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The Significance of Time Reversal Invariance of the Quantum Free exp(i p dot r) Francesco R. Ruggeri Hanwell, N.B. Oct. 16, 2025 In previous notes (1), we argued that one may introduce a probability into two body elastic scattering in Newtonian mechanics. In such a case, both energy and momentum should be conserved. (This may be considered both relativistically and non-relativistically.) This leads to: exp( i E) and exp(i (px)) (for momentum in the x direction). A complex number with unit modulus is used because there is no real value weight for a free particle, unlike a particle in an ideal case which has p(ei). We then noted that exp(i p) and exp(-ip) do not have the same value if one considers one representing the usual x axis, and the other, the reversed x axis, and so extended the probability to exp(i (px) x) so that (px)--> -(px) and x→-x yields the same probability. We note that such a transformation is equivalent to watching a movie played backward, i.e. to time reversal. In such a case, the minus values correspond to time moving backwards. We argue that this is a key feature of exp(i p dot r) as one really does not know about time in this function not simply because time is not present mathematically, but because it is time reversal invariant. We suggest that this has important consequences. In particular, in various quantum mechanical time-independent problems, one removes time and writes wavefunctions for probabilities linked to different physical times in the same equation. (These events, however, must be linked to each other probabilistically.) Two examples are scattering from a potential V(x) with exp(ipx) + f(theta)/r exp(ikr) and one dimensional reflection-refraction at an n1-n1 index of refraction junction: Aexp(ipx) + Bexp(-ipx) = Cexp(ip2x) at x=0 and Ap exp(ipx) - Bp exp(-ipx) = Cp2 exp(ip2x) at x=0. One might try to justify such equations mathematically (continuity etc), but we argue that from a physical point of view one should not be combining probabilities (at least in the classical sense) for events that occur at different times. One cannot simply state that a problem is time independent when a single particle scattering against V(x) or a single photon reflecting or refracting is clearly time-dependent and the time-independent approach yields a solution which describes the single particle time-dependent result. We suggest here that time reversal invariance of exp(i p dot r) means one does not know what time one has and so this allows one to add probabilities representing events at different times at the same x. Given that one has probabilities exp(ipx) or f(theta) exp(ipr)/r for different time events at the same x, one must be aware of conservation of material probability. If the modulus of the wavefunction represents material or classical probability in space, then there must be “interference” (i.e. addition and subtraction of the wavefunction as various x points) to allow for consistency with the classical result. In (1), we argued that adding x and t to create exp(-iEt+i p dot r) (Lorentz invariant) represented adding probability conservation to the already present E and p conservation. Here we suggest that this happens due to the time reversal invariance of exp(i p dot r). Thus, exp(i p dot r) seems to be a dynamic probability (it physically differs for p and -p in a way which shows direction of motion). It shows chunkiness of impulse hits (wavelength = hbar/|p|) and it is time reversal invariant so one can use it and add probabilities linked to events which occur at different times, meaning that interference occurs to allow for probability conservation. For example, in the case of a single particle scattering off of V(x), one initially (t=0) has exp(ipx)
which has a modulus of 1. This is the classical amount, i.e. 1 particle. The scattering must preserve this number 1. If one writes exp(i px) + f(theta) exp(ipr)/r, the modulus must involve interference because one must remove some “amount” from the incoming particle and assign it to the scattered one. The same type of argument applies to one dimensional refraction-reflection as we show. The Creation of the Free Particle Quantum exp(i p dot r) and its Meaning In previous notes, we argued that Newtonian two body elastic scattering introduces the notion of probability if one argues that a given (e1,e2) (p1, p2) vectors has the same probability of forming any (ei,ej) (pi,pj) vectors with the same product product probability as long as energy and momentum are conserved. This, at first, seems to imply the probabilities: exp(iE) and exp(i (px)) for p along x ((1)) Now exp(i (px)) and exp(i (-px)) should have the same value if one considers a usual x axis and the reflected x axis axis, but they don’t. Thus, we added x, i.e. exp(i x (px)) ((2)) We then generalized to a Lorentz invariant probability exp(-iEt+i p dot r) ((2)) We note two features of exp(i p x ). First, it is not the same for p and -p meaning that it distinguishes between direction of motion. We called this a dynamical probability in previous notes. The second feature is the “chunkiness’ of momentum hits because one has wavelength regions hbar/|p|. This has important consequences because interactions do not occur at the center-of-mass point as in Newtonian physics, but over a wavelength region, allowing a particle to interact with both slits of a 2-slit apparatus if they are separated by about a wavelength. Here we wish to stress a third important feature, namely that of time reversal invariance. If one writes: P→ -p and x—> -x then exp(ipx) → exp(i px) ((3)) We note that the above transformation is equivalent to time reversal, i.e. running a movie forwards and backwards. This means that one has the same probability exp(ipx) whether the particle is appearing in a regular movie or the time reversed one as seen from the regular system. In other words, t is moving backwards in the transformation of ((3)). We argue that this has dramatic consequences, because one may combine exp(i p dot r) probabilities for events occurring at different physical times because exp(i p dot r) does not represent a specific time. We suggest that this is the reason that one may combine wavefunctions for events in the same x at different times. It is not simply a math argument stating exp(ipx) does not explicitly contain time. It is linked with events which occur in time. We consider two examples.
Wavefunctions for Events Which Occur at Physically Different Times If one considers a particle scattering from a potential V(x), the usual approach in quantum mechanics is to state that this is a time-independent problem because V(x) does not contain time. One then writes: exp(ipx) + f(theta)exp(ipr)/r ((4)) The first term represents the incident particle (t=0) and the second, the scattered at some different time. We argue that there must be justification for combining events that occur at different times in ((4)). Physically, one would not do this, but if the wavefunctions are time reversal invariant, then there really is no sense of time and one may combine them, we argue. The key point is that one must still have conservation of “material” or classical probability. In other words, exp(ipx) may be considered to represent one particle, i.e. the incident one. This particle may or may not scatter, but if it does, by conservation of material probability, the modulus of ((4)) must show material removed from the incident beam to account for scattered results. This requires interference of the two wavefunctions (representing different physical times) in ((4)). Thus, in this example, quantum interference is linked with removing material from one time and passing it to a result at a later time. This is why we argue that the presence of x in exp(ipx) is linked with probability conservation. In other words, interference seems to be a mathematical feature. The second example is one dimensional reflection-refraction at an n1-n2 junction. In such a case, one has a single photon which either reflects or refracts. This suggests different times for these events. Normally, one would not add probability (wavefunctions) for different times, but if one has time reversal invariance, we argue that one can. A second interesting feature appears in this problem. The n1-n2 junction is at x=0, but the chunkiness of the wavelength means that one should physically not deal with a single point. We suggest that one should balance wavefunction probabilities at both x and x+dx to deal with this chunkiness. This leads to: A exp(ipx) + B exp(-ipx) = C exp(ip2x) where n1=1 and p2=n2p ((5a)) and A exp(i p (x+d)) + B exp(-ip (x+dx)) = C exp(ip2 (x+dx)) or pA exp(ipx) -pB exp(-px) = Cp2 exp(ip2x) ((5b)) Both ((5a)) and ((5b)) are evaluated at x=0. This allows one to solve for B and C by setting A=1. For n2>n1=1 one has a negative B meaning that there is interference removing some of A to allow for the creation of C. This is very similar to the scattering from V(x) scenario. Thus, a time reversal invariant probability in space allows one to combine events which occur at different times, but are interlinked. The conservation of material probability (modulus squared of both sides of ((5a))) means that one must have interference between probabilities representing different times.
Conclusion In conclusion, we argue that the quantum probability exp(-iEt+i p dot r) which is Lorentz invariant and conserves E and p (both relativistically and nonrelativistically) has other features. In particular, exp(i p dot r) indicates the direction of motion as p and -p give different results. We thus call exp(i p dot r) a dynamic probability. Secondly, there is a chunkiness region for p impulse hits given by the wavelength hbar/|p|. This allows a particle to interact with both slits of 2-slit apparatus if they are about a wavelength apart. Here, we stress a third important feature, namely that of time reversal invariance of exp(i p dot r). In particular for exp(i p x) with p→ -p and x→-x one still has exp(ipx). This is expected because if one has the x-axis pointing in one direction, one should not have a different probability from having it point in the opposite direction. The direction is which the x-axis points is arbitrary. We suggest that these two scenarios represent a movie being run forwards and another, run backwards. One does not know what time one has in exp(ipx). This, we argue, is the reason that one may call exp(ipx) time-independent. One cannot simply state that exp(ipx) is time independent just because there is no t present. Physically t is very much present in problems that are solved using the time-independent approach, such as scattering from V(x) and one dimensional reflection-refraction at an n1-n2 junction. We argue that the probability exp(ipx) may be used for events at different times if they are linked probabilistically. Linked means that some probability from one may be used to create the other as in 1-D reflection-refraction or scattering from V(x). What seems like it has no justification physically is allowed for using time reversal invariant probabilities, i.e. exp(ipx + f(theta) exp(ipr)/r for scattering off of V(x) and A exp(ipx) + B exp(-ipx) = Cexp(ip2x) at x=0 and Apexp(ipx) -pBexp(ipx) = Cp2 ex[(ip2x) at x=0. Given that material probability (the modulus of a wavefunction expression in x) must be conserved, one must have interference (addition and subtraction of the time reversal invariant probability). This accounts for material linked with one wavefunction and time being transferred to another in the same x space. Thus, this interference appears to be mathematical in nature. References 1. Ruggeri, Francesco R. Speculation on the Quantum exp(ip dot r) as a Sampling State (preprint, zenodo, 2025)
