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.
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 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 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 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 (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’,
Classical mechanics of a free particle is based on Newtonian mechanics and Lagrangian/Hamilton theories. A single frame of reference is chosen (the lab) and particles with the same mass mo (for example) moving at different velocities are considered in the free particle scenario (i.e in the absence of force (potentials)). We argue that the above is identical to considering a single particle with mass mo at time=to and x=0 viewed by various frames moving at different constant velocities -v. From the point of view of each frame, the particle moves at a velocity v. Thus the Newtonian picture of an mo particle moving at different speeds may be considered as a problem of special relativity. We argue (as in (1)) that (x,t)=(0,t) transforms to (x’,t’) such that x’/t’=v. If one uses a metric (1 0) (first row) (0 -1) second row, one may derive special relativistic relationships. In particular, one may obtain the Lorentz invariant -Et + px if (p,E) and (x,t) transform in the same manner. We argue that the relationships obtained from special relativity pertain to classical mechanics and so one may mathematically obtain a function L(v) such that p=dL/dv and E=pv - L for a free particle. In other words, special relativity governs Lagrangian results in the free particle case and is not really needed (in this special case). More importantly, it is special relativity which links E with t and p with x. As noted above, if one considers Netwonian mechanics in a single frame, the notion of special relativity does not arise, but the scaling of time by E and of x by p does i.e. A= -Et+px is the classical action both relativistically and nonrelativistically. It is just not written in this way usually (i.e. A = -mo t sqrt(1-vv) (c=1) relatiivstically and A =t .5mo vv nonrelativistically) We argued in (2) that scaling must manifest itself physically and that it accounts for the physical wavelength in quantum mechanics. We argue that the quantum mechanics of a free particle ultimately follows from special relativity. As a result, ideas of quantum mechanics would not have appeared in Newtonian mechanics nor Lagrangian or Hamiltonian theories. When special relativity was developed it focused on transformations related to different frames, but these are equivalent to describing one frame with many different velocities (same mo). Thus the physics in one frame is governed by special relativity even if it is nonrelativistic. We argue that the notion of the wavelength which appears even in the Schrodinger formalism follows from scaling of x by p which ultimately originates from special relativity. In other words, both the Lagrangian formalism and quantum mechanics follow from special relativity in the free particle case. We argue that free particle quantum mechanics is the starting point of overall quantum mechanics.
In (1), the notion of a stochastic classical action is introduced and a probability is linked to exp( constant * action). The objective of (1) is not to derive quantum mechanics, but to derive stochastic dynamics based on maximum entropy in action space. We have argued in a number of previous notes (2) that one may use the classical action to create a quantum free particle probability. This classical action, however, is not chosen a priori, but rather arises due to other considerations. We suggest that stochasticity is present in Newtonian mechanics at some time scale because dt is finite in dp/dt= Force (Newton’s second law) and so postulate that force delivery at some x,t is probabilistic within some dt and dx. A priori, dt and dx are not known, but if there is stochasticity, there should be a free particle probability, i.e. a probability(x,t) where x,t represent an x, t interaction point. Furthermore, the notion of dx and dt should arise naturally. We note that the form exp(i function(x,t,p,E) may be introduced because free particles have no distributed variable (like energy in the Maxwell-Boltzmann case). exp(i function(x,t)) immediately introduces periodicity, but we have argued that stochasticity can only exist within tiny, fixed dx, dt linked to dp/dt=F. In particular, one classical Newtonian scales this stochasticity is not seen. The periodicity should be directly in x and t space, suggesting exp( i A1 t + i A2 x). This expression, however, should be Lorentz invariant, and A1 and A2 should be constants, at least for a fixed mo and v. The reason that the one must have periodicity is that the dx and dt ranges of stochasticity in dp/dt=Fr must appear naturally. At this point, one is led to exp(-iEt+ipx) with -Et+px being a Lorentz invariant which satisfies the above criteria. This Lorentz invariant happens to be the classical action in both the relativistic and nonrelativistic cases, hence giving rise to action as being a stochastic variable (of (1)) because it is a function of the two scholastic variables x and t. Alternatively, one may choose the exp(iA1x+iA2t) due to its conservation properties. One may then argue that A1 and A2 could also be conserved for x,t held constant, i.e. A1 and A2 could be linked to Newtonian energy and momentum in elastic collisions. This allows one to solve for the special relativistic g(v) =1/sqrt(1-vv/cc). Given the idea of a conserved quantity form for a probability, one may argue that exp(i Action) satisfies a maximization of Shannon’s entropy subject to a conserved constraint (which is again a link with (1).) The question then becomes: Why should the Lorentz invariant A= -Et+px be the classical action.One may note that x,t usually refer to trajectory values and so for a free particle, one has x=vt. This means that one may factor out t, i.e. A= t( -E+pv), but this is automatically the form of the Lagrangian with E being the Hamiltonian. This holds relativistically and non-relativistically with the form: dL/dv partial = p. The remaining question is: Why should the classical action be Lorentz invariant? We consider E(v)-p(v) v = -L such that dE/dv - v dp/dv -p = -dL/dv partial. This means that dE/dv - v dpd/v = 0 ((1)). A Lorentz transformation, however, changes mo in a rest frame into p=mov/sqrt(1-vv/cc) and E=mocc/sqrt(1-vv/cc). One may see that this solution may be obtained from ((1)) showing a direct link between the classical action approach and special relativity. The connection may be carried further as Lt = -Et+px is a Lorentz invariant, but also the classical action (in both the relativistic and nonrelativistic cases). We argue that the use of the action in (1) as a stochastic variable is directly related to special relativity. In summary, from finite dt in dp/dt=F, t and x are stochastic variables within some naturally appearing dx and dt. One may write a linear combination of x and t and to create a stochastic Lorent invariant variable which should appear in exp(i Lorentz invariant) and create the dx and dt units. It may then be shown that this Lorentz invariant in E,p,x,t should be equivalent to the classical action.