Time and Classical and Quantum Mechanics: Indeterminacy Versus Discontinuity

Peter Lynds

Peter Lynds

Zeno of Elea's motion and infinity paradoxes, excluding the Stadium, are stated (1), commented on (2), and their historical proposed solutions then discussed (3). Their correct solution, based on recent conclusions in physics associated with time and classical and quantum mechanics, and in particular, of there being a necessary trade off of all precisely determined physical values at a time (including relative position), for their continuity through time, is then explained (4). This article follows on from another, more physics orientated and widely encompassing paper entitled "Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity" (Lynds, 2003), with its intention being to detail the correct solution to Zeno's paradoxes more fully by presently focusing on them alone. If any difficulties are encountered in understanding any aspects of the physics underpinning the following contents, it is suggested that readers refer to the original paper for a more in depth coverage.

Jan Hilgevoord

Peter Lynds

It is postulated there is not a precise static instant in time underlying a dynamical physical process at which the relative position of a body in relative motion or a specific physical magnitude would theoretically be precisely determined. It is concluded it is exactly because of this that time (relative interval as indicated by a clock) and the continuity of a physical process is possible, with there being a necessary trade off of all precisely determined physical values at a time, for their continuity through time. This explanation is also shown to be the correct solution to the motion and infinity paradoxes, excluding the Stadium, originally conceived by the ancient Greek mathematician Zeno of Elea. Quantum Cosmology, Imaginary Time and Chronons are also then discussed, with the latter two appearing to be superseded on a theoretical basis.

Francesco R. Ruggeri

Newtonian mechanics deals with smoothness in time and space through x(t) where the “details” occur within dx and dt which tend to zero. Thus there are no discontinuities in x and t. We have argued that the quantum free particle wavefunction pieces exp(-iEt) and exp(ipx) deal with discontinuities and that smoothness is due to a “modulus” of two dimensional probability e.g. exp(ipx)exp(-ipx) which yields 1 or determinism. Both exp(-iEt) and exp(ipx) contain finite measures in time and space, namely hbar/E and hbar/p and so there may be discontinuities. For example, exp(ipx) contains probability cos(px) and sin(px) within a wavelength so x=vt is only an average type of motion. Thus using exp(ipx) in a problem may result in time discontinuities because the particle is not moving as x=vt within a wavelength. In Part I we examined one-dimensional reflection/refraction of light from an interface of two media with n1=1 and n2>n1. We concluded that the quantum solution of this problem involves two equations, one linked to probabilities and the other to impulse. Multiplying the two equations together (with a special grouping, namely a spatial grouping) yields a single equation which may either be interpreted as a pressure balance equation. Thus one starts with two equations which are both grouped in time, but with the second demonstrating vector directions and obtain a single equation which is only grouped in space (which is linked with spatial grouping). By rearranging this equation, one may obtain a time grouping i.e. energy conservation equation, but the two groupings cannot exist together. This is the problem with the classical pressure picture linked to a steady state scenario and not a single photon process which displays a discontinuity in time i.e. a single photon may either reflect or refract. In this note we try to consider why classical energy conservation/ pressure balance only leads to one equation, while the quantum mechanical approach leads to the two required equations at the price of introducing a square root flux type of probability. We argue that there seem to be two factors. First there is the notion of scalars and vectors and a space versus time picture. The single conservation of energy equation AA/c1 = BB/c1 + CC/c2 where AA,BB,CCC represent the incident, reflected and refracted beams is expressed in terms of time and so vector information (i.e. -p for the reflected beam is not conveyed). If one rewrites this as AA/c1 -BB/c1 = CC/c2 then one obtains a pressure balance type of equation, but time is removed and one is grouping by space. The second and more important feature we argue is time discontinuity. If one considers a single photon, it may reflect or refract, but not both. Thus there is a time period in which to observe both reflection and refraction and more than one incident photon is needed. Newtonian mechanics in the form of steady state treats time as smooth and so everything must happen instantaneously. This is the idea behind pressure. One considers an impulse 2p multiplied by flux i.e. velocity. The flux is used to find the number of particles which strike a surface per second as if this were a continuous smooth process in time. For the case of reflection/refraction it is not. It is discontinuous because either a reflected ro a refracted photon is created each time. Thus the notion of pressure averages this over and loses some of the information in the problem. This, we argue, is why a pressure balance/ energy conservation equation is not sufficient to solve the reflection/refraction problem even though pressure is a physical concept and may be measured (on average) just as x=vt may be measured on average.

David Glick