Uncertainty Relation for Photons

Abstract

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.The uncertainty relations that first appeared in the fundamental Heisenberg paper on the conceptual content of quantum mechanics [1] play an important role in elucidating specific properties of the quantum world. The aim of this Letter is to derive in the framework of quantum electrodynamics a sharp uncertainty relation for photons treated as three-dimensional objects. The photon wave function that saturates our inequality has already appeared in different contexts.An uncertainty relation for the momentum and the center of mass coordinate along a given direction can easily be obtained from the commutators between the generators of the Poincaré group. It follows from general principles [2] that the commutators between the components of the momentum and the Lorentz boost are:Therefore, each component of the center of mass coordinate (the boost operator divided by the energy operator with proper symmetrization to secure Hermiticity),is a canonically conjugate variable to the corresponding component of momentum, [R i ,P j ] = i δ ij . Hence, the standard Heisenberg relation ∆x 2 i ∆p 2 i ≥ 2 /4 follows separately for every direction, with Gaussian functions saturating each inequality. These simple one-dimensional uncertainty relations hold for every relativistic system. In the special case of photons they were obtained by Holevo [3] in the framework of estimation theory. Holevo also proved that "the transversality condition precludes the three inequalities from becoming equalities simultaneously." This was to be expected because the existence of one wave function that saturates all three standard Heisenberg uncertainty relations would mean that photons are not any different from nonrelativistic particles. In a general approach based on the properties of the Poincaré group, the impossibility to saturate simultaneously the three uncertainty relations follows from the noncommutativity of the components ofR,The expression in parentheses on the right side represents the spin: the total angular momentum minus the orbital part. Schwinger [4] has shown that for massless particles with helicity λ the right-hand side in (3) becomes:He also found a very rough estimate of the lower bound in the three-dimensional uncertainty relation.In what follows we fully realize the Schwinger program by deriving a precise form of the uncertainty relation for phot...

Knots in electromagnetism

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Heisenberg uncertainty relations for relativistic bosons

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This work completes the program started in [1][2][3] to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived.

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