Théorème

Lemma G — Hilbert reconstruction

The reconstructed QFT's Hilbert space is $\mathcal{H}_\infty = \varinjlim \bigotimes_{p \mid m_K} \mathcal{H}_p$.

Statement

The Hilbert space of the QFT reconstructed (by Osterwalder–Schrader, cf. Lemmas E and F) is:

\boxed{\mathcal{H}_\infty = \varinjlim_{K \to \infty} \bigotimes_{p \,\mid\, m_K} \mathcal{H}_p,}

the inductive limit of the per-CRT-factor tensor products, where each $\mathcal{H}_p$ is the finite-dimensional Hilbert space associated with the mod- $p$ sieve.

Théorème

Plain reading. The state space of quantum mechanics (the “Hilbert space”) is not arbitrarily posited in PT. It is constructed as a particular combination of small spaces, one per active prime, assembled by tensor product. No hidden representation choice.

Why it matters

Lemma G completes the (E, F, G) reconstruction triad:

E gives the coupling,
F gives the spacetime metric,
G gives the quantum state space.

With these three lemmas, the QFT reconstructed from the sieve is completely characterised: observable algebra ( $\mathcal{A}$ ), metric (by F), Hilbert space (by G), and coupling (by E). No external element added.

This justifies saying PT “reconstructs physics” rather than “identifies the sieve with physics”.

Proof — outline

Step G.1: for each prime $p$ , the space $\mathcal{H}_p$ is the finite-dim Hilbert space associated with the transfer $T_p$ .
Step G.2: the CRT decomposition gives $\mathcal{H}_m = \bigotimes_{p \mid m} \mathcal{H}_p$ for every modulus $m$ .
Step G.3: the inductive limit $\varinjlim$ exists and is well-defined by OS compatibility.

Detailed proof

Step G.1 — Per-prime Hilbert space

For each prime $p$ , the Hilbert space $\mathcal{H}_p$ associated with the transfer $T_p$ has dimension $p - 1$ (number of non-zero residue classes mod $p$ after eliminating 0). Finite-dimensional, equipped with the inner product defined by $T_p$ ‘s stationary measure.

For active primes $\{3, 5, 7\}$ : $\dim \mathcal{H}_3 = 2$ , $\dim \mathcal{H}_5 = 4$ , $\dim \mathcal{H}_7 = 6$ .

Step G.2 — Tensor CRT decomposition

By CRT, for $m = p_1 p_2 \cdots p_k$ a product of distinct primes:

\mathcal{H}_m = \bigotimes_{p \mid m} \mathcal{H}_p.

This is the isomorphism between the Hilbert space on $\mathbb{Z}/m\mathbb{Z}$ and the tensor product over prime factors. Consistent with the tensor factorisation of the transfer matrix $T_m = \bigotimes_p T_p$ .

For the arithmetic torus $\mathbb{T}^3 = \mathbb{Z}/105\mathbb{Z}$ (with $105 = 3 \cdot 5 \cdot 7$ ):

\mathcal{H}_{105} = \mathcal{H}_3 \otimes \mathcal{H}_5 \otimes \mathcal{H}_7,

of dimension $2 \cdot 4 \cdot 6 = 48$ .

Step G.3 — Inductive limit

The family $\{m_K\}_{K \geq 1}$ with $m_{K+1} = m_K \cdot p_{K+1}$ (adding a new prime at each step) generates an increasing chain of spaces $\mathcal{H}_{m_1} \hookrightarrow \mathcal{H}_{m_2} \hookrightarrow \cdots$ .

The inductive limit:

\mathcal{H}_\infty = \varinjlim_{K \to \infty} \mathcal{H}_{m_K}

exists in the sense of pre-Hilbert spaces. The completion gives a separable infinite-dimensional Hilbert space, which is exactly the OS-reconstructed QFT state space.

Compatibility with OS axioms

The inclusions $\mathcal{H}_{m_K} \hookrightarrow \mathcal{H}_{m_{K+1}}$ are isometric (preserve inner product) because the transfers $T_p$ are stochastic. This ensures the limit is well-defined and the observable algebra $\mathcal{A}$ extends continuously.

Structural implications

Per-prime finite dimensionality: each factor remains finite-dimensional, making explicit computations possible.
Separable limit: $\mathcal{H}_\infty$ has a countable basis, consistent with standard quantum mechanics axioms.
CRT-factorisable: any operator on $\mathcal{H}_\infty$ decomposes over prime factors — the technical foundation of Pontryagin (BA5).

For the complete derivation, see chapter 9 of the monograph, section Hilbert Reconstruction (Lemma G).