In previous notes we argued that two flow/flux equations d/dx (partial) A(x,t)=p and d/dt (partial) A(x,t) = E for free particles (where x/t is set to v after the derivatives are taken) may be used to describe classical mechanics, both relativistic and nonrelativistic quantum mechanics (for a free particle with rest mass) and lead to the form of special relativity for a particle with rest mass. In this note we consider (as in (1)) the situation for a particle with no rest mass. We argue that the two flow/flux equations still pertain, but that there is a different equation for momentum (which as a function of E(v) and v) is related to energy flow. For a particle with no rest mass, v=E/p while for one with mass p=E(v)v. How is this possible? We argue that v=E/p describes an object with energy which is entirely active in creating motion i.e. energy flow. In other words for pv=E, p→0 and v→0 means E→0 which does not allow for rest energy. This describes a classical wave and also the photon or phonon. Such an object cannot be at rest because its energy is equivalent to motion so to speak. It is possible to trap such a particle in a box so it bounces back and forth. In such a case, there is rest energy and a different notion of energy flow, namely p=E(v)v. Thus, we argue that there is no contradiction in having two equations for momentum, one pertaining to particles with rest mass and the other to those without.
In previous notes we argued that two flow/flux equations d/dx (partial) A(x,t)=p and d/dt (partial) A(x,t) = E for free particles (where x/t is set to v after the derivatives are taken) may be used to describe classical mechanics, both relativistic and nonrelativistic quantum mechanics (for a free particle with rest mass) and lead to the form of special relativity for a particle with rest mass. In this note we consider (as in (1)) the situation for a particle with no rest mass. We argue that the two flow/flux equations still pertain, but that there is a different equation for momentum (which as a function of E(v) and v) is related to energy flow. For a particle with no rest mass, v=E/p while for one with mass p=E(v)v. How is this possible? We argue that v=E/p describes an object with energy which is entirely active in creating motion i.e. energy flow. In other words for pv=E, p→0 and v→0 means E→0 which does not allow for rest energy. This describes a classical wave and also the photon or phonon. Such an object cannot be at rest because its energy is equivalent to motion so to speak. It is possible to trap such a particle in a box so it bounces back and forth. In such a case, there is rest energy and a different notion of energy flow, namely p=E(v)v. Thus, we argue that there is no contradiction in having two equations for momentum, one pertaining to particles with rest mass and the other to those without.
In Part I we argued that two flow equations for a free particle, namely: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1a)) may be applied to both particles with rest mass and those without. In fact we argue that there are at least three scenarios associated with ((1a)) and ((1b)). The first is classical mechanics for a particle with rest mass mo and constant velocity. Using ((1a)) and ((1b)) with p=E(v)v this scenario yields special relativity. A second solution is to write ((1a)) and ((1b)) as eigenvalue equations: -id/dx (partial) exp(iA) = p exp(iA) and id/dt (partial) exp(iA)= E exp(iA) where A=-Et+px. This yields a quantum mechanical solution for a particle with rest mo if one assumes p=Ev. In such a case one may use the special relativistic result EE = pp + momo and establish the Klein-Gordon equation. A third solution pertains to objects with zero rest mass. As argued in Part I, these do not follow p=E(v)v, but rather the classical wave result: v=E/p. In this note we examine massless particles, namely waves on a string, photons and sound, in more detail. For the zero mass case, dA=0 so there is motion in such a way as to keep A constant. If one considers periodic motion and uses ((1a)) and ((1b)), a solution which is real and periodic is: d/dt d/dt cos(A) = vv d/dx d/dx cos(A) ((2)) A second order partial differential equation is needed as one insists the solution be real. This differs from the quantum mechanical solution of exp(-iEt+ipx) which is complex for a reason. The modulus must be linked to a probability of 1 for a free particle. The zero mass solution is one of a smeared-out physical wave. We note that ((1a)) and ((1b)) deal strictly with E energy and p momentum, but cos(A)=cos(-Et+px) is associated with a period proportional to 1/E and a wavelength proportional to 1/p. This raises two questions. First, what physical entity is periodic? Secondly, how does the energy appear in the problem as energy and not just 2*3.14 frequency of the physical entity? We try to consider this issue for a wave on a string, photon and sound wave.
In a previous note (1), we argued that one may obtain either the Schrodinger or Klein-Gordon equation from the classical free particle Action = Integral dt L = T L(v) by writing v=X/T and using: p= dAction/dx partial and -E = dAction/dt partial. This may be used together with an energy-momentum relationship E=pp/2m (nonrelativistic) or EE=pp + mm (relativistic) to create an eigenfunction equation by using exp(i Action) as a solution. This is equivalent to exp(-iEt+ipx). In (2), we argued that if one generalizes momentum p to Ev where E(v) is energy and uses nonrelativsitic Lagrangian equations: p=dL/dv and H=E= pv-L, one may find the relativistic Lagrangian for a free particle which in turn yields p= d/dx (TL) partial and -E= d/dt (TL) partial i.e. the form of energy and momentum for a free particle in special relativity. In this brief note, we argue that the Action/Lagrangian formalism for a free particle seems to follow directly from a flow(flux) type equation namely: d/dx Action partial = p = Ev (= mo v nonrelativistic limit). If energy is flowing out of a region given by the usual Gaussian flux equation d/dx function (i.e. grad dot vector) represented by d/dx Action partial, then one is losing energy E(v) in this region. Thus d/dt Action partial = -E. As a result, one should analyze the problem in special relativity i.e. use p=E(v) v (but no explicit form for E(v)) and then take the nonrelativistic limit to get the nonrelativistic approximation. As a result, assuming a flow problem with a function Action(x,t) leads to the Lagrangian formalism. From this one may obtain the forms of special relativity, classical equations of motion and quantum mechanics for a free particle (relativistic or nonrelativistic).
In previous notes (1)(2), we argued one may write two flow/flux equations for a free particle using an unknown function A(x,t) i.e. d/dx (partial) A = E(v) v ((1a)) and d/dt (partial) A = -E(v) ((1b)) where E(v) is energy which is a function of rest mass and velocity v=constant. A(x,t), but after taking derivatives, x/t is set to v. As shown in (2), one may write A=tL(v) and find E(v)=mo/sqrt(1-vv) (c=1) (and p=Ev) i.e. the result of special relativity. Thus special relativity seems to follow directly from a flow/flux equation where motion v increases energy. Alternatively, one may obtain both relativistic/nonrelativistic quantum mechanics for a free particle, by converting ((1a)) and ((1b)) into differential equations i.e. id/dt exp(iA) = Eexp(iA) and -id/dx exp(iA) = pexp(iA) with A=-Et+px yielding the same result. In both cases, physical motion v has physical implications. For special relativity, energy changes from mo to mo/sqrt(1-vv) and there are effects on clocks and lengths. For quantum mechanics there is time and space resolution linked to dx→1/p and dt→1/E. In this note we try to investigate if there is a connection between the two. In particular we try to see why -Et+px within the free particle wavefunction exp(-iEt+px) is a Lorentz invariant.
In a previous note (1), we argued that both free particle quantum and classical mechanics follow from an energy flow/flux scheme based on: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) A(x,t) turns out to be the classical action tL(v) (L=Lagrangian) with x/t=v and the above hold for both the relativistic and nonrelativistic cases. We further argued that ((1a)) and ((1b)) lead to two sets of solutions, one classical and the other quantum. The classical solution assumes dt→0 and dx→0 i.e. flows may be followed at each point in space (and time). In (2) we argued that ((1a)) and ((1b)) are consistent with p=dL/dv (partial) and H=E= pv - L. From these together with p=Ev one may find: E=mo/sqrt(1-vv) (c=1). For the nonrelativistic case, E= mocc + .5mvv. In this classical analysis one may resolve each t and x, thus if one has a function V(x) (potential) representing energy, the nonrelativistic result is: (E-V(x)) = .5mvv. The quantum solution to ((1a)) and ((1b)) is very different. It establishes eigenvalue equations i.e. id/dt (partial) exp(iA) = E exp(iA) and -id/dx (partial) exp(iA) = p exp(iA) for the free particle. The importance of this is that new physical mechanisms seem to come into play. A= -Et+px so exp(-iEt+ipx) suggests time and spatial regions proportional to 1/E and 1/p in which there is some ambiguity i.e. there are two dimensions and there seems to be some kind of probability for the particle to be in one dimension or the other at x (and similar results for t) i.e. cos(px) and sin(px). As a result, one may not have dx→0 and dt→0. Furthermore, it is argued that a two dimensional associated probability scheme must accompany energy flow. At first this may seem strange, but if one considers a photon which has an energy flow Ev, one has electric and magnetic fields perpendicular to each other and perpendicular to the motion which undergo periodic motion linked to cos(wt) and sin(wt). Thus, the quantum particle periodic probability mimics an already known result from Maxwell’s equations. In fact, we have argued that for fermions there is additional angular motion i.e. spin linked with the recognition of 3-space. Thus we argue that quantum mechanics is involved with periodic type probability motions which are needed to resolve dx and dt (linked to exp(-iEt+ipx) and periodic probability motion (spin) and are also needed for fermions to resolve 3-space as two p vectors may have point in different directions (i.e. spin). Quantum mechanics therefore seems to be a very different theory from classical mechanics, but both appear to be based on energy flow/flux. As a result, the average of a quantum bound state should be linked to a classical system we argue in this note as one only deals with energy flow/flux.
Date on paper should be 2022 not 2021. In a previous note, we argued that for a free particle there exists a function A(x,t) such that d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)). In particular, we used p=E(v) v for the relativistic case and p=mov. We showed that A(x,t) is the classical action for a free particle Integral dt L = T L with v=x/t before the derivatives are taken. After, x/t is set to v. As a result, there is a nonrelativistic A(x,t)= T .5mvv and a relativistic one A= -mo sqrt(1-vv) (c=1). We argued that ((1a)) and ((1b)) are in the form of flow equations. Energy E flows through space out of a dx region (i.e there is an energy flux Ev) so if x is held constant, the change in time is -E. We also noted that one may obtain both the Schrodinger and Klein-Gordon equations from ((1a)) and ((1b)) (see reference in (1)). As a result, A(x,t) is linked to both the nonrelativistic and relativistic free particle actions, yields free particle quantum mechanical equations and may also be used to obtain the form of special relativity (as discussed in the reference in (1)). In this note, we wish to consider spin which is not present in ((1a)) and ((1b)). For both relativistic and nonrelativistic classical dynamics which are linked to A(x,t) one deals with numbers (scalars) and so no notion of spin arises. In obtaining both the relativistic Klein-Gordon equation and nonrelativistic Schrodinger equation from ((1a)) and ((1b)), one creates an eigenvalue equation i.e. id/dt exp(i Action) = E exp(iAction) ((2a)) and -id/dx exp(i Action) = p exp(i Action) ((2b)) which only hold for the free particle. One may then use E and p in expressions EE = pp + momo (where p=Ev= dAction/dx partial) and E = pp/2m + V(x) where p=mov=dAction/dx (partial). For the free particle, one has ((2a)) and ((2b)) without establishing the Klein-Gordon equation. Given that p=Ev is a vector, one may try to write ((2b)) as a vector, but given that one has operators acting on exp(i Action) one does not use a basis vector ei (ei dot ei=1), but rather a “basis” operator or a spin matrix. Thus, we argue that spin emerges in this way without having to linearize the Klein-Gordon equation. Interestingly, this approach only holds for fermions. For the photon, which is a boson, a Dirac-like equation may be obtained directly from a d/dt (partial) EL EL + BB + b grad dot (El x B) = 0 as shown in (2) (with a,b being constants). This again follows from a flow scheme.
In classical physics it is well known that a stationary solution of the classical action Integral dt L ,where L is the Lagrangian, leads to a differential equation whose solution x(t) describes classical mechanical motion. In previous notes we have argued that two free particle flow/flux equations d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E lead to A(x,t) being the classical free particle action with a solution: A(x,t)= -Et+px. In this note, we consider two stationary solutions i.e. dA=0 which differ from the classical approach. One is for a particle with rest mass for which we assume p=E(v)v. The stationary behaviour leads to the form (Lorentz matrix) of special relativity and also to relativistic/nonrelativistic quantum mechanics for a free particle. The second dA=0 solution leads to a zero rest mass particle for which the Lorentz transformation of the first dA=0 solution also holds because (x,t) transforms in the same manner for both zero and nonzero rest masses. We argue that dA=0 i.e. the action is stationary because A=-Et+px is a Lorentz invariant i.e. a dot product.