Creation and Annihalation Operators for Fock Spaces

Start with a Hilbert space for the quantum field from a classical field $ψ \in ℝ^{3} \to ℝ$

ℋ = ((ℝ^{3} \to ℝ) \to ℂ)

So that $| ψ ⟩ \in ℋ$ is the position basis vector for a classical field configuration. Then the Fock space is like

ℱ = ℂ \oplus︎ ℋ \oplus︎ (ℋ \otimes ℋ) \oplus︎ (ℋ \otimes ℋ \otimes ℋ) \oplus︎ \dots

For a symmetric of antisymetric product, doesn't matter. Staying with the regular Hilbert space, the position operator at position $\vec{x}$ is $\hat{ψ} (\vec{x})$ defined by

⟨ ψ | \hat{ψ} (\vec{x}) | ψ ⟩ = ψ (\vec{x})

Define a translation operator

T (Δ x) | ψ (\vec{x}) ⟩ = | ψ (\vec{x} + Δ x) ⟩

We consider a finite volume cube of length $L$ , and impose periodic boundary conditions on so that the translation operator

T (L) = T (0) = I

And the translation operator is the exponential of the momentum operator

T (Δ x) = e^{- \frac{i}{ℏ} Δ x \hat{p}}

e^{- \frac{i}{ℏ} L \hat{p}} = I

Now lets say that there's an eigenstate with eigenvalue

\hat{p} | ψ ⟩ = p | ψ ⟩

And cuz matrix exponents do that it does

e^{- \frac{i}{ℏ} L \hat{p}} | ψ ⟩ = e^{- \frac{i}{ℏ} L p} | ψ ⟩ = | ψ ⟩

e^{- \frac{i}{ℏ} L p} = 1

So the solutions for the eigenvalues are

\frac{p L}{ℏ} = τ n, n \in ℤ

And for 3 dimensions its just

\frac{\vec{k} L}{ℏ} = τ (n_{x}, n_{y}, n_{z}), n \in ℤ^{3}

So the momentum eigenvals in each direction are

\vec{k} = \frac{ℏ τ}{L} (n_{x}, n_{y}, n_{z}), n \in ℤ^{3}

THen u can just take the limit as the length goes to infinity to go back to normal. Now we can just work in momentum basis instead of position basis. In the Fock space at a level can be defined with

| π_{k_{1}}, π_{k_{2}}, \dots π_{k_{j}} ⟩ \in ℱ

for $j$ particles each with their corresponding momentum ${\vec{k}}_{i}$ . Next define an operator

⟨ {π_{j}} | \hat{a} (\vec{k}) | {π_{i}} ⟩ = \sqrt{{| π_{i} |}} δ_{{π_{j}} = {p i_{i}} + π_{k}}

then we make the "real" position and momentum operators equal to

\hat{ϕ} (\vec{x}) = \int \frac{d^{3} \vec{k}}{τ^{3}} \frac{1}{\sqrt{2 ω_{k}}} (\hat{a} (\vec{k}) e^{i \vec{k} \cdot \vec{x}} + {\hat{a}}^{†} (\vec{k}) e^{- i \vec{k} \cdot \vec{x}})

\hat{π} (\vec{x}) = \int \frac{d^{3} \vec{k}}{τ^{3}} (- i) \sqrt{\frac{ω_{k}}{2}} (\hat{a} (\vec{k}) e^{i \vec{k} \cdot \vec{x}} - {\hat{a}}^{†} (\vec{k}) e^{- i \vec{k} \cdot \vec{x}})

ω_{k} = \sqrt{| \vec{k} |^{2} + m^{2}}