Zeta Regularization & Temporal Charge

Abstract

Divergent series like 1 + 2 + 3 + 4 + ... do not converge in the traditional sense, yet through analytic continuation of the Riemann zeta function ζ(s), they can be assigned finite, meaningful values. In physics, zeta-regularization appears in quantum field theory, string theory, the Casimir effect, and cosmology. We show how divergent sums in energy and mass can be reframed as divergent sums in temporal charge τ, with ζ-regularization yielding finite τ-invariants that connect to observable physics.

1. What is Zeta Regularization?

The Riemann zeta function is defined for Re(s) > 1 as:

Through analytic continuation, ζ(s) extends to the entire complex plane (except s = 1, where it diverges). This extension allows us to assign finite values to divergent series:

This is not ordinary summation. It is a mathematically consistent reassignment that preserves analytic structure and appears throughout modern physics.

Why This Matters

In quantum mechanics, infinite sums appear everywhere:

• Zero-point energy: ∑ ½ℏω_n = ∞
• Vacuum fluctuations in QFT
• Mode sums in string theory
• Casimir force between conducting plates

Without regularization, these infinities make predictions impossible. Zeta-regularization provides a principled way to extract finite, observable values.

2. Mathematical Foundation

2.1 Analytic Continuation

The zeta function can be extended using the functional equation:

This allows evaluation at negative integers:

2.2 Relation to Bernoulli Numbers

For negative integers, the zeta function relates to Bernoulli numbers B_n:

This connects divergent series to deep number-theoretic structures.

2.3 Emmy Noether and Symmetry

Emmy Noether's 1918 theorem on symmetries and conservation laws provides the foundation for understanding why zeta regularization works. Noether showed that continuous symmetries of physical systems generate conserved quantities. The analytic structure of ζ(s)—its symmetries under functional equations—is what allows divergent sums to be consistently regularized.

When we extend ζ(s) from Re(s) > 1 to the entire complex plane, we preserve the underlying symmetries of the mathematical structure. This is analogous to how conservation laws persist across different reference frames. The regularized values aren't arbitrary—they respect the symmetries encoded in ζ(s), just as physical conservation laws respect the symmetries of spacetime.

2.4 Julia Robinson and Diophantine Equations

Julia Robinson's work on Hilbert's Tenth Problem revealed deep connections between number theory and computability. Her 1961 paper with Martin Davis and Hilary Putnam, "The Decision Problem for Exponential Diophantine Equations," showed that certain infinite sets of integers can be characterized through polynomial equations.

This connects to zeta regularization through the Riemann hypothesis. The zeros of ζ(s) encode information about the distribution of prime numbers, which in turn relates to solvability of Diophantine equations. Robinson's insight—that infinite discrete structures (like primes) can be captured by finite algebraic conditions—parallels how ζ-regularization assigns finite values to infinite sums. Both reveal that infinity, when structured correctly, contains finite information.

2.5 Example: Sum of Natural Numbers

Consider S = 1 + 2 + 3 + 4 + ..., which clearly diverges. Using Ramanujan summation:

This value is not the limit of partial sums, but a regularized assignment consistent with analytic properties.

3. Zeta in Physics

3.1 Casimir Effect

The Casimir force between two parallel conducting plates arises from vacuum energy differences. The calculation explicitly uses ζ(−3):

This force has been measured experimentally and agrees with the ζ-regularized prediction to high precision.

3.2 Quantum Field Theory and Karen Uhlenbeck's Gauge Theory

Zero-point energy of quantum fields sums to infinity:

Using ζ-regularization, this becomes finite and Lorentz-invariant. The regularized vacuum energy enters calculations of:

• Anomaly cancellation in string theory
• Conformal field theory central charges
• Effective actions in curved spacetime

Karen Uhlenbeck's 1982 paper "Removable singularities in Yang–Mills fields" established rigorous methods for handling singularities in gauge theory. Her work showed that apparent point singularities in Yang-Mills fields with finite energy can be removed by gauge transformations—the field extends smoothly despite initial divergences.

This parallels ζ-regularization: both deal with apparent infinities that, when handled correctly, reveal finite physical content. Uhlenbeck's gauge-fixing techniques in four-dimensional Yang-Mills theory provide the analytic framework for regularizing divergences in quantum field theory. Her methods ensure that regularization procedures (including ζ-regularization) respect gauge invariance and produce physically meaningful, finite results.

3.3 String Theory

The critical dimension of bosonic string theory (d = 26) emerges from ζ-regularization of the infinite sum of oscillator modes:

This appears in the mass-shell condition and requires d = 26 for anomaly cancellation.

3.4 Cosmological Constant Problem

Vacuum energy density from quantum fields is formally:

Ζ-regularization provides a finite value, but the discrepancy with observed dark energy density (Λ) remains one of physics' deepest puzzles.

4. Bridging ζ to τ

From the τ framework, we have:

Energy and mass are inversely proportional to temporal charge. This means:

4.1 Divergent Energy Sums → Divergent τ Sums

Consider an infinite sum of energy modes:

If E_n ~ n (linear spectrum), then τ_n ~ 1/n, and:

The divergent energy sum becomes a ζ-regularized τ-sum:

4.2 Physical Interpretation

The ζ-regularized sum assigns a finite effective temporal charge to the infinite tower of modes. This τ_eff is observable through:

• Casimir force (effective τ of vacuum between plates)
• Lamb shift (effective τ of electron-photon interaction)
• Running coupling constants (effective τ at different energy scales)

4.3 Operational Formula

For a divergent sum ∑ E_n, the ζ-regularized temporal charge is:

where ∑_reg denotes the regularized sum.

5. Test for Nuclear Chemistry

For Dr. Simona Mastroianni: Does plotting half-life vs τ = ℏ/Q across all decay modes (alpha, beta, gamma) reveal a universal curve? Different isotopes decay via alpha, beta, or gamma emission with vastly different half-lives. If you plot half-life vs τ = ℏ/Q across all decay modes, does a universal curve emerge? Does temporal charge unify decay systematics that look chaotic in energy space?

Hypothesis

Different decay modes (alpha, beta, gamma) have different divergent sums:

• Alpha decay: Infinite sum over tunneling paths through Coulomb barrier
• Beta decay: Infinite sum over phase space of final states
• Gamma decay: Infinite sum over electromagnetic mode densities

If ζ-regularization is universal, applying it to each mode's divergent sum should yield finite τ_eff values that collapse all decay modes onto a single curve when plotted against half-life.

Procedure

1. Collect decay data for 100+ isotopes across the periodic table (mix of α, β, γ decays)
2. For each decay:
   • Record Q-value (energy released)
   • Record half-life t_1/2
   • Calculate τ = ℏ/Q
3. Plot log(t_1/2) vs log(τ) for all isotopes, color-coded by decay mode
4. Fit to power law: t_1/2 = Aτⁿ
5. Check if:
   • All decay modes collapse onto single curve (universal n)
   • Exponent n is consistent across modes
   • Residuals are random (no systematic mode-dependent deviations)

Prediction

If ζ-regularization is mode-independent: All decay modes will show t_1/2 ∝ τⁿ with the same exponent n ≈ 5–7 (from dimensional analysis and Fermi's golden rule), and scatter around a universal curve.

If energy is fundamental (not τ): Different decay modes will have different exponents, and the plot will show distinct curves for α, β, γ decays with no universal structure.

Data Sources

• NNDC (Brookhaven): www.nndc.bnl.gov
• ENSDF: Evaluated Nuclear Structure Data File
• IAEA Nuclear Data: www-nds.iaea.org

Why This Tests ζ-Regularization

Each decay mode has a different microscopic mechanism and different divergent sum:

If ζ-regularization works the same way for all three, then τ_α, τ_β, τ_γ should all relate to t_1/2 identically, producing a universal curve. This is a direct test of whether ζ(s) regularizes different quantum processes uniformly.

Expected Outcome

A single power-law curve spanning 20+ orders of magnitude in half-life (from microseconds to billions of years) would be strong evidence that:

• Ζ-regularization is universal across decay modes
• Temporal charge τ is more fundamental than energy E
• The infinite sums in different decay channels all regularize to the same τ-structure

References

1. Hardy, G.H. (1949). Divergent Series. Oxford University Press.
2. Elizalde, E. (1995). Ten Physical Applications of Spectral Zeta Functions. Springer.
3. Polchinski, J. (1998). String Theory, Vol. 1. Cambridge University Press.
4. Casimir, H.B.G. (1948). On the attraction between two perfectly conducting plates, Proc. Kon. Ned. Akad. Wet. 51, 793.
5. Riemann, B. (1859). Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsberichte der Berliner Akademie.
6. Hawking, S.W. (1977). Zeta function regularization of path integrals in curved spacetime, Commun. Math. Phys. 55, 133.
7. Dowker, J.S. & Critchley, R. (1976). Effective Lagrangian and energy-momentum tensor in de Sitter space, Phys. Rev. D13, 3224.
8. Noether, E. (1918). Invariante Variationsprobleme [Invariant Variation Problems], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 235–257.
9. Robinson, J., Davis, M., and Putnam, H. (1961). The Decision Problem for Exponential Diophantine Equations, Annals of Mathematics 74: 425–436.
10. Uhlenbeck, K.K. (1982). Removable singularities in Yang–Mills fields, Communications in Mathematical Physics 83(1): 11–29.

Appendix — Worked Examples

A.1 Sum of Natural Numbers

Consider S = 1 + 2 + 3 + 4 + ..., which diverges. Using the zeta function:

From the functional equation or Bernoulli numbers:

Therefore: 1 + 2 + 3 + 4 + ... = −1/12 (regularized value).

A.2 Sum of Cubes

Consider ∑ n³ = 1³ + 2³ + 3³ + ..., which also diverges:

This is the regularized value used in string theory calculations.

A.3 Casimir Energy

The energy per unit area between Casimir plates separated by distance d:

This diverges, but using ζ-regularization:

The force is F = −dE/dd:

This matches experimental measurements.

A.4 Τ-Space Interpretation

For a harmonic oscillator spectrum E_n = ℏω(n + ½), the temporal charges are:

The sum ∑ E_n diverges, but in τ-space:

This gives a finite effective temporal charge for the infinite oscillator tower.

A.5 Connection to QFT Vacuum Energy

In quantum field theory, the vacuum energy of a free scalar field is:

This integral diverges. Using ζ-regularization in momentum space and converting to τ:

The regularized vacuum has a finite effective temporal charge, which contributes to the cosmological constant.