In a previous note, we argued that exp(i Action) for a nonrelativistic particle may be varied independently in T and X, where velocity = X/T. This variation indicates that energy is not kept constant. The solution exp(i Action)= exp(i .5m XX/TT) (for this case) leads to the nonrelativistic Schrodinger equation. In this note, we consider the free particle relativistic Lagragian (1): mo sqrt(1-vv) where v=X/T. We argue that the equation describing independent fluctuations of X and T is the Klein-Gordon equation.
In a previous note, we argued that exp(i Action) for a nonrelativistic particle may be varied independently in T and X, where velocity = X/T. This variation indicates that energy is not kept constant. The solution exp(i Action)= exp(i .5m XX/TT) (for this case) leads to the nonrelativistic Schrodinger equation. In this note, we consider the free particle relativistic Lagragian (1): mo sqrt(1-vv) where v=X/T. We argue that the equation describing independent fluctuations of X and T is the Klein-Gordon equation.
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 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).
The classical action: Integral (0,T) dt1 L(v(t1), x(t1),t1) may be varied to find a stationary solution which yields Newton’s second law, a standard procedure in classical mechanics. Is it possible to obtain quantum mechanical motion from this same classical action? We argue that one may write the action for a relativistic/nonrelativistic particle moving at constant speed in terms of X and T i.e. v=velocity=constant=X/T. Varying with respect to X and T yields: d/dT Action = -E and d/dX Action = p. Thus, changing X by delta X yields a change in the action of p delta X which holds for all X because of the constant velocity. If one envisions a periodic function such as exp(i Action), this behaves as exp(ipx) in terms of -id/dX. Thus, there is an intrinsic wavelength related to 1/p. This follows directly from small changes of the Action. Alternatively, one may create a differential equation such as the Klein-Gordon or Schrodinger equation from classical conservation of energy-momentum using id/dT exp(iAction) = E exp(i Action) and -id/dX exp(iAction) = p exp(i Action). We note that exp(ipx) which describes the fluctuational behaviour of the classical Action may have positive and negative values showing that fluctuations may be both positive and negative These negative and positive fluctuations are, however, periodic so there is some systematic behaviour occurring, it seems.. We argue that exp(ipx) is statistical as its modulus is 1, indicating that a free particle has equal probability to be at any x point. For a system with a potential V(x), one may create an ensemble describing fluctuations of all possible p values i.e. W(x)=wavefunciton= Sum over p a(p) exp(ipx). The fluctuations are regular positive and negative, but there is different wavelength behaviour for each p i.e. 1/p so there is interference to yield an average fluctuation pattern i.e. the wavefunction.
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.
In Part I of this note, we argued that d/dX Action (where Action = Integral dt L, L being the Lagrangian) for a particle with constant velocity is p whether the classical L=.5mvv or the relativistic L=-mo sqrt(1-vv) (c=1) is used. Thus, a fluctuation or error of delta X leads to a change of (delta X) p in the Action. We tried to argue that this led to the form exp(ipx). In this note, we revisit this scenario. We argue that if one wishes to have a constant error due to a fluctuation/error in X in the Action regardless of p, then delta x = 1/p is a characteristic length. At first one may expect a relative error to be of interest, but we argue that a potential may knock a particle from one p value to another. In such a case, it may be useful to have the error, and not the relative error, remain constant. Thus, one already has a wavelength, but this suggests a periodic function to describe the error of the action throughout space, and there are many possibilities. A periodic function may have positive and negative values corresponding to positive and negative X fluctuations in a regular way. We ultimately argue that exp(ipx) is ideal and is also linked to spatial density through exp(-ipx) exp(ipx)=1. Thus, we argue that two ideas are involved. First, the idea of an error/fluctuation in the Action is used to develop the idea of a wavelength. Secondly, one tries to find a periodic function associated with the wavelength which is also linked to spatial density. Finally, one tries to generalize these ideas to the nonrelativistic case of potential present. This seems to present a statistical picture.
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.