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.
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.
In a previous note (1) we argued that two flux/flow equations d/dx partial A(x,t)=p ((1a)) and ddt partial A(x,t) = -E ((1b)) lead to the Lorentz invariant solution A(x,t)= -Et+px. These have two stationary solutions (dA=0), namely dE/dv= dp/dv with x/t=v and p=E(v)v (rest mass nonzero) and dx/dt = E/p (rest mass i.e. a classical wave). The first solution leads to relativistic/nonrelativistic quantum mechanics if ((1a)) and ((1b)) are written as eigenvalue equations. The second represents classical waves and their eigenfunction equation is the classical wave equation d/dt d/dt cos(A) = 1/vv d/dx docs(A)/dx. A periodic solution, we argue, implies action-reaction suggesting force or tension in the system. This seems to be the case for a classical wave on a string with fixed ends or for a sound wave in a gas with pressure. Light treated as a classical wave, however, is more complicated because there is no medium with tension, but rather one has periodically varying electric and magnetic fields. An electric field, however, is considered a force field and so there is an analogy with a periodically changing pressure. A classical wave requires a real periodic solution. A quantum mechanical free particle solution (nonzero rest mass) does not require a real periodic solution because there is no medium or obvious real entity changing periodically, but one may establish a complex solution using eigenvalue equations:id/dt partial exp(i A) = E exp(iA) and -id/dx partial exp(iA) = p exp(iA). At first a complex solution seems unphysical, but one may note that for a fixed t (for example), exp(ipx) is 1 at x=0 and again at pL=2*3.14. Thus there is a special length in the problem which yields the classical mechanical result d/dx partial A = p namely the wavelength 2.*3.14/p. In other words the quantum solution describes the classical solution at each x= n * wavelength. Within each wavelength region one has an exp(ipx) solution which consists of two out of phase periodic functions such that the modulus is 1. We try to argue that like in a classical wave there should be some impedance to a disturbance (like tension of a string or pressure of a gas). This impedance, however, is not represented by one physical entity (e.g. pressure in a sound wave) as one has a complex solution. A complex solution exp(ipx) is not alarming because light treated as a classical wave has two perpendicular entities, namely the electric and magnetic fields, that are both periodic. One may write a classical wave equation for each, but one may write a combined quantum-like wave equation for a photon: i d/dt partial Wi + eijk (-id/dxj) Wk = 0 (where eikj is the Levi-Civita symbol) which has a vector wavefunction Wi= Electric field (i) (x,t) + i Magnetic field (i) (x,t). The only impedance in the zero rest mass problem seems to be the rest mass (inertia) which resists motion. To be more general, it seems that momentum overall should be an impedance to motion. The entire momentum (E(v)v) is associated with an impact and a change in the momentum of a quantum particle (with rest mass) may be resisted by the entire momentum. In classical physics constant velocity motion seems to be a simple translation from one x point to another. The exp(ipx) solution to ((1a)) and ((1b)) shows that this interpretation only holds for n *wavelength points. Within each wavelength region out of phase two dimensional periodic motion occurs which suggests a kind of action-reaction. This behaviour occurs for various objects such as an electron, nucleon, nucleus, atom or even fairly large molecules etc. It has been argued that this quantum motion may apply for even objects in the macroscopic world e.g. mirrors. Impedance for nonzero rest mass quantum objects is velocity dependent as one may many different velocities associated with a nonzero rest mass momentum while classical waves have one velocity (for a given tension scenario). E.g. light always as a velocity of c because it has only one tension scenario i.e. electric and magnetic fields. Changing wavelengths and frequencies are linked to changing energy and momentum, but not velocity. We try to investigate some of these ideas.
It is known that classical waves such as waves on a string, sound and light (from Maxwell’s em equations) take the argument -Et+px where E is associated with a frequency and p with 1/wavelength. We argue that these classical nonrelativistic treatments already bring in features of special relativity (as does Schrodinger equation quantum mechanics for a free particle) as -Et+px is a Lorentz invariant. In fact we argue that Lagrangian formalism: E=pv-L and p=dL/dv is compatible with special relativity. In a previous note (1) we argued that d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E for a free particle with A=tL(v). We also noted that A= -Et+px is a solution, but that one may also write E(v)= E(x/t) and p(x/t) and take the derivatives d/dx partial A and d/dt partial A and obtain the same results. In this note we show how this follows from regular Lagrange theory and note that one already associates d/dx partial with p and d/dt partial with -E i.e. the quantum theory for a free particle is present.
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 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 (1) a Galilean transformation x’ = x-vt and t’=t is applied to the Schrodinger equation id/dt (partial) W(x,t) = -1/2m d/dx d/dx W(x,t) for a free particle, where W(x,t) is the wavefunction, and it is argued that W’(x’,t’) does not equal W(x(x’t,t’),t(x’,t’)). Instead it is shown that W’(x’,t’) = W(x,t) exp( -i/bar [mvx’+.5mvvt’] with x=x’+vt’ and t=t’. The author’s of (1) suggest this is due to remnants of special relativity creeping into the nonrelativistic situation. They then argue that this issue is already found in nonrelativistic classical mechanics, in particular in the Hamiltonian-Jacobi equation H(x, dS/dx) + dS/dt = 0 where S is the free particle action= t *Lagrangian and H is the Hamiltonian i.e 1/2m dS/dx dS/dx. They show that S’(x’,t’) = S(x,t) - (mvx’ +.5mvvt’) and also argue that this is due to a remnant of special relativity creeping into nonrelativistic classical mechanics. In this note, we consider the free particle action S= -Et+px nonrelativistically. (This form also holds for the relativistic free particle). It may be seen that such a solution satisfies the Hamilton-Jacobi equation, but the catch seems to be that one must transform both (x,t) and (p,E) to find a solution in a moving frame. The Galilean transformation leads to two different transformations, one for (x,t) and another for (p,E). A Lorentz transformation on the other hand treats (x,t) and (p,E) in the same way and -Et+px behaves as a dot product with a (1,-1) type metric. We argue that given the physical example of a particle with rest mass mo at x=0 at t=to, a moving frame transforms this into a new energy (including mo and kinetic energy), a p and x’,t’ such that x’/t’=v where -v is the velocity of the frame. One may suggest a more general transformation than the Galilean i.e. replace t’=t with t’=g(v) t. This then leads to -Et+px written in terms of E’,t’, p’,x’ equalling -E’t’ + p’x’ if g(v) = 1/sqrt(1-vv/bb) where b is a constant to make v/b dimensionless. Applying the transformation, which we assume holds for both (x,t) and (p,E) to light yields b=c. One may then show that -Et+px, written in terms of E’,t’,p’,x’, yields -E’t’ + p’x’,
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.