Analytical Information Theory

A topic of interest to sensory physiology and quantum physics is addressed, viz: the quantification of forms of information available for measurement. In the derivation of different forms of information “quanta”, the author has demonstrated that the modulating envelopes for the wave packets representing these quanta are functional solutions to the wave equation for electromagnetic forms of energy [1]. The wave equation involved is the WEBER equation for parabolic cylinder coordinates. The question thus arises: why should the wave equation, which determines the forms of informational quanta, be based on a parabolic cylinder structure?

An answer lies inherent in the author's suggestion to represent information as a number in HILBERT space [2], viz: if Δf = signal bandwidth (cycles), Δt = signal duration (seconds), f ₀ = signal mid-frequency (Hz) and t₀ = signal midperiod (s/cycle), and if Δf Δt + j f₀ t₀ = α, α α* gives a HILBERT space representation of the total information content of any signal (Δf Δt = 1/2; at minimum and is equivalent to one quantum; f₀ t₀ = 1/2; at minimum and is equivalent to one quantum). A direct solution to the question may be taken from an analogy with structural chemistry by postulating an electromagnetic resonance effect between the Δf Δt and f₀ t₀ structures of the signal. Thus, by analogy, a “total quantum number” is given by αα*, a “magnetic quantum number” by Δf Δt and an “azimuthal quantum number” by f₀ t₀. According to this viewpoint, which may be considered “analytical information theory”, the quantum mechanics of an electromagnetic field is based on a WEBER equation with parabolic cylinder coordinates (wave equation) because there are two modes of activity involving four dimensions of any electromagnetic stimulus. This hypothesis has implications for theoretical physics as well as for sensory physiology which must possess a clear definition of informational structure before the information transmission properties of a sensory pathway can be described.