The Primal Manifold (\(\mathcal{P}\)) and its Hilbert Space \(L^2(\mathcal{P})\)

The Primal Manifold (\(\mathcal{P}\)) as a compact topological group and the corresponding Hilbert space \(L^2(\mathcal{P})\) of square-integrable functions.

1. Primal Manifold (\(\mathcal{P}\)) as a Topological Group

The Primal Manifold (\(\mathcal{P}\)) is formally defined not as a traditional Riemannian manifold, but as a specific object from number theory:

• The Idele Class Group of \(\mathbb{Q}\) is defined as \(C_{\mathbb{Q}} = J_{\mathbb{Q}} / \mathbb{Q}^\times\).
• The Primal Manifold (\(\mathcal{P}\)) is then defined as the kernel of the norm map \(|\cdot|\) on this group: \[\mathcal{P} := \ker( |\cdot|: C_{\mathbb{Q}} \to \mathbb{R}_+^\times )\]
• Carries a fundamental isomorphism from class field theory: \[\mathcal{P} \cong \hat{\mathbb{Z}}^\times\] where \(\hat{\mathbb{Z}}^\times\) is the group of profinite units.

Conclusion: The space \(\mathcal{P}\) is a compact, Hausdorff, and totally disconnected topological group. Starting point for defining an integral and a Hilbert space.

2. Defining the Hilbert Space \(L^2(\mathcal{P})\)

For a space of "square-integrable" functions, what does "integrate" mean here? - ya well for a locally compact topological group like \(\mathcal{P}\), it is gonna be the Haar measure.

Four-steps:

1. The Measure (\(d\mu\)):

   By Haar's theorem, every locally compact topological group admits a unique (up to a positive constant) non-trivial Borel measure, \(d\mu\), that is translation-invariant. This means for any measurable set \(E \subset \mathcal{P}\) and any element \(g \in \mathcal{P}\), the measure is unchanged by a "shift": \(\mu(gE) = \mu(E)\).

   Since our group \(\mathcal{P} \cong \hat{\mathbb{Z}}^\times\) is compact, its total measure is finite. We can therefore normalize this measure so that the total volume of the space is 1: \[\int_{\mathcal{P}} d\mu(x) = 1\]

2. Space (\(L^2(\mathcal{P}, \mu)\)):

   So now the normalized Haar measure \(d\mu\) established, the Hilbert space \(L^2(\mathcal{P})\) is formally defined as the set of all complex-valued, measurable functions \(f: \mathcal{P} \to \mathbb{C}\) that are "square-integrable" with respect to this measure.

3. ”Inner Product” (\(\langle \cdot, \cdot \rangle\)):

   The space is a Hilbert space because it is equipped with a well-defined inner product. For any two functions \(f\) and \(g\) in \(L^2(\mathcal{P})\), their inner product is: \[\langle f, g \rangle := \int_{\mathcal{P}} f(x) \, \overline{g(x)} \, d\mu(x)\] (where \(\overline{g(x)}\) is the complex conjugate of \(g(x)\)).

4. The Norm (\(||\cdot||_2\)):

   The "square-integrable" condition is exactly that the norm induced by this inner product is finite: \[||f||_2^2 = \langle f, f \rangle = \int_{\mathcal{P}} |f(x)|^2 \, d\mu(x) < \infty\]

Summary

So we equip the Primal Manifold, \(\mathcal{P} \cong \hat{\mathbb{Z}}^\times\), with its unique normalized Haar measure, \(d\mu\). The Hilbert space \(L^2(\mathcal{P})\) is the space of square-integrable functions with respect to this measure.

This provides the formal foundation for discussing the next steps: the harmonic analysis (e.g., character theory) and the spectral properties of operators on this space.

93E3 BEBC C164 D766