The Fock Space Euler Product¶

Jean-Paul Niko · Sole Author

Theorem by @D_GPT (A2c/Task 3, 2026-03-09)

Theorem¶

Let \(\mathfrak{h} = \ell^2(\mathcal{P})\) with orthonormal basis \(\{e_p\}_{p \in \mathcal{P}}\). Define the prime Hamiltonian \(h\) by \(he_p = (\log p)\,e_p\). For Re\((s) > 1\), set \(T_s = e^{-sh}\) (contraction: \(T_s e_p = p^{-s}e_p\)). Then:

\[\text{Tr}_{\Gamma_s(\mathfrak{h})}\, \Gamma(T_s) = \prod_p \frac{1}{1-p^{-s}} = \zeta(s)\]

where \(\Gamma_s(\mathfrak{h}) = \bigoplus_{n \geq 0} \text{Sym}^n(\mathfrak{h})\) is the bosonic Fock space and \(\Gamma(T_s)\) is the second quantization of \(T_s\).

Proof. On the occupation basis \(\{|n_p\rangle\}\), \(\Gamma(T_s)\) is diagonal with eigenvalue \(\prod_p p^{-sn_p}\). The trace factorizes: \(\sum_{(n_p)} \prod_p p^{-sn_p} = \prod_p \sum_{n=0}^\infty p^{-sn} = \prod_p (1-p^{-s})^{-1}\). Absolute convergence for Re\((s)>1\). \(\square\)

In operator language: \(\text{Tr}_\Gamma\, \Gamma(T_s) = \det(I - T_s)^{-1}\).

Source-Space Embedding¶

If \((S^2)^\mathcal{P}\) is the RTSG source space, choose the \(l=0\) constant mode \(\eta_p \in L^2(S_p^2)\) at each prime. Define:

\[\mathfrak{h}_{\text{arith}} = \bigoplus_p \mathbb{C}\eta_p \cong \ell^2(\mathcal{P}), \qquad h\eta_p = (\log p)\eta_p\]

The Fock trace gives \(\zeta(s)\) exactly. The sphere contributes the rank-one local state; arithmetic contributes the prime Hamiltonian.

Dirichlet L-Functions¶

For a Dirichlet character \(\chi\), define \(M_\chi e_p = \chi(p)\,e_p\). Then:

\[\text{Tr}_\Gamma\, \Gamma(M_\chi T_s) = \prod_p (1-\chi(p)p^{-s})^{-1} = L(s,\chi)\]

More generally, for a \(d\)-dimensional local operator \(F_p\) with eigenvalues \(\alpha_{p,1}, \ldots, \alpha_{p,d}\):

\[\text{Tr}_\Gamma\, \Gamma(F_p \cdot p^{-sN}) = \det(I - p^{-s}F_p)^{-1} = \prod_j (1-\alpha_{p,j}p^{-s})^{-1}\]

This is the general local \(L\)-factor mechanism.

What This Is and Isn't¶

Is: An exact theorem giving \(\zeta(s)\) from prime-labeled bosonic Fock space. Standard in mathematical physics (second quantization = Euler product is well-known).

Isn't: A proof of anything new about \(\zeta\). The Euler product is Euler (1737). The Fock realization is the operator-theoretic packaging.

RTSG contribution: The source space \((S^2)^\mathcal{P}\) provides a geometric home for the one-particle modes, and the BRST filter selects the rank-one \(l=0\) mode at each prime. The prime Hamiltonian \(h\) is external arithmetic input, not derived from sphere geometry.

Jean-Paul Niko · RTSG BuildNet · smarthub.my · March 2026