In Newtonian mechanics a force changes momentum, for example it may change its magnitude only or its direction only or both. In the case of the Bohm Aharonov effect for a particle with rest mass there exists a potential which does not change momentum, but introduces a phase shift. Thus exp(ipx) for say an electron becomes exp(ip(x-xo)). Given that the form exp(-iEt+px) holds for both particles with rest mass and photons one might expect to find a Bohm Aharonov effect for photons. In (1), it is stated that a photonic Bohm Aharonov effect may be “induced by a dynamic modulation of a material’s permittivity”. In this note we argue that the refraction of light seems to have the same form as a Bohm Aharonov effect in the sense that if one begins and ends in the same medium, the particle has the same energy E, momentum p, speed of light c and makes the same angle with the normal. In a Newtonian sense there has been no change to momentum, but the ray is shifted in the x direction from one that does not refract. It seems that this may be described by exp(i py*y) exp(i px(x-xo)) i.e. a phase shifted form suggesting that for the x direction one has a Bohm Aharonov effect. We try to link this phase shift in the x direction to more general ideas. We argue as in (2) that a lack of information related to speed (i.e. the index of refraction) in the x direction leads to a conservation law related to speed in the x direction. This speed, however, is a hypothetical one such that it acts as a hypotenuse with c/n(i) being a projection. This differs from the actual x component of velocity which is a projection of c/n(i). Thus this actual x speed is not conserved meaning that light moving in a new medium will struck the outer bound at a different x than it would have it had not moved in a new medium. Thus the beam coming out of the new medium has the same E and p vector, but a shifted x i.e. there is a phase shift of the Bohm Aharonov type we argue even though we do not formally deal with a potential.
In Newtonian mechanics a force changes momentum, for example it may change its magnitude only or its direction only or both. In the case of the Bohm Aharonov effect for a particle with rest mass there exists a potential which does not change momentum, but introduces a phase shift. Thus exp(ipx) for say an electron becomes exp(ip(x-xo)). Given that the form exp(-iEt+px) holds for both particles with rest mass and photons one might expect to find a Bohm Aharonov effect for photons. In (1), it is stated that a photonic Bohm Aharonov effect may be “induced by a dynamic modulation of a material’s permittivity”. In this note we argue that the refraction of light seems to have the same form as a Bohm Aharonov effect in the sense that if one begins and ends in the same medium, the particle has the same energy E, momentum p, speed of light c and makes the same angle with the normal. In a Newtonian sense there has been no change to momentum, but the ray is shifted in the x direction from one that does not refract. It seems that this may be described by exp(i py*y) exp(i px(x-xo)) i.e. a phase shifted form suggesting that for the x direction one has a Bohm Aharonov effect. We try to link this phase shift in the x direction to more general ideas. We argue as in (2) that a lack of information related to speed (i.e. the index of refraction) in the x direction leads to a conservation law related to speed in the x direction. This speed, however, is a hypothetical one such that it acts as a hypotenuse with c/n(i) being a projection. This differs from the actual x component of velocity which is a projection of c/n(i). Thus this actual x speed is not conserved meaning that light moving in a new medium will struck the outer bound at a different x than it would have it had not moved in a new medium. Thus the beam coming out of the new medium has the same E and p vector, but a shifted x i.e. there is a phase shift of the Bohm Aharonov type we argue even though we do not formally deal with a potential.
The Bohm Aharonov phase shift for a free quantum particle with momentum p moving through a magnetic potential A(x) with no magnetic force is well known as exp(i Integral dx1 (a,x) e/cA(x1)dx1. In the literature this has been obtained in two very different ways. The first way (standard) (1) is to use the classical equation for a charged particle in an electromagnetic field with electrostatic potential i.e. (p-e/cA)(p-e/cA)/2m = E with p=-id/dx representing an overall momentum, not just that of the particle. In such a case, there is coupling at the momentum level, but E is the energy of a free particle with momentum p1=p-e/cA (classically) i.e. p1p1/2m. In other words, momentum considerations are treated differently from energy ones. The time dependent wavefunction is exp(-iEt) where E shows only a noninteracting free particle. The second approach (2) is the opposite. It ignores the Schrodinger equation and the momentum picture, but retains the idea that for a nonacclerating free particle p/m = velocity=dx/dt. It then considers an interaction which changes E namely delta E = p e/cA(x)/m. This leads to Integral pA(x)/m dt = Integral pA/m dx/v which yields the same Bohm Aharonov phase shift. In a previous note (3) we considered a free particle classical action as A=-Et+px. We argued that Et and px should be kept separate leading to exp(-iEt) and exp(ipx) (not combined) if one creates an eigenvalue equation. We argued that scaling exists in both time (by E) and space (by p). At the same time, the particle moves classically with a speed v=p/m, but has an associated exp(ipx) momentum wave (and independently an energy wave exp(-iEt)). Even if one uses v=dx/dt as in (2) these waves are different i.e. Et= Integral E dt = Integral .5pv dx/v = .5 px while px is the phase from the momentum consideration. Thus one might argue that it is unusual that a change in E i.e. delta E = p/m A(x) from (2) applied to exp(-iEt) yields the same Bohm Aharonov phase as the Schrodinger equation with p, A coupling, but E left untouched In this note, we consider the classical action A= -Et+px and try to explain this feature directly from this expression.
In Part I we noted that Fermat’s least time principle was equivalent to conservation of momentum in the direction of an invariant spatial variable and that this principle is contained in the dispersion relation E=pc for a light which is strictly a wave relationship. We then argued that this wave relationship should somehow be extended to a particle with rest mass because such a particle may bounce off a mirror like light. To make the connection we wrote 0 = -Et+(px)x +(py)y for light with r/t=c and constant = -Et+(px)x + (py)y for a free particle with rest mass. In both cases Fermat’s principle is contained and is equivalent to conservation of momentum in the x direction. The form -Et+px, however, suggests a wavelength proportional to 1/p and a frequency proportional to 1/E for both light and a free particle with rest mass. We argue in this note that the wavelength represents spatial information which may show conservation (e.g. of momentum in a particular direction). Thus the focus of Part I was to link a free particle with rest mass to the wave formalism of light. Newtonian mechanics describes a free particle with rest mass as well as one moving in a potential without any need for a wavelength or frequency. In fact, for light E=pc means p→pn and c→c/n where n is the index of refraction means that E remains constant. For a particle with rest mass (nonrelativistic) E = pp/2m + V(x). A change in p means one needs the potential to absorb or yield energy to keep E constant which is different from light. Given that Newtonian mechanics describes particles with rest mass one may ask: Why bother to try to link a free particle with light and Fermat’s principle as done in Part I? The answer is that a free particle (with rest mass) diffracts or shows interference just like light in experiments. Furthemore -Et+px which suggests a wavelength proportional to 1/p and frequency proportional to 1/E is equivalent to the free particle action: -mo t sqrt(1-vv/cc) (relativistic) and .5 mo t vv (nonrelativistic) with v=x/t. Thus if Lagrangian/Hamiltonian formalism is identical to Newtonian mechanics we argue that the idea of wavelength and frequency is already present. It is not manifest because one deals with averages in pp/2m + V(x) = E. To see how wavelength may be incorporated in Newtonian conservation of energy, consider a free particle in a potential i.e. E= pp/2m + V(x). Next consider knocking out the particle so it has constant momentum. We argue that such a particle should exhibit wave-like behavour i.e be described in a manner linking it with light/Fermat’s principle e.g. be described by exp(ipx). Fermat’s principle already suggests that d/dx → p so -id/dx exp(ipx) = p exp(ipx) is an eigenfunction. We then argue that if exp(ipx) describes the particle when it is knocked out of the bound state, this information should still exist while it is within the bound state. Thus we suggest the form: W(x)= Sum over p a(p)exp(ipx) as describing the particle within a bound state. On average one has Newton’s conservation of energy result, but the idea that a particle may be knocked out and should behave as a wave is also present. Furthermore the various exp(ipx) interfere so the wave nature is also present in the bound state.
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 Part 1, we argued that it is no coincidence that the photon free particle wavefunction exp(-iEt+ipx), which we call a complex probability that conserves energy and momentum, has a real part which matches the electric field of a photon. In particular, we argued that even if one did not know Maxwell’s electromagnetic laws, one would postulate a force linked to a photon which behaved as cos(-Et+px). Here, we reexamine Maxwell’s electromagnetic equations. These are presumably deterministic differential equations and the photon solution with perpendicular electric and magnetic fields in a plane perpendicular to motion with each field changing as cos(-Et+px) for motion in the x direction, a seemingly deterministic solution. This deterministic solution, however, seems to be problematic for the following reason. If one considers one-dimensional reflection-refraction according to the prescription of (1), one writes a continuity equation for the electric field of an incoming photon with a reflected and refracted one, and another continuity equation for the first derivative. Physically, this appears to be a steady stream approach as well as a mathematical continuity one. The solution in (1) is obtained without any notion of probability until the last line: 1= Probability(reflect)+Probability(refract). The situation, however, is completely probabilistic from the beginning we argue and so one should understand how a deterministic electric (and magnetic field) is actually associated with probability. (Furthermore, photon interference and diffraction are also probabilistic.). We have argued in previous notes that examination of special relativity, in particular, the Lorentz invariant -Et+px = constant(x,t), leads to a probability for interaction of exp(-iEt+ipx) for both a particle with rest mass and a photon. This probability conserves E,p, but at first glance does display the inherent force in the problem. It is the inherent force which delivers E,p. Probabilistically, there is an uncertainty associated with the interaction time of hbar/E and with interaction position, hbar/p. These should be associated with the inherent force. As a result, the photon electric field, which acts as a force, cannot be precisely as described by cos(-Et+px) because this violates the notion of hbar/p and hbar/E. Rather, this formula is an expression of probability of the action of the force. One ultimately does not have a deterministic situation, we argue. In Part 1, we noted that a rest mass mo is associated with a force (inertia) as well as energy and momentum, which are proportional to mo. A photon has E and p and so should be associated with a force as it may render its energy to an object. The interesting thing about the photon is that it has no rest frame, even for the force. Given that exp(-imocc t) describes the probability associated with a rest force mo, we suggest that for a force linked with a moving photon, one should consider cos(-Et+px). This was already proposed in Part 1. This means that instead of considering the momentum as delivering the force, as it is proportional to mo for a particle with rest mass, for a photon, the electric field represents probabilistic force rendering, even though it is real and appears to be deterministic. Given that the electric field (inherent force) changes as cos(-Et+px), one may see that this force may change sign, but momentum must be in one direction. This suggests the presence of a second vector, together with the electric field in a quadratic arrangement with both becoming negative at the same time, i.e. both following cos(-Et+px). One may then create energy and momentum using Electric field dot Electric field and second field dot second field and electric field x second field for momentum. In such a case, energy and momentum would be cycle averages as the electric field changes as cos(-Et+px).
In Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world. Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a to reflect/refract: 1= a + 1-a. Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of 2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance. Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength), one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.
In a previous note (1) we argued that Fermat’s principle of least time is present in the photon wave dispersion relationship E=pc which may be written as 0 = -Et+px with x/t=c. We argued that a nonzero constant = -Et+px applies to a free particle with rest mass, but that Fermat’s principle is still present. Furthemore the form -Et + px implies the existence of a frequency and wavelength for both a photon and free particle with rest mass. (Interestingly -Et+px is both the free particle relativistic action -mo t sqrt(1-vv/cc) and nonrelativistic .5mo t vv with c=1 and v=x/t. ) Fermat’s principle implies that d/dx → momentum (x component) as discussed in(1). An eigenfunction exp(ipx) demonstraties wave behaviour of a free particle with rest mass. In (1) we argued that if one knocks out a quantum bound particle it will be in a state exp(ipx). Thus this wave type information should also be included in the bound state so wepostulated a bound state represented by W(x) = Sum over p a(p)exp(ipx) (i.e. a wavefunction) such that the Fermat result d/dx → p still holds. In this note we wish to consider the idea of a statistical equilibrium and the concept of a constant energy. In the case of refraction it seems a complicated statistical scattering process occurs which slows down the motion of a photon i.e. c changes to c/n where n is the index of refraction. Thus one piece of information, namely n, describes the complicated statistical process as in thermodynamics. If E (energy) remains constant p→ pn. Thus in this complicated statistical scenario a cos(-Et+pn x) electric field pattern remains but p → pn i.e the wavelength has changed. The idea of a constant energy may even be applied to light scattering from a mirror moving with a constant v in the negative y direction. The actual energy and momentum of the incident and reflected rays are not equal, but for E1 special = p1 v(relative 1) and E2 special = p2 v(relative 2) with v(relative 1) = c- vcos(A) and v(relative 2)= c+vcos(B) where A and B are angles of incidence and reflection measured with respect to the y axis. E1 special t=E2 special. In this note we retain the idea that a particle knocked from a quantum single particle bound state is described by exp(ipx) in keeping with Femat’s principle. We consider the bound state to still have waves as Fermat’s principle suggests d/dx →p, but they cannot be of the same height and width because of the changing value of V(x) with x (i.e. the potential). According to Newton’s law one has conservation of energy i.e. pp/2m + V(x) = E, but we consider this to apply to a p (rms) = m v(rms) in the Schrodinger equation. If the forms exp(ipx)’s are to be retained within the bound quantum state (i.e. this information is not lost) and one has a constant E (energy) then the potential must be modified statistically to allow for the idea of conservation of energy for each p, but must also represent a constant potential to retain the form of exp(ipx). Thus in short the Schrodinger equation becomes a weighted sum of exp(i p(j) x) functions with each having a different wavelength (like refraction), but interacting with different constant potentials to ensure that E remains constant for each p(j). In other words we think of a potential V(x) as a statistical set of refraction discontinuities i.e. impulses which produce exp(ipx)’s which are like light moving through a homogeneous medium i.e. a momentum eigenstates. For this to happen, however, the various exp(ipx)’s need to combine with V(x) = Sum over k Vk exp(ikx) terms to create a Vmodified(p) exp(ipx) where Vmodiied(p) is a constant because each free particle exp(ipx) is associated with a different effective potential and a kinetic energy pp/2m, but the same energy En. We argue that this results from applying ideas of Femat’s principle to a particle with rest mass. Newton’s picture of a particle accelerating i.e. x(t) emerges if one considers rms values for which all information of exp(ipx) and crests/troughs of W(x) are ignored. In particular, W(x)=wavefunction may have crests and troughs, but -1/2m d/dx dW/dx/ W = KE(x) Newton which shows no crests or troughs, but simply balances V(x) in KE(x)+V(x) = E.