Zeta-Photon Conjecture
Roughed in at 2:37 AM. Needs formalization.
The Intuition
Three things that might be deeply connected:
- Riemann zeta zeros - mysterious distribution tied to primes
- Conic sections - geometry of slicing continuous structures
- Photon envelopes - discrete quanta from continuous fields
Thread 1: Zeta in Cryptography
Question: Can the zeta function’s unprovable nature protect cryptographic systems?
- RSA = factoring is hard
- Zeta-based = understanding prime distribution is hard
- GUE hypothesis: zero spacings behave like random matrix eigenvalues
- Could hash outputs be required to conform to GUE statistics?
Barriers:
- We can compute zeros; difficulty is proving structure
- Need computational hardness, not proof-theoretic hardness
Thread 2: Light Cones as Conic Sections
The spacetime light cone, when intersected by hyperplanes at various angles, produces conic sections:
- Timelike slice → ellipse
- Null slice → parabola
- Spacelike slice → hyperbola
The geometry of what’s allowed to communicate (causality) is literally conic.
Thread 3: Hilbert-Pólya and Quantum Hamiltonians
Conjecture: Zeta zeros = eigenvalues of some Hamiltonian
Berry-Keating proposed: H = xp (position × momentum)
This operator has:
- No ground state (continuous spectrum)
- Conformal symmetry
- Connection to chaotic dynamics
If true: The distribution of primes is encoded in a quantum mechanical system.
Thread 4: Photon Quantization
- Classical EM field is continuous
- Photons are discrete excitations
- Energy: E = hν (integer-like quantization via Planck)
- Wave packet envelope is continuous (Gaussian)
- But phase encodes discrete frequency
Parallel to primes:
- Number line is continuous
- Primes are discrete “punctures”
- Distribution follows continuous asymptotic laws (PNT)
- But individual locations are stubbornly discrete
The Synthesis (Speculation)
What if:
- The zeta zeros encode “allowed modes” of some wave-like structure
- These modes have geometric representation as conic sections
- The Hilbert-Pólya Hamiltonian describes photon-like excitations in a number-theoretic “field”
- Prime distribution = quantum vacuum fluctuations of this field
Crypto angle: A system where valid states must conform to these “allowed modes” would be protected by:
- The difficulty of computing the mode structure (zeta zeros)
- The impossibility of predicting without understanding (RH)
Research Threads
- Berry-Keating Hamiltonian papers
- Montgomery pair correlation / GUE statistics
- Connes’ approach to RH via noncommutative geometry
- Light cone quantization in QFT
- Modular forms and physics connections
Related
- Quantum Biology Musings
- Physicists corpus: Search for Riemann, zeta, quantum chaos
“The primes are the photons of arithmetic.” — ?? (someone should have said this)