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.
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.
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.
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 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 previous notes (1), we argued that stochasticity (in x and t for a fixed E energy and p momentum) may occur at a single particle level based on the Lorentz invariant A = -Et+px ((1)). This invariant is also the relativistic/nonrelativistic classical action for v=x/t. The argument made in (1) was that a fluctuation in t of constant/E could cancel one in x of constant/p. We further argued one could create eigenvalue equations using id/dt and -id/dx and obtain the two dimensional probabilities exp(-iEt) and exp(ipx), i.e obtain a description of free particle quantum mechanics. Experimentally, both exp(-iEt) and exp(ipx) have been verified through interference patterns. Theoretically, however, ((1)) was known about 1905 and no single particle stochasticity was associated with it. In this note, we try to argue why there should be an associated stochasticity associated with ((1)) based on motion through a constant potential for a particle with rest mass and motion through a medium with an index of refraction for a photon. We in fact argue that one cannot solve certain photon problems such as one dimensional reflection/refraction without explicitly considering the stochasticity which is usually left out when describing p2=pn2 and c2=c/n2 where n2 is the index of refraction in the second medium and p and c are values in the first medium with n1=1. We further consider exp(ipx) and exp(-iEt) as arising from the maximization of Shannon’s entropy subject to certain constraints. The product of these two two-dimensional periodic probabilities is exp(-iEt+ipx) which we argue must hold as seen from observers moving with constant velocity. Thus we argue that the Lorentz invariant -Et+px follows from statistical considerations alone. Given this expression (a subset of the whole picture), one may try to create a deterministic viewpoint by setting x/t=v=constant. This allows one to consider transformations which define E and p in terms of mo and v and yields the invariant EE = pp+momo (c=1). Thus we argue that the conservation of energy equation ultimately follows from a deterministic view of the argument of a statistical probability. In other words, it is only a subset of the full picture i.e the quantum mechanical statistical one.
Correction June 28, 2022: The classical wave-equation should read d/dt d/dt partial W(x,t) = vv d/dx d/dx partial W(x,t) Historically two types of “free” mechanical motions were considered, that of Newtonian mechanics x=vt (v=constant) and classical waves such as waves on a string which were described in terms of Newton’s laws for a continuous medium. It seems these two were considered as fundamentally different phenomena. Equations of motion were developed for classical mechanics as well as electromagnetic phenomena. Maxwell discovered that electromagnetic equations allowed for a classical wavelike solution which he identified as light. In the early 1900s it was realized that to retain electromagnetic equations invariance as seen from a constant moving frame moving, Lorentz transformations were needed. In 1905 Einstein obtained these transformations in a different manner considering that light has the same vacuum speed when viewed from different constant moving frames. Later in the 1920s quantum mechanics was developed using the idea of a probability wave exp(-iEt+ipx) for a free particle beginning with the work of deBroglie who made use of ideas of special relativity. In this note we suggest that there seems to be one fundamental “free” mechanical motion, namely that associated with a wave exp(-iEt+ipx) whether one considers a classical wave or a quantum free particle. Classical mechanics is a mapping into a continuous x-t space, we argue, and describes energy and momentum motion in this continuous space. This same description also occurs for classical waves. The medium undergoes a somewhat complicated nonlocal change, but the “velocity” of the wave maps to a continuous local space time i.e. v=x/t=dx/dt where E= pv. E=pv, however, is only a description of energy momentum propagation given in a simple form. It does not describe the overall picture. We argue the same is true for the simple x=vt (v=constant). It seems that an overall wave structure is associated with energy and momentum and for free motion this is linked with exp(-iEt+ipx). If this wave is fundamental one may argue that its phase may remain constant when viewed from different frames moving at constant v. Special relativity would then follow from this requirement and so is in a sense a consequence of wave behaviour. We investigate these ideas.
In Part 1, we argued that P(x)dx=dx/L (where L is an arbitrary length) applies to a particle at rest. If one views such a particle from a frame moving at constant -v, it is moving. We argued that in such a case, the relevant probability is not simply linked to the position x, but also to the momentum p at x because this delivers an impulse hit. In particular, probability is presumably to be used in a physical problem with interactions and probability and a dynamical particle requires a dynamical probability describing both x,t and E,p. This led to exp(-iEt+px) which actually predicts intervals in space hbar/p and time hbar/E. In other words, a probabilistic view point leads to these results. Here, we ask the question: What if one simply wishes to use deterministic physics to describe a moving particle? After all, Newtonian mechanics is deterministic and one does not really need to focus on P(x)=1/L or its modification for a moving particle. In such a case, a particle moving with constant v (starting at x=0, t=0) is described by x/t=v both in Newtonian mechanics and special relativity. There is no uncertainty in x and t here which begs the question: Why does it appear in the probabilistic formalism of Part 1? In particular, exp(-iEt+ipx) derived in Part 1 implies physical regions: dx = hbar/p and dt=hbar/E and these seem to be absent in x/t=v and this is a problem. We suggest that a probabilistic approach is buried in the Lagrangian formalism which leads to the result d/dt v = 0, i.e describes the particle’s motion in space. This motion is actually x/t=v, but this equation only presents a subset of the description of a particle that is moving with constant speed, even in a deterministic Lagrangian/Action approach. We have already pointed this out in (1). We briefly explain it again here, but the real question is why does one need to consider the Lagrangian in order to describe x/t=v? In other words, the probabilistic approach associated with v=x/t only emerges in the Lagrangian formalism using a relativistic action Lt = -Et+px, with x/t=v. Ignoring the action and Lagrangian leads to an absence of hbar/p and hbar/E and one is only concerned with x/t=v. We note that the Lagrangian is primarily linked to the result: p = dL/dv partial and so argue that one is not simply interested in motion (x/t=v), but also in interaction because p is associated with an impulse hit. Thus, the Lagrangian approach is more than simply a mechanism which describes x,t motion, it is associated with p and E (through Hamiltonian = E = pv-L.) Using the Lagrangian approach suggests that it does not suffice to consider x and t by themselves when p and E are present. One must consider all four variables x,t,E,p which is the same point made in Part 1. The physics of the problem dictates that it is not enough to simply think in terms of x,t,v. The Lagrangian approach, which is constructed to ultimately yield x/t=v, when written in terms of E, p, i.e. Lt = -Et+px demonstrates uncertainty in x, hbar/p and t, hbar/E as shown in (1). This uncertainty must then be written in terms of a probability which is Lorentz invariant and leads to exp(-iEt+ipx). The point is that one would not even consider a probability if one did not consider E,p and x,t together as a complete treatment of a moving particle. In other words, x/t=v is an incomplete description of the state of a particle moving with constant speed, even in a deterministic framework.. At first this might seem surprising, but it is well known that energy and p are linked to interactions. In special relativity, one does not consider x,t,v alone and then compute p and E from these, but considers four vectors and Lorentz invariant equations. It is possible to create a Lorentz invariant equation involving all four state variables x,t,E,p and it turns out that this is the relativistic Lagrangian which describes motion, so this approach is more general than simply stating x/t=v. Thus, delta x = hbar/p and delta t= hbar/E appear in both the probabilistic approach of Part 1 and the deterministic one of the Lagrangian used here.