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 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).
We have argued in previous notes that special relativity is often developed from the point of view of a particle with rest mass, leading to the Lorentz invariant -EE + cc p dot p = -momo cccc ((1)). Then, as a second step, one may obtain the equation E= [p] c by setting mo=0. This appears to be a straightforward math step to obtain a photon relevant result, but we have suggested in (1) that mo represents an inherent force, called inertia (Newton). Thus, special relativity for a particle with rest mass is really linked to this inherent force. In fact, we have suggested in (1), that spin ½ is obtained by finding an equation linear in a particle’s inherent force (rest mass) which is linked to E and p from special relativity and geometry. This, we suggest, is why Dirac linearized ((1)) to find spin ½. The reason for linearization is to have an equation linear in the inherent force, because force appears in F=dp/dt in a linear fashion. The rest mass case seems to be largely based on mo, inertia, which we call the inherent force. We further argued in previous notes that free particle quantum probability arises from -Et+px = Lorentz invariant ((2)) because x,t and x+hbar/p, t+hbar/E give the same result and we consider -Et+px to act somewhat like a free particle potential. This suggests that a photon should also have a probability exp(-iEt+ipx), which is verified as both a photon and particle with rest mass, with the same p, give rise to the same 2-slit interference. The problem in the photon case is that mo=0 and so no inherent force appears in special relativity. We argue that physically, there must be an inherent force as interactions (as in the 2-slit case) depend on it. In (1) we argued that the inherent force of a photon is the electric field, but this does not appear in ((1)) or ((2)), but emerges from Maxwell’s electromagnetic equations. This is why the photon has spin 1 and not spin ½ even though it has the probability exp(-iEt+ipx). (See (1) for details on the specific equation which gives rise to spin 1.) Here, we argue that for a particle with spin ½ (i.e. rest mass), exp(-iEt+ipx), which is a probability which conserves energy and momentum, the momentum and energy are proportional to rest mass (inertia or inherent force) and so exp(ipx) is really a probability distribution for inherent force (augmented to E,p due to the Lorentz boost). In the case of the photon, one still has exp(ipx), but the force is due to the electric field. This begs the question: What is the spatial distribution of the electric field? Is it free to be any function? We argue that given that exp(-iEt+ipx) is the energy/momentum conserving probability from special relativity, the inherent force cannot contradict this distribution. The electric field, which is a real valued function, must be cos(-Et+px). We note that one may formally obtain a classical wave equation from Maxwell’s EM equations to find El proportional to cos(-Et+px), but this does not explain the complex probability exp(-iEt+ipx) which holds for both a photon and particle with rest mass. Our objective is to argue that given exp(-iEt+ipx) for a photon, the electric field must behave as cos(-Et+px). Interestingly, the photon momentum density 1/uo Electric field x Magnetic field is proportional to cos(-Et+px) * cos(-Et+px). This does not follow Real(exp(-iEt+ipx), but it is the electric field and not momentum in the photon case which is linked to force. We argue that ultimately it is force which governs interactions, even though momentum (not momentum density) is conserved.
In previous notes, we presented a simple argument for obtaining the quantum free particle wavefunction exp(-iEt+ipx). We suggested that there are situations for which one cannot predict the outcome of an elastic collision and that given initial e1,e2 (energies) and p1,p2 (momentum vectors in 1-direction for simplicity), any ei,ej and pi,pj which conserve energy and momentum have the same product probability. Even in Newtonian elastic collisions of free particles, one sees calculations which allow any outcome which conserves momentum and energy. If the probability has the same real value for any free particle as no variable is being distributed and one wishes to impose Lorentz invariance, then exp(-iEt) and exp(ipx) represent the probabilities. This automatically gives rise to units in time and space, hbar/E and hbar/p (the hbar comes from considerations linked to a photon). One may ask, however: What is the real physics behind dx and dt? We argued in (1) that this physics is associated with force, in particular an intrinsic force. We considered a particle with rest mass mo and argued that this rest mass represents intrinsic force as it is proportional to Newton’s inertia and the force needed to set it in motion. If viewed from a moving frame, (0, mocc) maps to (E,p) with E containing the intrinsic inertial force as well as enhancements due to motion and p containing the intrinsic mo as well as the velocity vector. For a particle with rest mass, we suggest that a key idea is this notion of intrinsic force. We argued that one may consider the Lorentz invariant -Et+px = -mocc t1 and introduce dx=hbar/E, dt=hbar/p, dt1=hbar/(mocc) and dx1-> infinite as p->0. This leads to -E dt + p dx =0 or -E/dx + p /dt = 0. We then argued in (1) that p/dt is like an average force and from this we introduced exp(-iEt+ipx). Thus, dt and dx are linked with the notion of an average force. We suggest that this notion of intrinsic force is the reason behind linearizing -EE + p dot p = -momo cccc ((1)). -Edt+pdx=0 is too simple to show the geometry linked with rotational invariance etc in ((1)), yet at the same time ((1)) is not linear in the intrinsic force mo. This is the physical reason why ((1)) should be linearized via Dirac’s process, giving rise to spin ½. As a result, we postulate that a particle has an intrinsic force, in this case rest mass, which is directly linked with the probability exp(-iEt+ipx) and that one should be able to write an equation containing operators and the intrinsic force (in linear form) which also shows geometry. It is this equation which gives rise to spin. At this point, this is a postulate, i.e. speculation. It would be good if one could find an example in which these ideas arise naturally, i.e. a case in which one did not have to linearize an equation like ((1)) or make special arguments in order to obtain exp(-iEt+ipx). In that case, an equation linear in the intrinsic force would exist a priori and a direct link to exp(-iEt+ipx) would also naturally exist. We suggest that such an example does exist, namely for a photon. One may use the special relativistic equations developed for a particle with rest mass and set mo=0 to obtain E=[p] c for a photon. This is a fine equation, but unlike the rest mass case mo (proportional to the intrinsic force), there is no rest mass and no intrinsic force present for the photon. Certainly p (momentum) is linked with a force as both a particle with rest mass and a photon impart an impulse on a target, but momentum is not an intrinsic force, it is a consequential one. In the case of a rest mass, it arises because the rest mass (the intrinsic force) is seen to be moving. For a photon, one would then expect the existence of an intrinsic force which gives rise to momentum. We suggest that the electric field (together with the magnetic) represents this intrinsic force and that Maxwell’s equations allow one to directly write a linear equation in El (electric field) (no linearization required) which shows the geometry in the system. One may also explicitly show that El behaves as exp(-iEt+ipx) if one extends cos(-Et+ipx) to a more general complex number. In other words, the postulates made for the particle with rest mass mo seem to naturally arise in the case of the photon. This, we argue, is why the procedures for finding spin for a spin ½ particle with rest mass mo and a photon are so different even though both a photon and rest mass particle satisfy special relativity and are associated with exp(-iEt+ipx). It is the difference in the intrinsic force which is the key to finding spin, we speculate.
In Part 1, we tried to understand why spin arises. The basic conclusion was that it is linked with an intrinsic force which in turn is associated with E,p (energy momentum) and geometry (rotational symmetry and other symmetries). In practical terms, this means finding an equation linear in intrinsic force (as force appears linearly in nature), but still tied to E,p and geometry. Interestingly, this leads to two very different results for particles with rest mass (spin ½) and photons, although both satisfy the same special relativistic equations (albeit the photon with rest mass mo=0) and both have exp(-iEt+ipx) free wavefunctions. In the photon case, the Electric field El is the intrinsic force (together with the magnetic B) which gives rise to both energy and momentum density. Specifically, energy density= .5eo El dot El + .5/uo B dot B and momentum density = 1/uo El x B. In the photon case, however, there is no rest frame and so the classical vector El is directly associated with E (energy) and p (momentum) without one needing to consider special relativity. Instead one directly uses Maxwell’s electromagnetic equations in order to find a linear equation describing the classical vector El. This equation involves d/dt partial and grad partial which are linked to the quantum operators for E and p (id/dt, -igrad) and El itself is proportional on exp(-iEt+ipx) (actually cos(-wt+kx)), also from Maxwell’s equations. Given this scenario, one has a classical vector El which is perpendicular to B and both lie in a plane perpendicular to motion. For circular polarization, El moves in a circle and given that it gives rise to energy, this energy circulates like classical angular momentum. For the 3-vector for El, one says that one has spin 1, but for circular polarized light, this is classical angular momentum. As a result, spin in the photon case follows from the behaviour of the intrinsic force (a linear equation in this force as shown in Part 1), but one must also have a clear link between the intrinsic force and the consequential energy and force, because it is the energy which actually has to rotate. It is not enough to have El rotate. In the case of a particle with rest mass, we argued in Part 1 that mo (Newton’s inertia) is the intrinsic force which together with velocity gives rise to the consequential energy E and momentum p. The situation is that one actually has two physical frames, the rest frame with the intrinsic force mo and the moving frame in which one has E,p. In order to link the intrinsic force mo to E, p, one must use a Lorentz invariant. We suggested in Part 1 that -Et+px = -mocc t1 is associated with -E dt + p dx = -mocc dt1 + (p=0) * (dx1->inifite) =0. Then dx=hbar/p, dt=hbar/E and E/dx and p/dt are average forces as is mocc/dt1. Tthe idea of force is very much the cornerstone of exp(-iEt+ipx) we argue. In order to see spin arise, one needs an equation linear in mo (the intrinsic force), but linked to E,p and showing geometrical invariances, according to the recipe of Part 1. -EE + cc p dot p = -momocc ((1)) links E,p, mo and shows invariances, but is not linear in mo. One must then linearize it. This means that one compares to Lorentz frames, the rest frame containing mo and the moving containing E,p. Both mo and p are linked to forces, but mo is the intrinsic force and p, the consequential. As a result, they must somehow be separated as they are in ((1)), in particular they are perpendicular if one considers: EE = pz pz + momo (c=1, p->px). In order to linearize without using unit vectors, one cannot use numbers, but must use matrices (as Dirac showed). The operator linked with pz must anticommute with that linked with mo. The operators associated with the consequential and intrinsic forces differ and in fact anticommute in the rest mass case. One may show that one requires at least a 2x2 matrix (the full Dirac treatment uses a 4-vector made of 2 spinors), which means that there is a two vector state. This is presumably associated with motion, if exp(ipz z) is linked to motion as well. In order to analyze this 2 vector state (spinor), we first compare the situation with the photon case. There, one had a 3-vector El which means that one knows an exact position in the plane. The second piece of information must be the direction in which this vector is moving in a circle (for circularly polarized light). There are two a priori given pieces of information which together with the link between El and energy density yields classical angular motion. In the rest mass case, one again expects rotational motion in a plane perpendicular to the z axis (because pz describes all motion along z). (1,0) and (0,1), however, describe classical rotational direction, clockwise and counterclockwise, but this spin ½ case cannot be classical rotation because the physical vector in the plane is not given (as it is by El the intrinsic force). Thus, one has only the direction of rotation. We argue that if exp(ipx) means that one has p hits at different x within hbar/p with probability exp(ipx), one may imagine different x,y positions of energy in the plane (with probabilities) as long as the rotation for (1,0) for each is in the clockwise direction (for example). This means that a physical vector may appear at some x,y in the plane and then at an x1,t1 which would be considered a classical time in the past as long as the motion is clockwise. In this way, one may have an angle of 720 degrees (instead of 360 degrees) describe the motion. In terms of interaction with a B field, one has the clear two directions (spin up clockwise and spin down counterclockwise as examples). Thus, spin must exist in the rest mass case, because one must have mo perpendicular to p if special relativity is to hold. If one does not want to have spin, then one must disregard the equivalence of frames moving at constant speeds with respect to each other and that seems unphysical. (Note: There are spin 0 objects (Higgs particle), but we do not consider that here.
The Lorentz transformation of special relativity is based on velocity. Although this velocity may be differentiated by time, the notion of force is not needed to obtain the Lorentz matrix for a particle with rest mass. In particular, in special relativity one considers frames moving at constant velocities. In the case of a photon, the Lorentz transformation preserves the form of Maxwell’s electromagnetic equations. As a result, there does not seem to be a focus on force, rather the emphasis is on the two 4-vectors (p,E) and (x,t) for a particle with rest mass and photon. Nevertheless, a particle with momentum (and the capability to interact) imparts an impulse. In fact, one may think of energy changing due to an impulse (which includes force). Thus from an information point of view, one may ask: Does this force which creates a change in energy/momentum “disappear” as the particle/photon moves at constant speed only to “reappear” when it strikes a surface? If not, where is the information of this force stored? We argue it is associated with a change in energy and momentum of a free particle which in turn is linked to an internal ruler/clock i.e. the quantum frequency and wavelength from exp(-iEt+ipx) which preserves the force because a spatial probability distribution is created (for the wavelength). This force is linked with a direction i.e. the direction of p through the two dimensionality of exp(ipx). Thus, we argue that the force does not disappear, rather it is manifest in the internal clock and ruler which define spacetime for the free quantum particle, but in a probabilistic manner with the distribution being deterministic to ensure it maps to spatial invariance. One might at first find it surprising that force, usually associated with acceleration (i.e. deterministic motion) is now associated with probability cos(px) in space (, but deterministic probability, not random), but E,p and x,t are intertwined through the Lorentz invariant -Et+px for both a particle with rest mass and photon, so if E and p change, it is not necessarily surprising that t,x also change. Thus an impulse may change p and E, but then remains manifest in the system through the internal clock and ruler gradations in such a way that speed is still constant on average.
We have argued in previous notes (1) that the “wave” nature of quantum mechanics follows from the Lorentz invariant for a constant p: A= -Et+px which holds for both a particle with rest mass and a photon. In A, t and x are separated indicating independence which violates the Newtonian view x(t). “A” leads to dA/dx partial = p and dA/dt= -E. Creating eigenvalue equations yields: -id/dx exp(ipx) = p exp(ipx) and id/dt exp(-iEt) = E exp(-iEt). Thus, x and t may be separated. Given that exp(ipx) exists in all of space (because there is no time), it represents a somewhat spatially invariant probability. In the case of a photon, Maxwell showed one may write wave equations for the electric El and magnetic B fields using his electromagnetic equations with no charge or currents.. He showed the wave travels at the speed of light in a vacuum and thus characterized light as an electromagnetic wave. This is not the same statement as exp(ipx) which is a probability wave. One may note that Maxwell’s equations are Lorentz invariant, so the ideas of the first paragraph should hold. In this note, however, we consider the Poynting equation: d/dt partial (.5( eoElEl+1/uoBB)) + 1/uo grad dot El x B = function(charge and current). 1/uo ElxB is considered to be a momentum density. We note Maxwell’s equations contain d/dt partial and d/dx partial, so time and space are considered independent (unlike x(t)), and act on El and B. The presence of charge and current densities breaks spatial invariance i.e. creates local regions. We argue in this note that d/dx and d/dt are generators of translations in space and time. We further argue that Maxwell’s theoretical creation of a photon follows from introducing spatial invariance into the problem, i.e. removing all charge and current densities. Spatial invariance in Newtonian mechanics, however, is associated with a constant i.e. a constant speed and momentum and equal probability to be at any x. Given that d/dx and d/dt appear, one has a function of space and time which seems to be completely at odds with spatial invariance and the idea of a constant momentum. We have argued in (2), however, that constant motion, even though represented by v=constant, is associated with the force that created the momentum, and this manifests itself in a spatial density which is probabilistic. Thus the presence of d/dt and d/dx in Maxwell’s electromagnetic equations, even when charge and current densities are removed, implies the same thing. Thus we argue that Maxwell’s electromagnetic equations are probabilistic. We investigate these ideas in this note. We further argue that it is spatial invariance, together with the presence of d/dx partial which yields the periodic form exp(ipx). Thus it is not a coincidence that wave equations appear for El and B.
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.