The Significance of Time Reversal Invariance of the Quantum Free exp(i p dot r)

 Addendum  Oct. 17, 2025  In this note, we argue that exp(i p dot r) is time reversal invariant because p-->  -p and r--> - r leaves it invariant. We state this means that exp( i p dot r) holds for a forwards running movie, but also for a backwards running one. This means that ti --> tf and tf--> ti are assoicated with the same exp(i p dot r). This meands that there is no flow of time when one uses the probability exp(i p dot r). It is this lack of flow of time which allows one to add wavefunctions (exp(i p dot r)'s etc) for processes which occur at different times. This then allows one to link these processes together with a time-indpendent probability, we argue.

Addendum Oct. 16, 2025  In the case of 1-D reflection-refraction at an n1-n2 index of refraction junction, Aexp(ipx) for the incoming photon means that A*A is linked with flux because the refracted photon has p2 not= p. Thus, one requires two equations, one in Aexp(ipx) etc and one in p A exp(ipx) to create photon number which is flux/c = flux p/E where E is the same for the incident, reflected and refracted photons. In the case of elastic scattering from V(x), one does not have to worry about fluxes and may see interference showing how matter is removed from the incident beam to account for the scattered one.

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.