Optics Express

Action on pulse position and momentum using dispersion and phase modulation

Matthew E. Grein, H. A. Haus, Leaf A. Jiang, Erich P. Ippen

Published Jun 2, 2001• 10 citations

Abstract

The timing jitter and frequency jitter of quantized optical pulses obey Heisenberg's uncertainty principle. We show how one jitter may be reduced at the expense of the other, using dispersion and phase modulation.

DOI: 10.1364/oe.8.000664

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PUH Theorem 188 — The Heisenberg Uncertainty Principle from E8 Lattice Geometry: delta_x * delta_p >= hbar/2 as the Minimum Action of the E8 Phase I Condensate, and [x, p] = i*hbar from the E8 Lie-Poisson Bracket

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In 1927 Werner Heisenberg derived the uncertainty principle delta_x * delta_p >= hbar/2 from the commutation relation [x, p] = ihbar and the Cauchy-Schwarz inequality. He motivated it through the gamma-ray microscope thought experiment: measuring position precisely requires high-energy photons that impart momentum kicks. But the principle holds even for ideal measurements that cause no disturbance — it is a property of quantum states, not of measurement apparatus. The true origin is deeper: the commutation relation [x, p] = ihbar itself. What is its physical origin? Why ihbar and not zero? For 100 years this has had no mechanical answer. This paper establishes that within the Photonic Universe Hypothesis, the Heisenberg uncertainty principle is a geometric property of the E8 Phase I condensate. Position x and momentum p are canonically conjugate variables in the E8 Lie-Poisson bracket. The commutator [x, p] = ihbar is the quantisation of the classical Poisson bracket {x, p} = 1, with hbar = E_Pt_P being the minimum E8 lattice action (Theorem 176). The uncertainty product delta_x * delta_p >= hbar/2 is derived three independent ways: (1) from the E8 Lie-Poisson bracket and canonical quantisation; (2) from the Fourier duality of E8 bow waves (Theorem 187): sharp position requires a localised bow wave packet and sharp momentum requires a plane wave — these are mutually exclusive; (3) from the U(1) phase-number uncertainty of the E8 phase mode. The factor hbar/2 (not hbar) comes from the QHO ground state energy hbaromega/2, arising from the canonical commutation relation [a, a+] = 1 (Theorem 174). The energy-time uncertainty delta_E * delta_t >= hbar/2 follows from the E8 phase-mode dispersion relation E = hbar*omega. The uncertainty principle is not about measurement disturbance. It is about E8 lattice geometry. Four falsifiable predictions follow.

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