The unknown amplitude law q(x) defining an observed effect may be found using the principle of Extreme Physical Information. EPI is derived as follows. The observations follow an information flow J --> I, with J the information intrinsic to the source and I the Fisher information level in its data, obeying (i) I=4 Integral dx q' -squared. Here q'= dq/dx and p(x) = q(x)-squared is the probability. It was previously shown, using L. Hardy's 5 axioms defining physics, that I = max. Therefore, its variation (ii) delta I = 0. Note that I is generic, obeying (i) for all source effects, whereas J is specific to the particular effect. Hence, rather than having form (i), J obeys (iii) J = Integral dx j[q(x),s(x)] with j some function of its arguments and s(x) a known source, such as of mass, biological fitness, etc. Information I decreases under any irreversible operation such as measurement, so that I l.e. J or, equivalently, I = kJ where 0 l.e. k l.e. 1. Then the variation delta I = k delta J so that property (ii) gives (iv) delta J = 0 as well. Then combining (ii) and (iv), delta(I - J) = 0. Or, I - J = L = extremum. What kind of extremum? Eqs. (i) and (iii) give (v) L=4q'^2 - j[q(x),s(x)]. Differentiating (v), (d^2 L )/(dq'^2) = +8. Then by the Legendre condition the extremum is a minimum. The unknown source effect obeys (vi) I - J= minimum, EPI.
The unknown amplitude law q(x) defining an observed effect may be found using the principle of Extreme Physical Information. EPI is derived as follows. The observations follow an information flow J --> I, with J the information intrinsic to the source and I the Fisher information level in its data, obeying (i) I=4 Integral dx q' -squared. Here q'= dq/dx and p(x) = q(x)-squared is the probability. It was previously shown, using L. Hardy's 5 axioms defining physics, that I = max. Therefore, its variation (ii) delta I = 0. Note that I is generic, obeying (i) for all source effects, whereas J is specific to the particular effect. Hence, rather than having form (i), J obeys (iii) J = Integral dx j[q(x),s(x)] with j some function of its arguments and s(x) a known source, such as of mass, biological fitness, etc. Information I decreases under any irreversible operation such as measurement, so that I l.e. J or, equivalently, I = kJ where 0 l.e. k l.e. 1. Then the variation delta I = k delta J so that property (ii) gives (iv) delta J = 0 as well. Then combining (ii) and (iv), delta(I - J) = 0. Or, I - J = L = extremum. What kind of extremum? Eqs. (i) and (iii) give (v) L=4q'^2 - j[q(x),s(x)]. Differentiating (v), (d^2 L )/(dq'^2) = +8. Then by the Legendre condition the extremum is a minimum. The unknown source effect obeys (vi) I - J= minimum, EPI.
Consider a finite-sized, multidimensional system in parameter state a. The system is either at statistical equilibrium or general nonequilibrium, and may obey either classical or quantum physics. L. Hardy's mathematical axioms provide a basis for the physics obeyed by any such system. One axiom is that the number N of distinguishable states a in the system obeys N=max. This assumes that N is known as deterministic prior knowledge. However, most observed systems suffer statistical fluctuations, for which N is therefore only known approximately. Then what happens if the scope of the axiom N=max is extended to include such observed systems? It is found that the state a of the system must obey a principle of maximum Fisher information, I=I(max). This is important because many physical laws have been derived, assuming as a working hypothesis that I=I(max). These derivations include uses of the principle of extreme physical information (EPI). Examples of such derivations were of the De Broglie wave hypothesis, quantum wave equations, Maxwell's equations, new laws of biology (e.g., of Coulomb force-directed cell development and of in situ cancer growth), and new laws of economic fluctuation and investment. That the principle I=I(max) itself derives from suitably extended Hardy axioms thereby eliminates its need to be assumed in these derivations. Thus, uses of I=I(max) and EPI express physics at its most fundamental level, its axiomatic basis in math.
The principle of extreme physical information (EPI) places physics on an information-theoretic footing. It yields many of the fundamental wave equations of physics (of Klein-Gordon, Dirac, etc.). EPI is based upon the measurement of particle four-vectors ${\mathbf{\ensuremath{\theta}}}_{\mathit{n}}$ with random errors xn, n=1,...,N. This leads to an additive Fisher information form whose extremization provides the derivations. However, the model measurement procedure was, in the past, overly restrictive: the real and imaginary parts qn(x), n=1,...,N of the probability amplitude functions for the fluctuations were required to have nonoverlapping support regions. Here we rederive the requisite information form without this restriction. Our model measurement procedure is simply the efficient collection of four-data, that is, N independent measurements of four-coordinates of particles of the field. In effect, each measurement defines a different degree of freedom qn(x) of the scenario. \textcopyright{} 1996 The American Physical Society.
For the case of a free particle, it is known that one may extremize a Lagrangian L=.5mvv, where v=dx/dt, to obtain: d/dt dx/dt = 0. In this classical case, one is interested in x and t or x(t). One may think of moments i.e. powers of derivatives of x(t). Given the first moment is equal to a constant, all higher moments are 0. Each moment yields a piece of information. Thus there is one piece of information, namely the constant velocity v=dx/dt. A second derivative d/dt dx/dt =0 demonstrates there is no further information. The form d/dt dx/dt, however, is associated with an extremization of a function, namely L=.5mvv or Integral .5mvv dt. Thus we argue that extremization is really linked to information found in the problem for the free particle case. Next, we consider .5 Integral dt vv = .5 Integral v dx. This is proportional to Integral p dx where p represents a physical impulse. The equation d/dt dx/dt =0 shows there exists only one piece of information, regardless of the value of v. In other words, all v values satisfy this equation (where v=constant). In analogy, if one wishes all p values to satisfy Integral p dx, p = constant, without distinguishing p, then x= constant/p. This, however, is the quantum wavelength. If it is repeated, in space, one desires a periodic function in px which has a modulus of 1, i.e. an eigenfunction of -id/dx namely exp(ipx). This does not follow from a Fisher’s information type variation, but rather from the idea of information and two quantities of interest, namely a kind of probability exp(ipx) and x. The modulus of 1 is equivalent to the single piece of information present i.e. v=constant. To see Fisher’s information enter the quantum picture, one may consider a particle in a box with infinite walls. In such a case, exp(ipx) and exp(-ipx) are present or cos(px) if a(p)=a(-p). There is one piece of information, so d/dx cos(px) = function(px). function(px) is the one piece of extra information like v. Finding a second moment cannot yield any further information i.e. d/dx d/dx cos(px) = constant cos(px). If it did not, there would be extra information in the system, but v=constant is the only information. d/dx d/dx cos(px), however, follows from the extremization of Fisher’ s information subject to the constraint Integral cos(px)cos(xp) CC = 1. Thus we argue that in the free particle case, extremization really represents moments (shown by derivatives) and information. We also consider the case of a potential V(x) present.
Aims of the book The primary aim of this book is to develop a theory of measurement that incorporates the observer into the phenomenon under measurement. By this theory, the observer becomes both a collector of data and an activator of the phenomenon that gives rise to the data. These ideas have probably been best stated by J. A. Wheeler (1990; 1994): All things physical are information-theoretic in origin and this is a participatory universe … Observer participancy gives rise to information; and information gives rise to physics. The measurement theory that will be presented is largely, in fact, a quantification of these ideas. However, the reader might be surprised to find that the “information” that is used is not the usual Shannon or Boltzmann entropy measures, but one that is relatively unknown to physicists, that of R. A. Fisher. The measurement theory is simply a description of how Fisher information flows from a physical source effect to a data space. It therefore applies to all scenarios where quantitative data from repeatable experiments may be collected. This describes measurement scenarios of physics but, also, of science in general. The theory of measurement is found to define an analytical procedure for deriving all laws of science. The approach is called EPI, for “extreme physical information.” The secondary aim of the book is to show, by example, that most existing laws of science fit within the EPI framework. That is, they can be derived by its use. (Many can of course be derived by other approaches, but, apparently, no other single approach can derive all of them.) In this way the EPI approach unifies science under an umbrella of measurement and information.
Seeking the unknown dynamics obeyed by a particle gives rise to the de Broglie wave representation, without the need for physical assumptions specific to quantum mechanics. The only required physical assumption is conservation of momentum μ. The particle, of mass m, moves through free space from an unknown source-plane position a to an unknown coordinate x in an aperture plane of unknown probability density pX(x), and then to an output plane of observed position y=a+z. There is no prior knowledge of the probability laws p(a,M),p(a) or p(M), with M the particle momentum at the source. It is desired to (i)optimally estimate a, in the sense of a maximum likelihood (ML) estimate. The estimate is further optimized, by minimizing its error through (ii) maximizing the Fisher information about a that is received at y. Forming the ML estimate requires (iii) estimation of the likelihood law pZ(z), which (iv) must obey positivity. The relation pZ(z)=|u(z)|*2≥0 satisfies this. The same u(z) conveniently defines the Fisher channel capacity, a concept central to the principle of Extreme physical information (EPI). Its output u(z) achieves aims (i)–(iv). The output is parametrized by a free parameter K. For a choice K=0, the result is u(z)=δ(z), indicating classical motion. Or, for a finite, empirical choice K=h¯(Planck’s constant), u(z) obeys the familiar de Broglie representation as the Fourier transform of the particle’s probability amplitude function P(μ) on momentum μ. For a definite momentum μ, u(z) becomes a sinusoid of wavelength λ= h¯/μ, the de Broglie result.
In (1) the Schrodinger equation is derived using the extremization of Fisher information in a certain way formulated in (2) i.e. Extreme Physical Information. Fisher information, however, is defined in terms of P(x) which in quantum mechanics is W*(x)W(x) where W(x) is the wavefunction. As a result, the Schrodinger equation is derived in terms of P(x) and then the mathematical substitution P(x)=W(x)W(x) for a bound state “simplifies” the form of the equation. Otherwise there would be no need to introduce the wavefunction. Furthermore P(x) = constant for a quantum free particle which is also a solution of the Schrodinger equation. For P(x)=constant, however, one would normally not consider using the Fisher information. We further note that in a physical example, namely that of the reflection/refraction of light from an interface of media one is forced to introduce the concept of a wavefunction i.e. the square root of flux, otherwise the problem cannot be solved. Thus in this situation one does not solve the problem in terms of P(x) and then introduce W(x) to simplify the form. In particular, if one considers a single photon moving to the right, it either reflects or refracts (with probability), but the ensuing equations involve the weight of the square root fluxes (wavefunctions) of all three. How can this be unless there is uncertainty in time and space which allows one to have all three together? Thus the reflection/refraction problem depends on square root probability or flux and introduces the idea of uncertainty in x, t. Given the translational invariance in the problem, it is natural to introduce the translation operator d/dx. In fact, one may define the uncertainty in x such that one has an eigenfunction of -id/dx i.e. exp(ipx). This is complex and so its modulus respects P(x)=constant which would yield a Fisher’s information of 0, meaning that there is no probability and hence no need for Fisher’s information. There is, however, probability in x, but a different level than the deterministic x=vt where v=constant. Thus we argue that one may obtain exp(ipx) strictly from the notion of translational invariance. In other words, Fisher information is not required to find this form and the quantum physics associated with it. Nevertheless Fisher information and EPI (Extreme physical Information) may be used to derive the Schrodinger equation (applied to a bound state) and one may deduce exp(ipx) from this. We note, however, that Fisher information makes use of d/dx which is the translational generator and so the idea of translational invariance and Fisher’s information/EPI may be linked.