Classical mechanics defines the concepts of work = Integral dx F = change in kinetic energy and impulse= Integral dt F = change in momentum (vector). We argue that these two concepts are associated with different physical processes and contain different information. For example, a change in momentum is associated with an impulse e.g. the striking (and possibly breaking of a target) and there is a degeneracy in that p=m1v1=m2v2. Kinetic energy, on the other hand, describes a different physical process, namely the “climbing of a potential hill” and one may have different (mo, p) sets yielding the same kinetic energy, just as the same p (same impulse) may be associated with different kinetic energies. Thus there are two distinct physical processes (impulse and interchange of kinetic energy or energy) and each of these are associated with different information. In quantum mechanics, exp(-iEt) describes the time dependent part of a free particle wavefunction (special probability) while exp(ipx) describes spatial. We have argued in previous notes that this probability should be associated with entropy, but in that case there should be an entropy in time and one in space. Given that E and p differ numerically, one needs a reason for the existence of two different entropies. We argue that they are associated with different physical processes and information and so two different entropies may be possible. (We use nonrelativistic expressions here, but the same ideas apply to the special relativistic case.)
Classical mechanics defines the concepts of work = Integral dx F = change in kinetic energy and impulse= Integral dt F = change in momentum (vector). We argue that these two concepts are associated with different physical processes and contain different information. For example, a change in momentum is associated with an impulse e.g. the striking (and possibly breaking of a target) and there is a degeneracy in that p=m1v1=m2v2. Kinetic energy, on the other hand, describes a different physical process, namely the “climbing of a potential hill” and one may have different (mo, p) sets yielding the same kinetic energy, just as the same p (same impulse) may be associated with different kinetic energies. Thus there are two distinct physical processes (impulse and interchange of kinetic energy or energy) and each of these are associated with different information. In quantum mechanics, exp(-iEt) describes the time dependent part of a free particle wavefunction (special probability) while exp(ipx) describes spatial. We have argued in previous notes that this probability should be associated with entropy, but in that case there should be an entropy in time and one in space. Given that E and p differ numerically, one needs a reason for the existence of two different entropies. We argue that they are associated with different physical processes and information and so two different entropies may be possible. (We use nonrelativistic expressions here, but the same ideas apply to the special relativistic case.)
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).
Classical mechanics introduces both the concepts of work and impulse i.e.: Work= F dot dr and Impulse = Fdt= dp. As we have noted previously, there seems to be a tendency in classical mechanics to focus only on the work definition which is linked to energy and time. Thus one solves Newton’s equation to find x(t) or writes a conservation of energy equation. This approach seems to be carried over into thermodynamics as the first law is written as: dE=TdS - dW. We have argued in (1) that in the case of quantum mechanics, both the impulse-momentum picture and work-energy-time picture are needed. (In particular, the one dimensional description of a photon reflecting and refracting cannot be solved without the exp(ipx) impulse scenario.) Here we suggest that the notion of entropy should also be associated with impulses. For example, consider the classical case of a particle moving in a circle with a central inward force. The kinetic energy does not change and work is zero, but the momentum vector changes and so there is an impulse. Furthermore, information changes because one needs to continuously know the momentum coordinates. We suggest that this is an example of impulse being associated with entropy. A second example we suggest is a classical particle bouncing elastically off a wall. (One may argue, however, that these notions of entropy are hand-wavy.) In the quantum world, however, we suggest that impulse plays a major role in creating entropy which is clearly defined. We suggest the example of a particle or photon with a fixed momentum passing through a tiny slit, or n-slits. In each case, one obtains a different pattern on a screen which leads to a different entropy expression. The photon or particle does not change the magnitude of its momentum i.e. there is no energy change or work, but there are impulses which are different in the various n-slit scenarios and lead to different entropies. Thus we suggest that impulses and information associated with them may lead to an increase in entropy. This suggests that the first law of thermodynamics is not a complete description of changes in entropy as it bypasses the idea of impulses and only considers work/energy changes.
Classical mechanics is often concerned with the variables x and t, especially their relationship x(t) in describing a moving particle. Change in p(momentum)= Integral Force(x) dx and Change in KE =Integral dt Force(t) also appear, however, and each of these integrate out one of these variables, so p is linked with x only and KE with t only. Furthermore there is information loss as KE=pp/2m is the same whether the particle moves to the left or right and p=m1v1=m2v2 so with time gone one does not know the velocity. The association of KE with t for a free particle and p with x may be made stronger if one thinks in terms of probabiliites i.e. P(p1,x)P(p2,x) = P(p1+p2,x) and P(p,x1)P(p,x2)=P(p,x1+x2). We argue this leads to P(x,p)=exp(ipx) and a similar result of exp(iEt) or exp(-iEt). One may then note the similarity of these two terms px and +/-Et with the free particle action=Lagrangian*t = -Et+px which holds for both a relativistic and nonrelativistic particle, where v=x/t. Does the loss of information appear in nature? The (p,x) loss i.e. p=m1v1=m2v2 does in a two-slit experiment whose solution does not require time and is calculated using exp(ipx). Thus we argue that interactions are based on p and do have a loss of information. A bound quantum state, however, has interactions with V(x) and so exp(ipx) with its information loss should be present. A bound state, classical or quantum mechanical, however, has the same Etotal at each x, so (Etotal, t) is again a pair with x gone and loss of information means that an energy description (i.e. the time-independent Schrodinger equation) does not distinguish between forward and backwards motion. This is not needed to calculate energy values, but physically this distinct motion may exist, especially for high energy values.
In physics (including classical physics), conservation of momentum is always considered, but one must be more careful with conservation of energy as it may include forms like sound etc. As a result, there is a tendency to focus on conservation of momentum all of the time, and conservation of particle energy only in elastic collisions. In addition to this, one may think of a particle striking a target, in which case two equal momentum (but different energies due to different v speed and mo rest mass) cause the same impact. In the case of an interaction with a potential, however, it is strictly energy considerations which are important classically, i.e one uses p dot p /2m + V(x) = E. Knowing momentum without rest mass is not sufficient. One may consider it unusual that one may consider two different scenarios, one which focuses only on momentum (impulse) or conservation of momentum in collisions while only energy is the focus in classical bound states or motion in V(x). Given the classical focus on momentum, imagine that one wishes to describe all free particles with different momenta as a uniform probability distribution such that a product of two probabilities for p1,p2 is the same as for p3,p4 if p1+p2=p3+p4. (We use 1-dimension for simplicity.). One cannot use P= 1/n because p is absent and n->infinite. In one sense, all particles must have the same probability, but at the same time must contain extra information related to p for the conservation of p part of the problem. The notion of a uniform distribution is independent of conservation of momentum, hence of p momentum. Conservation of momentum requires an explicit presence of p. As a result, there seems to be a paradox. One may, however, resolve this paradox by arguing that a probability is a mathematical way of presenting a subset of the information present. If one argues for a uniform probability, then it does not need to contain p, and one would have 1 as an unnormalized probability for any free particle p. A different probability (representing a different subset of information) which respects all free particles as having the same weight, but includes p can also be constructed, namely exp(i C p). C is a constant introduced for unit issues. In this case, the modulus gives all free particle probabilities with p the same weight, but the presence of p allows one to have conservation of momentum. Given that momentum is the key variable that is conserved in classical physics, one should be satisfied with exp(i C p). One might argue that there is no need to consider energy E. We note, however, that if one has p, E1 and -p, E2 in a frame, then in a frame moving with -v, one has gp+gvE1= p1’ and p2’ = -gp + gvE2. Adding yields gv(E1+E2) and so energy automatically appears and cannot be discarded. Furthermore for p, E1 and -p, E2, exp(i C 0) is exp(i C gv(E1+E2)) as seen in the moving frame and so one is forced to consider conservation of energy together with momentum and include both p and E in the more detailed probability. In other words, special relativity automatically links p and E in a 4-vector, leading to the idea that both are important. We argue that one cannot simply consider exp(i C p) due to special relativity. Both E and p must be considered together and this is a key idea, we argue, because if one wishes to have a Lorentz invariant probability, it must be in E and p, i.e. -Et+px. Instead of having a probability for an interaction with p given by exp(i C p) , one has exp(-iEt+ipx) and we argue that this must have the same value in all frames for a particular interaction. One may note that E and p are, on the one hand, separated allowing for impulse hit versus E related interactions, but that overall the probability for interaction includes both and it is the full -Et+ipx which must be invariant. This further suggests that if one does not artificially force an interaction to occur at the center-of-mass, then x,t and x+hbar/p, t+hbar/E are equally good points for the interaction. We argue that there is no need to force x=vt because exp(-iEt+ipx) * exp(iEt+ipx) already shows that one has uniformity, not only for any p,E, but also for any x, t, as there are now four variables. Thus, there is no reason to enforce x=vt in exp(-iEt+ipx), when considering an interaction, we argue. This means that one has two possible interactions at different x,t values, i.e. x,t and x+hbar/p, t+hbar/E. If one is unable to follow the interaction in time, but may follow it in space as in the case of 2-slits about hbar/p apart or for a potential V(x), then one may use the exp(ipx) part of the interaction probability piece by itself. Thus, we argue that special relativity does not allow one to treat p independently of E. For example, for p values that are the same in one frame, but have E1, E2, will be different in another frame. One must consider E and p together and so instead of creating a probability for p alone, one must create one for E,p and then impose Lorentz invariance which leads to different x,t pairs yielding the same probability value for an interaction, i.e. free particle quantum mechanics.
Quantum mechanics is concerned with measurement and we suggest that in a bound problem impulses due to momentum are the form of measurement which occurs in x space. The Lorentz invariant A= -Et+px, which is also the free particle relativistic or nonrelativistic action with v=x/t, associates energy = pp/2m with time and p with x. Creating eigenvalue equations yields id/dt exp(-iEt) = E exp(-iEt) and -id/dx exp(ipx) = p exp(ipx). P (momentum) is associated with physical impulses and with its own two dimensional probability distribution exp(ipx). E is associated with t and has its own distribution in time exp(-iEt) which implies that the same E holds for each x point without any notion of x probability associated with it. In classical physics, a momentum p exists in a tiny dx (dx→0). For exp(ipx), a p impulse may occur within hbar/p with different probabilities, so one cannot restrict the measurement of p to a tiny dx unless dx is of the order of hbar/p. Nevertheless energy exists at all x points. Given that p measurements occur through specific impulses at different x points with exp(ipx) probabilities, one must calculate an average energy using {Sum over p pp/2m a(p) exp(ipx)} / {Sum over p a(p)exp(ipx)} using the probability associated with the measurement of p in space i.e. exp(ipx). This adds to V(x) to yield a constant E at each x point. We suggest that weighted exp(ipx)s are used to calculate average energy at each x, because it is through impulse measurements that one determines p at different x values and hence the average energy. One may note that one does not calculate {Sum over p a(p)exp(ipx) p } / {Sum over p a(p)exp(ipx)}. An impulse hit is associated with a certain probability and this measurement in x yields information about the corresponding energy pp/2m. Each impulse hit also yields information about a time probability exp(-i ppt/2m), but only if the particle is knocked out of the bound system i.e. only if pp/2m is the only energy present, which is not the case for a single particle bound system.
In previous notes, we have argued that one may obtain exp(ipx) (the spatial free particle probability) from the maximization of P1(x) ln(P1(x)) + i px P1(x), and a similar expression for P2(t). This yields exp(-iEt+ipx). If one integrates i Integral px exp(ipx) dx from px=0 to 2*3.14/p, one obtains i 2*3.14/p as a spatial entropy and i2*3.14/E as a temporal one associated with the complex probability exp(-iEt+ipx). If one considers classical motion of a free particle in space, then there is no uncertainty in time or space and so one might create a classical level entropy of: -E dt + p dx, where dt=hbar/E and dx=hbar/p.This yields -Edt+pdx=0 and one might equate this with a classical entropy dS as done in (1). This overall classical entropy is different from the spatial and temporal wave entropies dx and dt. If one does not wish to work at the classical level, one might try to use the hbar/p, hbar/E entropies in a problem requiring exp(-iEt+ipx), such as 1-D reflection-refraction at an n1-n2 index of refraction junction (x=0). In such a case, it seems that one does not use entropies, but rather their reciprocals which indicate the amount of resolution information present, i.e. L/dx = Lp/hbar. As a result, conservation of E and p may be reformulated in terms of a balance of resolution information which is the reciprocal of dx, i..e p (in the spatial case). For example, a particle at rest may break into two pieces with p and -p. The resolution is proportional to p initial =0 and after one has p and -p (i.e. conservation) of momentum which may be argued to be equivalent to conservation of resolution information. Using the notion of a balance of the reciprocal of entropy (i.e. p which is linked to conservation of momentum), one might interpret: Ap - Bp = C p2 where p2=n2 p in terms of resolution and not momentum conservation, i.e. A-B = C p2/p where p2/p = n2 is the greater relative resolution of the p2 photon. The idea of both a wave entropy (both temporal hbar/E and hbar/p) and its reciprocal resolution information seem to arise as well an overall classical entropy -Edt+pdx = 0 if dt=hbar/E and hbar/hp = 0. dt=hbar/E and dx= hbar/p We have argued in previous notes that one may obtain: dt=hbar/E and dx=hbar/p ((1)) as independent dx and dt fluctuation from the Lorentz invariant. One may show that -Et+px is equivalent to the classical relativistic or nonrelativistic action Lt if Lt=.5movv t (nonrelativistic) or Lt = -mo sqrt(1-vv/cc) t (relatvistic) and x/t=v ((2)) The reason that one recasts Lt as -Et+px is to use relativistic variables. The notion of an action remaining unchanged under a fluctuation delta x(t) , i.e. at t one considers x1 instead to x=vt, leads to: 0 = -E delta t + p delta x = 0 which has a solution of ((1)). In (1), however, the notion of an overall entropy is introduced such that: dS = 0 = -E delta t + p delta x ((3)) Given that one really has a probabilistic scenario (free particle quantum mechanics) associated with ((1)), i.e. exp(-iEt+ipx) = probability which ensures ((1)) and also spatial and temporal invariance by the unit modulus (less information than exp(-iEt+ipx)), then one might consider the notion of an overall entropy.