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The Significance of Time Reversal Invariance of the Quantum Free exp(i p dot r) Part 3 Francesco R. Ruggeri Hanwell, N.B. Oct. 18, 2025 Classical probability is usually a number assigned to an event, but given Sum over events i P(i) = 1, and the fact that events occur at different times, probability behaves like an additive amount without time being a major factor. For example, an incident photon which may either reflect or refract at an n1-n2 junction has the equation: Probability(incident photon) = Probability(to reflect) + Probability(to refract). All of these events are linked to different times, but they nevertheless appear in the same equation”. In Parts 1 and 2, we argued that one may introduce a probability to describe two body elastic scattering of free particles. A free particle cannot carry a real value weight because all free particles are equally likely. Probability arises because given an initial (e1,e2) energy and (p1,p2) momentum vectors, one does not know the outcome and may assign equal product probability to any (ei,ej) and (pi,pj) pairs which conserve energy and momentum. If this is the case, this probability should apply to all reactions, not just two body scattering. If energy is not conserved, then one may use a probability for momentum. This probability seems to be linked then to that of having an e and p vector. A free particle, however, has an e and p even if it is not interacting and so seems to be associated with a probability. We argued in Part 1, that one is forced to use: exp(-iEt+i p dot r) to describe this probability. As a result, time and space appear, but not in a manner linked to constant velocity for a particle with rest mass, i.e. E dt + p dx = 0 does yield a correct dx/dt. Furthermore, we showed in Part 2, that exp(i p x) is time reversal invariant meaning that it has the same value for a movie played forwards and backwards. We suggested that this means there is no time flow linked with exp(ipx) and that exp(ipx)s for different time events may be added. At first this may seem unusual, but in the first paragraph we noted that probability is about an “amount” linked with an event and not with time. exp(ipx) contains p and handles p and -p differently as they are different momenta and also allows for addition and cancelation in x. Nevertheless, it seems that exp(ipx) should be linked with the classical notion of finding a particle in a certain dx. To see this, one may consider a particle which has p and -p at the same time (i.e. no momentum). Then, exp(-ipx)exp(ipx) = 1, but this is the classical probability for a particle with no motion sitting somewhere in a length L=1. Thus, starting with a desire to have a probability which describes e and p conservation, one is led to an exp(i p dot r) which is linked to spatial presence. Interaction properties (conservation of momentum) and the notion of probability in interactions seems to govern the presence of a particle in x. Given that reality is time dependent and that exp(i p dot r) contains p, it seems that one might be tempted to identify a given exp(ip dot r) with its appropriate time interval. Furthermore, classical physics does not allow for interference of probabilities, while exp(ip dot r)’s interfere. We suggest, however, that this interference follows directly from the notion that free particles carry the same real weight (so a complex number of unit modulus is needed) as well as the form exp(-iEt+i p dot r) to maintain Lorentz invariance. Thus, it seems that there is a unique solution for a probability of x presence which is linked with e and p conservation. At the least, p conservation must exist in an interaction and so it seems that exp(ipx) must be the probability to use, if in fact there is probability in an interaction. Given the notion of AND and OR for
probabilities, it seems one has no choice but to accept interference of exp(ip dot r)’s for events which exist at different times. The fact that one adds these in the same equation is completely consistent with what one does with classical probabilities, it is just the interference which differs, but this comes about due to trying to account for interactional issues such as conservation of energy and momentum. This is due then because exp(ipx) accounts for momentum conservation, i.e. an interactional property that is not considered in usual classical probabilities. Classical Probability Classical probability seems to be linked to assigning a positive real value (between 0 and 1) to an event. Different events may be linked, but may occur at different times. For example, an initial coin may be tossed producing an outcome. This outcome exists at a different time than the initial state, but one still links them in the same equation: 1 = .5 + .5 = Probability(heads) + Probability(tails) ((1)) Thus, conservation of probability is not based on time of the events. Similarly, a photon which refracts or reflects at an n1-n2 (index of refraction junction) has: 1 = Probability(reflect) + Probability(refract) ((2)) Physically all three terms are linked with different times. Probability Linked to Conservation Laws in Interactions Conservation laws exist in Newtonian interactions. These constrain the dynamics. As shown in Part 1, it is possible to link probability with such conservation, in particular energy e and momentum p. In Part 1, we argued that this requires that one have: exp(-iEt+ i p dot r) ((3)) ((3)) is Lorentz invariant and means that any (ei,ej) (pi,pj) set with the same sum as the initial (e1,e2) and (p1,p2) vectors) has the same product probability. As a result, if one accepts this, one must use ((3)) as a probability. Given that a free particle always has an e and p, even if it is not reacting, ((3)) seems to be associated with it. In Part 2, we showed that exp(i p dot r) is time reversal invariant, meaning that there is no flow of time linked with it. Physically, however, events occur at different times. This might at first be considered a paradox, but ((1)) and ((2)) show that this occurs for classical probabilities and so there is no reason why it should not follow for ((3)). ((3)), however, is different form ((1)) and ((2)) in that it describes interactional constraints, i.e. conservation of E and p. This begs the question: How are ((1)), ((2)) linked with ((3))? In Part 2, we talked of multiplying exp(ipx) by its complex conjugate to obtain 1, but here we use a different argument. We consider an AND probability situation and create
exp(-ipx)exp(px). In other words, the particle is first given p and then -p so that it has both. This then is a particle at rest, i.e. there are no dynamics and hence no interactions. Then: exp(-ipx)exp(ipx) = 1 ((4)) ((4)) is the classical probability for a rest particle that may be anywhere in a length L=1. Multiplying exp(ipx) by its complex conjugate yields a real value which no longer contains interactional information, but only the x information in exp(ipx). Interference Given that ((3)) is the only possible free particle probability which: gives equal product weight to any (ei,ej) (pi,pj) which conserve momentum and energy with (e1,e2) (p1,p2) vectors ((5)), and the condition that AND and OR operations must hold, then there must be inteference. We have already argued that exp(ip dot r) is time reversal invariant and so there is no time flow. This makes exp(ipx) suitable as a probability and so one may add various exp(ipx) values for events occurring at different times. This leads to a prescribed a priori interference (addition and subtraction) at x, something that does not occur for classical probability. The underlying driver of ((3)), however, is conservation of momentum and energy, special relativity and the condition ((5)). It seems that there is no choice but to accept this interference as long as these are the only conditions present. exp(ipx) is a probability which combines the usual time mixing of classical probability with interactional features (conservation of momentum and energy). We showed in Part 2, that this may be used to diminish an exp(ipx) linked with an initial event to free up probability for a final event. The fact that exp(ipx) is suitable is based on the derivation of ((3)), i.e. there is no other possibility given the assumptions used to create ((3)). We note that physically there is time dependence in a problem and so one should consider using exp(ipx)s as a math calculation, but based on probabilities which contain the physics of interactions (conservation of momentum and energy) as well as a very specific probabilistic condition ((5)). Conclusion In conclusion, we argue that classical probability conservation statements contain probabilities for events which occur at different times. According to Newtonian mechanics, one would follow x(t), but if there is probability in an interaction, one may also write a probability conservation equation which mixes time events. For example, if one has a photon which may reflect/refract at an n1-n2 index of refraction junction, one may separately describe the incident, reflected and refracted photons at different times using Newtonian equations. There is, however, a certain probability to reflect and refract and one may write a conservation equation which mixes these events which occur at different times, i.e. ((2)).
The question then becomes: What governs the probability for a photon to reflect or refract? Is there some special condition linked with an n1-n2 junction, or is there some universal probability at play? We argued in ((1)), that there is a universal condition based on the idea that for a given particle with initial (e1,e2) (p1,p2), one may postulate that the product probability for all (ei,ej) (pi,pj) sets which conserve e and p is the same. This is a basic assumption which may then be applied to all types of interactions, including 1-D reflection-refraction. In Part 1, we argued that condition ((5)) means having a probability exp(-iEt+i p dot r). We also note that exp(i p dot r) is time reversal invariant which is consistent with the idea of adding exp(ipx)’s for events occurring at different times. Given that a probability should respect AND and OR conditions and that one ignores time, then one may add exp(ipx)s and observe interference. In Part 2, we argued that such interference is required to allow for probability to be removed from an incident exp(ipx) and have it added to an outcome exp(ipx). We suggest that if one accepts the assumptions that lead to ((3)) (obtained in Part 1), then one must use this probability in all interaction cases and accept interference. In other words, exp(ipx) introduces interference which is consistent with conservation of momentum and exp(-iEt+ipx) with conservation of energy and momentum in a Lorentz invariant manner.
