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.
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= n2p, 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=n2E, 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.
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.
The problem of a photon reflecting or refracting at an n1-n2 index of refraction is solved in (1) using the notion of continuity of the electric field of a photon and its d/dx derivative for the n1-n2 junction at x=0. The photon electric field behaves as cos(-wt+kx) or cos(-Et+px). We argue first of all that the photon is a quantum object and that this problem is really a quantum mechanical one as it may also be solved for a particle with rest mass using two different constant potentials V1, V2 instead of n1,n2. In this note, we consider the photon case for convenience. Given that the problem is really a quantum mechanical one (in our opinion), we ask: How far can one go by trying to solve it classically? We first point out that for a static uniform arrangement of objects in space, P(x)dx =dx/L, where L is an arbitrary length. In the case of particles moving at constant v, one may argue that each particle spends the same amount of time dt in each dx and that this also leads to a constant density in space. We suggest that for this to hold, one must consider a steady state problem and the notion of flux for physical reasons. This leads to writing the equation 1= Probability(reflect)+Probability(refract) as AA/c = BB/c + CC/c2 where AA, BB, CC are the incident, reflected and refracted fluxes and c1=c, c2=c/n2. We have shown in previous notes that one may mathematically write this single equation as: A+B=C and Ap-Bp = Cp2 using E=pc (with E the same for the incident, reflected and refracted photon) and then solve the two equations for the two unknowns B,C, setting A=1. This approach yields a result which matches experiment, but from the standpoint of classical physics, there is no a priori reason to take the square root of flux. It simply arises as a math trick so to speak, we argue, and there seems to be no physical motivation. Here, we suggest approaching the problem differently. We note that if c1-c2<<1, then there should be almost no reflection, while if c2<<1, there should be almost no refraction. We ask: What happens if one suggests that in the c1-c2<<1 case that one insists on BB dropping as (c1-c2)*(c1-c2), i.e. to second power as in a stable equilibrium and for CC dropping as c2*c2 for c2<<1 in the other extreme. This is a possible approach that seems to be linked to some kind of physical principle, and one may consider the simplest solution and compare it with experiment and find that it holds. In other words, it yields the same solution as the above approaches. Moreover, this approach gives a link with the square root approach of the above paragraph, because now one is linking c-c2 and c2 directly with square root fluxes A,B,C so as to obtain the quadratic feature automatically for c-c2<<1 or c<<1 when one ultimately uses AA, BB, CC. In other words, the assumption of quadratic vanishing of BB and CC, automatically leads to c-c2 or c2 vanishing of B and C, suggesting that one may automatically achieve the quadratic vanishing form by working with A,B,C square root flux variables. We suggest that this idea of quadratic vanishing of BB or CC possibly places the approach of using A+B = C, pA-pB=p2C on more physical grounds and points to a new kind of physics which uses A,B,C, namely quantum mechanics.
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 previous notes, we argued that quantum and classical features coexist in free particle quantum mechanics because exp(-iEt+ipx) (probability for an interaction at t or x) follows from -Et+px, but x=vt is a deterministic equation that applies to the center-of-mass motion. In other words, interactions involve stochasticity. One might argue that this entire notion follows from the use of -Et+px (Lorenzt invariant). If one did not use this invariant, one might argue that an association between E,p and x,t might not exist. Here we argue to the contrary. While Newtonian mechanics uses KE = pp/2m so that v does not need to appear explicitly if one uses E=KE and p, such is not the case in special relativity. There, one has E and p and might argue that physically these are variables linked with an interaction. At first, v does not appear explicitly if one simply writes E and p, but it does appear if one divides p/E for a particle with rest mass. Now v is directly linked to the deterministic x=vt (for xo=0, to=0) and so we argue that spacetime is inherently present if one links E to p. For a photon, E=pc and c=x/t and so one cannot escape linking E,p with spacetime as well. Given that E and p are measures in E,p space, we suggest that there must be corresponding measures in x,t space (i.e. hbar/p, hbar/E). Moreover, we suggest that there might be two kinds of information present. It is well-known that delta t = hbar/E and delta x = hbar/p (i.e. resolutions), but given that p = E/cc v one might consider two particles with the same p, one with vo, the other with 2vo. Then quantum time resolution information is (T/(hbar/E)), while for a fixed x, 2v only experiences half the time (half a different kind of time information) or deterministic time information is proportional to 1/v. The ratio of the two is constant for p constant. For E=p/v (c=1), one may have two particles with the same E, but vo, 2vo (and hence po, 2po). Quantum spatial resolution information is L/(hbar/p) while for a fixed time 2vo sees twice the space, i.e. twice the information. Again, the ratio of the two types of information is a constant.