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.
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.
Note: Nov. 21, 2022 One may note that in Fermat's least time principle one extemizes time=distance/speed of light. This is often applied to rays of light for which the momentum vector is parallel to the displacement vector. Thus |p| distance = (px)x + (py)y for a two dimensional problem. Thus one could write |p|time which would contain (px)x + (py)y. This expression is part of -Et+p dot r which appears in the description of a photon. In addition, both relativistic and nonrelativistic classical actions (with v=x/t) are equivalent to -Et+p dot r for a free particle. In quantum mechanics x and p are considered conjugate variables and d/dx, the generator of translations in x, is an operator representing p (up to the multiplicative factor -i). In classical mechanics, however, p is not thought of in terms of the operator d/dx (even though p and x appear in Hamilton’s formalism and phase space in statistical mechanics). We argue that the idea of p→ (-i) d/dx is present in Newtonian mechanics and leads to Fermat’s least time principle for light. We try to think in terms of cause and effect and invariance or lack of information. A key idea we argue is to have a function f(x=invariant variable) with value fo at x such that df/dx = (px)/constant. In other words, the generator of translations of an invariant variable x acting on a special function f(x) yields the information describing motion in that direction, namely (px). A geometric way to achieve this is with a triangle which is similar to the momentum triangle such that d/dx of the hypotenuse (where x is an invariant variable and d/dx the translation generator) yield x/hypotenuse . This, when multiplied by a constant, creates a conserved quantity, namely momentum in the x direction. From this one may create Fermat’s principle of least time for light. Furthermore, one may find another function A = -Et+ (px)x + (py)y + (pz)z which behaves in a similar manner i.e. dA/dx partial = (px). This is the classical free particle action (both relativistic and nonrelativistic) and the creation of an eigenvalue equation leads to a quantum mechanical form.
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.
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
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.
It was argued in (1) that Fermat’s least time principle for light (in a two dimensional (x,y) reflection or refraction scenario) minimizes time by actually varying it in terms of a spatially invariant variable i.e. x (i.e. taking d/dx) with the flat face of the mirror/medium along the x axis at y=0. This is equivalent to conserving momentum in the x direction. Thus d/dx is associated with (px) which may be associated with the Lorentz invariant -Et+p dot r. In other words, d/dx emerges as a momentum operator (up to a factor) and d/dt as an energy operator, yet both are generators in time and space linked with invariance of constant values i.e. conservation. Furthermore an eigenvalue equation: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt) automatically (by the nature of eigenvalue) leads to orthogonal eigenfunctions linked to conservation of energy and momentum. In special relativity rest mass mo represents a constant in time which may exist at any point x. In fact, it is often thought of as a point, for example a center of mass point. The actual object with mo, however, may be large with moving parts all within a “container”. What then does x,t in -Et+px mean for constant E,p? It seems that x,t are center-of-mass coordinates for the overall system. We argued in previous notes that 1/p is proportional to a wavelength so for p→0, mo may be at any x point with the same probability. Within the system with moving parts, however, there is motion and an x scale for each moving piece. For a single particle bound system, there is a single energy at each x which is like mo and a potential V(x) and KE(x). In (2) we argued that given the free particle relationship -EE+pp= -momo (c=1) one may retain a constant E value by replacing E with E-V(x) and pp with <pp> (and not ). How do the probabilities appear? They are linked to -Et+px with frequency hbar/E and wavelength hbar/p and are specifically manifested by exp(-iEt) and exp(ipx). For and <pp> we use spatial probabilities i.e. W(x) = Sum over p a(p)exp(ipx). Each exp(ipx) is linked to the operator -id/dx and invariance, but pp is linked with the operator -d/dx d/dx and one does not use an eigenfunction of this operator. Thus the notion of conservation of momentum is retained in the exp(ipx)s and their linear combination W(x) yields a function such that an average kinetic energy exists at each x point i.e KE(x)=KEclassical = -1/2m d/dx dW/dx / W. This is combined with a V(x) to ensure constant E at each x point i.e. create a kind of mo piece linked with id/dt. We argue that E being constant at each x is a statistical equilibrium type of result. Each exp(ipx) is associated with an average motion of x=p/m t and given that exp(ipx) and exp(-ipx) are included, there is also the notion of which is positive and negative in different regions and integrates to 0. prms(x) from <pp> becomes the classical p(x). Thus bound state quantum mechanics seems to be based on a free particle treatment which follows from the special relativistic Lorentz invariants -EE+pp=-momo and -Et+px = constant, directly linked to conservation ideas. Conservation of momentum of an internal bound particle is maintained through exp(ipx), while <pp> by itself is not linked with conservation as there is no eigenstate, but is combined with a V(x) to create a constant En linked with the entire system (and P=0) which then becomes associated with invariance in time and the generator id/dt. In this note, we try to argue that the notion of W*(x)W(x) (where W(x)=wavefunction= Sum over p a(p)exp(ipx)) is spatial density and P(p)=a*(p)a(p) based on spatial invariance and conservation of momentum for a single particle bound state even though there seems to be no spatial invariance or notion of conservation of momentum in W(x) which is simply a solution of the time-independent Schrodinger equation. One may move from a W(x) scenario (i.e. Schrodinger equation) to a W*(x)W(x) picture, by trying to “pick out” invariant (p only terms) using the orthogonality of the exp(ipx)s. Thus W*(x)W(x) (spatial density) emerges naturally in this approach.
In a previous note (1) we argued that the translational generator d/dx being associated with momentum (px) already appears in Fermat’s principle for both reflection and refraction. We suggested that d/dx is linked to an invariant spatial variable which in turn is associated with momentum conservation in the same direction. This may be written as: d/dx (px)x = (px). Alternatively (px)x = -i ln(exp(i px x) ) with exp(i px x) being a directional probability distribution (i.e. values for p and -p differ) and ln(exp(i px x)) being of the form of information in information theory. One may then ask: How does one link average momentum in the case of a bound state with a potential V(x)? In general, such a bound system is solved using the time-independent Schrodinger equation i.e. conservation of energy using d/dx twice and a p(rms) not a p average obtain from -id/dx ln(W) = p(x) where W(x) is the wavefunction and pave= W dW/dx. After all, the form exp(i (px) x) may be obtained from considerations of d/dx and p, not energy conservation (even though E=pc for light). We argue that exp(ipx) is a directional probability. If both p and -p are present then: p exp(ipx) -p exp(-ipx) is not zero because the spatial probability distinguishes between the two. Only by multiplying exp(ipx) and exp(-ipx) does one remove motion. If p and -p are present, however, as in an OR situation, i.e. exp(ipx) + exp(-ipx), then it is as if the particle is bound or localized and P(p,x)+P(-p,x) = C cos(px). We consider the two components of exp(ipx) in: Integral dx cos(px) sin(px). Why does this equal 0 from a momentum perspective? sin(px) = 1/p d/dx cos(px). Thus the integral is: Integral dx { P(p)+P(-p)} momentum(P(p)+P(-p)) } or the average momentum at x. Overall the particle is bound and so there can be no momentum globally so the integral must be zero and hence cos(px) and sin(px) must be orthogonal over a period even though they represent the same p value. This same idea holds for a linear combination of {P(p,x)+P(-p,x)} sums for different p’s. (We consider overall positive or negative parity.) Thus: Integral dx W dW/dx = 0 holds for even-odd reasons, but also for the physical overall zero momentum. Even with a potential present: -i WW d/dx ln(W) = Pave(x). In other words, bound state quantum mechanics is linked to W dW/dx integrating to 0 even though there are positive and negative pockets at various x values. This follows from the orthogonality of cos(px) and sin(px) over a wavelength from a free particle which conserves momentum. This result holds for any V(x) even though V(x) fixes the coefficients of the linear combination of Sum over p a(p) {P(p,x)+P(-p,x)} using energy considerations. Zero average momentum follows from momentum conservation found in exp(i (px) x). In fact we have argued that the unusual probability wave form exp(i (px) x) exists to ensure constant momentum (momentum conservation) for a free particle. We argue that considering average momentum equal to 0 overall leads to ideas of zero point energy, orthogonal energy states and a crest/trough pattern without considering the time-independent Schrodinger equation which is based on energy conservation. Thus we argue that ideas related to d/dx and momentum carry over from Fermat’s principle into a bound quantum state and its average momentum which is often not considered.