Essay Abstract
We ask the question in the title in the context that the two postulates leading to spacetime seem to have clear physical content. However, spacetime does not appear to discover or foreshadow quantum mechanics. We question the reason for this by examining a model clock.
Author Bio
Garnet Ord is a Faculty member in the mathematics department at Ryerson university. Much of his research looks for a statistical mechanical layer beneath quantum mechanics.
"in finite frequency limit"
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Garnet,
You wrote: "The concept of spacetime is a case in point. We can access it efficiently through ﾅinsteinｧs two postulates and it is convincing because the postulates themselves seem straightforward and unambiguousｮ ﾉt is so convincing in fact that we might wonder ｢ﾗhat is wrong with the two postulates that their invocation fails to uncover quantum mechanicsｿ｢ｼｯpｾｼpｾﾔhe second postulate is falseｬ ﾇarnetｮ ﾔhe speed of light does depend on the speed of the light sourceｺｼｯpｾｼpｾhttpｺｯｯwwwｮamazonｮcomｯﾒelativityｭﾉtsｭﾒootsｭﾂaneshｭﾈoffmannｯdpｯｰｴｸｶｴｰｶｷｶｸ ｼｯpｾｼpｾﾒelativity and ﾉts ﾒootsｬ ﾂanesh ﾈoffmannｺ ｢ﾍoreoverｬ if light consists of particlesｬ as ﾅinstein had suggested in his paper submitted just thirteen weeks before this oneｬ the second principle seems absurdｺ ﾁ stone thrown from a speeding train can do far more damage than one thrown from a train at restｻ the speed of the particle is not independent of the motion of the object emitting itｮ ﾁnd if we take light to consist of particles and assume that these particles obey ﾎewtonｧs lawsｬ they will conform to ﾎewtonian relativity and thus automatically account for the null result of the ﾍichelsonｭﾍorley experiment without recourse to contracting lengthsｬ local timeｬ or ﾌorentz transformationsｮ ﾙetｬ as we have seenｬ ﾅinstein resisted the temptation to account for the null result in terms of particles of light and simpleｬ familiar ﾎewtonian ideasｬ and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an etherｮ｢ｼｯpｾｼpｾﾐentcho ﾖalev pvalevﾀyahooｮcom
Dear Dr. Ord,
I found your essay extremely interesting and will return to it for a second read to better understand the concepts you presented. Upon the first reading, I have the impression that there may be certain similarities in our ideas at which we may have arrived from completely different perspectives.
On the apparent disconnect between QM and the two postulates of SR, have you considered the possibility that they may only seem disconnected because we have not yet generally recognized certain inherent connections without even considering the novel ideas you propose?
I'd like to point out what I see as a connection, namely the fact that objects traveling at the speed of light 'observe' in their frames a zero duration of existence in spacetime, since a zero proper time between coming into existence and going out of existence (e.g. emission and absorption) implies exactly that. Given that naively a zero duration of existence would seem to imply non-existence, yet that it is incontrovertible that objects traveling at the speed of light do exist, a way to reconcile this apparent paradox is to posit that they do not exist "in" spacetime, but "outside" (4 years ago, I turned this argument around to present a derivation of the speed of light postulate based on this idea, see http://fqxi.org/community/forum/topic/329.)
Now, the question naturally becomes, if such objects do not exist in spacetime then where do they exist? I did not answer this in my original essay but in subsequent work, I attempted to show that if objects exist in a 2+! analog of spacetime then one may assume that the manifestation of their worldlines to spacetime observers is in terms of a superposition of all the worldlines of the spacetime objects into which they can emerge (a spacetime event currently identified as "wave function collapse"). This already takes us halfway to the path integral.
The other part, which is more directly related to your paper, is that a postulated symmetry indirectly provides a means to compare the passage of time along the worldline in the 2+1 analog (which I prefer to call areatime)with the passage of time along each worldline that is part of the object's spacetime manifestation can be decomposed into two complex conjugate phase factors of the form
[math]e^{\tau/\tau_A}[/math]
where tau is the proper time associated with the spacetime worldline manifestation and tau_A is the proper time associated with the object in areatime. There are two aspects of this idea which seem directly related to what you present in your paper:
1) the phase factors effectively function as "clocks"
2) at least for a single free particle, the term that needs to be substituted for tau_A to obtain the correct relativistic phase factor of the Lagrangian formulation is its inverse Compton frequency multiplied by the imaginary unit.
I normally don't talk about my ideas at length in other people's threads, I just wanted to show why I think that there may be similarities between our ideas. However, I will need to re-read your paper in some more depth to be able to better pinpoint the similarities.
Should you be interested to find out more about, my last fqxi essay talked in some depth about the phase factor (http://fqxi.org/community/forum/topic/954) and there is also a conference talk that puts it all together http://youtu.be/GurBISsM308.
My current essay may be less related to your topic, as it concerns the relationship between QM and GR, but I would certainly appreciate if you did take a look (http://fqxi.org/community/forum/topic/1431).
All the best,
Armin
Dear Mr. Ord,
I have read your paper - at least in parts. I was impressed by your approach: It was fascinating to see how you could reveal parts of reality that are in a way obscured by the maps we use. You are asking: What is wrong with the two postulates of special relativity that their invocation fails to uncover quantum mechanics?
To demonstrate this failure you have interestingly made use of a tool that you have described as Particle-as-Clock. Using this particle-like tool you have found a room for a wave-particle duality, that is obviously obscured by the relativistic spacetime.
I've attacked the problem defined in your question from a more principal point of view and discovered a part of reality (i.e. spacetime) that is still hidden. If you look at the second postulate of special relativity you can see it is restricted to a sort of wave-like face of c. But if the wave-particle duality is a central feature of light it would be natural to assume that the speed of light has Two Faces as well, that is, a wave-like face of c and a particle-like face of c, but the latter one is still missing. It is not part of Special relativity. (See my paper: Is the speed of light c of Dual Nature?)
I think you have touched this still hidden particle-like face of c just by a systematic use of your particle-as-clock tool.
I wish you good luck for your interesting and clear-minded paper.
Kind Regards
Helmut
Dear Garnet Ord,
In Coherently-cyclic cluster-matter model of universe two types of space-time is expressional; one is to describe the motion of orbital-matters in an eigen-rotational string in that the space-time is continuum and the other is to describe the motion of projectile trajectory of independent macro objects of dense tetrahedral-branes, in that the space-time is discrete.
With best wishes,
Jayakar
Dear Garnet Ord,
Thank you for your essay. Your 'particle-clocks' idea provides an interesting approach to 'what is wrong with spacetime'. Let me suggest another reason for this. Compatible with your keen mathematical analysis. As you point out in your essay, "The [Spacetime] requirement that particles move on smooth worldlines violates the uncertainty principle." and "Since progression through time is particularly suspect, we consider the idea that a particle is really a small digital clock".
The Spacetime continuum assumes physical time to be 'instantaneous', t=s. Thus, physical events depicted as points in the Spacetime continuum are assumed to be 'instantaneous' at each instant of time t=s. The Second Law of Thermodynamics, however, requires that "every physical event takes some positive duration of time to occur". (see my Chapter, "The Thermodynamics in Planck's Law") Thus, physical time is 'duration', t-s. And not 'instantiation', t=s. And so physical events require some positive 'duration of time' to occur. Rather than occurring at each instant t. When a particle moves along a smooth worldline, each moment of its movement is a 'physical event' which requires a positive duration of time, according to the Second Law. This, in my opinion, would explain why Cosmology (based on GR and Spacetime) is in conflict with Thermodynamics.
Best regards,
Constantinos
essay: "The Metaphysics of Physics"
Hello,
I agree that the second postulate is non-intuitive and difficult to reconcile with the Newtonian picture that seems so natural at terrestrial speeds. However the essay is less about the validity of the postulate and more about a what-if scenario where we relax the classical picture of smooth worldlines between events. By examining how a simple digital clock works under these circumstances one can see interesting connections to quantum mechanics.
Hi Armin,
thanks for the comments. I agree there are definite similarities with the ideas you suggest. I shall read your work on QM and relativity with interest!
Hello Helmut,
thanks for the comments. I look forward to reading your paper.
Hello Jayakar,
thanks for the link.
Hi Constantinos,
thanks for the comment. I would be very interested to see the second law used to consider the picture of events as having finite duration vs events as a 'points'. I will read the chapter with interest.
Dear Garnet,
Thanks for your interest in my Chapter (and my essay). There is much in it and can be a little confusing upon the first reading to unravel the results. But specifically on your stated interest of 'my take' on the Second Law. Let me give you a little guidance on how this comes about.
One of the many results to come out of my derivation of Planck's formula for blackbody radiation is a direct proportionality between entropy and time. The actual result is as follows: ΔS = kνΔt , where k is Boltzmann's constant and ν is frequency (or 'rate of evolution' is more closer to the 'truth'). From this, it is obvious to me that the Second Law is not really about 'entropy' (though it can be thought that way) but really about 'physical time'. Thus, the Second Law can be restated to say, "every physical event takes some positive duration of time to occur'". It is curious to me why there is no Basic Law in physics pertaining to probably the most fundamental of all physical quantities; "physical times" associated with physical events. I like to suggest the Second Law is just that Law of Nature.
With this understanding, many mysteries in Physics and Cosmology clear up. Foremost, the misuse of Spacetime to represent physical events.
Best regards,
Constantinos
Dear Garnet,
Your idea is very interesting. It is a mystery that matter has a frequency but its amplitude has only a probabilistic meaning. I share the same idea of a clockwork in a particle. In fact, I have used a similar idea but assuming the particle has a vibration of time rate in my essay and is able to obtain the properties of a particle field. I am reading other papers in the contest but your paper is one that I read a number of times. Our approach seems to have many common points. I hope we can continue to communicate even in the future to get your feedback and ideas.
To better understand your idea, I have two questions:
1. The Compton clock has 4 states. The clock rotates around these states with spatial displacement. Since it has spatial displacement, will the clock run slower than the clock in an inertial frame? From relativity, the particle should be running along a time-like geodesic and has the same time rate of an outside inertial frame.
2. The states of the clock is real. However, the overall phase of the matter wave is unobservable. The phase can be shifted without changing the probability density. Can the field generated by the Compton clock have the same property?
Sincerely,
Hou Ying Yau
Hello Yau,
thanks for the comments. Please send me an email if you would like more discussion.
In answer to the first question, the 'moving clock will run slow'. This feature is controlled by the fixed spacetime area between audible events. It is actually the fact that the invariant is an area, that leads to wave propagation and the uncertainty principle in the 'chessboard' limit.
The phase of Compton's clock in the chessboard limit is similar to what one would expect in quantum mechanics as the model faithfully produces the Dirac propagator. However, although the model hints at measurement features, there is as yet no immediate implication of the Born postulate. One could simply invoke it, but the hope is that it will ultimately follow through a statistical argument.
Dear Garnet,
You have written a very interesting and relevant essay. I have a few questions:
1. Do the causal dymanical triangulations people know about your work? I notice that you referenced the paper by Ambjorn, Jurkiewicz, and Loll.
2. Two different possible "methods of quantization" are confusing me. Following your discussion of continuous spacetime as an infinite mass limit (up to the middle of page 5), one might reasonably think about "classical discrete spacetimes" built from the Alexandrov-type "areas," somewhat in the spirit of CDT. One could then "sum over geometries" to get a quantum theory. Alternatively, you mention the chessboard model with stochastic edge lengths and Feynman sums over a particular spacetime. I am not sure how much of your discussion in this particular paper is intended to apply to SR only, and how much should be expected to generalize.
3. On a related note, I am trying to compare the type and amount of local information you use with the corresponding approaches in CDT and causal sets. It seems that you have a local scale (via the clock), local dimension, and local orientation?
4. I note from your bio that you study statistical structure underlying quantum theory. The only classical statistical structure I see here is the stochastic edge length. Does such structure play any other role in your view of fundamental spacetime?
5. Near the beginning of your essay, you express the opinion (with which I wholeheartedly agree) that the real numbers are not real (i.e., physical), but that they are convenient for calculus, etc. Now it seems to me that the taking the continuum for granted can have very serious effects that have nothing to do with harmless interpolation. For instance, the whole of quantum field theory is based on the representation theory of the Poincare group, which falls apart once you shed the continuum. I think that you and others are very right to keep the fiction of the continuum front and center.
By the way, if you want to know the motivation for my questions about emergent spacetime, you may read my essay here: On the Foundational Assumptions of Modern Physics.
Take care,
Ben Dribus
Dear Garnet,
Your model has very interesting features using 4 states for the internal clock that lead to some familar properties of Dirac propagator. The model that I developed in my essay has fluctuations in time rate which can produce the quantum field for a zero spin particle. It also has properties that I hope can resolve some fundamental questions for non-locality. The future step will be to extend it to 1/2 spin particle field. It seems our approaches have some common points that may supplement each other to get the Dirac field. I hope I can get your feedback on my essay.
Sincerely,
Hou Yau
Hi Ben,
thanks for your comments and questions. To save space, I use your numbering system below.
1) I am not sure, but in general I think the relevant binary relation is not symmetric.;-)
2) Up to the middle of page 5, the infinite frequency limit has been taken as in Fig. 3a. The result is Equation (7). The lowest order approximation of this clock is (6) and because it is translation invariant, the resulting 't' could well represent Newton's absolute time.
(7) as it stands, as you suggest, has a form of quantization in it. The quantization however just scales the unit vectors of the `spacetime' for which the 'clock' is a representative time-keeper. In taking the limit we see a link between the initial finite areas of the Compton clock, and the infinite frequency limit that shrinks those areas to zero while filling in the events on the t-axis. This leads to Minkowski spacetime with its odd signature, but by shrinking the areas to zero we have supressed quantum propagation.
The second limit is more interesting because by building in a Poisson process for events, infinite frequency is allowed, but the probabilistic weight favours a mean finite area between events. The result is the chessboard model that gives the Dirac propagator! By allowing the finite area, the multiple-history picture of the path-integral survivies and it is this that gives wave propagation. The fact that wave propagation survives in this limit and not the original is I think the most interesting feature of this sketch. It shows that SR is conventionally classical because it starts in the continuum in such a way that quantum propagation is surpressed. If you back away from the continuum, as per this clock, you see spacetime as an infinite mass limit and Dirac propagation as continuum limit in which mass is kept finite.
I do expect this picture to generalize. The above calculation shows that relatiity and quantum mechanics are siblings rather than marriage partners. We should be looking for the parent from which they emerge, and to do that we may have to forgo the initial convenience of the calculus.
(3) Yes on all counts. The scale is determined by event frequency (ultimately mass). The local dimension that characterizes clock frequency is two (spacetime area), and the oriented area (in the sense of geometric algebra) is what is averaged by the stochastic process in the chessboard limit.
(4) My background in stat mech colours my view of both relativity and quantum mechanics. In the sketch, I only mention the Poisson process that, when added to the clock, gives the chessboard model. However structural similarities between, for example, the diffusion and Schroedinger equations and the telegraph and Dirac equations are, to me, too direct to be a coincidence. The classical equations emerge from an underlying statistical mechanics, and as a result the partial derivatives involved are an approximation valid on restricted scales. I would expect the same to be true for our 'fundamental' quantum equations too.
(5) I agree. Scale dependence is common in macroscopic physics and I suspect that to resolve some of our foundational problems we shall have to be more circumspect about the continuum.
Dear Garnet,
Thanks for the detailed reply! By the way, have you studied Sorkin's classical sequential growth for causal sets at all? The reason I ask is because this is a stochastic model which is as "primitive as possible," and I wonder how it compares to your view of an underlying statistical structure. I don't agree with all of Sorkin's postulates, but my own ideas are fairly close to his in spirit; in particular, theories like CDT assume a lot of structure that I would prefer to try to explain. Take care,
Ben
I agree with Pencho. It is not about intuition, it is about logic. I disagree with Pencho in that the situation requires the "ballistic interpretation." Whereas a pulse of light seems to act like a particle in that it has a beginning and an end, and if emitted from a coherent collimated laser, stays relatively compact as it radiates, a strobe flash radiates in a hemispheric pattern, any given observer only intercepting a portion thereof. The wave interpretation has more going for it.
See: A Challenge to Quantized Absorption by Experiment and Theory
See also my essay: A Logical Analysis of Albert Einstein's Mirror-Light-Clock Gedankin
