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.
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.
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).
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.
A process exists in the literature (1) of creating an information term based on Fisher’s information using spatial density P and adding it to a classical Lagrangian based on a velocity field i.e. velocity=dS/dx. Maximization of S and P is performed and a differential equation in terms of spatial density and S is obtained. Finally, it is noted that a change of variables: W= sqrt(p) exp(iS/hbar) may simplify the differential equation so that it becomes the Schrodinger equation with W as the wavefunction. In this note, we consider the wavefunction as appearing “naturally” as a normalization constant when using conditional probability and argue that W is the velocity field. Given that the Schrodinger equation includes a term of the form -1/2m d/dx d/dx W (in 1-D) i.e. the first derivative is related to the average velocity and the second to the ensemble average of p*p. This follows because unnormalized conditional probability f(p,x) is the Fourier basis function i.e. exp(ipx). The second derivative term - 1/2m d/dx d/dx W may be obtained from a Lagrangian containing (d/dxW) (d/dx W), thus formally one may use maximization to obtain the Schrodinger equation at a level using W directly. The maximization form, however, is identical to the “classical translational” form (2) of Fisher’s information. Thus, it seems that the Fourier basis exp(ipx) and the maximization of classical translational Fisher’s information are linked. In this note, we wish to investigate this in further detail. We also use translational properties together with conditional probability to show that f(p,x) where P(p/x)= a(p) f(p/x) / [Sum over p a(p) f(p/x)] is exp(ipx). In particular, we obtain an equation taken from the translation of the conservation of energy equation at point x to x+delta x. This links df(p,x)/dx with -dV/dx (the force). If the force is stochastic and yields “k” momentum hits, -dV/dx= Sum over k kVk(x). We argue that df(p,x)/dx also brings out this k term i.e. is an eigenfunction of k i.e. -idf(p,x)/dx =k f(p,x) yields exp(ipx) as a solution. This f(p,x) is automatically an orthonormal basis (multiplied by an appropriate constant) and helps to show why the “momentum operator” is linked to a “translation operator” i.e. d/dx. The bound state Schrodinger equation is considered in this note.
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.
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.