Dear Akinbo,
Thank you for your patience, I had some other issues to take care of before I could continue our debate, and I knew this post was going to be a long one.
I will get to your question towards the end of this post, but I would like to first address the issues surrounding the questions I asked of you.
First, let me at least give a sketch of an intuitive qualitative justification for my claim that the non-commutativity relations are equivalent to the uncertainty relation, and that rejecting them amounts to rejecting all of QM.
The way I think of this is that the commutator of the operators is a measure of the independence of each other of the states that result when two different operators act on a particular quantum state, in the following sense: if the resultant states are completely independent, then they can be separately specified to arbitrary precision in terms of either observable, which is mathematically signified by the fact that the resultant states are eigenstates that both operators have in common. If they are not completely independent, then acting on the particular quantum state with one operator gives a resultant state that can no longer be specified to arbitrary precision in terms of the other observable. In that case, the order of operation matters: If you have two operators, A and B, and the resultant states when they act on a particular quantum state S are not completely independent, then acting on S with A and then B gives a different resultant state than acting on S with B and then A. Thus the difference between the two coupled operations, which is just the commutator, will be non-zero.
On the other hand, lack of "arbitrary precision" in the specification of a state just means there is some uncertainty in the specification of a state, so that if the resultant states of the operation of two operators on S are not completely independent, then there will be a statistical spread of the product of the observed measurement outcomes associated with the two operators. Mathematically, this is signified by the fact that the product of the standard deviation of the outcomes associated with operations is greater than zero. The statement that it is equal to or greater than hbar/2 is just the Heisenberg uncertainty principle.
Why does rejecting these amount to rejecting quantum mechanics in its entirety? Because the entire mathematical apparatus of quantum mechanics, and in particular the mathematical identification of quantum states as rays in Hilbert space is essentially built to accommodate the consequences of these relations: The fact that any pure quantum state at all that is definite in some measurement basis can be expressed as a superposition of more than one state in some other measurement basis reflects mathematically the uncertainty, the limit in precision in being able to express the same state in terms of different observables. If all operators in quantum mechanics commuted with each other, then the whole formalism would all be completely unnecessary and could be discarded; we could just get by with a description in terms of classical physics, where we can specify the same state in terms of, say, position or in terms of momentum to arbitrary precision.
Now, let me address my reason for asking you the two questions:
1. Do you believe that the introductory textbook definition of a wavelength is exact?
2. Do you accept the non-commutativity relations?
You said "I smell a 'dialectic' bait meant to entrap me" and I confirm that you smelled correctly. I normally do not try to trap people, but your original comment "It gives me a kind of pleasure when my "opponents", wriggle and wreathe trying to explain a question in a convoluted kind of way because of a belief that they must not let go a dogma they are holding on to." made it fair game for me to give you a reality check.
The trap was that answering each question with yes, which is by far the reasonable answer in each case, leads to a contradiction. To see why, consider first how we go about measuring wavelength in an idealized sense: We measure the distance between two points, which is to say, we measure two positions in space and subtract them from each other, and then assign to each a point on one of two successive waveforms such that if the two waveforms were superimposed, the two points would be right on top of each other, and then say that the distance between the two points in space is equal to the distance between the two points on the waveform.
Notice that if this idealization were exact, we could arbitrarily re-express any distance between two points in space in terms of a wave length and vice versa, which implies that you could re-express position measurements in terms of wavelength measurements and vice versa, which implies that you can express either measurement in terms of the other to arbitrary precision, or in short, that position measurements and wavelength measurements are completely independent of each other
However, if you accept the non-commutation relations, of which the non-commutativity of position measurements and momentum measurements is the best known example, then you will run into the following problem: The commutator of x and p (where to keep things simple I am using 1D) is
[math][x,p]= i\hbar[/math]
But the DeBroglie relation says
[math] p=h/\lambda=h\lambda^{-1}[/math]
where
[math]\lambda[/math]
is the wavelength associated with any object described by QM. Dividing
[math]h=2\pi\hbar[/math]
out of both sides of the first equation gives
[math][x,\lambda^{-1}]=i/2\pi[/math]
which means that position measurements and wavelength measurements are not completely independent, in contradiction to the implication of our first answer. The factor of 2pi signifies that a given measured position could correspond to a a point anywhere within that waveform.
Now, for most purposes this does not matter, and in most contexts, we can, in fact, arbitrarily interchange distance measurements in space with wavelength measurements without getting into any kind of trouble. That is why you find this definition without any further qualification in freshman physics textbooks.
But remember, we were talking about a specific situation where this distinction does matter, namely whether for light particles wavelength times frequency is the same as distance traveled in space per unit time. It turns out that the relationship is very subtle and my argument is not as strong as I would like it to be because there is a compensating uncertainty between Energy and time that hides the distinction. In particular, for photons, because
[math]E=pc[/math]
if we divide the expressions for the two uncertainty principles by each other, we get
[math]\frac{\sigma_x \sigma_p}{\sigma_t \sigma_E}=\frac{\sigma_x \sigma_{\lambda^{-1}}}{\sigma_t \sigma_f}=1[/math]
where the sigmas are the standard deviations associated with each set of measurement outcomes. If you wanted to mount an attack on my view that photons do not travel in space you might be better advised to use this as an argument against it than to accuse me of dogmatism.
I think it is true that most physicists today, along with "Roemer, Galileo, Newton, Maxwell" would disagree with my view, but there are two simple explanations for that: First, I think it is fair to say that most physicists do not spend a lot of time thinking about the ontological status of photons, and second, they have not had the opportunity to seriously examine the arguments that led me to my position. Of course, just saying "photons do not travel in space" and nothing more sounds crazy, but I believe that seriously examining the arguments that I have gathered so far would make that position far more plausible.
I admit that the arguments are not yet conclusive, but the collection of different hints from two seemingly independent theories (SR and QM) pointing to the same idea is strongly suggestive. The non-conclusiveness of my arguments leaves me open to the charge of having a prejudice towards the position I hold, and that is true, but I think this is no different from the ordinary prejudices towards speculative ideas that any scientist holds as he or she is pursuing research into that idea. You need a modicum of inductive commitment to certain tentative ideas in order to be motivated to gather evidence for or against it. At this stage, this has little to do with dogma. Dogma comes in when 1) one regards non-conlusive evidence as conclusive and/or 2) one ignores contradictory evidence.
In fact, let us see what dogma looks like in a concrete example. You successfully avoided my trap by answering yes to the first question and no to the second. But you did so at a steep cost, namely by massively damaging your credibility, as I will now show.
I cannot tell whether you really meant your "no" answer to my second question or not, so I will consider both cases.
Let me first suppose that in fact you accept the canonical commutation relations, but answered "no" to my second question only because you sensed that I was trying to trap you. I think it is instructive to relate to you my own past experience of finding myself in your situation: It has happened to me more than once that, in discussing some math/physics matter I could sense that there was a trap was a trap waiting for me, but that I willingly stepped in it just so I could find out how the opponent would "pull it off" to be able to, say, find a contradiction in my argument that I could not see. In short, getting closer to the "truth" was more important to me than winning the argument.
In contrast, if I suppose that you answered "no" to my question but in fact believe the opposite, then I have no choice but to conclude that not losing an argument is more important to you than getting closer to the truth. If we considered our discussion a debate, then that would in an of itself not be a big problem, after all the point of engaging in a debate is usually to win. But this is not how you frame your discussions with others, you always frame them as "Dialectics". A dialectic is not a debate, the point of engaging in a dialectic is not to win an argument but to get closer to "the truth".
So if I assume that you did not really mean your answer, that you answered contrary to what you really believe in order to avoid losing an argument, then I am also forced to conclude that your talk of a "dialectic" with me was not sincere.I could then not escape the conclusion that the real point of your "dialectic" with me is just to convince me of your beliefs. But the desire of convincing others of one's own beliefs while pretending otherwise presupposes that one is utterly convinced, unshakable of one's own beliefs. For some beliefs, this is still perfectly alright because there is evidence supporting the belief that can be considered, for all practical purposes, to be conclusive.
But our "dialectic" is about the edge of our knowledge (and beyond) pertaining to fundamental aspects of reality, is it not? If it is, then it is an arena in which our beliefs, whatever they are, are not supported by conclusive evidence. If they were, then the arena in which they play out would not be be considered the "edge of our knowledge". But if it is the case that you hold beliefs of which you are utterly convinced, as though they were supported by conclusive evidence, when in fact, they are not (by virtue of the subject matter), then you have just satisfied the first criterion of dogmatism I gave above: one regards non-conclusive evidence as if it were conclusive.
So let me instead suppose that your "No" answer to my question was sincere, you really reject the canonical commutation relations, and, therefore, in fact, all of quantum mechanics. Here it is useful to compare what evidence you base your beliefs on vs. the belief that we should accept quantum mechanics as a fundamental description of nature.
You helpfully provided some reference articles as evidence for rejecting the non-commutation relations or, equivalently, the Heisenberg uncertainty principle, and I did look at them. Here is what I found:
Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements
Let me quote from the introduction of the article:
`Heisenberg wrote, ''At the instant of time when
the position is determined, that is, at the instant when the
photon is scattered by the electron, the electron undergoes
a discontinuous change in momentum. This change is the
greater the smaller the wavelength of the light employed,
i.e., the more exact the determination of the position'' [1].
Here, Heisenberg was following Einstein's example and
attempting to base a new physical theory only on observable
quantities, that is, on the results of measurements. The
modern version of the uncertainty principle proved in our
textbooks today, however, deals not with the precision of a
measurement and the disturbance it introduces, but with
the intrinsic uncertainty any quantum state must possess,
regardless of what measurement (if any) is performed
[2-4]. These two readings of the uncertainty principle are
typically taught side-by-side, although only the modern
one is given rigorous proof. It has been shown that the
original formulation is not only less general than the
modern one--it is in fact mathematically incorrect [5].'
In other words, this article, which you marshaled as evidence for rejecting the uncertainty principle (as commonly understood) in fact supports it. What it does is to give an argument for the incorrectness of Heisenberg's older and less general original version of his uncertainty principle. While it may be the case that, as the paragraph claims, the older version is still being taught in textbooks today (In fact, none of the textbooks that I have used mention it), I would say that among quantum physicists, the more general version is exclusively the one which is used.
Experimental realization of Popper's Experiment: Violation of the Uncertainty Principle?
This paper realizes a thought experiment first proposed by Karl Popper to take advantage of entanglement to test the limits of the uncertainty principle. I was not familiar with Popper's thought experiment, and found the paper very interesting. However, the authors recognize that, though the experiment confirms Popper's prediction, it does not support his interpretation of it due to the fact that those aspects of the entangled pair the experimented tested were not directly applicable to the HUP for a single particle.
In fact, in the conclusion they state:
"Our experimental result is
emphatically NOT a violation of the uncertainty principle which governs
the behavior of an individual quantum."[Their Capitalization]
So, this article, which you advanced as a refutation of the HUP in fact denies doing exactly that.
Particle Measurement Sidesteps the Uncertainty Principle
This interesting article discusses a new technique, called compressive sensing, which, as I understand it, provides less information in a measurement of a particle's position than usual, so that its complementary property can be measured with greater precision. The concluding paragraph begins with:
The physicists stress that they have not broken any laws of physics. "We do not violate the uncertainty principle," Howland says. "We just use it in a clever way."
So the technique described in the article does not refute the HUP, it exploits it in a novel way. Once again, an article you gave as support for your beliefs in fact supports the opposite of your beliefs (If I take your answer to my second question to correctly reflect your beliefs)
Scientists Now Uncertain About Heisenberg's Uncertainty Principle
This low-quality article discusses the same paper as above, "Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements" but does so in a highly misleading way. I already mentioned what the result of that paper is, but somehow (probably due to the author's misunderstanding) he interprets their result as refuting "one prong" of the uncertainty principle.
I will grant you that this article makes a claim of the refutation of the HUP, but this claim is the opposite of what the authors of the experiment themselves say of their result (see the above paragraph), so in choosing whom to believe I would rather go with the scientists who did the experiment than the journalist who reports on their work.
So, to summarize, if I take your "no" answer to reflect a genuine rejection of the canonical communication relations and thereby, all of quantum mechanics, then, at least going by the evidence you have provided me, your belief is based on essentially no evidence. There are of course beliefs that are held in the absence of any evidence, they are usually identified as "faith". What takes your purported belief in the present situation far beyond faith is the fact that the majority of technological innovations over the last several generations provide a massive body, a mountain of evidence that contradicts your belief. By choosing to ignore it, you satisfy the second criterion for dogmatism in a way that could hardly be satisfied any more strongly:one ignores contradictory evidence.
So no matter whether I suppose that your "no" answer was genuine or not, I am forced to conclude that whatever dogmatic beliefs I may hold, yours are manifold greater. By evading my trap, you won the battle but lost the war.
Let me now finally get to your question. I repeat it below:
"Given a light source, e.g. a pulsar say 10^3 light seconds away, and sending out pulses once every 60 seconds, such that the moment a pulse is detected, another is already emitted and on on its way and would be detected also after 60 seconds. So we have detections every 60 seconds. Now if, on detecting a pulse, the observer moves towards the next incoming pulse, can he reduce the detection time to 59 seconds? Again, if on detection, the observer moves away from the already incoming and in-flight photon, can he delay the detection time to 61 seconds?"
There are two factors to consider here: one is the fact that if the observer moves relative to the source, then successive waves crests travel different distances before they reach the observer (notice that here I am adopting the standard language of waves traveling in space in order to avoid the much more cumbersome language I would have to use if I wanted to express my beliefs about the existence of photons in spacetime. What licenses me to do this is the fact that the ontological status of photons is irrelevant to the present question). The other is the fact that if the observer moves relative to the source, then he will observe the emission events at the source to occur spaced out more due to the time dilation effect.
The first factor implies that the inverse relation between frequency of emission and emission period must be corrected by a factor derived from the relative speed. The corrected relation if the relative speed is such that the source and observer are moving towards each other is
[math]f=\frac{c}{(c-v)T}[/math]
where f is the frequency, T is the period, v is the relative speed. The second factor implies that we must substitute T_0, the period in the rest frame by
[math]T=\gamma T_0=\frac{T_0}{\sqrt{1-v^2/c^2}}[/math]
where gamma is the Lorentz factor. Substituting this into the first equation gives
[math]f=\frac{c}{(c-v)\gamma T_0}=\frac{c\sqrt{1-v^2/c^2}}{(c-v) T_0}=\frac{\sqrt{c^2-v^2}}{(c-v)T_0}=\frac{\sqrt{(c-v)(c+v)}}{(c-v)T_0}=\sqrt{\frac{(c+v)}{(c-v)}}\frac{1}{T_0}[/math]
but since in the frame of the source the inverse relationship between frequency and period still holds, we can substitute
[math]f_0=\frac{1}{T_0}[/math]
into the last equation to get
[math]f=\sqrt{\frac{(c+v)}{(c-v)}}f_0[/math]
If the source and observer are moving away relative to each other, the above relation becomes
[math]f=\sqrt{\frac{(c-v)}{(c+v)}}f_0[/math]
The reason I went through the derivation is because I suspect that you want to point out what you perceive as another "illogicality" of SR, and having the derivation handy will make it easier to pinpoint if what you wish to argue is based on a misunderstanding of some aspect of the situation.
Ok, now to your question. What you call the "Detection time" is in fact the inverse of the frequency of detections in the frame of of the observer (i.e. f), but remember the inverse of the frequency is NOT the period associated with the emission event in the frame of the moving observer because each successive wave travels a shorter distance to get to the observer than the previous one. So plugging in 1/59 for f and 1/60 for f_0 into the second last equation and using Wolfram Alpha, I get the answer that yes, it is possible if the relative speed is on the order of 5*10^6 km/s toward each other. Similarly, if I plug in 1/61 for f and 1/60 for f_0 into the last equation, then using WA again I get a solution if the relative speed is on the order of 4.9*10^6 away from each other.
I am not sure where you are going with this example, but my suspicion, based on the fact that you used the word "detection time" instead of frequency is that you wish to mount an objection based on the assumption that in the observer's frame the inverse association between frequency and period holds. But as I pointed out above and in the derivation, this assumption is false.
Already, this got to be a very long post, so I'll cut it off here.
Best,
Armin
