In previous notes (1) we discussed how (p=0, E=mocc,) (x=0, to) transforms to p’, E’, x’,t’ such that x’/t’=v. We noted that (x=0, to) is a special (x,t) vector linked to a stationary frame. We noted that a general vector (x,t) = (0,t) + (x,0) and that (x,0) Lorentz transforms such that the Lorentz invariantt -Et+px =0. We also noted that E scales t and p scales x and argued that this was the basis for the frequency and wavelength for a quantum free particle. In this note we investigate this idea further. In particular we argue that given -Et+px, it is not only the case that (x,0) transforms into (x1,t1) such that -E’x1+p’t1=0, but that 1/E=t and 1/p=x also yields -E 1/E + p/p=0. In other words there is freedom in both t and x proportional to 1/E and 1/p even though the special vector (0,to) defines the velocity i.e. x’/t’=v. This freedom may be associated with uncertainty in position and time even though there is a well defined x’/t’=v. This uncertainty has important consequences if interactions occur i.e. interference effects.These are usually associated with a wave which has the dispersion relationship E=pv, but we argue that they also hold for a particle with rest mass i.e p=E(v) v because it is the uncertainty in t and x which keeps -Et+px=0 which is the source. This also happens to hold for a photon which is a wave in the sense that it has no rest mass.The uncertainty itself yields the velocity i.e. x/t = 1/p / (1/E) = E/p = c. Given the periodic nature of -E 1/E + p 1/p=0 i.e. -E n/E + p/p = 0 where n=1,2,3… we argue that a periodic function be sought such that it describes a kind of probability due to uncertainty in x,t, but this uncertainty disappears when t=n/E and x=n/p.
In previous notes (1) we discussed how (p=0, E=mocc,) (x=0, to) transforms to p’, E’, x’,t’ such that x’/t’=v. We noted that (x=0, to) is a special (x,t) vector linked to a stationary frame. We noted that a general vector (x,t) = (0,t) + (x,0) and that (x,0) Lorentz transforms such that the Lorentz invariantt -Et+px =0. We also noted that E scales t and p scales x and argued that this was the basis for the frequency and wavelength for a quantum free particle. In this note we investigate this idea further. In particular we argue that given -Et+px, it is not only the case that (x,0) transforms into (x1,t1) such that -E’x1+p’t1=0, but that 1/E=t and 1/p=x also yields -E 1/E + p/p=0. In other words there is freedom in both t and x proportional to 1/E and 1/p even though the special vector (0,to) defines the velocity i.e. x’/t’=v. This freedom may be associated with uncertainty in position and time even though there is a well defined x’/t’=v. This uncertainty has important consequences if interactions occur i.e. interference effects.These are usually associated with a wave which has the dispersion relationship E=pv, but we argue that they also hold for a particle with rest mass i.e p=E(v) v because it is the uncertainty in t and x which keeps -Et+px=0 which is the source. This also happens to hold for a photon which is a wave in the sense that it has no rest mass.The uncertainty itself yields the velocity i.e. x/t = 1/p / (1/E) = E/p = c. Given the periodic nature of -E 1/E + p 1/p=0 i.e. -E n/E + p/p = 0 where n=1,2,3… we argue that a periodic function be sought such that it describes a kind of probability due to uncertainty in x,t, but this uncertainty disappears when t=n/E and x=n/p.
In Part I we suggested that free particle quantum mechanics seems to follow from special relativity. In this note, we focus on the idea of a rest frame defining a rest mass. In such a frame momentum is zero i.e. one has the 4-vector (0,mo)=(p,E). (x,t), however, is also a 4-vector which is usually assigned to a point in space time. In Part I, however, we considered (x,t)=(0,to) associating p=0 and x=0 and alternatively mo and to. Velocity is usually defined as dx/dt =v, but using the Lorentz transformation (1) (with v set to -v), (x,t)=(0,to) transforms to x’/t’ = v. The relationship between x,p and t,E is further manifested through the Lorentz invariant -Et+px in which E scales t and p, x. This we suggest breaks space into pieces proportional to 1/p and time (independently) into pieces proportional to 1/E We next consider particles “created” at different x points. In other words (0,mo) applies, but the particles are at different x positions. In order to keep the relationship p=0,x=0, one must have an x origin at all particle positions. Then spatial scaling of 1/p does not line up unless the particles are n/p (n=integer) apart. Otherwise the scaling in space is unaligned. For different p, scaling in space is unaligned already with (0,mo) and (0, to). Thus for two particles with the same p, but created at different x values at different times and arriving at x1, t1, if there is no alignment, special relativity suggests these two are distinguishable, while Newtonian mechanics does not distinguish. This distinguishability, however, depends on a physical mechanism i.e. a wavelength. Thus, even without considering different p (momentum) values, two identical momenta should be distinguishable if they are not aligned in terms of p scaling even for the same rest mass mo. This we argue is strictly a result of special relativity and is support for the exp(ipx) form of quantum free particle mechanics.
We take as our starting point the free particle classical action which may be written as A= -Et+px if t and x are treated as independent variables. It is possible to find two relationships linking v (velocity) to x and t from dA=0. The first is the classical wave result vp = E while the second is p=Ev where E=mo/sqrt(1-vv) (c=1). Furthermore one may write dA/dx partial = p ((1a)) and dA/dt partial = -E. ((1b)) This may be generalized to two eigenvalue differential equations, but this immediately begs the question: Why do this? We argue that Lorentz invariant form -Et+px represents scaling of both t and x. For example, p scales x and we argue there should be physical relevance to this. This physical relevance results in the wavelength proportional to 1/p. In other words, -Et+px represents more than velocity. The treatment of the scaling for a quantum wave and classical wave differ, we argue, in that the former keeps E,t and p,x separate while a classical wave does not. In other words pv=E applies to a classical wave which is described by cos(-Et+px). In other words, one takes ((1a)) and ((1b)) which separate E,t and p,x into two independent equations and takes account of scaling by combining the two (to include both x and t) i.e. creating the classical wave equation: 1/vv d/dt d/dt partial W(x,t) = d/dx d/dx partial W(x,t). We argue that quantum mechanics on the other hand keeps ((1a)) and ((1b)) separate. The velocity (relativistic) p=Ev is not a wave velocity. A solution to ((1a)) is exp(ipx) and to ((1b)) exp(-iEt). These are usually combined for a free particle to give exp(-iEt+ipx) and a wave is described with a phase velocity pv(phase)=E. We prefer to keep exp(-iEt) and exp(ipx) separate . For example, for two slit interference one does not make use of phase velocity or exp(-iEt). For a bound state one uses p=Ev, but a single free particle is represented by exp(ipx) in W(x)=wavefunction = Sum over p a(p)exp(ipx). exp(-iEn t) appears after one has found En, using the exp(ipx)’s in the sense that one does not include an exp(- i pp/2m t) for each exp(ipx). We argue that by separating E,t and p,x, the latter leads to a scheme for which there is “no time”. It is true that the particle moves at a speed x/t, but p=m1v1=m2v2 etc so p does not uniquely define velocity, rather it defines an impulse hit 2p = Integral F dt . Thus the scaling px which leads to an x, p only wave exp(ipx) is linked to an interaction. If the particle is moving freely and not interacting, then one has classical determinism i.e. v=constant = x/t. As soon as an interaction occurs, one changes from variables x,t which are linked to x, P conditional (x/p) and there is uncertainty in the position of x within a wavelength. The phase in exp(ipx) is only dependent on x not on the time the particle reaches x. exp(ipx) thus represents a probability and one must consider a sum of different possible outcomes (as in a two slit experiment) in order to describe an interaction. This is very different from the local interaction of classical mechanics. Nevertheless for a bound state one makes use of the two formalisms i.e. KE(x)= Sum over p pp/2m a(p)exp(ipx) / W(x) = KEclassical(x). We try to discuss some of these ideas in this note.
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.
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 general, x and t are considered free variables taking on all real values (with t>0). A particle’s velocity may be calculated from x(t). For the Lorentz invariant A= -Et+px one may consider E for a free particle to follow from Integral F dx and p from Integral Fdt. If one wishes to put these on an equal footing, then Fdx requires a dt and Fdt a dx. This means t is replaced with dt and x with dx in -Et+px. What does it mean to have t and x intervals associated with a specific E,p and continuous x(t) motion? There does not seem to be either an x or t interval. There are, however, dx and dt intervals associated with the creation of E and p from p=0.In other words, these dt, dx intervals are physically real properties just like p and E and are manifested experimentally. If one imposes time reversal symmetry, not only should these intervals represent a kind of probability, but they should be symmetric. Thus, x=0 and t=0 are starting points such that A=0. A is a Lorentz invariant and so the end point should also have A=0 for time reversal to apply. This time reversal leads to two symmetric distributions, one in time with hbar/E as a period and one in space with hbar/p as a wavelength.This, however, does not show the direction of motion, say in the spatial distribution. Thus the symmetric x-probability distribution should ultimately be linked to both p and -p i.e should have a vector nature, although this has to be formally proved. A full x-probability distribution linked with p should show the direction of motion. One may write F(px) for the even symmetric function described above and G(px) and G(-px) for skewed full probability distributions such that G(px) + G(-px) = 2F(px). This implies G(px) = F(xp) + B(px) where B is an odd function. We discussed the idea of G(px) being the sum of an odd and even function in previous notes, but did not try to give a reason why F(px) should be an even function with probability endpoints of 0. In this note, we argue that this is due to time reversal invariance. A free Newtonian particle follows x= vt and this should hold on average for a quantum particle which travels through space measured by an external ruler and clock. Above, however, we discussed probability distributions due to p. In fact their sharpness d/dx is proportional to p. How does one eliminate x dependence? As we argued in a previous note (2), one may take the AND condition G(px)G(-px) which should remove all motion.This should be a constant. It is not just motion which is removed, but overall impulse which is the cause of the x-distributions. The square nature of G(px)G(-px) explains why quantum mechanics deals with square root probabilities. These are the probabilities which are linked to impulse, hence to interactions, but the spatial distribution cos(px) from exp(ipx), does not describe the motion x=vt which is also real.The problem, as we show, is that G(px) cannot be real. This forces one to adopt a two dimensional probability and ultimately obtain exp(ipx) i.e. the two dimensional vector one might expect.
The classical action (relativistic or nonrelativistic) may be written with v=x/t and x and t varied independently to yield dA/dx partial = p and dA/dt partial = -E. Thus A = -Et+px is a solution. This expression also appears in classical wave forms and in the electromagnetic description of a photon (from Maxwell’s laws). -Et+px is a Lorentz invariant and so is directly linked to special relativity and different frames moving with constant velocity. In (1) we argued that x and t should be treated as independent in A. This suggests that p is a scaling in space and E in time. Thus there should be physical lengths proportional to 1/p and physical time intervals proportional to 1/E. Special relativity considers different frames (moving with constant velocity) and 4-vectors so it seems to be quite different from a scenario with t and x separated and scaling of x by p. We argue that although -Et+px is a Lorentz invariant and as such may be used to find E(v) and p(v), quantum mechanics is usually performed in a special frame, namey one in which V(x) or V(x,v) the potential is at rest. (For electromagnetism one treats (A, phi) as a 4-vector, but this is not the norm.) As a result, energy En found in a rest frame of the potential also represents “rest energy” so En is binding energy which is added to rest mass energy. In other words motion in the rest frame of a potential creates rest energy = rest mass. We argue that V(x) may be written as a sum over k Vk exp(ikx) and so no time related terms appear, only momentum linked ones. Thus momentum is the key feature for the interaction, even though velocity and time are still in the picture (just sometimes not directly in the interaction scheme). For example, a two slit interference or single slit diffraction does not depend on time or speed although these are certainly in the problem. For a bound state v and t are also in the problem as one calculates average kinetic energy using pp/2m = .5 p v which is rate dependent, but there is no acceleration which is a time-dependent phenomenon. Rather one deals with an ensemble of exp(ipx)’s because these interact with the potential in a wavelike manner to create the effect of acceleration. exp(ipx) a momentum wave linked to the scaling of x by p in px is not only involved in an interaction with V(x), but represents extra information from the classical mechanical picture. exp(ipx) for a known rest mass mo means one knows v=x/t=dx/dt, yet there is an extra piece of information namely the phase px which is involved in even “free” interactions such as two idendical particle with different phases arriving at the same x at the same t or the average kinetic energy in a bound state. Velocity v is converted into a momentum expression through p/mo in the norelativistic case and so exp(ipx) governs the whole scenario. Acceleration then becomes a mathematical abstraction of the motion of vrms(x) for any quantum bound energy level with the real physics associated with exp(ipx) being ignored.