Geometric Factors in Electromagnetic Field Commutators and the Quantum Information Control of Vacuum Fluctuations

The electromagnetic field commutation relations are defined in terms of geometric factors that are double averages over two finite four-dimensional space-time regions. The square root of any of the uncertainty relations derived from the aforementioned commutators is taken as a critical field, in the sense that any electromagnetic field much larger than it can be treated as classical. Another critical electromagnetic field associated with the quantum information control of vacuum fluctuations can be chosen as the square root of the mean quadratic fluctuation of each quantity of electromagnetic field, when the number of photons is defined and is equal to zero. Any electromagnetic field expectation value could be measured if it is much greater than the last critical field. This article covers a magnitude order comparison between the critical fields and its consequences for measuring the electromagnetic field information. 1. Geometric Factors in Electromagnetic Field Commutators From the quantum theory of the electromagnetic field formalism it is possible to obtain [1] the commutators in two different space-time points of two magnetic H or electric E components [Ei(P1), Ek(P2)] = [Hi(P1), Hk(P2)] = 4πi c ( ∂2 c∂t1∂t2 δik − ∂ 2 ∂xi1∂xk2 ) Δ (1) [Ei(P1), Hi(P2)] = 0 (2) [Ei(P1), Hk(P2)] = −4πi ( ∂2 ∂xl2∂t1 ) Δ , (3) where P1 = ( r1, t1), P2 = ( r2, t2), δ = Dirac’s function, δik = Kronecker’s delta, t = t2 − t1, r = | r2 − r1|, Δ = (δ(r − ct)− δ(r + ct))/(4πr), i = k, (i, k, l) forming an even permutation of x, y, z. Presented at the 38th Symposium on Mathematical Physics “Quantum Entanglement & Geometry”, Toruń, June 4–7, 2006.