Many references I know on QFT start the discussion of the Klein-Gordon field theory with some discussion about harmonic oscillators. One such reference is Folland's Quantum Field Theory book. The idea is usually describe a system with infinitely-many oscillators as a warming up for the real Klein-Gordon free field.
Instead of starting with the Klein-Gordon equation for the field, passing it to momentum space and then recognizing the obtained equation as a collection of harmonic oscillators, I was trying to follow a different approach which I describe as follows.
As in the case of many harmonic oscillators, the idea is to work on a discrete setting and obtain the Klein-Gordon theory as a limiting case. I want to think about the problem as a many-particle quantum mechanical problem of non-interacting free particles. Suppose we start if a finite region (actually a torus) $\mathbb{T}_{L} := [-L/2,L/2]^{d}$ for some $L>0$. The momentum space is just $\mathbb{T}_{L}^{*}:=\frac{2\pi}{L}\mathbb{Z}^{d}$. The one-particle space Hilbert space $\mathscr{H}$ is $L^{2}(\mathbb{T}_{L})$, which has basis $\{\varphi_{p}\}_{p\in \mathbb{T}_{L}^{*}}$, with $\varphi_{p}(x) = L^{-d/2}e^{i\langle p, x\rangle}$, with $\langle p,x\rangle$ denoting the usual inner product on $\mathbb{R}^{d}$.
The states $\varphi_{p}$ are to be interpreted as eigenstates of the Free Hamiltonian $H$ with well-defined position $p$. To describe many-particle states, we pass to the bosonic Fock space $\mathcal{F}(\mathscr{H})$.
Following my previous question, I want to define fields as prof. Susskind proposes: $$\Psi(x) := \sum_{p\in \mathbb{T}_{L}^{*}}\varphi_{p}a_{p} \quad \mbox{and} \quad \Psi^{\dagger}(x) := \sum_{p\in \mathbb{T}_{L}^{*}}\varphi_{p}a_{p}^{\dagger}$$ where $a_{p}$ and $a_{p}^{\dagger}$ are the (usual) annihilation and creation operators on the bosonic Fock spaces associated to basis elements $\varphi_{p}$. The above field operators are not self-adjoint, so not observables but we can consider a linear combination: $$\Phi(x) := \frac{1}{\sqrt{2}}(\Psi(x)+\Psi^{\dagger}(x))$$
Question: How should I adapt this scenario to obtain a discrete version of the Klein-Gordon equation? It does not seem that I am too far from it and, as far as I understand, the construction of the Klein-Gordon field theory really looks like this one: (in the continuum) one starts with $\mathscr{H} = L^{2}(V_{m},dp/\omega_{p})$, where $V_{m}$ is the mass shell and $dp/\omega_{p}$ is the Lorentz invariant measure on $V_{m}$, $\omega_{p} = \sqrt{p^{2}+m^{2}}$, and then defines the field quantization analogously to what is done before (known as Segal quantization). However, I haven't used any information about the Klein-Gordon equation, which is something that obviously has to be taken into account (but I don't know how). Also, to obtain the Klein-Gordon field expression, one has to change $a_{p} \to \frac{1}{\sqrt{2\omega_{p}}}$ and $a^{\dagger}_{p}\to \frac{1}{\sqrt{2\omega_{p}}}a^{\dagger}_{p}$ or something like this, because the expressions come with this extra factor in the denominator. However, it does not seem natural to introduce these objects in the above setting. How to change things properly?
