Imaging Inside a Wavelength

Abstract

Classical physics assigns sharp positions x,y,z to objects, but it is well-known in quantum mechanics that this description must give way to a blurrier picture with x,y,z promoted to operators in some Hilbert space. The photon poses a subtler problem as Newton and Wigner pointed out many years ago, and in fact there is no suitable position operator with all of the qualities one would expect-something new must arise. That new ingredient is that the various components xcirc,ycirc,zcirc of a position operator of rcirc no longer commute. The commutation relations are reminiscent of those of the x,y,z components of spin, and in fact the picture that emerges is one where the photon can have a well defined z, for example, representing the front of a light wave, while the and y components of its position become blurry in a wavelength-dependent fashion. We review these results and discuss possible experimental setups and expected results that may occur for the position of a photon inside its wavelength. Quantum theory predicts that a noncommutative effective geometry to emerge-a new prediction which present-day technology should just make it feasible to test