandoned within the algebraic approach to Quantum Field Theory. In fact, when conditions are good, Fock space emerges as the GNS Hilbert space for some privileged vacuum state of the algebra of observables. We briefly describe how this emergence occurs before proceeding to raise some problems for the naive Fock space story.

The algebraic reconstruction of Fock space arises from the algebraic version of canonical quantization. Suppose that S is a real vector space (equipped with some suitable topology), and that σ is a symplectic form on S. So, S represents a classical phase space . The Weyl algebra U[S,σ] is a specific C∗-algebra generated by elements of the form W(f), with f ∈ S and satisfying the canonical commutation relations in the Weyl-Segal form:

W(f)W(g) = e−iσ(f,g)/2W(f + g)

Suppose that there is also some notion of spacetime localization for elements of S, i.e. a mapping O → S(O) from double cones in Minkowski spacetime to subspaces of S. Then, if certain constraints are satisfied, the pair of mappings

O → S(O) → U(O) ≡ C{W(f) : f ∈ S(O)},

can be composed to give a net of C∗-algebras over Minkowski spacetime. (Here C∗X is the C∗-algebra generated by the set X.)

Now if we are given some dynamics on S, then we can — again, if certain criteria are satisfied — define a corresponding dynamical automorphism group αt on U[S,σ]. There is then a unique dynamically stable pure state ω0 of U[S,σ], and we consider the GNS representation (H,π) of U[S,σ] induced by ω0. To our delight, we find that the infinitesimal generators Φ(f) of the one-parameter groups {π(W(f))}t∈R behave just like the field operators in the old-fashioned Fock space approach. Furthermore, if we define operators

a(f) = 2−1/2(Φ(f) + iΦ(Jf)),
a∗(f) = 2−1/2(Φ(f)−iΦ(Jf)),

we find that they behave like creation and annihilation operators of particles. (Here J is the unique “complex structure” on S that is compatible with the dynamics.) In particular, by applying them to the vacuum state Ω, we get the entire GNS Hilbert space H. Finally, if we take an orthonormal basis {fi} of S, then the sum

i=1 a∗(fi)a(fi),

is the number operator N. Thus, the traditional Fock space formalism emerges as one special case of the GNS representation of a state of the Weyl algebra.

The Minkowski vacuum representation (H00) of A is Poincaré covariant, i.e. the action α(a,Λ) of the Poincaré group by automorphisms on A is implemented by unitary operators U(a,Λ) on H. When we say that H is isomorphic to Fock space F(H), we do not mean the trivial fact that H and F(H) have the same dimension. Rather, we mean that the unitary representation (H,U) of the Poincaré group is a Fock representation.



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