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)/2}W(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 α_t 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 {f_i} of S, then the sum

∑_i=1^∞ a∗(f_i)a(f_i),

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 (H₀,π₀) 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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Fock Space

Fock space is just another separable infinite dimensional Hilbert space (and so isomorphic to all its separable infinite dimensional brothers). But the key is writing it down in a fashion that suggests a particle interpretation. In particular, suppose that H is the one-particle Hilbert space, i.e. the state space for a single particle. Now depending on whether our particle is a Boson or a Fermion, the state space of a pair of these particles is either E_s(H ⊗ H) or E_a(H ⊗ H), where E_s is the projection onto the vectors invariant under the permutation Σ_H,H on H ⊗ H, and E_a is the projection onto vectors that change signs under Σ_H,H. For

present purposes, we ignore these differences, and simply use H ⊗ H to denote one possibility or the other. Now, proceeding down the line, for n particles, we have the Hilbert space Hⁿ ≡ H ⊗ · · · ⊗ H, etc..

A state in Hⁿ is definitely a state of n particles. To get disjunctive states, we make use of the direct sum operation “⊕” on Hilbert spaces. So we define the Fock space F(H) over H as the infinite direct sum:

F (H ) = C ⊕ H ⊕ (H ⊗ H ) ⊕ (H ⊗ H ⊗ H ) ⊕ · · · .

So, the state vectors in Fock space include a state where there are no particles (the vector lies in the first summand), a state where there is one particle, a state where there are two particles, etc.. Furthermore, there are states that are superpositions of different numbers of particles.

One can spend time worrying about what it means to say that particle numbers can be superposed. But that is the “half empty cup” point of view. From the “half full cup” point of view, it makes sense to count particles. Indeed, the positive (unbounded) operator

N=0 ⊕ 1 ⊕ 2 ⊕ 3 ⊕ 4 ⊕···,

is the formal element of our model that permits us to talk about the number of particles.

In the category of Hilbert spaces, all separable Hilbert spaces are isomorphic – there is no difference between Fock space and the single particle space. If we are not careful, we could become confused about the bearer of the name “Fock space.”

The confusion goes away when we move to the appropriate category. According to Wigner’s analysis, a particle corresponds to an irreducible unitary representation of the identity component P of the Poincaré group. Then the single particle space and Fock space are distinct objects in the category of representations of P. The underlying Hilbert spaces of the two representations are both separable (and hence isomorphic as Hilbert spaces); but the two representations are most certainly not equivalent (one is irreducible, the other reducible).

Natural topography of Hilbert Spaces as a generator of free will?

There is a theory that goes by the name Margenau’s Theory, which tries its hand at reducing human consciousness to a field of probabilities in what is termed a Fock Space, which is an algebraic construction in quantum mechanics to construct the quantum states space of a variable or an unknown number of identical particles from a single particle Hilbert Space, H.

Hilbert space extends the notion of Euclidean geometry from a two-dimensional space to a three-dimensional space, and is talked about as an abstract vector space possessing the structure of an inner product that allows the length and the angle to be measured. Now, such a space could be created by an electrical activity at the synaptic level. Normal behaviour could therefore be seen as the elasticity of the field, and free will as a rupture within it. But, in what topology begs the question? Since, there is nothing in the natural topography of Hilbert Spaces that could be the generator of free will….

Three states of the Hilbert space of the quantum dimer model. There are off-diagonal matrix elements in the effective Hamiltonian which connect state ͑ a ͒ to state ͑ b ͒ , and state ͑ a ͒ to state ͑ c ͒ , by a resonance between pairs of horizontal and vertical dimers around a plaquette. The latter matrix element differs from the former because only the latter has a diagonal link across the resonating plaquette. Also shown are the corresponding values of the heights h a on the sites of the dual lattice. Credit