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Fisher's Information and Probability Dictated by Physics and Not Vice Versa?

Francesco R. Ruggeri

Published Mar 18, 2023• 0 citations
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Abstract

Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0<=P(x)<1 and Integral P(x)dx =1. The wavefunction, exp(ipx)/sqrt(L), is comprised of two probability distributions and ADD and MULTIPLIES statistically (i.e. OR and AND ) situations, but is involved in relative average calculations because it is not normalized. It is also different because it allows for positive and negative values. These are properties which do not follow from statistics using a classical P(x).

DOI
10.5281/zenodo.7748929

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Given a coin, one has two complementary pieces of information and a corresponding probability of .5 and .5. Similar considerations apply to a die. As a result, one may get the impression that there exists a unique probability distribution in a probabilistic system and that this single probability scheme is linked to the constrained maximization of Shannon’s entropy or Fisher’s information. We argue however, that there may be different probability schemes in the same problem associated with different pieces of information which are relevant to the physics which occurs in the system. Thus, we argue that it is the physics which selects the probability distribution to be examined. For example, one may loosely think in terms of a detailed probability scheme versus a kind-of coarse grained one in the same system. For example, for a quantum free particle, P(x)=1 i.e there is the same probability for a particle to be at any x point. This, however, is based on velocity, i.e. the relationship between dx and dt. There is another piece of information in the system, namely mo which together with v yields momentum (either relativistic or nonrelativistic) .This need not be measured using a clock (which velocity requires), but by the impact of an impulse on a target. Thus, nonrelativistically p=m1v1=m2v2 give the same impulse, but have different velocities. Quantum mechanics shows that a free particle is governed by exp(ipx) which contains two unnormalized probability distributions cos(px) and sin(px). These are key for impulse interactions (i.e. the physics of interactions such as with a 2-slit apparatus etc), yet exp(ipx)/sqrt(L) maps into the “coarse-grained” P(x)=exp(ipx)exp(-ipx) / L = constant. P(x) is one probability scheme in the problem and exp(ipx) another (for a free particle). Each answers some physical questions, but one needs to know which probability to use (i.e. what information is relevant) to a particular situation. P(x) is relevant to motion in space not involving interactions. exp(ipx) is relevant to interactions (even with a magnetic vector potential causing no magnetic field) or with a 2-slit apparatus. Furthermore, in a quantum bound state, one has W*(x)W(x) = P(x) (W(x)=wavefunction =Sum over p a(p)exp(ipx)) and a*(p)a(p).=P(p). Each probability scheme corresponds to the analysis of a different kind of measurement, but various schemes may also be linked. Thus one may have to start with one to create the others. For example, exp(ipx) is the base for creating W(x)=Sum over p a(p)exp(ipx), which ultimately yields P(x)=W*(x)W(x). As a result, we argue that if probability exists in a system, it is not necessarily the case that there is a unique probability distribution. There may be various probability schemes linked with different information and different measurements/interactions. Thus, physics leads one to focus on a specific probability scheme. As a result, we argue that one must be careful when discussing pure statistical quantities and their extremization, because these quantities, like Shannon’s entropy and Fisher’ s information, contain a classical probability distribution, but they do not indicate which distribution to use in the case that a system has more than one. . As a result, the analysis of a system requires knowledge of the physics (interactions) occurring.We argue against thinking in terms of physics following directly from statistical considerations. The constrained extremization of one statistical quantity may yield a probability distribution related to one piece of information, but not to another. Furthermore, Shannon’s entropy and Fisher’s information pertain to classical probability 0<=P(x)<1 and Integral P(x)dx =1. The wavefunction, exp(ipx)/sqrt(L), is comprised of two probability distributions and ADD and MULTIPLIES statistically (i.e. OR and AND ) situations, but is involved in relative average calculations because it is not normalized. It is also different because it allows for positive and negative values. These are properties which do not follow from statistics using a classical P(x).
In (1), a variational approach is applied to Fisher’s information (with constraints) to obtain a differential equation in P(x)=probability to be at x. It is suggested the equation may be simplified using P(x)=W(x)W(x) where W(x) is real. The result is the nonrelativistic Schrodinger equation with mass=1. As a first observation, we note that Fisher’s information, which is a purely statistical quantity, should not necessarily yield a nonrelativistic equation which is an approximation of relativistic results. In particular, we argue that Fisher’s information is used to find <pp> not necessarily 1/2m <pp>, but this is a minor point. A more important issue, we argue, is the meaning of P(x) in a bound system. A classical bound system is sometimes said to have a P(x)=C/v(x) i.e. P(x)dx is proportional to dt, the amount of time a particle spends in each equally sized dx. In the high energy limit, the envelope of a quantum bound probability P(x)=W(x)W(x) (W(x)=wavefunction) is said to roughly match C/v(x), yet this density does not follow from the extremization of Fisher’s information (together with two general constraints which yield E and V(x)=potential). This begs the question: What is P(x)? If one considers a particle moving at constant speed (i.e constant momentum) then P(x)=constant because each dx point carries the same probability. Classically a particle placed in a box with infinite potential walls would have a P(x) =constant and p and -p motion would be separated in time. Thus if P(x) is to have meaning in a box, one must conclude that the probabilities associated with p and -p are not constant, but rather functions of x. This then leads to the notion of a complex new kind of probability (which maps to P(x)=constant) for a free particle i.e. W(x) =cos(px) + i sin(px) so that the modulus W*(x)W(x)=P(x)= 1. These ideas seem to go beyond the notion of Fisher information. What then is the role of Fisher information? We argue it is related to two very specific moments of the distribution cos(px) or rather W(x) = Sum over p a(p)cos(px) with a(p)=a(-p), namely and <pp>. In classical physics there exists p and pp at each x point (but one is the square of the other), but with distributions <p does not equal to <pp> in general. Both and <pp> are linked to physical concepts which exist in Newtonian mechanics, namely impulse and work. With the presence of cos(px) one has a combination of Newtonian mechanics and a statistical distribution. We argue that the role of Fisher’s information is to allow one to transform from to <pp>. In particular, -idW/dx appears in Fisher’s information (and is linked to ), but the specific role of a variation is to transform it into <pp> which appears in a conservation of energy equation. We argue that this is why Fisher’s information may be used to derive the Schrodinger equation.
Quantum mechanics is based on two different probability schemes, namely W(x) =wavefunction and W*(x)W*(x)=P(x), with one being linked to the other. Why should there be two probability schemes? In (0) we argued that there are two spatial behaviour schemes because an impulse p (momentum) is automatically associated with two spatial distributions cos(px) and sin(px) in exp(ipx), yet a particle with constant p moves as x=p/m t. Thus if one is only concerned with motion, each x point has the same weight, and this is physically measurable as a photon or electron travels a certain distance in a certain time in free space. On the other hand, an impulse interaction is associated with exp(ipx) is a completely different spatial picture, and this too is measurable as in a 2-slit experiment. Thus it seems the two schemes are measured differently. We argue that this is an important issue because if there are two x-probability schemes there should be different entropies associated with each. Ttraditionally (1), one sees two quantum entropies: - W*(x)W(x) ln {W*(x)W(x)} and - a*(p)a(p) ln{ a*(p)a(p) } where W(x) = Sum over p a(p)exp(ipx). These are associated with the x=p/m t scheme or x-dependence based on motion or average motion. W*(x)W(x) represents spatial density which should be measured presumably by some sort of scattering, but we ask whether some information is lost in this approach? In particular, we note that a wavefunction may be obtained for a bound state. In the high energy limit, the envelope over the density W*(x)W(x) is said (2) to match C/v(x) from classical physics where .5mv(x)v(x) + V(x) = E. C/v(x), however, is proportional to delta t (with delta x being constant) and has nothing to do with the strength of an impulse hit. In the classical case, there is a single v(x) and so a single p(x) in each delta x thus no impulse entropy within a delta x i.e. at a point x. In quantum mechanics, however, we argue that a constant p automatically has an x probability distribution exp(ipx) associated with it and there might be an entropy linked to this.This entropy is associated with a probability to receive a p impulse at x and is so not linked to interference while the W*(x)W(x) is. In (3) we introduced the impulse or wavelength entropy (for a free particle |cos(px)|/C ln{| cos(px)|/C},, but here we wish to investigate its properties more closely.
It is well known that a Lagrangian (usually kinetic energy - potential) may be varied to produce Newton’s second law of motion. In (1) it is noted that kinetic energy is proportional to a Fisher’s information of the form b Integral dt dx/dt dx/dt (b=constant) and so in a sense Fisher’s information may be varied together with another piece i.e. -V(x). In (2), it is noted that Fisher’s information, this time with a probability P(x), i.e. b Integral dx P(x) {d/dx ln(P) } {d/dx lnP) } may be varied (with constraints) to yield the Schrodinger equation. (One may note that the two definitions of Fisher’s information are the same if P(x)=x(t)x(t) in the first case.) Here we consider the notion of impulse and pressure which depends on p. We argue that a free quantum particle with constant p undergoes x=p/m t on average, but is associated with a probability distribution exp(ipx) which contains uncertainty in x (within a wavelength) and no time. P, however, represents an impulse which the particle may impart. In the case of potential V(x) one may consider a linear (interfering) sum of exp(ipx)’s i.e. W(x)= wavefunction = Sum over p a(p)exp(ipx). This leads to an average impulse of -idW/dx / W. Classically pressure is created by multiplying impulse by velocity (flux). If time does not exist within a wavelength then there is no time within it. There is, however, still an external clock and p ave in a sense should correspond to an m v(ave) nonrelativistically. Thus WW (bound system) is in a sense like a pressure. One may then suggest that for an equilibrium scenario one extremizes this pressure with respect to constraints (EWW and V(x)W). The variation is with respect to W(x), the wavefunction. Given that a bound particle maps to an rms momentum such that KE(x) = prms(x)prms(x)/2m where prms(x) is the classical value, then at the classical level prms(x) vrms(x) should be pressure and should also satisfy an extremization equation. In fact, the extremization equation using WW should be very closely related to it. One is in fact the d/dx of the other. The classical notion of pressure, proportional to vv, may be extremized subject to constraints. This leads to Newton’s second law which contains d/dt explicitly. Changing to “x” concepts yields v(x)d/dx which may be integrated to yield an energy conservation equation matching the quantum one. Thus we try to think of extremization of Fisher’s information i.e. EPI (extreme physical information) in terms of pressure, in both the classical and quantum cases.
Shannon’s entropy is calculated using probabilities P(i) i.e. S= - Sum over i P(i) ln(P(i)). A probability is related to an uncertainty associated with event i, but this event may not be a physical event, but rather a measurement. In other words, measurements in a system may be made according to one distribution or another. This distribution may be converted into something which looks like a probability even though the system is completely deterministic. The entropy then calculated is based on the measurement distribution and not the physical system. As a first example, consider the completely deterministic motion of a bound classical system between two turning points. Person A may send values of x (one dimensional space) to person B, based on a certain distribution instead of sending the values based on time i.e. x(t1),x(t2) etc. One distribution which may be used is the uniform one i.e. P(x)= dx/L. Another distribution, used in the literature (1), is based on the time spent in each constant dx i.e. P(x) = Cdx/v(x) where v(x) is velocity. These two distributions lead to different Shannon’s entropies, but the underlying system is completely deterministic which begs the question: Why does one associate an entropy with it? The entropy seems to be associated with the sending of information. We try to apply these ideas to a quantum bound state. In the literature (2), one sometimes sees a Shannon’s entropy based on W(x)W(x) = P(x), where W(x) is the single particle bound wavefunction. This is a distribution, but is it based on uncertainty in the system? W(x)W(x) often consists of a set of crests of different heights and widths. We argue that the inherent uncertainty in x is due to the width of a crest. In other words each crest may be normalized to 1 and a Shannon’s entropy for each crest calculated. One must then weight the different crests. If one calculates the classical time associated with the base length of crest, this could be used as a weight which in the high energy level limit is associated with P(x)=Cdx/v(x) from (1), but it seems there is no reason to base the weight on time. The uncertainty in x is due to a superposition of exp(ipx)s which are not linked to time. Alternatively, one may use an unnormalized weight of the length of the base as uncertainty is only in x. One may then add a Shannon’s momentum entropy using a(p)a(p)=P(p) where W(x)=Sum over p a(p)exp(ipx).
Quantum free particles are represented by exp(ipx) which we argue is a dynamical probability which maps into a static probability exp(-ipx)exp(ipx) i.e. the usual spatial density. Shannon’s entropy = - Sum over i P(i) ln(P(i)) is often associated with the static probability P(x)=W*(x)W(x) (1), but in (2) we argue that one may introduce a different probability C1 Real(W(x)). In this note we consider both in different examples. In particular we argue that the two different probability definitions are ultimately linked with momentum conservation. The form W*(x)W(x) introduces local nonconstant spatial densities if there are mixed exp(ip1x) and exp(ip2x)’s within W(x). If there are not, this form W*(x)W(x) does not distinguish space i.e. exp(ipx)exp(ipx+ib) is still a constant. Nevertheless interference may still occur which may be linked with entropy. Thus we first examine the case of exp(ipx) + exp(ipx + ib) where b is a constant phase shift. W*(x)W(x)=constant which does not lead to a structure different from exp(-ipx)exp(ipx), yet W(x)=C2{exp(ipx) + exp(ipx+ib)} interfere. On the other hand |cos(px)| and | cos(px)(1+cos(b)) - sin(px)sin(b) | are not the same distributions (one is a parity eigenstate, the other is not) and so we argue there is a different local entropy density. Thus introducing extra information i.e. b changes local density and entropy density if one adjusts the definition of probability used. We next consider W(x)= exp(ip1x)+exp(ip2x). One may also use the probability C3 Real|W(x)|, but here we are interested in an association with conservation of momentum. For a given exp(ipx), exp(-ipx)exp(ipx)=1. We argue that this represents a steady state picture i.e. a kind of equilibrium in which all x points are treated the same. W*(x)W(x) breaks this equilibrium scenario by introducing a term 2 cos( (p1-p2)x ). Thus both local spatial density and local entropy (based on W*W) are different and do not represent the “equilibrium” scenario. We further argue that momentum conservation may be linked with this equilibrium type of density and entropy density at least in the case of the transition amplitude <p1 V(x) p2>. V(x)exp(ipx) yields a sum of exp(ipx)’s, but only exp(ip1x) survives the integral over dx. The others create spatial densities which are not constant and so represent local entropy densities which do not treat all x points the same i.e .is not an equilibrium one. Thus we try to use Shannon’s entropy with two different probability definitions to analyse different quantum interference scenarios and argue that momentum conservation (and possibly others) may be linked with local entropy density in some situations.
The notion of entropy originates historically in classical physics and is intertwined with thermodynamics. Shannon’s entropy: - Sum over i P(i) ln(P(i)), where P(i) is the probability for i, however, seems to apply to any probability, but in quantum mechanics appears to be particularly linked to spatial and momentum entropy (1) i.e. - Integral dx W*(x)W(x) ln[ W*(x)W(x)] and - Sum over p a*(p)a(p) ln[ a*(p)a(p) ] where W(x)= wavefunction = Sum over p a(p) exp(ipx). In this note we speculate that one may create an entropy from the real part of exp(ipx) i.e. cos(px) as this term is associated with probability within a wavelength even though the “center of mass” of the particle may be moving uniformly at v=x/t=constant and W*(x)W(x)=1 or a constant. The latter implies an entropy independent of p but there is uncertainty in a free particle due to its wavelength which is experimentally manifested. We argue there is no reason to limit entropy to x or p considerations. For a bound state, a pattern of crests and troughs of different lengths and heights may be produced and so again there is a pattern of unnormalized probability distributions with different weights. The absolute value may be taken so negative wavefunction values are not an issue. For both the free particle and the bound state two features emerge. First the shape of the distributions is linked to the curvature which in turn is associated with the kinetic energy. Thus the probability distributions which describe the quantum system are governed by kinetic energy. Secondly, a pattern of distributions, both negative and positive, ensures that exp(ipx) is perpendicular to other exp(ip1 x) functions and Wn(x) is perpendicular to other Wn1(x)’s. In other words energy identifies the statistical state which exists throughout space and one cannot confuse the identity of one state with another. Thus they are orthogonal throughout space and probability distributions (linked to entropy) seem to enforce this. The inner product of W*(x) and W(x) is like an inner product of a product probability of a state and its time reversed state. We suggest one should be able to define a Shannon’s entropy expression for both |cos(px)|/C1 from exp(ipx) over a half wavelength where C1 is a normalization factor and |W(x)|/C2 roughly over the length over which it has humps i.e. roughly the classical turning points for an oscillator for example. This entropy is not additive and does not map to classical entropy in the large n (energy level) limit. Furthermore it may not be linked to thermodynamics because it is not linked to x or p. Nevertheless, it might be used to describe uncertainty within the quantity system and equilibrium/stable scenarios which are observed in nature. For example, different Wn(x) states are statistically stable states, but one may knock out the particle and it will be in an exp(ipx) state which is also stable. Furthermore, a photon may excite Wn1(x) to Wn2(x), thus this new entropy seems to be linked to stable states allowed in nature. The “price to pay”, however, seems to be uncertainty in wavelength regions for a free particle and the crest/trough regions for a bound state. We also consider some properties of a similar entropy for bound states.