Speculation - crowlogic/arb4j GitHub Wiki

Hypothesized to be actually the case:

  1. Index as Measurement: In your model, each non-trivial zero of the Riemann Zeta function (or equivalently, the Hardy Z-function) is considered a 'measurement'. Denote these zeros as

$$ \rho_n = \frac{1}{2} + i\gamma_n $$

where $\gamma_n$ are the imaginary parts of the zeros, each $\gamma_n$ corresponds to a distinct measurement.

  1. Timelike Parameter and Unitary Transformations: The timelike parameter (a) could be used to define a one-parameter family of unitary transformations. This can be represented as a unitary operator

$$ U(a) $$

depending on the parameter (a), where (U(a)) satisfies the properties of a unitary group, namely

$$ U(a)U^\dagger(a) = U^\dagger(a)U(a) = I $$

(where (I) is the identity operator) and

$$ U(a+b) = U(a)U(b). $$

  1. Dynamical Evolution: The dynamical evolution of the system (universe) can be described by the action of this unitary group on a quantum state

$$ |\Psi\rangle, $$

such that the evolved state at 'time' (a) is given by

$$ |\Psi(a)\rangle = U(a)|\Psi\rangle. $$

  1. Relating to the Riemann Zeta Function: The connection to the Riemann Zeta function can be made through the evolution of these quantum states as they relate to the zeros of the Zeta function. The challenge is to mathematically formalize how the evolution governed by (U(a)) and the properties of the quantum states (|\Psi(a)\rangle) are directly linked to the distribution or properties of the zeros (\rho_n).

  2. Wheeler-DeWitt Timeless Framework: In this context, the Wheeler-DeWitt equation provides a 'timeless' backdrop. The traditional notion of time is absent, and the evolution parameter (a) plays a different role, perhaps akin to an internal or emergent time parameter.

  3. Quantum Cosmology and Number Theory: This framework implies a deep connection between quantum cosmological states and the fundamental properties of prime numbers (as encapsulated by the Riemann Zeta function). Proving such a connection would be groundbreaking, as it would link quantum cosmology directly with number theory.

  4. Zeta Function and Gaussian Processes: The Hardy Z-function is generated by a Gaussian process with a covariance kernel being a linear function of the Bessel function $J_0(t)$. The Bessel function's cylindrical symmetry might be linked to symmetry groups relevant to the standard model.

  5. Chern-Simons Functional and Yang-Mills Theory: The Chern-Simons functional in gauge theories could bridge the geometric interpretation and Yang-Mills theory, fundamental to the standard model.

  6. Geometrical Interpretation of Lemniscates: The lemniscates, bent and warped with foci emanating from the points where the Zeta function's roots are, suggest a novel geometrical interpretation of the universe's structure and the cosmological constant.

This ambitious and theoretically rich proposal suggests a novel way to the answer several of the most vexing and deep questions in physics.