In previous notes (1), we argued that stochasticity (in x and t for a fixed E energy and p momentum) may occur at a single particle level based on the Lorentz invariant A = -Et+px ((1)). This invariant is also the relativistic/nonrelativistic classical action for v=x/t. The argument made in (1) was that a fluctuation in t of constant/E could cancel one in x of constant/p. We further argued one could create eigenvalue equations using id/dt and -id/dx and obtain the two dimensional probabilities exp(-iEt) and exp(ipx), i.e obtain a description of free particle quantum mechanics. Experimentally, both exp(-iEt) and exp(ipx) have been verified through interference patterns. Theoretically, however, ((1)) was known about 1905 and no single particle stochasticity was associated with it. In this note, we try to argue why there should be an associated stochasticity associated with ((1)) based on motion through a constant potential for a particle with rest mass and motion through a medium with an index of refraction for a photon. We in fact argue that one cannot solve certain photon problems such as one dimensional reflection/refraction without explicitly considering the stochasticity which is usually left out when describing p2=pn2 and c2=c/n2 where n2 is the index of refraction in the second medium and p and c are values in the first medium with n1=1. We further consider exp(ipx) and exp(-iEt) as arising from the maximization of Shannon’s entropy subject to certain constraints. The product of these two two-dimensional periodic probabilities is exp(-iEt+ipx) which we argue must hold as seen from observers moving with constant velocity. Thus we argue that the Lorentz invariant -Et+px follows from statistical considerations alone. Given this expression (a subset of the whole picture), one may try to create a deterministic viewpoint by setting x/t=v=constant. This allows one to consider transformations which define E and p in terms of mo and v and yields the invariant EE = pp+momo (c=1). Thus we argue that the conservation of energy equation ultimately follows from a deterministic view of the argument of a statistical probability. In other words, it is only a subset of the full picture i.e the quantum mechanical statistical one.
In previous notes (1), we argued that stochasticity (in x and t for a fixed E energy and p momentum) may occur at a single particle level based on the Lorentz invariant A = -Et+px ((1)). This invariant is also the relativistic/nonrelativistic classical action for v=x/t. The argument made in (1) was that a fluctuation in t of constant/E could cancel one in x of constant/p. We further argued one could create eigenvalue equations using id/dt and -id/dx and obtain the two dimensional probabilities exp(-iEt) and exp(ipx), i.e obtain a description of free particle quantum mechanics. Experimentally, both exp(-iEt) and exp(ipx) have been verified through interference patterns. Theoretically, however, ((1)) was known about 1905 and no single particle stochasticity was associated with it. In this note, we try to argue why there should be an associated stochasticity associated with ((1)) based on motion through a constant potential for a particle with rest mass and motion through a medium with an index of refraction for a photon. We in fact argue that one cannot solve certain photon problems such as one dimensional reflection/refraction without explicitly considering the stochasticity which is usually left out when describing p2=pn2 and c2=c/n2 where n2 is the index of refraction in the second medium and p and c are values in the first medium with n1=1. We further consider exp(ipx) and exp(-iEt) as arising from the maximization of Shannon’s entropy subject to certain constraints. The product of these two two-dimensional periodic probabilities is exp(-iEt+ipx) which we argue must hold as seen from observers moving with constant velocity. Thus we argue that the Lorentz invariant -Et+px follows from statistical considerations alone. Given this expression (a subset of the whole picture), one may try to create a deterministic viewpoint by setting x/t=v=constant. This allows one to consider transformations which define E and p in terms of mo and v and yields the invariant EE = pp+momo (c=1). Thus we argue that the conservation of energy equation ultimately follows from a deterministic view of the argument of a statistical probability. In other words, it is only a subset of the full picture i.e the quantum mechanical statistical one.
The Klein-Gordon equation is often created by taking the deterministic relativistic invariant EE = pp+ momo (c=1), replacing E with E-V(x), E=id/dt partial and p with -id/dx partial and having the derivatives act on a function W(x,t). This seems to suggest that the starting point is deterministic classical mechanics (special relativistic form) E,p equation with stochasticity somehow introduced by having average E and p follow from the action of the operators id/dt partial and -id/dx partial on a stochastic function W(x,t). It also makes it somewhat difficult to understand, from first principles, why one would create a linear equation in E,p (compatible with EE=pp+momo), i.e. the Dirac equation. We suggest that the entire approach to Klein-Gordon equation is stochastic, with the EE=pp+momo equation appearing as an invariant expression of average values which describes motion as seen from different frames moving with constant average velocity v. In such a case, one is not limited to the form EE=pp+momo and there is nothing unusual about trying to create an equation linear in E and p. In particular, we argue that the free particle form exp(-iEt) exp(ipx) follows from two separate maximizations of Shannon’s entropy subject to complex constraints (for periodicity). As a result, free motion with a constant E and p is already stochastic as seen from a photon moving in a region with an index of refraction n or a particle moving in a region with constant potential. Changing to a potential V(x) leads to the notion of an average kinetic energy because a constant p is associated with a distribution in x i.e. exp(ipx). One may think in terms of these average values in order to consider frames moving with constant v. This leads to both -Et+px -EE+pp=momo being invariants. If a V(x) is present, one may replace E with E-V(x), but then needs a statistical average Sum over p a(p)exp(ipx) to describe pp.. The time portion may still be exp(-iEt) if one has a bound system characterized by a single average energy. This yields the Klein-Gordon equation.There is, however, no particular reason why one could not use an equation linear in E and p (consistent with -EE+pp=momo), because the latter follows form the an average v (deterministic view) of the stochastic exp(-iEt+ipx) two-dimensional probability.
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 a previous note (1) we argued that quantum square root probabilities follow from the consideration of light reflecting and refracting at a surface along the y direction at x=0. No knowledge of Maxwell’s periodic photon electric field is needed. The photon number and energy are conserved at a single photon level, but flux/intensity may be included through (2) AA/c1 = BB/c1 +CC/c2 where AA is the intensity of the incident beam (AA>0), BB the reflected and CC the refracted. There are two unknowns B,C if A is known and one equation which may be broken into two A+B=C and A/c1 -B/c1 = C/c2. For n1<n2 (n=index of refraction B<0). Thus we argue that there is an underlying square root probability. In this note we first heuristically suggest the form for the square root probability. A key point is that “particle” features such as E and p (for a photon) govern the dynamical wavelike probability. This also applies to a particle with rest mass. We also ask why this square root probability has not appeared in dynamical theories and argue that it appears in special relativity (i.e. a Lorentz boost) for a particle with rest mass mo at x=0 at t=to. As seen from a frame moving at constant speed -v, this particle moves with speed v. Thus x’=vt’, yet a 2x2 Lorentz transformation is applied to (x=0,t=to). Only one relation is needed because x’=vt’ and p’=E’/cc v suffice to yield the variables x’ and p’. Thus we argue that a Lorentz transformation describes more than is necessary for a simple particle. The 2x2 nature of the problem we suggest describes more than a simple particle. It leads to the Lorentz invariant -Et+px ,but x=vt so why should x and t appear as if independent? We argue that -Et+px=A is compatible with dA/dx |constant t = p and dA/dt |constant x = -E. P and E are “particle/photon” properties, but exp(ipx) or exp(-iEt) represent eigensolutions of -id/dx and id/dt. Thus if x and t are independent exp(ipx) may be interpreted as existing at all x points (as there is no time). This is the picture of an unusual probability which is two dimensional, with components which may be positive or negative as in the reflection/refraction problem. Thus we argue that the notion of a wave probability or square root probability not only exists in the reflection/refraction problem, but in the case of special relativity with what looks like an unnecessary 2x2 matrix from the point of view a particle (mo, x=0, t=to) describing an associated square root wave like probability completely dependent on the particle properties E and p.
A quantum particle (or photon) is described by the two dimensional probabilistic exp(-iEt+ipx), with t and x being treated as independent. exp(-iEt) suggests a stochasticity in time associated with energy E, but none in space. Similarly, there is a stochasticity in x associated with p but none in time. For -Et+px to be invariant as seen in different frames moving with constant average velocities, -EE+pp must also be an invariant i.e. equal momo (c=1). This leads to a contradiction. exp(ipx) suggests a stochasticity in x for a given p, but -EE+pp=momom converts a given p into an E, so there exists stochasticity in x for E as well (which contracts the notion that exp(-iEt) is only stochastic in time). The link between E and p, however, which breaks x-t independence is the notion of average velocity v=x/t, which is physical (as is exp(ipx)). How does one deal with this contradiction? We suggest that p and E are associated with different physical interaction processes. P is related to impulse hits, while E may be associated with loss of energy over a distance if a potential V(x) exists. Even though exp(ipx) suggests stochasticity in x, at the same time one considers fixed objects (targets, slits) and even potentials V(x) at well-defined deterministic x points. This seems to mean that if one wishes to make energy deterministic at each x, one must create an average of exp(ipx)s. We argue that there seem to be two important ideas associated with exp(ipx). First, it is a two-dimensional probability linked to a flux because it is due to p which is due to a v. Probability flux does not get created or destroyed (unless there are sinks or sources). This leads to issues when dealing with a bound state which should be localized. A flux probability is not localized, nor is an average in general (except in the case of an infinite potential). A bound state problem is really a flux problem consisting of a particle trapped in a potential together with it tunneling out or in i.e. one has a dynamic equilibrium picture somewhat like evaporation in equilibrium. (To see the clear difference between a state based probability and a flux-based one, consider the problem of a photon moving in the x direction and reflecting/refracting from an n1-n2 (index of refraction) y-plane at x=0. State probabilities (P(reflect), P(refract)) are not sufficient to solve the problem, but a flux-based one exp(ipx) which allows a single particle to appear as if in a steady stream scenario is.) Secondly, unlike evaporation, the guiding statistical condition for exp(ipx) is that its average makes E deterministic in x space. Thus E is like T (temperature) in that it is a statistical parameter, but it is not based on maximization of entropy (equilibration), but rather in creating E as an average which is then treated as deterministic (ignoring in a sense exp(-iEt) value characterizing an equilibrium system (not based on equilibration). This deterministic E system is linked with discrete values (due to the exp(ipx) equilibrium), but given that E is treated as deterministic, a stochastic process based on maximization of entropy is introduced to describe the absorption and emission of photons (which allow for jumps from one average-deterministic En to another). We argue that the flux-like probability exp(ipx) exists in order to deal with sharp interfaces (e.g. n1-n2 index of refraction) at a deterministic x=0 which probabilistically leads to different p values. It uses the reverse approach i.e. a deterministic p and a stochastic x. In such a way, one may use continuity (and first derivative continuity) at the sharp x with flux-probabilities containing the various allowed p values (due to the impulses). The continuity follows from no sinks or sources existing for the flux-like probability.
In a number of previous notes, we argued that free particle quantum mechanics, i.e. exp(-iEt+ipx), follows from the Lorentz invariant A = -Et+px. For t=0,x=0, A=0, but we argued that there is another solution, dx = hbar/p and dt=hbar/E (we use intervals because x=0, t=0 is a trajectory point). We then noted that this solution is linked with the probability exp(-iEt+ipx) which is Lorentz invariant and creates the above intervals in t and x. Finally, we remarked on the fact that exp(-iEt) conserves energy and exp(ipx), momentum, when one uses these for 2-body collisions, i.e. AND situations. Here we point out that there are numerous dt= f(p,E) and dx=g(p,E) (f, g are arbitrary functions) which yield A=0. For example, dt=p and dx=E is an example. Only dx=hbar/p and dt =hbar/E, however, lead to a probability form which conserves momentum and energy and this is something which we did not stress clearly in previous notes. Thus, there seem to be a number of issues involved in obtaining a free particle probability exp(-iEt+ipx) and these are independent requirements, which then lead to a particular solution. We argue that these requirements are: ((1a)) One wishes to preserve Lorentz invariance ((1b)) One desires discrete dx, dt intervals instead of Newton’s intervals which tend to 0. ((1c)) These intervals need to be linked with a particle related probability which defines them. This probability is critical physically because if the theory is to define photons and particles equally well ( as special relativity ((1a)) applies to both), one must account for the probabilistic effects of 2-slit interference and 1-D reflection-refraction at an n1-n2 index of refraction junction. ((1d)) One must have conservation of momentum and energy. We argue that it is the combination of these requirements that ultimately leads to the free particle probability exp(-iEt+ipx).
In a previous note (1), we argued that maximization of Shannon’s entropy -P(x)ln(P(x)) subject to the periodic (complex) constraint i p x P(x) yields the free particle two-dimensional probability exp(ipx). Similarly one may obtain exp(-iEt) (with a minus associated with E to allow for -Et+px to remain invariant under t->t+constant/E and x->x+constant/p), also for a free particle. We argued that multiplying the two together exp(-iEt+ipx) yields a phase which should be invariant when seen from frames moving at constant average velocity v=x/t.. This means that not only -Et+px, but -EE+pp= momo (c=1) is a Lorentz invariant. The latter, however, represents conservation of energy for no potential. We argue that conservation of energy for a free particle is linked to both the maximization of Shannon’s entropy for both E and p, i.e. it is an equilibrium in both time and space. We have argued before that in the presence of a potential V(x), one may replace E with E-V(x) in the Klein-Gordon equation. Here we wish to examine the statistical features of this situation in more detail. Finally, we note that the third statistical feature is present, associated with the absorption or emission of a photon. The photon wavefunction exp(-iEt+ipx) is subject to both E and p maximization of Shannon’s entropy and it is associated with an equilibrium E1 or E2 energy level of say an electron in an atom. The difference is that this absorption or emission seems to occur stochastically without being directly linked to the momentum impulse hits between the electron and V(x). In describing photon emission/absorption, one may maximize Shannon’s entropy using only conservation of energy, i..e. use the Maxwell-Boltzmann factor. The x and t notion of maximum entropy are still present as the photon wavefunction is exp(-iEt+px). The photon momentum, however, is conserved with the total momentum of the atom and not the impulse hits of the electron with V(x). The two stochastic features occur together.