In classical mechanics, a free particle moves through x and t with dx/dt=v=constant. Given that x and t may be shifted, it is dx and dt which are important. For fixed dx, dt(x) yields information about the “probability” for the particle to be in dx so to speak. Thus one does not really distinguish between deterministic and probabilistic descriptions in this scenario. A problem with the probabilistic picture, however, is that different v (or p values) have the same constant probability P(x)=constant for free motion. In other words, P(x)=constant does not distinguish either the magnitude or direction of v. Thus, information is lost. Is it possible to restore this information and still maintain a probabilistic picture? The answer, given by free particle quantum mechanics, seems to be yes. The probabilistic approach is to assign a probability P(x) to each x point. This probability must treat each x point as the same and yet give information about v or p. (We use p because it represents impulse and is linked to special relativity i.e. a 4-vector.) The key idea we argue is to consider the first order change in x space because one assigns a P(x) to each x and different p values are linked with different motion from one x to another.. This is the way to distinguish the direction and magnitude of p, but nothing else we argue. In such a case: d/dx f(px) = Constant p f(px) ((1)) ,we argue, where f is an unknown function. We show that this function, which is odd in x, cannot be real i.e. one needs at least a pair of real functions comprising f. We note that the derivative only gives information about p, thus df/dx /f = p. We argue this implicitly means each x point has the “same weight” even though the pair of functions f are not constants. We next analyze the behaviour of the two pairs of functions comprising f and conclude that in order for d/dx d/dx f = pp f, each function of the pair must be converted into itself. Thus: d/dx d/dx f1 = -pp Constant f1 and a similar equation for f2 where f1,f2 are the pair. These, however, are periodic equations. Thus the implicit idea of each x point carrying the same weight is equivalent to f1(x) and f2(x) being separately periodic. This is equivalent to constant modulus problem. How is this linked to Shannon’s entropy or Fisher information extremization? We argue that ((1)) means that ln(P) does not carry enough information in this situation, while for a Maxwell-Botlzmann gas it does. It is d/dx ln(P) which carries information about p and its direction. If one wishes to create a function which does not distinguish the sign of momentum, it still has to incorporate the information of motion in one direction i.e. d/dx ln(P) d/dxln(P). One may take the average of this (called Fisher’s information) instead of the average of ln(P) in Shannon’s entropy. To link with the above ideas, P(x)=f1(x)f1(x) if one considers a bound problem with infinite potential walls. Then Fisher’s information becomes df1/dx df1/dx. The extremization of Fisher’s information subject to Integral f1(x)f1(x)= 1 leads to the periodic condition described above. This means, we argue, that for the case of pp, one cannot sense any change in either f1(x) or f2(x). f1(x) and f2(x) are used to indicate differences to an odd power of d/dx and so for an even, there is no informational change at the f1(x) and f2(x) level. This we argue is what Fisher’s information extremization with Integral f1 f2 =1 gives i.e. periodicity.
In classical mechanics, a free particle moves through x and t with dx/dt=v=constant. Given that x and t may be shifted, it is dx and dt which are important. For fixed dx, dt(x) yields information about the “probability” for the particle to be in dx so to speak. Thus one does not really distinguish between deterministic and probabilistic descriptions in this scenario. A problem with the probabilistic picture, however, is that different v (or p values) have the same constant probability P(x)=constant for free motion. In other words, P(x)=constant does not distinguish either the magnitude or direction of v. Thus, information is lost. Is it possible to restore this information and still maintain a probabilistic picture? The answer, given by free particle quantum mechanics, seems to be yes. The probabilistic approach is to assign a probability P(x) to each x point. This probability must treat each x point as the same and yet give information about v or p. (We use p because it represents impulse and is linked to special relativity i.e. a 4-vector.) The key idea we argue is to consider the first order change in x space because one assigns a P(x) to each x and different p values are linked with different motion from one x to another.. This is the way to distinguish the direction and magnitude of p, but nothing else we argue. In such a case: d/dx f(px) = Constant p f(px) ((1)) ,we argue, where f is an unknown function. We show that this function, which is odd in x, cannot be real i.e. one needs at least a pair of real functions comprising f. We note that the derivative only gives information about p, thus df/dx /f = p. We argue this implicitly means each x point has the “same weight” even though the pair of functions f are not constants. We next analyze the behaviour of the two pairs of functions comprising f and conclude that in order for d/dx d/dx f = pp f, each function of the pair must be converted into itself. Thus: d/dx d/dx f1 = -pp Constant f1 and a similar equation for f2 where f1,f2 are the pair. These, however, are periodic equations. Thus the implicit idea of each x point carrying the same weight is equivalent to f1(x) and f2(x) being separately periodic. This is equivalent to constant modulus problem. How is this linked to Shannon’s entropy or Fisher information extremization? We argue that ((1)) means that ln(P) does not carry enough information in this situation, while for a Maxwell-Botlzmann gas it does. It is d/dx ln(P) which carries information about p and its direction. If one wishes to create a function which does not distinguish the sign of momentum, it still has to incorporate the information of motion in one direction i.e. d/dx ln(P) d/dxln(P). One may take the average of this (called Fisher’s information) instead of the average of ln(P) in Shannon’s entropy. To link with the above ideas, P(x)=f1(x)f1(x) if one considers a bound problem with infinite potential walls. Then Fisher’s information becomes df1/dx df1/dx. The extremization of Fisher’s information subject to Integral f1(x)f1(x)= 1 leads to the periodic condition described above. This means, we argue, that for the case of pp, one cannot sense any change in either f1(x) or f2(x). f1(x) and f2(x) are used to indicate differences to an odd power of d/dx and so for an even, there is no informational change at the f1(x) and f2(x) level. This we argue is what Fisher’s information extremization with Integral f1 f2 =1 gives i.e. periodicity.
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.
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.
The extremization of Fisher’s information subject to constraints usually containing V(x), the classical potential, appears in the literature (1), (2). V(x), however, is a concept introduced by Newton and associated with conservation of energy for a conservative potential. We argue the appearance of V(x) in an extremization approach to obtain an “equation” bypasses the already existing conservation of energy equation. If V(x) is used, we argue that one cannot dispense with conservation of energy and that it should be written a priori. If that is the case, what does extremization of Fisher’s information guarantee? In (1) a probability distribution P(x) is set equal to W(x)W(x) its square root. In (2), a complex function is defined: W(x) = sqrt(density(x)) exp(i S/hbar) where S is the classical action. Here density(x)=P(x). In both cases, the square root of the density appears. The problem, we argue, is that for a free particle with density(x)=P(x)=1, there is no Fisher’s information to extremize, yet the free particle is a key part of quantum mechanics. Its probability (square root probability) is the solution of an eigenvalue problem of the translation operator -id/dx and it is consistent with Newtonian mechanics i.e. conservation of energy with E= pp/2m. From this one may create an ensemble W(x) = Sum over p a(p) exp(ipx). The average kinetic energy is -1/2m d/dx/ dW/dx / W which happens to be the extremized value of Fisher’s information for P(x)=W(x)W(x). In other words, we argue that conservation of energy must occur and that one needs a solution for a free particle and this does not follow from Fisher’s information extremization. Extremization of Fisher’s information produces the form d/dx dW/dx which is taken to be linked to <pp> which already follows from -d/dx d/dx exp(ipx). One argument made in the literature (3) is that Newtonian conservation of energy also follows from Fisher’s information extremization i.e. the Lagrangian for a free particle .5m dx/dt dx/dt is of the form of a Fisher’s information with xx = probability (which is unusual). The strategy, it seems, is that one may apply extremization of Fisher’s information directly to a quantum problem by assuming a P(x)=W(x)W(x) and use d/dx instead of P=xx with d/dt. An issue, as noted, is that for a free particle W(x) exists, but extremization of Fisher’ information does not exist. Even if E=pp/2m (nonrelativistically) and this follows from the extremization of Fisher’s information, W(x)=exp(ipx) does not, but it is consistent with E=pp/2m. The reason for this consistency is that it contains p (momentum) which by its very definition must ensure E=pp/2m (nonrelativisitcally). Fisher information extremization, however, yields the bound state solution i.e. may be used to derive the Schrodinger equation. In (1) this is done for a bound state, in (2) for the more general case, but (2) defines W(x) = sqrt(density) exp(iS/hbar) where S= the classical action. For a free particle, the definition of W(x) is exp(-iEt+ipx) which is the solution of the free particle problem without using any equation. Thus the definition is the solution itself for the free particle case in (2). It seems that Fisher’s information, which uses density(x), does not include a scheme for describing motion in one direction in space without time present. exp(ipx) i.e. cos(px) and sin(px) distributions in two dimensions allow for this. This motion is associated with the translation operator d/dx which we argue is key in solving for the square-root probability of a free particle in keeping with E=pp/2m, which (3) argues follows from the extremization of Fisher’s information. Thus, even if one accepts the argument that Fisher’s information is responsible for the energy conservation equation, using Fisher’s information = Integral dt dx/dt dx/dt constant, does that allow one to immediately switch to a problem with P(x)=W(x)W(x) and the derivative d/dx and apply Fisher’s extremization there, we ask? We argue that there is an implicit assumption that: C1 dx/dt dx/dt = C2 d/dx dW/dx / W where C1 and C2 are constants This implies that for a free particle which treats each x point the same i.e. P(x)=constant, W(x)W(x) cannot be P(x). It also implies that d/dx dW/dx = pp W ((1)) where p=mov nonrelativistically. ((1)), however, is the solution to the free particle problem without performing any Fisher information extremization i.e. exp(ipx) is a solution. This solution may be used to create an ensemble W(x)=Sum over p a(p)exp(ipx) which may be used in a conservation of average energy equation. Thus we argue that Fisher extremization used to create the Schrodinger equation is based already on the idea that W(x)=exp(ipx) for a free particle and simply produces the conservation of energy equation which is already known to exist from Newtonian mechanics or Fisher’s information extremization which is argued to be the same as Lagrangian variation in (3).
Note March 9, 2023 At the same time that Fisher's information is linked with variance of an estimator, it is also linked with a classical notion of pressure for a particle in a box with infinite potential walls. Thus minimizing the variance subject to Integral P(x)dx=1 may also be thought of as extremizing pressure with respect to Integral P(x)dx=1. We argue that the cos(px), sin(px) distributions are linked with the impulse p and at the same time the particle moves with an average speed p/m (nonrelativistically). A quantum free particle is represented by the free particle wavefunction exp(ipx). This consists of two periodic distributions (in 2 dimensions) such that their modulus is 1. sin(px) and cos(px) are a possible solution set. sin(px) and cos(px) are, however, linked to a unit radius which rotates in 2 dimensions. Alternatively one may note that d/dx cos(px) = -p sin(px) and d/dx sin(px) = p cos(px). Thus one may determine the form of a free particle “probability” using the notion of periodicity, 2 dimensions and a constant modulus. No concept of extremization is needed. It is interesting to note, however, that d/dx d/dx cos(px) = pp cos(px) and d/dx d/dx sin(px) = pp sin(px). Each distribution sin(px), cos(px) by itself represents a solution of extremization problem, namely the extremization of Fisher’s information defined by I= Integral dx P(x) {d/dx ln(P)} {d/dx ln(P)} where P(x)=W(x)W(x) and W(x)=cos(px) or sin(px). Then I = Integral dx dW/dx dW/dx. Extremizing with respect to the constraint Integral P(x) = 1 yields: d/dx d/dxW = constant W. cos(px), sin(px) and exp(ipx) are solutions of this extremization equation. In order to have a solution which treats all x points the same in a certain sense, one may choose exp(ipx). The point is that even for a free particle, the two distributions involved are linked with extremization of Fisher’s information (with Integral W(x)W(x)dx=.5Length), although one may obtain exp(ipx) with no extremization and no knowledge of Fisher’s information. Why should Fisher’s information be extremized? Fisher’s information is a statistical quantity which appears in the Cramer-Rao bound: var(estimator) >= 1/ I. One may take the inverse of this equation and consider variance(estimator) to be linked to a spread or uncertainty in x. If one wishes to minimize this, one may maximize I subject to constraints. Thus there seems to be a purely statistical approach (i.e the extremization of Fisher’s information) even for a free particle. This extremization approach, however, is linked directly to the fact that one has two probability distributions in a 2-dimensional object with a constant modulus. This does not have the appearance of any extremization, but is in fact equivalent to one for each distribution function. In fact, we argue that if P(x)=W(x)W(x) with Integral P(x)dx=1, then if W(x) is cyclical/periodic i.e. d/dx d/dx W = constant W, then W may be obtained through the extremization of Fisher’s information subject to the constraint Integral P(x)dx =1. Thus periodicity and extremization are linked, but in a somewhat unusual manner.
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).
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.