In Search of Continuity: Thoughts of an Epistemic Empiricist by Ian Durham

Dear Ian,

I onjoyed reading your essay. Several of the issue that you discuss have been paradoxical for ages and have not been resolved satisfactory. There is, however, a way to get around those issues, which I will explain below by addressing them specifically.

1. Page 4. the second expression for the average speed makes the assumption that the limit can be determined, because dt (read d as Greek delta) can be made arbitrary small. This is not the case. As I describe in my essay, for electrons, dt has a lower bound equal to 10 exp (-20) sec. A The electron would cease its existence by 'making it smaller'. In other words, mathematically, one can determine the stated limit, but it is unphysical. The velocity of a massive particle is well-defined, even although dt is different from zero. The velocity of a particle does not need to be described in terms of a mathematical limit at all: a definition in terms of a discrete ratio is sufficient, i.e. v = dx/dt, where dx and dt are constrained by quantum condition (4) stated in my essay. In this way, an electron can be assigned a velocity without measuring this (see page 6 of your essay where you discuss the issues around this).

2. Page 5. Classical Light. I assume you mean light described in terms of classical theory.

3. Page 5. "by Brukner and Zeilinger to argue that the continuum is nothing but a mathematical construct, a view I wholeheartedly endorse". I do not necessarily agree with this view. As I describe in my essay, continuity needs to co-exist with discreteness. In the theory I describe, two underlying continuous fundamental fields are needed to explain the existence of particles, interaction between particles, and dynamically emergence of local discrete space and time.

4. Page 5. "So what happens in the limit as dt --> 0 for classical light?" As I indicated under 1., dt cannot be smaller than 10 exp (-20) sec. Light cannot be attributed a discrete time, unless one wants to define it as wavelength/c. The latter is not very useful, since the time would be wavelength dependent.

5. Page 5. "Suppose we decrease dt while leaving dx unchanged. As dt gets smaller and smaller, it implies we are measuring the difference between x1 and x2 more and more rapidly. Lest we forget, classical physics limits how rapidly information can propagate. At some point, without changing dx, we will be empirically prevented from further reducing
dt since the ratio of
dx to dt cannot exceed the speed of light. So, if we wish to take
dt --> 0, we must take dx--> 0 in order to keep the ratio at or below the speed of light."

There is an implicit assumption made that dt can be made arbitrary small, which is not the case as I explained earlier. For a stationary electron dx=0, while dt=10 exp (-20) sec, such that the ratio dx/dt=0 and there is no issue with violating the speed-of-light as one would get by assuming that dt can be made arbitrary small. When the speed of an electron increases, both dx and dt increase, but their ratio cannot exceed the speed-of-light c. In my essay, I explain that this is due to the fact that the internal speed of random spatial motion of an electron is equal to the speed-of-light. The details can be found in the second report on my website.

6. Page 5. "The classical theory of light assumes light is a wave which is an inherently non-local phenomenon". Indeed. This result in a paradoxical behavior. However, is is also known that light consists of photons, which propagate at the speed-of-light, and posses a kind of corpuscular behavior when detected. When one assumes that photons are oscillating blobs, then they do not behave as a classical wave. Still, a 'wavelength' can be assigned, which is equal to the length of the oscillation.

I have a few more comments, which I may write up later.

Dear Dr. Durham,

Thankyou for the reply. In my essay is a defintion of 'the Light,' which is a generalisation of the energy of a photon. Historically and conceptually this definition precede's the advent of QM.

Based upon what you said above, 'the Light' is both epistemic and ontic. Therefore the reason knowledge of 'the Light' is discontinous (at the subatomic level)is because 'the Light' itself is discontinous. Perhaps you could read my essay and let me know if you think that is correct.

Robert

Dear Ben Baten,

For an electron at rest Dx (D is Delta) is not zero, as you say, but

Dx ~ (hbar/mc),

with m being the electron mass. This is the equation (13) in the Reference 6 cited in my Essay.

Reference 6 rigorously revises these and other topics (for instance, the equation (14) gives the value of Dx for an ultra-relativistic electron with momentum p), explains why those limits Dx and Dt are not fundamental but arise only under certain approximations in the propagators (as the approximate propagators used in relativistic QFT), and corrects other claims that you and Ian are doing here.

Dear Ian Durham,

as explained in my previous posts above, the existence of a lower limit to Dt (D is Delta) does not imply that the instantaneous velocity for light is not defined as, however, you believe.

As I showed, when one considers also the lower limit to Dx one obtains the exact equation for photons

Dx/Dt = c = dx/dt

which implies that expressions as (dx/dt) are perfectly well-defined and measurable.

You correctly point out Baten's mistake about Dx for electrons. However, again the lower limits for both Dx and Dt for relativistic electrons (or other particles) does not imply that the instantaneous velocity for those particles is not defined.

A rigorous analysis of relativistic localization for electrons was given in the reference 6 cited in my . One can easily obtain the next exact equation for fermions

Dx/Dt = (c alpha) = dx/dt

where alpha is one of the Dirac matrices.

Precisely the instantaneous velocity (c alpha) is used in QED to obtain the current density

j = e Psi* (c alpha) Psi

where e is the particle charge and Psi the field

Or in a more standard form

j = e c \bar{Psi} gamma Psi

with \bar{Psi} the adjoint field and gamma another of Dirac matrix.

There are other claims that you do that are corrected in the same reference 6.

Ian,

Indeed. My knee-jerk reaction is a case in point. This part of relativity is abused so widely and often that it's blinded me to checking terms more carefully for definition.

Tom

Dear Juan,

See my last reply to Ian in which I have included an answer to the issue you bring up.

Dear Dr. Ian Durham,

Iam a little bit confused of your thoughts on digital and analog nature of reality.Which one is more basic than the other one,analog or digital? Or do you want to say that it lies in our way of perception of reality. Any way historic background upon which you have based your essay is really absorbing.

But,I have other thoughts on the above problem in my essay.Why dont you,please,go through it and have a different view of the problem? Expecting your openion on it.

Best regards and wishing success in the competition.

Sreenath B N.

Dear Ben,

You are right on that we are talking about different things.

By "x" both Ian and me are referring to the position as used, for instance, in QED. This is the instantaneous position of "point particles" as described in a quantum field theoretic framework. There is not such a thing as "internal motion" for point particles. Moreover, the nonzero Dx (D is delta) is not associated to real motion of any kind (althouth I know that some few references claim otherwise for the Zitterbewegung).

If you want compute velocities/speeds you must use compatible positions and times. If "x" denotes position in QED, then "t" denotes time in QED and ratios as Dx/Dt and dx/dt are mathematically well-defined and with physical meaning. If by "x" you mean otherwise (as it seems that you mean with your "externally observable motion"), then you must also change "t" from that on QED to that in your own model.

Finally, I want to emphasize again that the Dx ~ (hbar/mc) written in my above message is valid only for a stationary electron in QED and that it is neither the De Broglie wavelength lambda nor the Einstein-de Broglie 'wavelength' postulated in 1924 by de Broglie in his unfounded mixing of non-relativistic wave quantum mechanics with special relativity.

To avoid further misunderstandings, I would add that the instantaneous velocity (c alpha), used in QED to obtain the fermion current densities, is not observable, although the speed and one of the components are observable.

The reason which the vector (c alpha) is not observable has nothing to see with the existence of a lower limit for Dt, but is a direct consequence that the x in QFT is not Hermitian. This is the true reason which there is not position operator in QFT and position is downgraded to unobservable parameter, as emphasized in many texts.

Dear Ian,

I have been analyzing another kind of statements done by you both in your Essay and in this forum. Relations as (Dx/Dt = c = dx/dt) given by me in this forum are independent of the "given existing technology".

Another issue is our current ability to measure arbitrary physical, chemical, or biological quantities using the technology at our hand. This would be the debate of the difference between (dx/dt)_th and (dx/dt)_exp, for instance.