Quantum Phase Transitions in Cyclotomic Fields: A Spectral Approach to the Riemann Hypothesis

Introduction

The intersection of quantum statistical mechanics, algebraic number theory, and the distribution of prime numbers represents one of the most profound frontiers in mathematical physics. The source paper, arXiv:hal-00001235, titled "On the Cyclotomic Quantum Algebra of Time Perception," provides a unique framework for understanding these connections. At its core, the paper explores how the Galois group of the cyclotomic extension of the rationals, denoted as G = Gal(Q^cycl/Q), acts upon a C*-algebraic system to manifest properties reminiscent of the Riemann zeta function.

The specific problem addressed involves the construction of a quantum dynamical system where the partition function is identified with the Riemann zeta function, ζ(β). This is not merely a formal coincidence; it arises from the spectral properties of a Hamiltonian H₀ defined over a Hilbert space of number states. The contribution of this analysis is to bridge the gap between the "time perception" model presented in the source paper and the rigorous requirements of the Riemann Hypothesis. By examining the phase transition of KMS (Kubo-Martin-Schwinger) states at the critical temperature β = 1, we can gain insight into the distribution of primes and the potential location of the non-trivial zeros of the zeta function.

Mathematical Background

To analyze the connections to the Riemann Hypothesis, we must first define the key mathematical objects introduced in arXiv:hal-00001235. The fundamental structure is the Hilbert space l²(N), where the basis states are labeled by natural numbers |n>.

The Hamiltonian and Partition Function

The Hamiltonian H₀ is defined such that its action on a state |n> is given by H₀|n> = log(n)|n>. Under this definition, the thermal evolution of the system is governed by the operator exp(-β H₀). Applying this to the basis states yields exp(-β H₀)|n> = n^-β|n>.

The partition function Z(β) is the trace of this Boltzmann-weighted operator: Z(β) = Trace(exp(-β H₀)) = ∑ n^-β. This sum is the definition of the Riemann zeta function ζ(β) for Re(β) > 1. The pole at β = 1 corresponds to a thermodynamic critical temperature.

The Cyclotomic Galois Group

The source paper emphasizes the role of the Galois group G = Gal(Q^cycl/Q). The field Q^cycl is generated by all roots of unity. The group G is isomorphic to the profinite completion of the integers, which acts as the symmetry group of the algebra. This group permutes the extremal KMS states of the system, creating a bridge between arithmetic symmetry and physical equilibrium.

Main Technical Analysis

Spectral Properties and Zero Distribution

The relationship between the Riemann Hypothesis and the cyclotomic algebra lies in the spectral interpretation of the zeta function. In arXiv:hal-00001235, the phase operator E is defined such that its eigenvalues are roots of unity. Specifically, for a given integer q, the operator acts on states related to the group (Z/qZ)*.

The distribution of these phases, when analyzed across all q, reflects the distribution of the primes. The paper notes that the thermal state φ_β(e_p) involves a product over primes p dividing q. This expression is deeply connected to the Euler product of the zeta function. The fluctuations of this state as β approaches 1 from above provide a spectral view of the prime density.

Algebraic Structures and L-functions

The source paper links the Galois group action to the permutation of extremal KMS states. For β > 1, the state is defined by a trace involving the representation of the algebra and the Hamiltonian. This allows for the construction of L-functions. Since G is the automorphism group of Q^cycl, the action of an element w on the roots of unity is equivalent to arithmetic automorphisms in global fields.

The tables provided in the source (e.g., the powers of 3 modulo 7 and 2 modulo 9) demonstrate the periodicity of these extensions. The Carmichael lambda function, λ(q), which represents the exponent of the multiplicative group, shows a fractal character in its summatory function. This fractal nature is linked to the 1/f noise observed in the power spectral density of the system, suggesting that the distribution of prime-related sequences is not random but governed by a specific scaling law.

Novel Research Pathways

1. Stochastic Phase Fluctuations and the Critical Line

A promising research direction is to model the non-trivial zeros of ζ(s) as resonances in the cyclotomic algebra. By defining a stochastic perturbation to the Hamiltonian H₀, one could analyze the variance of the KMS states under the action of the Galois group. The goal is to prove that the state remains extremal only when the real part of the parameter is 1/2, establishing a thermodynamic requirement for the critical line.

2. Galois Symmetry Breaking and Thermodynamic Stability

The Riemann Hypothesis may be equivalent to the statement that no spontaneous symmetry breaking of the Galois group occurs in the thermodynamic limit for temperatures corresponding to the critical strip. Investigating the stability of these KMS states under small perturbations that break the Galois symmetry could reveal why the zeros must be aligned symmetrically.

3. Carmichael Lambda Scaling and Fractal Dimensions

Using the theory of Arithmetic Phase Systems, researchers can calculate the Hausdorff dimension of the set of frequencies generated by the λ(q) function. There is a known link between the fractal dimension of prime distributions and the error term in the Prime Number Theorem; reconciling the 1/f noise value of 0.70 with the Riemann Hypothesis would provide a new bridge between number theory and signal processing.

Computational Implementation

(* Section: Carmichael Lambda and Zeta Partition Function *)
(* Purpose: Demonstrate the fractal growth of the Carmichael function 
   and its relationship to the thermal zeta state as described in 
   arXiv:hal-00001235. *)

Module[{maxT = 2000, lambdaData, summatoryLambda, normalizedLambda, zetaValues},
  
  (* Calculate the Carmichael Lambda function for each n *)
  lambdaData = Table[CarmichaelLambda[n], {n, 1, maxT}];
  
  (* Compute the summatory function: S(t) = Sum[lambda(n), {n, 1, t}] *)
  summatoryLambda = Accumulate[lambdaData];
  
  (* Normalize the summatory function to observe fractal character *)
  (* Normalization t^1.90 is used as a scaling heuristic *)
  normalizedLambda = Table[
    {t, summatoryLambda[[t]] / (t^1.90)}, 
    {t, 1, maxT}
  ];
  
  (* Calculate Zeta values for the partition function Z(beta) *)
  zetaValues = Table[{beta, Zeta[beta]}, {beta, 1.1, 3.0, 0.1}];

  (* Plot the results *)
  Print[ListLinePlot[normalizedLambda, 
    PlotLabel -> "Normalized Summatory Carmichael Lambda Function", 
    AxesLabel -> {"Order t", "S(t)/t^1.90"}, 
    PlotStyle -> Blue]];

  Print[ListLinePlot[zetaValues, 
    PlotLabel -> "Zeta Partition Function Z(beta)", 
    AxesLabel -> {"beta", "Z(beta)"}, 
    PlotStyle -> Red]]
]

The investigation into the cyclotomic quantum algebra of time perception reveals a sophisticated mapping between the Galois theory of cyclotomic fields and the thermodynamics of the Bost-Connes system. The primary finding is that the symmetry group G acts as a fundamental regulator of the KMS states, with the Riemann zeta function emerging naturally as the partition function of the system.

The most promising avenue for further research lies in the physical realization of this algebra via superconducting circuits or phase operators. By mapping the Hamiltonian eigenvalues to the energy levels of a quantum device, the distribution of the Riemann zeros could be studied as a tangible spectral phenomenon. Specific next steps should involve a more rigorous derivation of the noise exponents and the transition of the Mobius function from the low-temperature limit to the high-temperature fluctuations near the critical point.

References

• arXiv:hal-00001235: On the Cyclotomic Quantum Algebra of Time Perception.
• Bost, J. B., & Connes, A. (1995). Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory. Selecta Mathematica.
• Connes, A. (1999). Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Mathematica.