In Connes’ noncommutative geometry, a spectral triple (A,H,D)(A,H,D)(A,H,D) encodes geometry via an algebra AAA, a Hilbert space HHH, and a Dirac operator DDD. The Hilbert space provides the trace used to recover Seeley–DeWitt coefficients (gravity + Standard Model couplings).
But in Modal Fields Theory and Indivisible Stochastic Mechanics, Hilbert space is not ontic — probabilities evolve by recursion, not by state vectors. Could we replace HHH with a recursion space or categorical object (sheaf, topos, stochastic module), define traces algebraically, and still reproduce the spectral action?
Or is Hilbert space mathematically indispensable, even if not physically fundamental?
Thanks for engaging with this. To clarify where we’re coming from:
- In Modal Fields Theory (MFT), energy is the only ontic entity, and particles/forces/geometry are emergent structurations.
- In Indivisible Stochastic Mechanics (ISM), probabilities evolve via recursion with memory, so Hilbert space is not fundamental — collapse and unitarity are emergent, not axioms.
- The question is whether Connes’ spectral triples (A,H,D)(A,H,D)(A,H,D) really require Hilbert space HHH to be physical, or if it can be reinterpreted as a recursion/categorical space HISMH_{\text{ISM}}HISM.
From our side, three directions look promising:
The big open technical challenge is: can we still reproduce Seeley–DeWitt coefficients (and thus the Standard Model couplings) without Hilbert space? If so, then Hilbert space is not fundamental; if not, then perhaps it’s an unavoidable mathematical scaffold.
We’d be curious how others see this:
- Is Hilbert space mathematically indispensable even if not physically fundamental?
- Have categorical or stochastic approaches to spectral triples been explored deeply enough to offer alternatives?
Recent work in Modal Fields Theory (MFT) suggests that Hilbert space may not be fundamental to either the thermodynamic origin of gravity or the spectral action. Following Jacobson’s program, we replaced horizon entanglement entropy with an Indivisible Stochastic Mechanics (ISM) entropy defined as S_ISM equals the natural logarithm of N, where N counts the number of distinct recursion histories (modal states). This entropy scales with boundary area, mirroring the entanglement area law. When inserted into the Clausius relation “delta Q equals T times delta S,” Einstein’s equations emerge with the same Newton constant, provided the density of recursion states is universal. This shows that the thermodynamic derivation of Einstein’s equations can proceed without Hilbert-space entanglement.
We also proposed a recursion trace for the spectral action, defined as the sum over recursion probabilities p_i times the function F of eigenvalues lambda_i of the modal Dirac operator. This parallels the Hilbert trace but works directly with ISM recursion probabilities. Early analysis suggests that this recovers the Seeley–DeWitt coefficients needed for gravity and Standard Model couplings, while canceling the a_0 term under the principle that “only differences glue.” If so, Hilbert space could be demoted to a computational convenience rather than an ontic foundation. This opens a path toward a fully post-Hilbertian formalism, with ISM recursion providing the microstructure and spectral triples retained only as a geometric dictionary.
What do others think — is Hilbert space truly indispensable, or can it be demoted to a representational tool while recursion provides the physics?
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