Conservation of energy and momentum are fundamental ideas of physics which appear in Newtonian mechanics and as such are considered to be deterministic. One may follow the center-of-masses of two particles r1(t), r2(t) up to an interaction and pinpoint energy and momentum at a specific x,t. Special relativity is also deterministic (based on x=vt) and one would expect that conservation of energy and momentum should be enforced in a deterministic manner there as well. In every frame, one could write: Kronecker delta (E1-E2) Kronecker delta (p1x-p2x) … Kronecker delta (p1z-p2z). Such determinism would bypass quantum mechanics completely because quantum mechanics involves a non-deterministic approach to conservation of E,p. We argue, however, that it is actually deterministic special relativity which gives rise to non-deterministic conservation of E,p for a specific reason. We suggest that in a rest frame, a particle with rest mass mo is associated with a running time t, but no notion of x space or momentum. One might be tempted to set p=0, but in principle we argue that there is no notion of momentum for a rest particle and no notion of space because it is simply sitting at one point. Momentum only arises if one views the particle at rest from a frame moving at -v, leading to p = mov/sqrt(1-vv/cc). It is only in Newtonian mechanics that one introduces a priori a momentum which may then be applied in any frame. Such is not the case in special relativity. If one takes this idea seriously, it has strong consequences because one cannot have a set of Kronecker deltas in each frame as listed above. One Kronecker delta in the rest frame (mo1-m2), would have to transform into Kronecker(E1-E2) Kronecker (p1x-p2x) for motion in the x direction alone. Mathematically a single Kronecker delta does not transform into two Kronecker deltas. One requires a mathematical function which describes energy conservation in the rest frame and then transforms naturally to describe both energy and momentum conservation in a moving frame. Given that mocc and t are the only variables in the rest frame, we suggest a function based on the Lorentz invariant -Et+px. If this is the case, then t and x must act as independent variables which at first seems like a problem because it is understood that x=vt. One may show, however, as we do in detail, that this invariant allows for fluctuations to t and x and so one may in fact create the function exp(-iEt+ipx) (which is like a probability) and enforces non-deterministic conservation of energy and momentum. Non-deterministic conservation means that one must interact exp(-ip1x) exp(ip2x) over some space to obtain 0. Physically, this is linked to the notion that p hits occur probabilistically in space (2-slit experiment) and so this same probabilistic notion applies to conservation. Deterministic conservation at only one x,t is at odds with 2-slit interference which shows that one may interact probabilistically at two points in space. Thus, one must account for conservation of p at both.
Conservation of energy and momentum are fundamental ideas of physics which appear in Newtonian mechanics and as such are considered to be deterministic. One may follow the center-of-masses of two particles r1(t), r2(t) up to an interaction and pinpoint energy and momentum at a specific x,t. Special relativity is also deterministic (based on x=vt) and one would expect that conservation of energy and momentum should be enforced in a deterministic manner there as well. In every frame, one could write: Kronecker delta (E1-E2) Kronecker delta (p1x-p2x) … Kronecker delta (p1z-p2z). Such determinism would bypass quantum mechanics completely because quantum mechanics involves a non-deterministic approach to conservation of E,p. We argue, however, that it is actually deterministic special relativity which gives rise to non-deterministic conservation of E,p for a specific reason. We suggest that in a rest frame, a particle with rest mass mo is associated with a running time t, but no notion of x space or momentum. One might be tempted to set p=0, but in principle we argue that there is no notion of momentum for a rest particle and no notion of space because it is simply sitting at one point. Momentum only arises if one views the particle at rest from a frame moving at -v, leading to p = mov/sqrt(1-vv/cc). It is only in Newtonian mechanics that one introduces a priori a momentum which may then be applied in any frame. Such is not the case in special relativity. If one takes this idea seriously, it has strong consequences because one cannot have a set of Kronecker deltas in each frame as listed above. One Kronecker delta in the rest frame (mo1-m2), would have to transform into Kronecker(E1-E2) Kronecker (p1x-p2x) for motion in the x direction alone. Mathematically a single Kronecker delta does not transform into two Kronecker deltas. One requires a mathematical function which describes energy conservation in the rest frame and then transforms naturally to describe both energy and momentum conservation in a moving frame. Given that mocc and t are the only variables in the rest frame, we suggest a function based on the Lorentz invariant -Et+px. If this is the case, then t and x must act as independent variables which at first seems like a problem because it is understood that x=vt. One may show, however, as we do in detail, that this invariant allows for fluctuations to t and x and so one may in fact create the function exp(-iEt+ipx) (which is like a probability) and enforces non-deterministic conservation of energy and momentum. Non-deterministic conservation means that one must interact exp(-ip1x) exp(ip2x) over some space to obtain 0. Physically, this is linked to the notion that p hits occur probabilistically in space (2-slit experiment) and so this same probabilistic notion applies to conservation. Deterministic conservation at only one x,t is at odds with 2-slit interference which shows that one may interact probabilistically at two points in space. Thus, one must account for conservation of p at both.
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.
We have argued in Parts 1 and 2, that the free particle quantum form exp(-iEt+ipx), which we call a complex probability, arises from special relativity in order to allow for conservation of energy and momentum (in an elastic collision). One may see from single particle bound states how exp(-iEt) plays the part of a binding energy. We have noted in Parts 1 and 2, that exp(-iEt+ipx) follows from the Lorentz invariant -Et+px, but that one does not consider the deterministic x=vt equation which applies to the center-of-mass, but rather treats t and x as independent variables which force the independence of E and p (two different interaction variables) and also force separate E and p conservation. The question becomes: How is it possible to have x,t play two such different roles in special relativity? Here we try to answer this question. We argue that special relativity is based on determinism, i.e. x=vt. One may create a Lorentz transformation for a particle with rest mass mo at x=0 with t, a running variable. As seen from a frame moving with -v, one has x’=vt’. In other words, a separate x’,t’ is associated with each v. In Newtonian mechanics, it is possible to have mov1 and mov2 in the same frame (i.e. a frame with a ruler and clock), but in special relativity, each mo in the rest frame maps to a different x1’,t’ and x2’,t2’. This, however, seems to lead to a contradiction. If one considers two particles p1,E1 and p2,E2 (one dimensional motion) which elastically collide, then before the collision, they may be thought of as a composite particle (for more than a single x,t point) with energy E1+E2 and momentum p1+p2. Given the validity of special relativity, we argue that one may associate a Lorentz invariant -(E1+E2)t + (p1+p2) x with this composite particle. Furthermore, any outcome Ei+Ej = E1+E2 and pi+pj = p1+p2 would satisfy this same invariant. At this point there is no contradiction. The contradiction arises if one considers a single t and x for a composite particle, while internally there exist t1’,x1’ and t2’, x2’. The composite particle, in terms of special relativity, is not compatible with each -E’t’+p’x. The energies and momenta, however, add for each particle (in conservation equations) and so we argue that one should have a relativistically compatible approach which allows one to link each particle to a composite particle and see conservation of E and p. We note that it is precisely x=vt for each particle which leads to x1’,t1’ and x2’,t2’ and so argue that if one gives up this notion of determinism for x and t and introduces independent x, t values (fluctuations), then one can indeed retain the -Et+px form for each particle and use it to create composite particles. This seems to give rise to an AND situation which does not exist for both p,x and E,t in the deterministic relativistic approach. In that approach, one can only think of p’s and E’s adding separately, without any linking of x’s and t’s. In the fluctuation approach, it is possible to have the same x and t fluctuation points for each particle. We argue that it is this kind of thinking which leads to the probabilistic form exp(-iEt+ipx) which still respects the Lorentz invariant form and at the same time allows for conservation of E,p with the same t,x fluctuation points allowed. x=vt does not allow one to combine x1’,t1’ and x2’,t2’ for a composite particle because these may coincide for only one x,t point and one wishes to describe a composite particle with an x,t which exists over more than one x,t. This is why we suggest the physical fluctuation scenario which exists together with x=vt. x=vt describes the deterministic motion of the center-of-mass, while exp(-iEt+ipx) describes probabilistic hits in x and energy in t. It is special relativity which forces one to associate time fluctuations with E and spatial ones with p. As a result, we argue that special relativity gives rise to both probabilistic and deterministic versions, both of which respect the form -Et+px. The deterministic version is linked with x=vt center-of-mass motion, while exp(-iEt+ipx) represents probabilistic appearances of E in t and p in x, i.e. there is a spatial probability for impulse hit delivery.
We have argued in previous notes that free particle quantum mechanics arises from special relativity and is dependent on both the deterministic x=vt and fluctuation/probabilistic delta x= hbar/p and delta t = hbar/E. In other words, both determinism and probability are inherent to special relativity and it is also special relativity which defines energy and momentum. At first, it may seem that these are rather strong suggestions. We argue, however, that perhaps these ideas are already present in classical Lagrangian/Stationary Action theory. This theory was developed to describe Newtonian mechanics, but we argue that because it uses the notions of E (energy) and p (momentum), it must be linked to special relativity which yields the full expressions for these functions. The interesting feature of stationary Lagrangian theory is that a notion of fluctuations in path (x,v,t) are used to ultimately obtain the deterministic path x=vt for a free particle as the stationary one. The notion of a stationary, deterministic path x=vt is present, but so is the notion of fluctuations about this path. The problem is that the stationary action process is a mathematically one and does not attribute and physical features to the fluctuations about the stationary path, whereas we argue that these fluctuations are linked with probabilistic impulse hits or E interactions in time and are directly associated with the quantum wavefunction (complex probability) exp(-iEt+ipx) which physically manifests itself (as in 2-slit experiments etc). In order to connect stationary classical action theory to special relativity, we use H = pv - L -> -Hdt + pdx = L dt. If one considers fluctuations of this equation, the RHS set to 0 leads to the stationary deterministic x=vt, while the LHS leads to delta t = hbar/E and delta x = hbar/p. In other words, there is a delta x and delta t not linked to v = delta x / delta t. We suggest that this approach shows both probabilistic and deterministic features present side by side and is directly linked to special relativity because -Hdt + pdx is of the form of the Lorentz invariant -Et+px.
In previous notes, we introduced a complex probability into Newtonian two-body elastic scattering to ensure that for a given (e1,e2) and (p1, p2) (momentum vectors), any (ei, ej) (pi, pj) have the same probability to occur. This ultimately led to the Lorentz invariant probability exp(-iEt+i p dot r). Here we begin with the delta functions delta( pinitial - pfinal) and delta( Einital - Final) and see that one may write these as Integral da C exp(-i pfinal + i p initial) and a similar expression for E. Here a is simply a math variable. (Note: This is only one math form for a delta function, but this particular form will prove useful from physical considerations later.) The delta function may be used in an expression Integral delta(pinitial - pfinal) dpfinal and only pfinal values which satisfy conservation of momentum yield a nonzero integral. Furthermore, this integral may be multiplied by various functions and still maintain conservation of momentum. At this point, the above is simply a calculation formalism. The question we ask is whether one may make the argument that a particle itself is responsible for conservation of momentum in an interaction. We suggest this may be the case based on Newton’s notion of action-reaction. It is not simply that one applies a force to a particle and it changes. The particle itself resists and this is the idea of inertia as well as action-reaction. The particle itself is strongly involved in a reaction and we suggest that a particle carries a property of ensuring conservation of momentum and energy itself. It is not something that is imposed by the interaction, otherwise there would be no need for the notion of action-reaction. The particle reacts and the manner in which it reacts leads to conservation of momentum and energy. We suggest that there seem to be grounds for considering a particle to actually be the driver of energy and momentum conservation and that this actually extends to further features, namely probabilistic results in interactions which do not appear in Newtonian mechanics. This property might then be taken as exp(i p a) and exp(i E b), where a and b are math variables used to establish the delta function. One, however, cannot really establish a property of a particle if math running variables a and b are present. The entire expression should consist of particle state variables. This seems to suggest the consideration of Lorentz invariance because E and p change when the particle is seen from a frame moving with -v. One may create a state based exp(-iEt+ipx) energy-momentum conserving property of a particle which is Lorentz invariant. We next note that p and E are linked to interactions, but ones that differ. In particular, p is linked with an impulse which has nothing to do with velocity , i.e. E. On the other hand, E = .5movv in the nonrelativistic limit and is dependent on v, i.e. a m1v1= m2v2 have the same impulse, but different kinetic energies. Thus, we suggest that exp(-iEt) and exp(ipx) be used separately depending on the type of interaction one considers. In a quantum single particle bound state, one is interested in impulse hits with the potential we argue and so does not need to consider exp(-iEt) as one cannot measure the particle moving through the potential in a classical manner. We argue that a constraint (momentum and energy conservation) may actually be due to properties of the particle and that this ultimately follows from Newton’s action-reaction. In (1), we argued that exp(-iEt+ipx) follows from Lorentz invariance which is based on the Newtonian idea of work increasing energy, i.e. 1/sqrt(1-vv/cc) and A= -Et+px have minus signs for this very Newtonian reason. We argued that dx = hbar/p and dt=hbar/E follow as fluctuations which leave A unchanged due to the presence of the minus sign.Here we again argue that a Newtonian idea is responsible for exp(-iEt+ipx), namely action-reaction. We show that the two approaches are identical.
In classical physics, one states in words the existence of conservation of momentum and energy. An example would be an elastic collision of two particles. In particular, this conservation is imposed, it does not mathematically appear unless one describes the process by say a Kronecker delta, i.e. delta( momentum vector before, momentum vector after) or delta(energy before, energy after). Even in such a case, the Kronecker delta is isolated and so one still has to manually ensure that energies and momenta chosen in a calculation ensure conservation. A Kronecker delta is mathematically equivalent to the dot product of two orthonormal states in linear algebra. This implies an operator which may obtain the particular value in question, momentum vector or energy, when operating on the state. A Hermitian matrix operator is guaranteed to have real eigenvalues and orthogonal eigenvectors, which is required. Thus, mathematically, one may postulate energy and momentum states and corresponding operators because this automatically creates the Kronecker delta which describes conservation in classical processes. This approach holds both nonrelativistically and relativistically. The question becomes: How does one specifically describe the operator and its state? For both a relativistic and nonrelativistic free particle classical action, which describes motion, one may write: A= -Et+px where E,p are the relativistic free particle values in one case and E=p dot p/2mo and p=mov in the other. One may note that E and p are values which may be conserved in interactions. The action shows that the operators -d/dt partial and d/dx partial yield E and p respectively, but one does not have an eigenstate. This may be obtained using id/dt partial exp(-iEt+ipx) = E exp(-iEt+ipx) and -id/dx partial exp(-iEt+ipx) = p exp(-iEt+ipx). Because one uses partial derivatives, t and x are treated as independent relative to the time and space translation operators d/dt partial and d/dx partial. As a result, we argue that by mathematically linking a Kronecker delta associated with classical conservation with linear algebra, i.e. momentum and energy states and finding a corresponding operator by examining the classical action leads to the quantum free particle wavefunction exp(-iEt+ipx). One may apply this to an elastic two particle collision to see that the Kronecker delta and its associated state/operator form describe conservation of both energy and momentum.One may ask whether exp(ip dot r) has any physical relevance. Two-slit interference experiments using particles with rest mass shows that it does.
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.