In a previous paper, “On the Applicability of the Lorentz Transformations”, the issue of the physical vs. apparent velocity of a photon, c vs. (c±v), was examined in the cases of the emitter moving relative to the detector and vice versa. In the present paper, the case is examined of both the emitter and detector moving relative to the origin of a third coordinate system, the case used by Einstein for his derivation of the Lorentz transformations. It is shown that both Einstein’s derivation and a more general derivation are inconsistent with the behavior of the photon, and that the physical (not apparent) velocity of the photon as measured in the stationary coordinate system in this case must be (c±v). It is also noted that because the velocity of the photon is independent of its momentum and energy, the dispersion relations between conjugate Fourier variables of energy and time, ∇E∇t ≥ ħ/2, and momentum and position, ∇p∇x ≥ ħ/2, do not affect its velocity. As a result, unlike for non-zero rest mass quanta, the position of a photon as a function of time can in principle be determined with an arbitrary accuracy.
In a previous paper, “On the Applicability of the Lorentz Transformations”, the issue of the physical vs. apparent velocity of a photon, c vs. (c±v), was examined in the cases of the emitter moving relative to the detector and vice versa. In the present paper, the case is examined of both the emitter and detector moving relative to the origin of a third coordinate system, the case used by Einstein for his derivation of the Lorentz transformations. It is shown that both Einstein’s derivation and a more general derivation are inconsistent with the behavior of the photon, and that the physical (not apparent) velocity of the photon as measured in the stationary coordinate system in this case must be (c±v). It is also noted that because the velocity of the photon is independent of its momentum and energy, the dispersion relations between conjugate Fourier variables of energy and time, ∇E∇t ≥ ħ/2, and momentum and position, ∇p∇x ≥ ħ/2, do not affect its velocity. As a result, unlike for non-zero rest mass quanta, the position of a photon as a function of time can in principle be determined with an arbitrary accuracy.
The Lorentz transformation of special relativity is based on velocity. Although this velocity may be differentiated by time, the notion of force is not needed to obtain the Lorentz matrix for a particle with rest mass. In particular, in special relativity one considers frames moving at constant velocities. In the case of a photon, the Lorentz transformation preserves the form of Maxwell’s electromagnetic equations. As a result, there does not seem to be a focus on force, rather the emphasis is on the two 4-vectors (p,E) and (x,t) for a particle with rest mass and photon. Nevertheless, a particle with momentum (and the capability to interact) imparts an impulse. In fact, one may think of energy changing due to an impulse (which includes force). Thus from an information point of view, one may ask: Does this force which creates a change in energy/momentum “disappear” as the particle/photon moves at constant speed only to “reappear” when it strikes a surface? If not, where is the information of this force stored? We argue it is associated with a change in energy and momentum of a free particle which in turn is linked to an internal ruler/clock i.e. the quantum frequency and wavelength from exp(-iEt+ipx) which preserves the force because a spatial probability distribution is created (for the wavelength). This force is linked with a direction i.e. the direction of p through the two dimensionality of exp(ipx). Thus, we argue that the force does not disappear, rather it is manifest in the internal clock and ruler which define spacetime for the free quantum particle, but in a probabilistic manner with the distribution being deterministic to ensure it maps to spatial invariance. One might at first find it surprising that force, usually associated with acceleration (i.e. deterministic motion) is now associated with probability cos(px) in space (, but deterministic probability, not random), but E,p and x,t are intertwined through the Lorentz invariant -Et+px for both a particle with rest mass and photon, so if E and p change, it is not necessarily surprising that t,x also change. Thus an impulse may change p and E, but then remains manifest in the system through the internal clock and ruler gradations in such a way that speed is still constant on average.
Momentum and velocity are two well-known physical observables in classical physics (and quantum physics as well because quantum particles have impulse and move through space in a certain amount of time based on an external clock and ruler). They are, however, measured in different ways. Velocity describes motion through x in some time t, while p is measured through impulses. Thus one does not need a clock to measure momentum. It may be measured in terms of the damage done to a foil, compression of a spring etc. Thus p=m1v1=m2v2 etc., i.e. a numerical number associated with momentum does not automatically yield the velocity unless one knows rest mass. In many situations rest mass is known (i.e. one deals with an electron etc) and so there seems to be a tendency to consider momentum and velocity to be almost the same thing when they are distinctly different. In fact, we argue that the notion of wavefunction W(x) and density P(x)=W*(x)W(x) are associated with momentum and velocity respectively. In fact, for a free particle these two measurables are linked with completely different spatial probabilities. Constant velocity impies P(x)=constant while exp(ipx) describes the two-dimensional x-probability associated with p which is based on a constant mo and v. In special relativity for a free particle (i.e. p constant, v constant), there are two well known Lorentz invariants: EE = pp + momo (c=1) and A = -Et + px. One may note that v does not appear explicitly even though one may write E and p in terms of v i.e. E = mo/sqrt(1-vv) and p=Ev. The first invariant describes E in terms of rest mass and p (measured through impulse). The second involves p,E and x,t with x and t appearing as independent even though ultimately x(t) and v=dx/dt=constant = x/t if x and t both start at 0. This independence is crucial because it means one may consider fluctuations which keep A constant separately. For example, if t→ t+ hbar/E and x→ x+hbar/p, then A remains constant. This leads to the idea of the quantum wavelength and a spatial probability independent of t, namely exp(ipx). One average, however, x=vt, but how does this appear in Lorentz invariants? There is no explicit v. We argue that the Lorentz invariant A = -Et + px must convert E from a function of p to one in v and that this occurs through a Legendre transformation. Thus the form of the invariant is itself a Legendre transformation with E being transformed into L, the Lagrangian. There is, however, a second important feature because p=dL/dv (partial) and given that p is constant, dp/dt=0 so d/dt dL/dv partial =0 which is in the form of an extremization. We argue that this is not an accident. dL/dv=p follows from E being constant, so, in other words, it follows from the px term. This same term leads to exp(ipx) which means that one does not have smooth x=vt motion within a wavelength. Thus, we argue that when one transforms to the second measurable v, the notion of this fluctuation behaviour must also be manifested. In other words, v=constant or dv/dt is not enough. The extremization behaviour of the Lagrangian, we argue, is thus not a mathematical convenience, but gives an indication that even though the extreme path x=vt is followed on average, other paths are sampled. We argue that this idea is incorporated into special relativity, in fact within a single Lorentz invariant A = -Et + px.
We have argued in previous notes that special relativity is often developed from the point of view of a particle with rest mass, leading to the Lorentz invariant -EE + cc p dot p = -momo cccc ((1)). Then, as a second step, one may obtain the equation E= [p] c by setting mo=0. This appears to be a straightforward math step to obtain a photon relevant result, but we have suggested in (1) that mo represents an inherent force, called inertia (Newton). Thus, special relativity for a particle with rest mass is really linked to this inherent force. In fact, we have suggested in (1), that spin ½ is obtained by finding an equation linear in a particle’s inherent force (rest mass) which is linked to E and p from special relativity and geometry. This, we suggest, is why Dirac linearized ((1)) to find spin ½. The reason for linearization is to have an equation linear in the inherent force, because force appears in F=dp/dt in a linear fashion. The rest mass case seems to be largely based on mo, inertia, which we call the inherent force. We further argued in previous notes that free particle quantum probability arises from -Et+px = Lorentz invariant ((2)) because x,t and x+hbar/p, t+hbar/E give the same result and we consider -Et+px to act somewhat like a free particle potential. This suggests that a photon should also have a probability exp(-iEt+ipx), which is verified as both a photon and particle with rest mass, with the same p, give rise to the same 2-slit interference. The problem in the photon case is that mo=0 and so no inherent force appears in special relativity. We argue that physically, there must be an inherent force as interactions (as in the 2-slit case) depend on it. In (1) we argued that the inherent force of a photon is the electric field, but this does not appear in ((1)) or ((2)), but emerges from Maxwell’s electromagnetic equations. This is why the photon has spin 1 and not spin ½ even though it has the probability exp(-iEt+ipx). (See (1) for details on the specific equation which gives rise to spin 1.) Here, we argue that for a particle with spin ½ (i.e. rest mass), exp(-iEt+ipx), which is a probability which conserves energy and momentum, the momentum and energy are proportional to rest mass (inertia or inherent force) and so exp(ipx) is really a probability distribution for inherent force (augmented to E,p due to the Lorentz boost). In the case of the photon, one still has exp(ipx), but the force is due to the electric field. This begs the question: What is the spatial distribution of the electric field? Is it free to be any function? We argue that given that exp(-iEt+ipx) is the energy/momentum conserving probability from special relativity, the inherent force cannot contradict this distribution. The electric field, which is a real valued function, must be cos(-Et+px). We note that one may formally obtain a classical wave equation from Maxwell’s EM equations to find El proportional to cos(-Et+px), but this does not explain the complex probability exp(-iEt+ipx) which holds for both a photon and particle with rest mass. Our objective is to argue that given exp(-iEt+ipx) for a photon, the electric field must behave as cos(-Et+px). Interestingly, the photon momentum density 1/uo Electric field x Magnetic field is proportional to cos(-Et+px) * cos(-Et+px). This does not follow Real(exp(-iEt+ipx), but it is the electric field and not momentum in the photon case which is linked to force. We argue that ultimately it is force which governs interactions, even though momentum (not momentum density) is conserved.
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.
Abstract We present a reformulation of Bohr’s analysis of Einstein’s thought experiment with the photon box, which hopefully settles some unclear issues. The box is weighed by two position measurements, the first immediately before the emission of the photon and the second after a quarter period of oscillation of the box, or alternatively after a half period. These times are precisely defined with respect to the moving clock installed in the box, but vary statistically when measured with a stationary clock outside the box, due to gravitational time dilation in accordance with Bohr’s original idea. Moreover, we will discuss two other proposals to solve the apparent inconsistency without invoking the theory of general relativity.
Special relativity seems to be associated with linking physical relations as seen in one frame with those as seen in a frame moving at a speed relative to such a frame. A constant speed is often used for the moving frame. This suggests that classically there is a special frame for a particle with rest mass called a rest frame. In such a case, one simply introduces a parameter for rest mass and does not worry about the complicated motion and interactions which actually create the rest mass. It seems that even quantum mechanics, which considers a particle to be moving as one does not know its location, one uses the idea of a potential V(x) which in principle is associated with a very large rest mass, presumably at rest as no variables of such a rest mass appear in the Klein-Gordon or Dirac equations. It seems that one uses a preferred frame when solving relativistic quantum mechanical equations. Furthermore, given V(x) and a bound state with exp(-i E bound t), E applies to the overall bound system, not to a possible exp(ipx) constituent. The special relativistic invariant equation seems to exist at the system level, like with the rest mass and one may apparently calculate E in the special rest frame of the large mass and hence find a rest mass of the bound system, i..e large rest mass*cc + mo cc + E, mo is the rest mass of the single bound particle. As a result of the use of the special rest mass frame for the V(x) producing large rest mass, it seems that one does not have to worry about special relativistic constraints on the interaction with V(x), as only one frame is involved. In other words, a bound calculation is about finding a new rest mass for an overall new system. Then this system may be used in Lorentz boosts. One simplifies by working in the rest frame of the mass creating V(x). Perhaps surprisingly then, we note that exp(-iEt+ipx) for a free particle is Lorentz invariant and in fact must be Lorentz invariant for the following reason. Imagine that one wishes to create a probability for free particles which is linked with conservation of momentum. Any p must have the same real value weight because for free particles there is no bias for one p or the other. There is, however, a bias for a p1 and p2. It should have the same probability as a p3 and p4, if p1+p2 = p3+p4. (We use one dimension for simplicity.) As a result, exp(ip) seems to be a probability which respects these conditions. The problem is that one may have p=m1v1=m2v2 hence (E1,p) and (E2,p). With respect to momentum, it is irrelevant if one particle has E1 and the other E2, but not if one does a Lorentz boost. Then, in one case: p→ g(v) p + g(v) v E1 and in the second, p→ g(v) p + g(v)vE2. Here g(v) =1/sqrt(1-vv/cc). In other words, p is the same for each particle in one frame and not in the other which violates conservation of momentum because in frame 1 both particles can complete the same conservation of momentum equation, but in the second frame this is not the case, because each has a different momentum. Thus, exp(ip) cannot hold. One requires a linear form in p which is Lorentz invariant, i.e. exp(-iEt+ipx), we argue (and have done so in previous notes). We see now that momentum probability is linked with space in exp(ipx). The idea seems to be that for free colliding particles there is no preferred frame and the probabilities used must reflect this Lorentz invariance. As a result, we argue that a free particle probability in E,p arises because of Lorentz invariance, but then apply it to a V(x) problem and no longer worry about Lorentz invariance at the interaction level. V(x) with no associated rest mass variables assumes a rest frame for the V(x) producing mass. A Lorentz boost which then invalidates exp(ip), would only apply to the total rest mass of the bound particle system which has no overall momentum. It seems that one does apply the Lorentz transformation to the interaction of exp(ipx)’s with V(x). One is only interested in the overall bound system as a single entity for special relativity. As a result, one may only impose momentum considerations as V(x) is a spatial function and one must link it with exp(ipx). In fact, one may write V(x) = Sum over k Vk exp(ikx) to associate it with momentum hits linked with Vk energies, which already moves away from the idea of special relativity as Vk and k are not linked as a 4-vector. The idea of exp(ip1x)exp(i k1x) = exp(ip2x)exp(ik2x), however, seems to still hold (as it does for free particles), linking these two values (i.e. Vk1 and Vk2) with (p1+k)(p1+k) exp(i (p1+k)x). Lorentz invariance for a particle with rest mass seems to be linked with the idea that one ultimately has a rest frame for the particle. One then establishes a relation between the rest mass in one frame and its new E and p as seen by another frame. One does not, however, need to impose Lorentz invariance conditions on the interaction which creates the rest mass it seems. For example, exp(-iEt+ipx) is the Lorentz form used to describe a free particle, but exp(ipx) is the only portion used to describe its interaction with V(x) in the rest frame of the large mass creating V(x). Furthermore, exp(-iEt+ipx) ensures conservation of energy and p for a free particle, but in an interaction case, that does not happen as the potential may absorb energy.