In the classical electromagnetic theory the wave-vector k = (2 π / λ ) σ underlies the Fourier space of propagating (or radiative) fields. The k -vector combines into a single entity the wavelength λ and the unit vector σ that signifies the beam's propagation direction. The Fourier transform relation between the three-dimensional space of everyday experience and the space of the wave-vectors (the so-called k -space) gives rise to relationships between the two domains analogous to Heisenberg's uncertainty relations. Considering that in quantum theory the electromagnetic k -vector is proportional to the photon's momentum ( p = ħ k , where ħ = h /2 π , h being the Planck constant), one should not be surprised to find relationships between dimensions of a beam in the X Y Z -space and its momentum spread in the k -space. Such relationships impose fundamental limits on the ability of measurement systems to determine the various properties of electromagnetic fields. In this chapter we address two problems that have widespread applications in optical metrology, spectroscopy, telecommunications, etc., and discuss the constraints imposed by the uncertainty principle on these problems. The first topic of discussion is the separation of two overlapping beams of identical wavelength having slightly different propagation directions. This will be followed by an analysis of the limits of separating co-propagating beams having slightly different wavelengths. Angular separation and the limit of resolvability Figure 19.1 shows an aperture of diameter D , which transmits two plane waves of the same wavelength λ propagating in slightly different directions.
The main purpose of this paper is to show that the notions of frames and fusion frames introduced in non-harmonic Fourier analysis are also very natural in discussion of some basic problems in theory of open quantum systems and, in particular, in quantum optics. Frames are collections of vectors in a Hilbert space which assure a natural representation of each vector in the space, but may have infinitely many different representations for any given vector. For a given quantum system represented in a Hilbert space the question of minimal number η of observables Q 1 ,..., Q η , whose expectation values at some instants t 1 ,...,t p determine the statistical state of the system is discussed. We assume that the time evolution of the system in question is governed by a semigroup of linear transformations with generator .
The origin of physical dynamics and the reason of existence of special relativity are explored. This endeavour is started by analysing the logic of nature. Next, only mathematics is used in order to explore the dynamics of this model of physical reality. The model that is described here annihilates the old reality and creates a new reality at each dynamic step. Hilbert space cannot treat dynamics. It contains nothing that supports dynamics. In the contrary, dynamics manages the Hilbert spaces. Like traditional quantum logic, Hilbert space cannot treat physical fields. By embedding the separable Hilbert space in a rigged Hilbert space, it can house fields by representing them as blurred sets of Hilbert vectors. The field is the convolution of the blur with a set of Dirac delta functions that represent Hilbert vectors. When the blur is differentiable, then the field is differentiable as well. The field values are attached to the Hilbert vectors. In this way traditional quantum logic can be expanded, such that it also treats fields. This extended quantum logic still cannot handle dynamics. The logic only describes a static status quo. Dynamics let nature step from one status quo to the next. It does that by letting nature transform from configuration space to Fourier space. There the fields control the difference between the past and the future status quo. The Fourier transform converts the rather complicated differentiation into a simple multiplication and since the multiplication factors are close to unity, this comes down to still simpler addition. After the confrontation in Fourier space, nature returns back to configuration space. Feynman's path integral approach exploits this fact. The up and down Fourier transforms reshuffle the Hilbert vectors. All Hilbert vectors are affected. The Hilbert vectors represent virtual or actual quanta and present themselves as shot noise.