Addendum Nov. 27, 2022. In Newtonian mechanics, one may slice the x axis into equally spaced dx with a particle being in one dx at time t1 and in the next at t2. In this note we suggest describing directional constant motion with a probability density P(p,x). If the particle moves out of x we suggest that;. P(p,x) dP(p,x)/dx=0 or P(p,x) must be at least two dimensional i.e. P(p,x) = F(p,x) + iG(p,x) such that F dF/dx + G dG/dx =0. This is equivalent to: FF + GG = constant for any x which shows all x points being the same even though P(p,x) is a function of x which distinguishes direction. Finally, with P(p,x)P(-p,x) = constant because this represents no motion and so no distinguishing density is needed one has P(p,x) = exp(ipx). In Part I we argued that directional motion should be associated with a different spatial probability densities. dtime(x)= dx/velocity(x), however, taken as proportional to a density in the literature (1) does not distinguish. We suggested that Newton’s first law is equivalent to Fermat’s least time principle which for reflection from a fixed mirror is equivalent to d/dx (px) x = (px). (px)x, however, may be written as -i ln[exp(i (px) x) ] which has the form of constant multiplied by ln probability in information theory. Thus we suggested in Part I that exp(ipx) is a kind of probability density which distinguishes direction in motion. We also noted that exp(-ipx) exp(ipx) = 1 which is linked to the dt(x) density for constant motion and that Integral exp(-i p1 x) exp(ipx) dx = constant delta (p1, p). In this note we wish to examine in more detail the notion of x, t and wavelength. exp(ipx) depends on x and p=mo dx/dt or (p=mo v / sqrt(1-vv/cc) with v=dx/dt). X and t are continuous variables linked to the idea of invariance in space for constant motion. exp(ipx) and exp(-iEt) on the other hand define built-in particle space and time intervals for a given p,E. Thus exp(ipx) defines constant motion in space without the notion of time (i.e. t and x are independent within a built-in ruler gradation proportional to 1/p (two humps-one positive one negative). This leads to a contradiction if the particle is tiny compared to 1/p. On the one hand the particle has smooth motion v=x/t, but on the other a “created” particle length unit must be defined suggesting a probability density over the length proportional to 1/p. To allow for the two, one must assume a constant average motion over the length 1/p as measured by an external ruler and clock and at the same time abandon the notion of following the particle in x,t (infinite resolution) within the length proportional to 1/p even though its probability distribution exp(ipx) is well defined at each x. Thus the identity of the particle is defined by the probability exp(ipx) within 1/p regions even though a direct hit with another particle within this region involves p. Furthermore given that the spatial probability defines the particle it is not enough to state that for two particles one has p1 and p2, the spatial probabilities must distinguish by being orthogonal i.e. having a 0 dot product: Integral dx exp(-i p1 x) exp(ip2 x). We argue in the note that not only is particle identity described by exp(ipx) and exp(-iEt), but also conservation of energy and momentum in various problems. In Part I we suggested considering the product of P(p,x) and P(-p,x) in order to obtain a constant value linked to spatial variance throughout x even though exp(ipx) and exp(-ipx) are not spatially invariant. We first note that exp(ipx) and exp(-ipx) are the time reversal of each other and then consider exp(ipx) AND exp(-ipx) together within a little region of the order of 1/p. exp(ipx)exp(-ipx)= 1, but each moves with its own density profile expx(ipx) and exp(-ipx). Thus constant density ignoring the direction of motion seems to be identical to the product of the time reversed density states. Even though constant motion is only a translation when viewed on average, we suggest that directional motion must be linked with skewed density and be based on internal distances and clocks. This contradicts the idea of a smooth translation and leads to an unusual probability description within hbrar/p wavelength units. The catch seems to be, however, that one needs to introduce two dimensions in order to demonstrate the skewness linked with directional motion. By creating a spatial wavelength, the region within is one in which one may not follow the particle in x and t i.e. x and t are independent as in the classical action: A = -Et+px. dA/dx partial = p, but one cannot write this unless x and t are independent and only represent constant v on average i.e. x/t. Finally the wavelength and interval in time, which are physical, are directly linked to ensuring conservation of energy and momentum in the direction of an invariant variable. We argue this by consider a two dimensional reflection problem i.e. Fermat’s principle.
Addendum Nov. 27, 2022. In Newtonian mechanics, one may slice the x axis into equally spaced dx with a particle being in one dx at time t1 and in the next at t2. In this note we suggest describing directional constant motion with a probability density P(p,x). If the particle moves out of x we suggest that;. P(p,x) dP(p,x)/dx=0 or P(p,x) must be at least two dimensional i.e. P(p,x) = F(p,x) + iG(p,x) such that F dF/dx + G dG/dx =0. This is equivalent to: FF + GG = constant for any x which shows all x points being the same even though P(p,x) is a function of x which distinguishes direction. Finally, with P(p,x)P(-p,x) = constant because this represents no motion and so no distinguishing density is needed one has P(p,x) = exp(ipx). In Part I we argued that directional motion should be associated with a different spatial probability densities. dtime(x)= dx/velocity(x), however, taken as proportional to a density in the literature (1) does not distinguish. We suggested that Newton’s first law is equivalent to Fermat’s least time principle which for reflection from a fixed mirror is equivalent to d/dx (px) x = (px). (px)x, however, may be written as -i ln[exp(i (px) x) ] which has the form of constant multiplied by ln probability in information theory. Thus we suggested in Part I that exp(ipx) is a kind of probability density which distinguishes direction in motion. We also noted that exp(-ipx) exp(ipx) = 1 which is linked to the dt(x) density for constant motion and that Integral exp(-i p1 x) exp(ipx) dx = constant delta (p1, p). In this note we wish to examine in more detail the notion of x, t and wavelength. exp(ipx) depends on x and p=mo dx/dt or (p=mo v / sqrt(1-vv/cc) with v=dx/dt). X and t are continuous variables linked to the idea of invariance in space for constant motion. exp(ipx) and exp(-iEt) on the other hand define built-in particle space and time intervals for a given p,E. Thus exp(ipx) defines constant motion in space without the notion of time (i.e. t and x are independent within a built-in ruler gradation proportional to 1/p (two humps-one positive one negative). This leads to a contradiction if the particle is tiny compared to 1/p. On the one hand the particle has smooth motion v=x/t, but on the other a “created” particle length unit must be defined suggesting a probability density over the length proportional to 1/p. To allow for the two, one must assume a constant average motion over the length 1/p as measured by an external ruler and clock and at the same time abandon the notion of following the particle in x,t (infinite resolution) within the length proportional to 1/p even though its probability distribution exp(ipx) is well defined at each x. Thus the identity of the particle is defined by the probability exp(ipx) within 1/p regions even though a direct hit with another particle within this region involves p. Furthermore given that the spatial probability defines the particle it is not enough to state that for two particles one has p1 and p2, the spatial probabilities must distinguish by being orthogonal i.e. having a 0 dot product: Integral dx exp(-i p1 x) exp(ip2 x). We argue in the note that not only is particle identity described by exp(ipx) and exp(-iEt), but also conservation of energy and momentum in various problems. In Part I we suggested considering the product of P(p,x) and P(-p,x) in order to obtain a constant value linked to spatial variance throughout x even though exp(ipx) and exp(-ipx) are not spatially invariant. We first note that exp(ipx) and exp(-ipx) are the time reversal of each other and then consider exp(ipx) AND exp(-ipx) together within a little region of the order of 1/p. exp(ipx)exp(-ipx)= 1, but each moves with its own density profile expx(ipx) and exp(-ipx). Thus constant density ignoring the direction of motion seems to be identical to the product of the time reversed density states. Even though constant motion is only a translation when viewed on average, we suggest that directional motion must be linked with skewed density and be based on internal distances and clocks. This contradicts the idea of a smooth translation and leads to an unusual probability description within hbrar/p wavelength units. The catch seems to be, however, that one needs to introduce two dimensions in order to demonstrate the skewness linked with directional motion. By creating a spatial wavelength, the region within is one in which one may not follow the particle in x and t i.e. x and t are independent as in the classical action: A = -Et+px. dA/dx partial = p, but one cannot write this unless x and t are independent and only represent constant v on average i.e. x/t. Finally the wavelength and interval in time, which are physical, are directly linked to ensuring conservation of energy and momentum in the direction of an invariant variable. We argue this by consider a two dimensional reflection problem i.e. Fermat’s principle.
Newton’s second dp/dt = Force where p is momentum is equivalent to pp/2m + V(x) =E for a conservative force= -dV/dx with p=mo v(x), v being velocity. In general one may divide the x axis into equally spaced dx units with dt(x) (i.e. time spent) differing in each unit. A main point we wish to make is that dt(x) is the same whether the particle moves to the right or to left. Thus using dt(x) as a kind of density (proportional to C/v(x)) as done in (1) means this spatial density does not distinguish between motion to the right or left. This is one kind of spatial density we argue, one that considers only the magnitude of velocity. Heuristically, however, one might argue that motion to the right or left involves a skewed spatial density in the direction of motion. This we argue represents a second kind of spatial density which follows from Newton’s first law and is associated with quantum mechanical motion. Ideally the two should be linked. Newton’s first law indicates that a particle at rest remains at rest and one in constant motion in a direction continues in this state unless acted on by force. This law is usually not associated with a mathematical equation and so a spatial density is not derived. In (2), however, we argue that Newton’s first law is associated with invariance in a spatial direction if there is no force in that direction, for example x. In such a case, d/dx is the translation generator in that direction and momentum is kind of “information”. We argued in (2) that one may write d/dx f(x) = p to describe this with the simplest solution being f(x)= px for constant p. We further argued that this is in fact linked to Fermat’s least time principle for light. Time =distance/speed of light, but momentum and displacement are parallel. Thus for a two dimensional problem: p dot r = (px)x + (py)y = p distance = p sqrt(xx+YY) and p sqrt( (L-x)(L-x) + YY) for two rays, one being the incident and the other the reflected. Given that there is conservation of momentum in the x direction one may write (px) incident = (px) reflected or d/dx (p distance incident) - d/dx (p distance reflected) = 0 which is Fermat’s principle for reflection with one index of refraction. For refraction p1=pn1 and p2=pn2, but (p1x) = (p2x). f(x)=px satisfies d/dx f(x) = p. This expression depends on x, but also on the direction of motion i.e. one may have p or -p. In (2) we further argued that this expression may be written in terms of: i px = ln(exp(ipx)) with exp(ipx) being periodic, but neither growing or tending to a constant zero for a constant p. As a result we identified exp(ipx) with a second type of spatial density or probability, one which accounts for the direction of motion. exp(ipx) is also associated with a quantum mechanical description of a free particle. Thus Newton’s first two laws give rise to two different spatial densities, but they are related. In particular, the modulus of exp(ipx) equals constant which is P(x) or spatial density from Newton’s second law. In this note, we start with the assumption that a velocity (not velocity magnitude) spatial density should exist and examine what properties of such a density/probability arise.
In part I we argued that Newton’s first law implies that constant momentum (i.e. momentum conservation) is associated with spatial invariance i.e. a lack of information in a certain direction. This is equivalent to stating there exists no force in that direction. We argued that one may apply the generator for translation in the spatial direction i.e. d/dx for x and apply it to a function f(x) such that df(x)/dx = (px). The simplest form is f(x)= (px)x. We also argued that it is precisely this function which appears in Fermat’s least time principle because time = distance/speed and one may multiply time by |p| = momentum magnitude. Given that momentum and displacement are parallel vectors for a light ray, p dot r = p distance = (px)x + (py)y for a two dimensional problem (e.g. reflection or refraction). “Distance d” is the key function to use because dd= xx + yy implies that for fixed y, d d/dx = x/d = sin(theta1) (angle measured from y axis) so p d d/dx = (px) which is a conserved quantity if x is invariant. Thus conservation of momentum for two rays is equivalent to extremization. Extremization, we argue, suggests there may be a built-in physical mechanism which allows a free particle to correct itself if perturbed slightly just like a Maxwell-Boltzmann gas which extremizes entropy subject to constant average energy does. This built-in mechanism should be linked to motion in space. Writing (px)x as ln(exp(ipx), one not only obtains an eigenvalue equation i.e. -d/dx exp(i (px) x) = (px) exp(i px x), but the notion of a relative change arises if exp(i px x) is interpreted as a kind of probability. A relative change is natural because an absolute one ignores the idea that probability may differ with x even though (px) is constant. Thus from the idea of invariance in space found in Newton’s first law, which is equivalent to conservation of momentum in that direction, the equation d/dx f(x) = (px) leads directly to -id/dx exp(i px x) = (px) exp(ipx x) where exp(i px x) represents a physical probability type of wave which stablilizes the constant (px) motion because this motion is equivalent to extremizing time or |p| time or p dot r / speed of light in Fermat’s principle. In other words, the quantum mechanics of a free particle is associated with stabilizing wave behaviour exp(ipx x) which was not seen experimentally at the time of Newton or Fermat, but seems to be part of their formalisms, with Fermat’s principle derivable from Newton’s first law.
Newtonian mechanics describes forward and backwards motion by x=vt and x= -vt. In order to understand this motion, one needs to create dx and dt units such that a particle with v moves from x,t to x+dx, t+dt. X and t are necessarily coupled and dx and dt are not zero (even in the dx->0, dt->0 limit). They are just very small. For a particle with v=0 (at rest), x and t are decoupled. Given that the particle may be at rest at any x in L, it is natural to define a probability P(x)dx = dx/L, even in Newtonian mechanics. For x=vt and x=-vt, however, x and t must be decoupled, but given different initial points, one can have different sets of x,t measurements. The key point we stress is that motion is linked with dx and dt and these do not have a fixed numerical value in Newtonian mechanics, but nevertheless the concept exists. In principle, Newtonian mechanics is probabilistic because one does not really know what occurs within dx and dt. It is this essential probability, we argue, that is manifested in quantum mechanics by defining actual dx and dt values, although in essence it is already present in Newtonian mechanics. Given that probability P(x) is well-defined and applies to v=0, one may postulate a dynamic probaiblity P1(v) as we have done in previous notes. If one wishes probability to be Lorentz invariant, one may write P1(-Et+px). At first, based on Newton’s x=vt and x=-vt, one would expect that x and t must be coupled through x=vt. The problem with this is that one may form an AND (product) scenario in probability theory. Furthermore, P1(-Et+px) does not contain initial point information.If one writes P1(v)*P1(-v), one should obtain P(x)=1/L because there is no motion. P(x) decouples x,t and so P1 must do the same. In other words, P1(-Et+px) contains decoupled x,t. This differs from Newton’s coupled x(t). If P1 does not increase or decrease indefinitely in x,t, then a solution is P1(-Et+px) = exp(-iEt+ipx). The question we ask is: How does exp(-iEt+ipx) contain more information than x=vt, x=-vt from the Newtonian scenario? In the Newtonian picture, in order to have a v and -v particle at the same x,t, they must be at different x,t values before and after. This suggests a disconnect in dx (and dt) for x and t. This disconnect is present in exp(ipx) and exp(-ipx), at least in the sin(px) term. The only new information contained in exp(-ipx-iEt) is that dx=hbar/p and dt=hbar/E. The uncertainty in x and t (due to dx and dt) is already present in Newtonian mechanics, but one ignores it by considering dx->0 and dt->0. The notion that dx and dt may be finite does not arise experimentally in classical calculations, but becomes apparent on atomic and nuclear scales. As a result, we argue that the probabilistic nature of a moving particle exp(-iEt+ipx) is already present in Newtonian mechanics, but one does not know how to fix dx and dt. As a result, it is not really the case that Newtonian mechanics is so greatly different from quantum free particle mechanics. The dx and dt lengths may be associated with uncertainty even in Newtonian mechanics and if this is considered, then Newtonian mechanics would not be dissociated with 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction interface.
Newtonian mechanics (from the 1600s) defines a quantity called momentum p in terms of mv where m is mass and v=dx/dt with dx and dt tending to zero. Newton’s first law is equivalent to conservation of momentum in the absence of a force. The second demonstrates how p changes under a force. If one has equal action/reaction then overall momentum is conserved. As a consequence the idea of a “wave” associated with p does not arise. The main focus is on the idea of force. Fermat’s Least Time Principle for light (1600s), however, is based on minimizing the overall time traveled by light which reflects or refracts. Young’s interference experiments did not occur until the very early 1800s and Newton considered light to be a particle at this time. In fact light bounces off a foil creating an impact like a particle. Fermat’s principle does not directly make use of momentum or its conservation. It is focused on minimizing time. To do so, however, one must take a derivative with respect to a certain variable and set it equal to 0. This variable is the spatially invariant one in a problem. For example consider reflection (stationary or moving mirror) or refraction in two dimensions x,y. Let the mirror or medium face lie along the x axis at y=0. There exists a constraint on y and given a fixed starting point one cannot change y. This leaves x to be varied because it is invariant. Thus the extremization of time is really a statement of invariance of a spatial variable. In other words, Fermat’s principle is linked to the physical variables t and x while Newton’s focuses on p (momentum). This invariance in turn gives rise to conservation of momentum in the direction of the invariant variable i.e. (px). There is still the idea of Newton’s change in momentum as (py) abruptly changes in refraction as it hits the interface of media. The point we wish to make is that from Fermat’s least time principle one sees that conservation of momentum in the absence of a force is completely intertwined with invariance in space. The two cannot be separated in this approach and there is nothing unphysical about t or x. In fact, one may create a function A= -Et + (px)x + (py)y such that dA/dx partial = px = conserved quantity. In this function (px) and x are on the same footing. One does not have one without the other. This idea is carried over into special relativity as -Et+p dot r is a Lorentz invariant. Given that p momentum is physical (it manifests itself in impulse hits) and p and x are inseparable in (px)x we suggest there should exist a physical manifestation in x i.e. in space linked to the maintenance of conservation of momentum in this direction even in the absence of a force. This we argue leads to the physical idea of a wavelength which follows from A=-Et+px with A=0 in all constant velocity frames leading to a frequency proportional to 1/E and wavelength to 1/p. We have discussed these ideas in previous notes, but the main point we wish to make here is that equivalence of spatial invariance in the x direction in Fermat’s principle with conservation of momentum in the same direction suggests that if p is physical, the spatial invariance must also be manifested physically. It is not simply a mathematical convenience to link t to x to px, we argue. Furthermore this wavelength should be linked to establishing conservation of momentum as it does in quantum mechanics where the orthogonality of exp(ipx) i.e. Integral exp(-i p1 x) exp(ip x) dx automatically ensures p1=p without imposing momentum conservation a priori as in Newton’s first law. Both Newtonian mechanics and Fermat’s principle suggest that p may change abruptly due to a force or an immediate change in medium. The idea of a physical wavelength, however, suggests there may be interactions completely different from an abrupt change in p due to an impulse. In particular there may be interference or Bohm-Aharonov interactions. Newtonian Mechanics
In (1) we argued that one may derive Newton’s second law (which is equivalent to an energy conservation equation) by using information arguments. We argued that V(x), a potential, is a fixed piece of spatial information linked to the x axis being divided into equal dx units. The variable spatial information is dt(x) i.e. the time spent in each dx or v(x) =dx/dt (to put this in the same form as the time displacement generator). One may not simply compare v(x) directly with V(x) because they change differently. v(x) changes in a weighted way i.e. v(x)d/dx or d/dt while V(x) changes as d/dx V(x). Thus we suggest d/dt v(x) = C V(x) which ultimately leads to Newton’s second law. Using d/dt → v(x)d/dx and integrating yields conservation of energy: p(x)p(x)/2m + V(x) = E. A similar argument was used in (2) to deal with the special relativistic case. There one cannot use d/dt → v(x)d/dx to obtain a conservation of energy equation. The point we wish to make is that for both the nonrelativistic and relativistic classical equations there are two types of “information” equations. The first is a “force” i.e. -dV/dx equation which is based on how V(x) changes with x. The LHS of this equation is linked to d/dt p(x). The second is a corresponding energy conservation equation. The quantum case is quite different, we argue, because a free particle is described by exp(-iEt+ipx). There is still the notion of p=mo dx/dt(x) as in the nonrelativistic case because p must be defined, but exp(ipx) is additional imposed piece of spatial information. -id/dx exp(ipx) = p exp(ipx) and -d/dx d/dx exp(ipx) = pp exp(ipx). This begs the question: How does one reconcile the two? For the nonrelativistic Newtonian and relativistic cases it is possible to have two equations (one with Force the other with energy) using one x dependent variable p(x). In the quantum case one may take an average to define a kind of complex = Sum over p a(p) p exp(ipx)/ Sum over p a(p)exp(ipx, but this is complex, not real. Furthermore both +/- is included in so there seems to be a problem linking it with changes in V(x) in one direction. Thus it seems that one is forced to consider the second kind of information equation from classical mechanics and define a <pp> = Sum over p a(p) pp exp(ipx)/ {Sum over p a(p)exp(ipx)}. In the nonrelativsitic case this may be multiplied by 1/2mo and then added to V(x) to obtain a constant E. If one only looks at <pp>/2mo as a spatial function then this is identical to KE(x) i.e. kinetic energy(x) for a classical system. Thus in the nonrelativstic case, Newtonian energy conservation is always embedded in the quantum system regardless of the energy. In other words the quantum bound equation i.e. the time independent Schrodinger is not obtained independently from the classical one in terms of information.One may ask: How does information related to exp(ipx) affect the quantum system? Two immediate ways are as follows. First <pp>/2mo = -1/2mo d/dx dW/dx / W where W= Sum over p a(p)exp(ipx). This leads to a differential equation which has discrete En values verified by experiment. Secondly W is linked to spatial density W*(x)W(x) which exhibits crests and near zero regions. Using <pp> suggests that p(x) in the classical case is linked to prms(x) quantum. Thus building quantum mechanics using “informational” ideas seems to involve embedding classical conservation of energy into the quantum equation a priori. This holds both nonrelativstically or relativistically (Klein-Gordon equation). This immediately raises the question: If classical mechanics (in the form of energy conservation) is embedded in quantum mechanics for any energy, why do people seek to take some kind of limit (3) to obtain classical mechanics from quantum? We argue that quantum mechanics has more spatial information than classical, but retains classical conservation of energy so there is no limit to take. One may simply ignore ideas related to the wavelength to obtain classical mechanics, we argue.
Classical mechanics samples space as being uniform i.e. each x point having the same weight. Thus the number of x states in a length L is proportional to L or in 3-space to volume V. The ln of this value then appears in classical entropy calculations. In the case of a particle moving in space the same idea holds, but this means the particle spends equal amounts of time in each dx region. Thus we argue this result of a constant probability in space is based on time sampling. Time sampling is the sampling done for an equilibrium gas. The system does not change on average in time, so it is as if one has a snapshot. Thus one has a constant probability in a length L in time sampling. One may argue that this constant is information so ln(probability(x))=constant1 or P(x)=constant 2. The relativistic/nonrelativistic action may be written as A = -Et+px if one uses v=x/t. This Lorentz invariant form allows one to clearly separate t and x with E and p as constants and may also be considered to be information. In such a case, if one ignores time, then px is the spatial contribution to A and this does represent a constant probability in space. Rather there is scaling of space by p. If x=constant/p, then each p has its own unique length, called a wavelength, in which space sampling is the same i.e. creates the same distribution form, but with 1/p as the wavelength. This we argue is space sampling of a spatial distribution and has physical consequences (i.e. two slit interference, measuring small regions etc). One may note that the wavelength=hbar/p depends on the magnitude of p. In actuality, p has a direction as this is space sampling. If px is considered to be information, then ln(probability(x))= px or probability x = exp(ipx). To obtain time sampling, one needs to freeze out motion i.e. exp(ipx)exp(-ipx) (AND situation). Thus the manner in which one samples an underlying distribution may give rise to a different distribution. Arguing that a particle spends the same amount of time in each dx region and so the spatial probability is constant completely ignores the form px from the classical action. exp(ipx) is associated with a wavelength, thus x samples are within different lengths for different p’s. This suggests an associated entropy which is not additive because one loses information when adding p1+p2. Probability exp(ip1x)exp(ip2x) = exp(i (p1+p2)x), however, is conserved as is the overall impulse p1+p2 which depends only on the net momentum, whereas p1 entropy depends on a specific value.