A photon has an energy which experimentally increases the temperature of a gas which absorbs it. This temperature may be measured by a thermometer. The point we wish to make is that there is no notion of spacetime present in this experimentally obtained idea of energy. This seems curious because accoriding to quantum mechanics, there is an uncertainty in interaction time given by hbar/E and so energy and uncertainty in time are linked. One may perhaps start to see a link between E,p and x,t through the special relativistic 4-vector (pc, E). In such a case, p and E don’t exist alone, but c=speed of light of a photon in a vacuum is present. For a photon, furthermore, one has E=pc and again E,p are linked to c and hence x/t. These ideas, however, do not imply quantum mechanics and resolutions in time and space linked to hbar/E and hbar/p. We note that special relativity is traditionally linked to Newtonian mechanics, but is an extension of this theory. We consider the one dimensional case of a photon moving from a vacuum into a region with an index of refraction of n2. According to traditional ideas, the energy of the photon does not change, but c2=c/n2 <c and p2= p*n2, i.e. momentum increases. Physically, one might consider the n2 region to contain atoms or molecules with which the photon may interact (without losing energy), but leading to time delays. In such a case, more time than expected by c would be required for a photon to travel a distance x1. This leads to c2<c and energy remaining the same. At first, one might expect that momentum remains the same as well (at least in the regions between interactions with molecules/atoms). This, however, is not the case. We suggest that if one considers the ideas of E and p being associated with Newton’s second law, i.e. dp/dt = F. As a result, one might expect F (force) to remain the same for a photon in n2 because when it is not interacting it is a vacuum photon. The problem is that dt must be much larger than the vacuum result, because time is wasted for interactions (and no motion occurs for these). For a fixed F, dt is larger in n2 and so dp must be larger. If dp = 2p, then p must be larger as well. In fact, for c2=c/n = dx/dt, where dx is fixed, dt= dx *n/c . Then, p must change to p*n. This, however, is still not quantum mechanical. This is known in classical physics (i.e. the Snell’s law case). One may, however, examine the new resolution of p*n. In the vacuum, on has p and a t means x=ct. In n2, t menas x= ct/n, i.e. a reduction in resolution length by 1/n. Now p2=p*n so resolution in n2 is given by constant/ p2 which is a quantum mechanical result. If one considers the Lorentz invariant -Et+px=0 for a photon, then using constant/p for a delta x, implies delta t = constant/E.
A photon has an energy which experimentally increases the temperature of a gas which absorbs it. This temperature may be measured by a thermometer. The point we wish to make is that there is no notion of spacetime present in this experimentally obtained idea of energy. This seems curious because accoriding to quantum mechanics, there is an uncertainty in interaction time given by hbar/E and so energy and uncertainty in time are linked. One may perhaps start to see a link between E,p and x,t through the special relativistic 4-vector (pc, E). In such a case, p and E don’t exist alone, but c=speed of light of a photon in a vacuum is present. For a photon, furthermore, one has E=pc and again E,p are linked to c and hence x/t. These ideas, however, do not imply quantum mechanics and resolutions in time and space linked to hbar/E and hbar/p. We note that special relativity is traditionally linked to Newtonian mechanics, but is an extension of this theory. We consider the one dimensional case of a photon moving from a vacuum into a region with an index of refraction of n2. According to traditional ideas, the energy of the photon does not change, but c2=c/n2 <c and p2= pn2, i.e. momentum increases. Physically, one might consider the n2 region to contain atoms or molecules with which the photon may interact (without losing energy), but leading to time delays. In such a case, more time than expected by c would be required for a photon to travel a distance x1. This leads to c2<c and energy remaining the same. At first, one might expect that momentum remains the same as well (at least in the regions between interactions with molecules/atoms). This, however, is not the case. We suggest that if one considers the ideas of E and p being associated with Newton’s second law, i.e. dp/dt = F. As a result, one might expect F (force) to remain the same for a photon in n2 because when it is not interacting it is a vacuum photon. The problem is that dt must be much larger than the vacuum result, because time is wasted for interactions (and no motion occurs for these). For a fixed F, dt is larger in n2 and so dp must be larger. If dp = 2p, then p must be larger as well. In fact, for c2=c/n = dx/dt, where dx is fixed, dt= dx n/c . Then, p must change to pn. This, however, is still not quantum mechanical. This is known in classical physics (i.e. the Snell’s law case). One may, however, examine the new resolution of pn. In the vacuum, on has p and a t means x=ct. In n2, t menas x= ct/n, i.e. a reduction in resolution length by 1/n. Now p2=p*n so resolution in n2 is given by constant/ p2 which is a quantum mechanical result. If one considers the Lorentz invariant -Et+px=0 for a photon, then using constant/p for a delta x, implies delta t = constant/E.
Under usual circumstances, i.e. a photon traveling in a vacuum, the notion of a spatial resolution of a photon does not arise (assuming that one does not know about photon interference). One simply focuses on x=ct. If one has an n1-n2 index of refraction junction (as in Part 1) with n1=1, and n2>1, the photon slows down in n2, has the same energy E and a greater momentum p2= n2*p, where p is the momentum in n1. If one considers that a photon in n2 interacts with atoms/molecules, i.e. wastes time that could be used for travel in x, then for a given t, the photon travels a shorter distance, hence c2<c. We suggested in Part 1 that this suggests a sharper resolution in x for p2, due to the shorter distance which we showed was proprotional to 1/p2. We pointed out a mathematical correspondence between the shorter distance x in n2 and 1/p2. Now p2 is a physical property of a particle, but in Part 1, we did not establish spatial resolution as a physical property. We simply noted that in n2, a time t is associated with a shorter x than in n1=1 and that this length in n2 is proportional to 1/p2. Furthermore, we did not suggest any notion of probability, only of spatial resolution. Here we suggest that the notion of probability exists if one considers that in a given time interval t, the time wasted may occur at any point in t or in x for that matter or even at multiple points if one assumes that while the particle is interacting with an atom/molecule, it is not interacting with a force because it is not traveling in x (or is trapped in the dx of the atom and so is not translating overall in one direction. One might argue that a hit from p2 in n2 occurs probabilistically because of these interactions with atoms/molecules, but physically, the idea of a finite fixed number spatial resolution has not been established. We suggest that ideas which apply for p2 in n2, should also apply for a photon moving in a vacuum. Afterall, the photon in n2 is described by E, c2 and p2. One does not really need to know about the interactions with atoms and molecules. This would suggest that the notion of a resolution proportional to 1/p2 should exist in the vacuum even though one cannot assign it to interactions with molecules/atoms. Basically, the interaction with atoms/molecules in n2 served to establish a different dt time for F=dp/dt, i.e. a larger one and hence a larger dp = 2p (for reflection). In other words, the physical implications of a sharper spatial resolution ( x smaller for a given t in n2) was directly linked to 1/p2. We consider E2=n2*E, p2, c for a photon in n1 and E,p2,c2=c/n2 for a photon in n2 and argue that it could make sense to have physical properties of delta t = hbar/E and delta x = hbar/p. We suggest that one may postulate a fixed value dx = constant/p for resolution where constant is a fixed number. There is no longer the idea of simply proportionality. We also suggest that spatial resolution, like p, must be a physical attribute. In n2, we saw that a changed spatial resolution, changed p2 (on average) from F = dp/dt arose. One might argue that one really has a photon with p and c in n2 when it is not interacting, but we suggest that this cannot be resolved. It is the p2 photon with its changed spatial resolution which would emerge in an experiment. If a photon in a vacuum already has a fixed spatial resolution equal to hbar/p, then shifting to p2 in n2 would mean a fixed physical resolution of hbar/p2, but also a resolution of hbar/p2 in a vacuum. In other words, the idea of spatial resolution linked to hbar/p seems to be a statistical approach to handling a problem with the idea that one does not have infinite spatial resolution in x.
Historically, special relativity followed as a generalization of Newtonian mechanics and later quantum mechanics (both nonrelativistic and relativistic) was developed as a somewhat independent theory with some connections to Newtonian/special relativistic concepts such as energy and momentum and their conservation. For example, the Newtonian concept of a potential V(x) is used in quantum mechanics. Here we wish to consider the notion of a system seen from different frames moving with constant speed (-v) together with some basic physical ideas that are presumed to hold at any scale such as the idea of a push or pull changing the speed of a particle and the notion of speed itself. We argue that one may develop special relativistic equations in such a manner, but at the same time consider notions of dp/dt and dE/dx linked not only to Newtonian mechanical ideas, but at the same time to quantum mechanics, by generalizing these forms. We argue that one may start with the notion of energy, momentum and their conservation together with the generalizations of a linear force linked to finite dt=hbar/E and dx=hbar/p. This allows one to a priori write, based on conservation of energy for v<
In Part 1, we argued that Newtonian mechanics essentially contains the notion of probability found in the free particle wavefunction exp(-iEt+ipx). The difference is that the wavefunction/probability quantifies dx=hbar/p and dt=hbar/E. The standard Newtonian approach is to have dx->0 dt->0 in order not to deal with the lengths of these intervals. This limiting feature may work in the classical world (but not for photons reflecting and refracting through windows), as the actual sizes of dx and dt may in general be very small in the classical world, but they are not small in the case of atomic or nuclear lengths. The question is: In Newtonian mechanics, without any reference to exp(-iEt+ipx), how would one obtain finite values for dx and dt? Furthermore, why should one wish to obtain dx,dt values associated with x,t in x=vt. We suggest that because one deals with a moving object, and not a stationary one, any x,t on a x=vt trajectory should be associated with a dx,dt to verify its existence. Unlike the stationary case, for which one may strike the particle and make it move, dx,dt for the moving particle should be dynamical and associated with Lorentz invariance. We suggest that it is meaningless to speak of x,t for a moving particle without a corresponding dx,dt. If one can obtain dx,dt values, then one can argue, as we did in Part 1, that Newtonian mechanics contains the notion of an uncertainty dx about x and dt about t for a free particle. This is essentially the uncertainty which allows the 2-slit interference to occur and solves the probabilistic reflection-refraction problem at an n1-n2 index of refraction interface. Of course, if probability must exist within dx and dt, one requires a math expression for such a probability (i.e. exp(-iEt+ipx). One cannot simply use Newtonian mechanics because it does not make use of such a probability, but at least it predicts that such a probability should exist, we argue. Here we show that one may directly use special relativity to show that dx=hbar/p and dt=hbar/E in Newtonian mechanics, such that one has a dx and dt associated with the same value of the classical action as x,t for which x=vt. The classical action describes, for various x,t values describes the x(t)) trajectory in spacetime and so having built-in dx,dt values in this same action ( which is the Lorentz invariant -Et+px both nonrelativistically and relativistically) shows that Newtonian mechanics must be associated with finite dx,dt. The reason that we consider a dx and dt in the first place is that v=dx/dt. The Newtonian approach is to take the limit dx->0 and dt->0. This, we argue, is associated with a coarse-grained equation x=vt which holds both relativistically and nonrelativistically. This is associated with a coarse-grained classical action.Each x,t on the trajectory x=vt yields a different action value for the action (x/v*.5mvv or -x/v mo sqrt(1-vv/cc) or -Et+px where x/t=v. One may consider x=vt and x+dx1 and t+dt1 as the next points in the trajectory. According to Newtonian mechanics, one may make these values arbitrarily small. Furthermore, x=vt holds both nonrelativsitically and nonrelativisitically. How then can one propose a finite valued dx and dt? We argue that the manner in which this may be achieved is to have a dx,dt associated with x,t which does not change the action value for x,t, i.e. does not change -Et+px. This set dx,dt is finite valued even though the next x,t considered may have a dx1, dt1 value such that dx1<<dx and dt1<<dt. If, however, dx and dt are to be taken seriously, the dx1 and dt1 less than dx,dt are simply math values, not physical ones. In other words, one needs a dx,dt to justify the existence of x,t and if x=vt allows any x+dx1 (dx1<<hbar/p), then this is a coarse-grained equation. The coarse-grained action, however, allows for the defining dx=hbar/p and dt=hbar/E even if the action itself is coarse-grained.
Newtonian mechanics (from the 1600s) defines a quantity called momentum p in terms of mv where m is mass and v=dx/dt with dx and dt tending to zero. Newton’s first law is equivalent to conservation of momentum in the absence of a force. The second demonstrates how p changes under a force. If one has equal action/reaction then overall momentum is conserved. As a consequence the idea of a “wave” associated with p does not arise. The main focus is on the idea of force. Fermat’s Least Time Principle for light (1600s), however, is based on minimizing the overall time traveled by light which reflects or refracts. Young’s interference experiments did not occur until the very early 1800s and Newton considered light to be a particle at this time. In fact light bounces off a foil creating an impact like a particle. Fermat’s principle does not directly make use of momentum or its conservation. It is focused on minimizing time. To do so, however, one must take a derivative with respect to a certain variable and set it equal to 0. This variable is the spatially invariant one in a problem. For example consider reflection (stationary or moving mirror) or refraction in two dimensions x,y. Let the mirror or medium face lie along the x axis at y=0. There exists a constraint on y and given a fixed starting point one cannot change y. This leaves x to be varied because it is invariant. Thus the extremization of time is really a statement of invariance of a spatial variable. In other words, Fermat’s principle is linked to the physical variables t and x while Newton’s focuses on p (momentum). This invariance in turn gives rise to conservation of momentum in the direction of the invariant variable i.e. (px). There is still the idea of Newton’s change in momentum as (py) abruptly changes in refraction as it hits the interface of media. The point we wish to make is that from Fermat’s least time principle one sees that conservation of momentum in the absence of a force is completely intertwined with invariance in space. The two cannot be separated in this approach and there is nothing unphysical about t or x. In fact, one may create a function A= -Et + (px)x + (py)y such that dA/dx partial = px = conserved quantity. In this function (px) and x are on the same footing. One does not have one without the other. This idea is carried over into special relativity as -Et+p dot r is a Lorentz invariant. Given that p momentum is physical (it manifests itself in impulse hits) and p and x are inseparable in (px)x we suggest there should exist a physical manifestation in x i.e. in space linked to the maintenance of conservation of momentum in this direction even in the absence of a force. This we argue leads to the physical idea of a wavelength which follows from A=-Et+px with A=0 in all constant velocity frames leading to a frequency proportional to 1/E and wavelength to 1/p. We have discussed these ideas in previous notes, but the main point we wish to make here is that equivalence of spatial invariance in the x direction in Fermat’s principle with conservation of momentum in the same direction suggests that if p is physical, the spatial invariance must also be manifested physically. It is not simply a mathematical convenience to link t to x to px, we argue. Furthermore this wavelength should be linked to establishing conservation of momentum as it does in quantum mechanics where the orthogonality of exp(ipx) i.e. Integral exp(-i p1 x) exp(ip x) dx automatically ensures p1=p without imposing momentum conservation a priori as in Newton’s first law. Both Newtonian mechanics and Fermat’s principle suggest that p may change abruptly due to a force or an immediate change in medium. The idea of a physical wavelength, however, suggests there may be interactions completely different from an abrupt change in p due to an impulse. In particular there may be interference or Bohm-Aharonov interactions. Newtonian Mechanics
In classical physics, velocity follows from exact measurements of space and time intervals (i.e perfect resolution of both). Inertial mass follows from a resistance of an object to a force (push or pull). Momentum, however, which nonrelativistically is mv and should be associated with ideas of inertial mass, and space time resolution may be instead linked to the physical idea of an impulse. An impulse seems to be like a hit and does not appear to be linked directly to space time physically, although its consequences are i.e. a particle receiving the impulse moves in space and time. In classical mechanics, x(t) and so one considers either equally spaced dt intervals and corresponding dx(t) or vice versa (constant dx) if an integral only involves space dependent values. In quantum mechanics, one has two uncertainty principles, one related to E and t and the other to p and x. We argue that p is linked to the notion of an impulse which does not on the surface involve space or time. Given that p, even in nonrelativistic quantum mechanics, is mv there should at least be some built in notion of dx and dt. From the uncertainty principles one sees that dx (i.e. the wavelength) is linked to hbar/p and dt to hbar/E. Thus neither dt nor dx are constants, but depend on the motion. In the high energy quantum limit dt —> W(x)W(x) dx →1/v(x) dx where v(x) is classical velocity and W(x) the bound state wavefunction. The bound state wavefunction, however, applies to the envelope of WW (not the actual crests and troughs) and so the dx-dt scheme which emerges is a smoothed over one and not the actual dx -dp of quantum mechanics. We try to examine some of these ideas in this note.
In Newtonian mechanics a particle with constant momentum moves deterministically according to x=p/m t. Even a quantum object such as a photon or electron should satisfy this equation on average to give meaning to the notion of the speed of light in a vacuum, for example. Thus it seems surprising that a quantum free particle is also associated with exp(ipx) which only contains momentum and is periodic and complex. Furthermore the physical consequences of exp(ipx) are real as shown in a two-slit interference experiment etc. In this note, we attempt to discover why exp(ipx) exists in the first place by examining the example of the reflection/refraction of light which occurs in the macroscopic classical world. Basically our argument is based on analyzing two important pieces of information in the system, namely the scalar energy (or particle count) and the vector momentum. A main point we make is that energy or particle number considerations are linked to determinism i.e. a specific time of interaction. One may note that the interaction x is fixed by the two medium barrier. This means a single incident photon exists in one time range and the reflected OR refracted photon in another. It is possible to compare the incident photon to either the reflected or refracted or to a sum of probabilities. In particular, a single photon has an energy E and a count of 1. If it reflects, it has an energy E and count of 1 so there is a balance, likewise for refraction. Thinking in terms of probabilities “a” and 1-a to reflect/refract: 1= a + 1-a. Can one perform such a matching for momentum, which is a physical piece of information? We argue one cannot. For example, a single incident photon has a momentum p which is linked to the impulse it may deliver. If it refracts, “all of a sudden” it is linked with momentum of p2 = n2 p where n2 is the index of refraction in the second medium and n1=1. This seems problematic because a physical impulse of 2p is all of a sudden an impulse of 2p2. Thus there exists a big problem which seems to require the notion of probability and all three momentum values p, -p and p2 considered together, unlike the situation of energy/particle number for which a comparison of the incident photon and one possible result yields a balance. Combining all three momenta, however, is also problematic in a deterministic picture because the incident photon exists in one time range and the reflected or refracted in another. How can one possibly compare them together? The solution seems to be to argue that the incident photon is only deterministic on average. In particular, if it has an x probability distribution over a certain length (called a wavelength), one does not know the precise time at which it hits “x” the medium interface. In such a situation, there is time uncertainty which allows all three momenta to be compared at the same x. This, however, leads to a major consequence, namely that so-called deterministic motion associated with constant p is deterministic on average, but contains repeated pockets of x probability distributions such that time is not deterministic within. We try to examine these ideas in this note.