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RECENT POSTS IN THIS TOPIC

Cristi Stoica: on 12/26/08 at 8:29am UTC, wrote Dear Peter Morgan, I like your approach. I consider that some of the...

Eckard Blumschein: on 12/22/08 at 1:54am UTC, wrote Dear Peter, This is not yet my reply to what you wrote on Dec. 18. I just...

Peter Morgan: on 12/20/08 at 14:19pm UTC, wrote Dear Eckard, I think that here you misspeak: "One must not simply add two...

Eckard Blumschein: on 12/20/08 at 1:01am UTC, wrote This leads to the attachment.

Eckard Blumschein: on 12/20/08 at 0:58am UTC, wrote Dear Peter, Thank you for your detailed reply. I will deal with it. In...

Peter Morgan: on 12/18/08 at 13:31pm UTC, wrote Dear Eckard, I'm sorry, but as far as I can tell your attachment says...

Eckard Blumschein: on 12/18/08 at 1:42am UTC, wrote Dear Peter, When Peter Lynd admitted his hope my criticism might be...

Peter Morgan: on 12/17/08 at 2:47am UTC, wrote Dear Eckard, Almost everyone who has entered this contest appears to...


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April 26, 2018

CATEGORY: The Nature of Time Essay Contest (2008) [back]
TOPIC: The direction of time in quantum field theory by Peter Morgan [refresh]
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Peter Morgan wrote on Oct. 27, 2008 @ 09:12 GMT
Essay Abstract

The algebra of observables associated with a quantum field theory is invariant under the connected component of the Lorentz group and under parity reversal, but it is not invariant under time reversal. If we take general covariance seriously as a long-term goal, the algebra of observables should be time-reversal invariant, and any breaking of time-reversal symmetry will have to be described by the state over the algebra. In consequence, the modified algebra of observables is a presentation of a classical continuous random field.

Author Bio

Peter Morgan has a degree in Mathematics, followed by a dozen years as a computer programmer, a couple of years dropped out, and an M.Sc. in Particle Physics. After a few years, it seemed interesting to ask what the differences, similarities, and relationships between quantum fields and classical statistical fields might be. In progressively different ways, he has asked the same questions for about a dozen years and has been publishing papers since 2004. He has been a research affiliate in the Physics department at Yale since 2004.

Download Essay PDF File

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Peter Morgan wrote on Oct. 31, 2008 @ 20:45 GMT
Some may consider this paper too mathematical, although I took some trouble to keep it relatively elementary. In any case, I have submitted the paper because it was the topic of the contest that caused me to write the paper, which derives from considerations about time-reversal but gives a new perspective on quantum field theory.

It's perhaps also rather against my paper that I don't say much about the Nature of Time directly, because I'm by now in the habit of not doing too much metaphysics in papers. I believe it hinders Physicists and Mathematicians accepting new work if one makes too much of it. Nonetheless, I believe that this mathematical and conceptual approach to quantum field theory may somewhat interest those who do engage in metaphysics.

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T H Ray wrote on Nov. 1, 2008 @ 13:13 GMT
Excellent take on techniques to guarantee continuous function physics (consistent with experience and experiment) using algebraic methods. I agree in principle, though I personally think that a continuous random field must be modeled as n-dimensional continuous in order to guarantee sufficient "room" to break time reversal symmetry; i.e., a dissipative model.

Thanks for a thoughtful and well constructed essay.

Tom

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Peter Morgan wrote on Nov. 1, 2008 @ 19:05 GMT
Thanks, Tom.

Having looked at your FQXi Essay Contest paper again, I'm curious how your comment (your 2nd sentence) could be realized in my QFT-motivated formalism? I /think/ I don't accept the premise of your comment, however, because I take dissipation to be an achievable feature of a state over a time-reversal invariant algebra. I think a state over the algebra of observables of the continuous random field as I have constructed it here *can* be dissipative, by contingently restricting to positive-frequency test functions when constructing a state by the action of creation operators on the vacuum (or at least by a contingent time-reversal asymmetry of the state).

Something I've only realized because of responding to your comment is that the vacuum is zero energy in the continuous random field formalism -- in the sense that it is time-reversal invariant -- unlike the QFT vacuum.

That said, I think I'll riff a little. The proposal here is only that we *might* require the algebra of observables of a quantum field theory to be time-reversal invariant, in line with conventional ideas of coordinate invariance. *If* we do, in a creation and annihilation operator formalism, then the vacuum state *must* have negative frequency components as well as the usual positive frequency components, and the algebra of observables is essentially classical (in a very clear but limited sense), but the algebra of positive-frequency modes of the time-reversal invariant algebra is essentially identical to conventional quantum optics. This last fact makes it /relatively/ difficult to make an empirical argument against this new formalism.

The interesting thing, to me, is that this approach to understanding QFT by comparison with classical random fields is rather different from other approaches to the comparison of quantum/classical. I take it that continuous random fields are of interest even if we can't construct models that are as empirically effective as those of the standard model of particle physics because of the light they shine on QFT.

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Dr. E (The Real McCoy) wrote on Nov. 1, 2008 @ 21:48 GMT
Hello Peter,

I enjoyed your paper!

But I was puzzled by your comment,

"It's perhaps also rather against my paper that I don't say much about the Nature of Time directly, because I'm by now in the habit of not doing too much metaphysics in papers. I believe it hinders Physicists and Mathematicians accepting new work if one makes too much of it."

I'm not sure how saying...

view entire post


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T H Ray wrote on Nov. 1, 2008 @ 23:13 GMT
Thanks for taking a look at my paper, Peter, though I didn't mean to be self-promoting. My comment--and it was a comment, not a criticism--was motivated by the thought that Poincare recurrence in such a closed system might obviate randomization of your continuous field, and therefore present the same (mathematical) problem of time reversal symmetry as with the classical field. I agree that QFT needs the kind of mathematical approach you bring to the party, though. (An algebraic basis for the continuous field was a major part of Einstein's quest.) We differ mainly in choice of space. Obviously, I am married to supersymmetry.

Dr. E, I have to take Peter's side in the question of physics vs metaphysics. Using your own example, though Faraday was one of the greatest experimentalists ever, can one claim to truly understand his results without knowing Maxwell's equations? I don't think Einstein would disagree; special and general relativity are mathematically complete.

Tom

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Peter Morgan wrote on Nov. 2, 2008 @ 01:38 GMT
DrE, I rather enjoyed most of your quotes, but I didn't say that I don't do metaphysics, nor that metaphysics are not important, ... I think I said that /this/ paper is metaphysics-lite. When /I/ go heavy on the metaphysics, /I/ don't get published, but of course there are some who do metaphysics well enough to be published --- and in ways that I find interesting.

For what it's worth, however, I'm /trying/ to do Physics in a way that resists the post-positivist critique of Positivism --- which I take to be substantial, albeit, lamentably, mostly nonconstructive --- by thinking about models in Physics and "bridge principles" from those models to the world in a way that I take to be somewhat associated with Lakatos's ideas on research programs. Alternatively, I would also want to say that models /mediate/ between ourselves and the world, which switches the emphasis in what I take to be a useful but not essential way (for this, see "Models as Mediators", an edited volume by Morgan(not me)&Morrison). Taking such views seriously, and trying to be honest to whatever, vaguely, they are, makes writing for publication in Physics journals a delicate matter.

Tom, I guess Poincare recurrence has no impact on a system that is essentially acausal. Random fields model correlations without a commitment to underlying causal models (but also without insisting that there cannot be underlying causal models). The kind of classicality that I've constructed, which in a sense is a half-way house, is not necessarily something that Einstein would have condoned.

Peter.

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T H Ray wrote on Nov. 2, 2008 @ 02:06 GMT
To do some riffing of my own:

I have been educating myself on your precise meaning of “continuous random field,” and so read your NKS 2007 paper. Very nice. Now that I have a better understanding of the probabilistic structure of the field (“fractal structure all the way down”) I think I have a better grasp of the dissipative mechanism though I would still want to argue that recurrence will obviate time asymmetry, for the reason that there are no boundary conditions on correlated causes from –oo to +oo. Yes, of course, I don't believe Einstein would approve of postulating continuous functions, even if probabilistic, without boundary conditions. I have to imagine that the system is closed, because correlation of causes implies endpoints which are correlated and time symmetry is preserved.

On the other hand, I appreciate that correlated observables in this random field model amount to a distributed cause, rather than a common cause between two particles—a large scale analog, I think, is laterally distributed control in complex systems science; e.g., Bar-Yam’s Multiscale Variety. Then we could speak of self organized critical correlations, and correlations of correlations—I think I am starting to grok your emphasis on “contextual measurement” in modeling experiments; quantum averaging would thus subsume the measurement problem in a very natural way, would it not?

As they say in show business, I think this approach has legs.

Tom

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Peter Morgan wrote on Nov. 2, 2008 @ 13:10 GMT
Many thanks, Tom. Keep in mind that this is **just a model**. What happens "at infinity" cannot be experimentally tested. There's just how far we've managed to look so far. In this class of models there is a boundary condition that is largely the same at infinity as for QFT: the vacuum state looks the same wherever you are, no matter how far you go. That's enough to make the mathematics well enough defined to do honest business. We also can't measure the infinitesimally small, which this model includes information about, but it's all only as a place-marker, on the doubtless faulty basis that small-scale details are the same as at scales we /can/ examine, until we find ways to examine and understand Planck-scale Physics and beyond.

I'm not quite clear what you mean by imagining the system to be closed. It's just a model, closed or open is billions of light years away, and GR/QG issues presumably become important much sooner.

Yes, insofar as causes can be inferred from correlations, they are distributed.

Yes, contextuality --- in a field sense, which can be elaborated as a conceptually unexceptionable classical holism (I like to think that Bohm would like this aspect), not in a contextual particle property sense, which goes too much against the grain of classicality --- is essential. The relationship between classical measurement theory and quantum measurement theory cannot be understood without recourse to something like contextuality. If you haven't read "The straw man of quantum physics", it elaborates on how to think about measurement as a coarse-grained equilibrium of the field (at a fine-grained scale, there are discrete, non-equilibrium measurement events happening all the time, but the statistics of the measurement events have to be time-translation-invariant, for the events to be taken to be an ensemble; equilibrium, even if only coarse-grained, is a holistic property of the whole experimental apparatus, which feeds into Bohr's views on measurement nicely); straw man is very short!

By "correlations of correlations", do you mean n-point correlations, n>2?

I'm also not quite sure what you mean by "quantum averaging". I think the measurement problem is not a problem in a random field approach, although I have so far failed to elaborate the story at all well.

If this has legs, the first decent mathematician who comes along will leave me in the dust. But I've been waiting for it all to fall apart for all of the dozen years I've been trying to bring it together, so I'd be all too happy to have company on the road even if it's in front.

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T H Ray wrote on Nov. 4, 2008 @ 12:18 GMT
I appreciate that it is just a model, Peter. Is it, or can it be made, a unitary model? Then, being algebraic—or at least using algebraic tools—it presents the possibility of a closed form expression that mediates a scale-invariant measure that would obviate the notion that the basis for Planck scale measure is “doubtless faulty.” We come back to Einstein’s show-stopping observation...

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T H Ray wrote on Nov. 4, 2008 @ 13:45 GMT
I messed up the link in ref 2. It's

necsi.edu/projects/yaneer/NECSITechnicalReport2003-11.pd
f

Tom

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Peter Morgan wrote on Nov. 4, 2008 @ 17:10 GMT
Tom, Thanks for the above. I'm sorry the perspective above doesn't seem to me to come through in your FQXi essay.

I think, without looking at it, that I understand your invocation of self-organization to be a way of understanding thermodynamic transitions of things like photographic emulsions and CCDs. For me, at this stage, these thermodynamic transitions are essentially given -- needing detailed explanation, but not getting it from me. Physicists take the "something" that causes the events to be "particles", even though they use the field formalism of quantum optics extensively and acclaim the empirical success of QED, etc. I've engaged with the field side of the relationship. I say that embedding thermodynamically nontrivial apparatus (stuff that's rather carefully engineered to be that way) in different fluctuating fields will result in different statistics of events.

On unitarity, a self-adjoint generator of time translation can be introduced consistently with the rest of the algebraic structure in the Lie field formalism (in section IV of "Lie fields revisited", particularly). Unitarity is shouted around the place so often, however, that I'm never sure that I've caught all the intended meanings. So is that enough, do you think?

I've never been very happy about scale invariance as a fundamental symmetry in Physics -- but then time-reversal invariance is about equally not part of our experience. We can require scale-invariance of the measurement algebra as a way of making scale-invariance breaking explicit in the state, I guess.

The one really hopeless aspect of talking to people is that sometimes you have to read stuff; in this case, your [2] and [3] above. My daughter Eleanor wants lunch, so bye.

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T H Ray wrote on Nov. 6, 2008 @ 18:43 GMT
Peter, my FQXI essay doesn’t capture this POV, because I assume quantum mechanical unitarity (the wave function and its complex conjugate) as a condition of event length 1 probability (p.5); i.e., because my model is discrete, I can only speak in terms of the simplest field (0,1). This was never very satisfying to me, because locality demands a field result—at least renormalizable if not...

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Peter Morgan wrote on Nov. 7, 2008 @ 15:21 GMT
Tom, I'm sorry to say that with your last I think we are, for now, approaching things too differently. I'm intending to work in a conceptual environment that is recognizable to a moderately conventional but open-minded Physicist, whereas the appearance to me is that you are too engaged with complex-systems ways of thinking for me to make more than superficial contact with you.

Please note that I make no claims for ultimate superiority of my approach; as far as I'm concerned it's just an accident of my intellectual history. I would say that a convergence or reconciliation of quantum/random field models and complex-systems modeling may well be desirable, but /I/ can't see how it would come about.

There are hints in what you say about scale invariance of what might be interesting mathematics, but I can't at present see how to do the mathematics myself. In my terms, I would like to see an explicit presentation of an algebra of observables that is scale-invariant, and a discussion of the differences, similarities, and relationships between a Lorentz/Poincare invariant quantum/random field and a scale-invariant quantum/random field. The trouble is perhaps that I've been thinking, haltingly, in terms of algebras of observables for long enough that I am now relatively deaf to ways of discussing traditional, very-poorly-defined Hamiltonian and Lagrangian formalisms for quantum theory.

You mention positivity of energy. I have some things to say about positivity of energy as a result of an e-mail correspondence with someone else that I will post separately, because their comments crystallized well.

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Peter Morgan wrote on Nov. 7, 2008 @ 15:59 GMT
Someone has been kind enough to point out to me in the last few days, and politely, which was nice, that "Of course one has to impose the positivity of the energy", and that anti-particles "are just free particle solutions running backward in time". I think the mathematics of my FQXi paper itself is intrinsically relatively indifferent to these questions, but they are important questions for...

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T. H. Ray wrote on Nov. 7, 2008 @ 16:08 GMT
Okay, then. I wish you success, Peter.

Tom

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Peter Morgan wrote on Nov. 12, 2008 @ 15:23 GMT
I point out, for anyone coming by here, that the current issue of "Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics", Volume 39, Issue 4, Pages 705-916 (November 2008), is a Focus Issue on "Time-Symmetric Approaches to Quantum Mechanics". A number of articles that anyone thinking about the direction of time might expect to read.

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Ryan Westafer wrote on Dec. 3, 2008 @ 00:29 GMT
Peter,

Great essay. I especially appreciate your statement about Hermiticity:

"In this Hilbert space formalism, in other words, the choice of a direction of time is the difference between classical and quantum fields property so commonly used in quantum mechanics."

It is unfortunate, however, that most have not fully appreciated the physical implications of complex analysis. We toss out reverse time and conjugate solutions when we are faced with a perplexing imaginary term. Indeed, in the view of unitarity- the new form of conservation under the Hilbert formalism- we finally see the universal truth through quantum mechanics. I hope you will read my essay too, where I reveal that our entropic viewpoint has blinded us from the anti-reality of time reversal, charge conjugation, and the truth of a harmonic universe.

Furthermore, I might suggest: perhaps it is not that time does not exist- it is, rather, that time maintains zero expectation value- just like everything else. In an instant, when we precipitate a quantum mechanical transaction in an expectation value integral, time may not exist. However, if we choose a different basis, as we may by a unitary transformation, we will see time "ebb and flow" periodically, to varying degree.

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Peter Morgan wrote on Dec. 3, 2008 @ 15:41 GMT
Hi Ryan, thanks for your comment. I've now commented on your essay on its comment stream. I often think that complex analysis is purely there because of the utility of algebraic completeness, and the notation of Fourier analysis is much easier if we do cos and sin in a unified way. If that means we have too many DoFs, however, there will presumably be better and worse ways to constrain the mathematics so that our experimental data is sufficient to determine parameters in our theory. Restricting to positive frequency works pretty well for predicting empirical data, but I consider that whether it's the most effective way should be up for discussion.

I believe that unitarity is not a good thing for everyone to be so obsessed about. It's enough for a presentation of an algebraic structure to be Lorentz/Poincare covariant. There are other important algebraic structures, for example associativity of a *-algebra of observables and positivity of a state over the algebra, which are enough to allow a probability interpretation (chapter III of Haag's "Local Quantum Fields" is the best way to this, but it's still relatively heavy math). An elementary paper that set my mind pretty much at rest over complex numbers and has been a motivating force for well over ten years even though I doubt I will never be able to cite it formally, is Leon Cohen, Foundations of Physics 18, 983(1988), "Rules of probability in quantum mechanics" --- a Hilbert space and operators are a mathematically effective way to generate probability densities, expected values, and correlations.

I'm afraid that I don't understand the last paragraph of your comment, either from my POV or as a statement in the conventional ways of talking about QM. I'm sorry not to be able to give you even a paraphrase of what I think it says to help you reformulate it.

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Ryan Westafer wrote on Dec. 3, 2008 @ 21:04 GMT
Thanks for elaborating on your approach.

I do agree with what you say! The point at which we differ, I think, is with reality vs. completeness. You refer to taking the positive frequencies only, and to Lorentz/Poincare covariance. Perhaps they are sufficient to describe what we observe, but a complete theory might also acknowledge things which we cannot observe directly (after all, we know there exist dark matter and dark energy). In CPT symmetry, how will you account for antiparticles without allowing negative frequencies. Even if we don't always use them in our day-to-day computations- they are there in the mathematics. Thus, the symmetry is important. If I face the wall of my bathroom and look into the mirror, I can now see myself. Furthermore, I can even see out the window: wonderful things may lie in the opposite direction.

Regarding your last paragraph about my last paragraph ;-)... I did not clarify the difference between time-independent and time-dependent. What if we let something other than time be our "external" parameter in Schrodinger's equation? I meant to say that we may change our basis so we don't "see" time. "Exist" is a stretch, but I do believe existence is revealed by observation.

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Peter Morgan wrote on Dec. 3, 2008 @ 22:29 GMT
I'm OK with negative frequencies, indeed my essay depends on taking a different view of them than is conventionally taken, but I differ from the conventional way of making negative frequencies correspond to anti-particles that have positive energy (but not for light, which has no anti-stuff). Perhaps see my comment in this thread of "Nov. 7, 2008 @15:59 GMT" for a clarification of the issue of negative frequency/ negative energy in my POV as I think of it now (except there are too many other papers to check out). I wish I had understood this particular aspect better before posting the essay, but there's always the future, right?

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matthew kolasinski wrote on Dec. 6, 2008 @ 06:57 GMT
hi Peter,

i enjoyed reading your paper. thanks.

iduhknow, something tells me you can cut a metaphysical two-step with the best of 'em and could have gone unscathed here.

re:

"I point out, for anyone coming by here, that the current issue of "Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics", Volume 39, Issue 4, Pages 705-916 (November 2008), is a Focus Issue on "Time-Symmetric Approaches to Quantum Mechanics". A number of articles that anyone thinking about the direction of time might expect to read."

thanks for that also, but i see they charge per article and there's lots of interesting reading here. i see they'd like us to vote. something tells me FQXi cold use some help. it'll take a while before i could get to the magazine.

regards,

matt kolasinski

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Peter Morgan wrote on Dec. 6, 2008 @ 13:56 GMT
Thanks Matt,

Some of the papers in SHPMP are probably obtainable in pre-print form, either from arxiv.org or from the Philosophy of Science archive in Pittsburgh. Search by author's name or by title. Occasionally authors post their papers on their personal web-sites, which is allowed by many journals; alternatively, e-mail to authors with a very brief, polite request for a PDF. Be clear what paper you want; don't try to engage them in conversation, but be friendly. The enclosure of published papers is part of how academia freezes out independent research, but you can develop strategies for obtaining papers. Public libraries sometimes have access to journals on a principle of public access, librarians often develop strategies for getting stuff for other research. There are local associations of independent researchers (usually mostly people on the arts side, but they know a lot about getting access). There's the National Coalition of Independent Scholars, to which many such associations are affiliated. These organizations are serious about getting (legal) access to library resources. Being an effective independent researcher is a serious, long-term business. If you can get your research to a high enough level, doors open more easily. Good luck!

There is perhaps a metaphysics to this? The Nature of Storage, Boundaries, and Access? Getting at it doesn't mean we understand it, sadly.

For something that could be seen as metaphysical, anyway it's a polemic against a metaphysical commitment to particles, see my "The straw man of quantum physics" pre-print on arxiv, cited in my FQXi essay. It's not much about time, however.

I enjoyed your essay a few weeks ago, but I regret that I can't see a way to use it --- I've become too focused on mathematics in the last few years. I'm likely going to vote for Gambini and for Jannes, but I haven't yet decided on a third (I commend these two papers to anyone who hasn't read them already). I can't see my paper making it into anyone's three restricted votes. A juried third prize also looks increasingly unlikely to me, which I want mostly for the associated invitation to FQXi membership. I'd better choose a third paper so I can stop being so interested in all this (or perhaps that should be obsessed), and get back to research.

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matthew kolasinski wrote on Dec. 10, 2008 @ 01:43 GMT
hello again Peter,

thank you for the suggestions on accessing academic papers.

i still have a bit of reading i have an interest in accomplishing here.

re:

I enjoyed your essay a few weeks ago,

thank you very much.

but I regret that I can't see a way to use it

lol, yes, no 'hammer', no 'pry bar', no utility in that sense. just happy to get it read. the only 'utility' i might have hoped for in it was the possibility of stimulation of creativity through offering a slightly different perspective.

--- I've become too focused on mathematics in the last few years.

math's quite a romance. i'm particularly fond of geometries. wish i had more of a background in it. looking forward to a new form in the not too distant future, more fluid, doesn't look so much like a log jam on the platte river trying to describe simple things. those 'simple' things can get remarkably challenging to describe.

it's also remarkably easy to get lost in the math 'hardware store'. big store these days.

i may not vote. there's several here i'd like to vote for; setting a criterion for selecting one over another is proving challenging. i can see why some are interested in the contest aspect, but with the exchange of ideas here, i figure we're all winners.

best wishes,

matt kolasinski

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Eckard Blumschein wrote on Dec. 16, 2008 @ 17:54 GMT
Peter Morgan,

Those who hope I might be wrong in my essay would certainly appreciate you providing ammunition against it from SHPMP (39)4 special issue.

Eckard Blumschein

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Peter Morgan wrote on Dec. 17, 2008 @ 02:47 GMT
Dear Eckard,

Almost everyone who has entered this contest appears to have spent many years thinking about Natural Philosophy in one tradition or another. I would not want to have hopes that anyone else who has devoted a significant part of their lives to their research might be wrong. I hope I don't. As I said on your comment thread, however, I regret that I can't see how to use your methods constructively in my mathematical context. I'm afraid that extends to my also being unable to see how you might apply the (many) ideas in the SHPMP Focus Issue on "Time-symmetric Approaches to Quantum Mechanics" to move your own approach forward constructively.

Best wishes,

Peter.

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Eckard Blumschein wrote on Dec. 18, 2008 @ 01:42 GMT
Dear Peter,

When Peter Lynd admitted his hope my criticism might be unfounded, he understood that it has serious consequences. If time symmetry is an artifact then one needs no constructive suggestion for improving it.

Please find attached how everybody might check themselves whether or not he uses complex domains properly.

Hints to errors of mine are always welcome.

Regards,

Eckard

attachments: 2_Microsoft_Word__How_do_negative_and_imaginary.pdf

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Peter Morgan wrote on Dec. 18, 2008 @ 13:31 GMT
Dear Eckard,

I'm sorry, but as far as I can tell your attachment says the same things as before, so I again do not see its application to my way of doing mathematics. I'm still not quite clear whether your comments are intended as Physics or as Mathematics.

If we are to have a conversation, I need you to engage with my constructive work --- cite sentences and equations from my FQXi paper and the papers that it cites and tell me why they are stinky bad --- because what I'm doing is not quite conventional. For example, I regard my algebraic approach purely as a way to generate probability densities, expected values, and correlations. I certainly intend my use of complex Hilbert spaces to be taken in an engineering way, much as complex numbers are taken by electrical engineers. Whether values in sample spaces associated with random variables are only positive-valued or may be negative-valued or complex-valued is not of much consequence for the role of an algebra for generating probability theory stuff for the sample spaces we use.

You might like Local Quantum Physics, because in that mathematics the (real-valued) sample spaces (that is, the eigenvalue spaces of operators) are bounded both above and below, not just bounded below (by zero), as you insist upon (although I'm not clear on the importance of this to you). The argument is that we only measure things on a scale, -5 to +5, say (or 0 to 10 if we add 5), +/- infinity are never experimental results. Boundedness of operator algebras plays a very important part in the analysis.

The relation of this kind of mathematical purism to the ordinary models of Physics is not completely understood, but, I would say, is only desultorily under investigation.

You clearly know something about signal processing, so let me ask you your opinion of Leon Cohen's work on Wigner functions? It's my impression that the mathematics of classical signal processing is rather firmly grounded in real numbers --- and when appropriate in only positive numbers, as when we deal with intensity rather than with field displacements --- with complex numbers only as a convenience. I look to this engineering tradition as much as I look to the mathematics of quantum theory.

Peter.

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Eckard Blumschein wrote on Dec. 20, 2008 @ 00:58 GMT
Dear Peter,

Thank you for your detailed reply. I will deal with it.

In the meantime, you might be surprised by implications shown in the attached part 2

Eckard

attachments: 4_Microsoft_Word__How_do_part_2.pdf

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Eckard Blumschein wrote on Dec. 20, 2008 @ 01:01 GMT
This leads to the attachment.

attachments: 5_Microsoft_Word__How_do_part_2.pdf

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Peter Morgan wrote on Dec. 20, 2008 @ 14:19 GMT
Dear Eckard, I think that here you misspeak: "One must not simply add two complex physical quantities as if they were real ones. Real and imaginary parts are orthogonal to each other. Instead, one has to add their squares according to sin^2 + cos^2 = 1." The addition of real and of complex numbers are identical in the mathematical sense that the respective definitions of the operations of addition and of multiplication for the real numbers and for the complex numbers form a commutative field (in the sense of elementary algebra). What you are referring to is, I think, the fact that the quadratic norm on the real numbers and on the complex numbers are different. One also has the fundamental algebraic difference that there is an element in the field of complex numbers that squares to -1 (of course, there are two, j and -j, to adopt the engineering notation you express a preference for), whereas there is no number that squares to -1 in the field of real numbers. The complex numbers are of course also isomorphic as a vector space to R^2, though not canonically; the latter, however, is relatively impoverished because it has no multiplication operation defined. The introduction of a norm is not strictly speaking essential to the complex numbers, although the topology it introduces is essential to complex analysis.

For what it's worth. I hope my doubts that the above will help are unfounded. Really abstract mathematical thinking is a foreign language that I can more-or-less understand, but that I'm not fluent in, so the above is not as nicely expressed as it would be by a native pure mathematician. Again, are you talking Mathematics quibbles or Physics quibbles? Peter.

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Eckard Blumschein wrote on Dec. 22, 2008 @ 01:54 GMT
Dear Peter, This is not yet my reply to what you wrote on Dec. 18. I just would like to clarify that your quibble detracts from the essence. Yes, I referred to a typical horror mistake of beginners to simply add measured magnitudes of two quantities, for instance voltage at R and at L.

I intended to say that one should use complex calculus with care. Fortunately nobody is a native pure mathematician who is unable to overlook things. When Bob the builder applies 345, he does not need the foreign language of allegedly abstract poor mathematics.

The next possible mistake I mentioned was to consider multiplication by j instead omega j as equivalent to d/dt. Doesn't one have to consider every frequency separately?

I only checked that Heisenberg, Dirac, and Schroedinger actually correctly considered n=1,2,3,...

I let it to everybody to ask for the meaning of x_4=ict.

Why do you call my criticism quibbles?

At first I found out that the inner ear performs a real-valued spectral analysis. I understood that the usual notion of time has been abstracted from the unilateral elapsed time. This turned out to be the key for unexpected simple explanations of seemingly murky matter in many disciplines including quantum physics.

By the way I came across with old deficits in the fundamentals of mathematics too.

May I ask you for help? I suspect that there is something wrong with the signal processing for single electron counting. Gompf et al. in PRL 1997 arrived at a result that was quite different from direct measurement with a streak camera and also unexpected. If you are interested, I will give you the details.

Eckard

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Cristi Stoica wrote on Dec. 26, 2008 @ 08:29 GMT
Dear Peter Morgan,

I like your approach. I consider that some of the consequences, especially the hope of avoiding the renormalization techniques, by using Lie fields, show that your continuous random fields formalism deserves to be taken seriously.

Best regards,

Cristi Stoica

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