Wave with Self-Spacetime Creating Notion of Average Particle in Wave-Particle Duality?

    A quantum particle is often described as exhibiting dual particle-wave nature. A particle seems to be an object which obeys Newton’s laws/special relativity with a wave also being a phenomenon existing in Newtonian mechanics (special relativity). In other words the particle and wave are governed by x-t external variables which have infinite resolution. It was originally thought that x and t in addition to being external were absolute, but special relativity changed this idea.

     We have suggested in previous notes that a free quantum particle with energy E and momentum p creates its own space time i.e  time intervals are proportional to 1/E and space to 1/p. This means no external clock is needed and different internal rulers and clocks exist for different E,p. It also suggests finite time and space intervals. In fact, an absolute is introduced namely A=0 = -Et+px with t and x being independent. This suggests that t→b/E and x→b/p. A is the free particle Lagrangian (both relativistic and nonrelativistic) with x and t being considered independent. Thus we suggest that the external x ruler and t clock of Newtonian mechanics represent an average of the internal picture and are also described by A, but with x/t=v and not x and t being treated as independent. In (1) we showed that having discrete x spacing with a running time (i.e. Netwon’s time variable) and insisting on spatial invariance leads to dA/dx partial = p.

  Thus instead of considering particle wave duality we suggest an underlying wave structure which creates its own spacetime intervals (1/E, 1/p), but may be mapped in certain situations to the external running x and t of Newton’s theory. For example, consider an electron (a quantum particle) with a fixed p moving over a distance much greater than its wavelength. It may be mapped  to Newton’s external running x,t values with x=vt, but this is an average mapping into continuous x and t. In reality the situation is more complicated because there may be a Bohm-Aharonov effect occurring. Thus the underlying wave picture is in fact physically important and linked to interactions. Thus a self-ruler is not simply a measuring device - interactions may occur unlike in the case of the external ruler.

  Given that p  defines a kind of spatial invariance approximated by a periodic function exp(ipx) (eigenfunction of -id/dx) and that Newton’s running time is not needed in a quantum bound state equilibrium, one may wonder what one should do in the case of a particle bound in a V(x) potential. In Newtonian mechanics V(x) suggests non-spatial invariance and v(x) velocity changes with x as well in contrast with exp(ipx) which is periodically invariant. In fact Newtonian mechanics (and Lagrangian/Hamiltonian as well) focus on d/dt i.e x(t). We have argued above that time intervals 1/E are independent of 1/p spatial ones. It may be noticed, however, that different p values lead to different wavelengths and so even though there is a kind of spatial invariance for each, when compared to each other this invariance is broken and superpositions of exp(ipx) may create local x probability densities. At the same they create spatial momentum densities because -d/dx (d/dx) 1/2m yields a kind of average kinetic energy when normalized. This seems to describe the x-p uncertainty relationship. In order to create spatial resolution (other than the built-in one in the wavelength) one needs to superpose different momenta, each of which are spatially invariant according to their periodic scheme exp(ipx). Thus one may create an average 1/2m <pp> and obtain a vrms. This may be described in terms of Newtonian external running x and t variables which have infinite resolution, but this vrms does not map to a physical state i.e to exp(i (mvrms) x) localized at x. Rather the various exp(ipx)’s which create vrms are the physical entities as may be seen by knocking a particle out of a bound state. The physical exp(ipx) momenta states have spatial invariance and so cannot be associated with a specific position in space. Only their superposition creates the notion of specific position in space, but then there is a distribution of p.  This seems to be the basis for the x-p uncertainty principle.

    Thus we suggest that the particle of particle-wave duality is created through a mapping of a physical wave scenario to a continuous (infinite resolution) x-t scheme which in reality is an abstraction because each p,E free particle state has it sown internal clock-ruler i.e. 1/E and 1/p time and space units. Given that 1/E and 1/p have finite values the idea of dt and dx tending to 0 only makes sense as an abstraction which only holds in certain situations.