Classical interacting random field models

[Peter Morgan]

This is an announcement, in the Physics Forums context, of my "Lie fields revisited", http://arxiv.org/abs/0704.3420" , which has recently been accepted by J. Math. Phys., and an attempt to elaborate on why you might read it and to elicit critiques.

For a little over ten years, I have been trying to understand and characterize in detail what differences there are between quantum fields and their closest classical equivalent, random fields. Random fields at finite temperature have nonlocal correlations, just as do quantum fields, so what is the difference?

I showed in "A succinct presentation of the quantized Klein–Gordon field, and a similar quantum presentation of the classical Klein–Gordon random field", http://dx.doi.org/10.1016/j.physleta.2005.02.019" and papers.

If you're wondering at this point what a classical random field IS, it's almost trivial, it's an indexed set of random variables (that's to say, if you know about random variables, a random field is easy, http://en.wikipedia.org/wiki/Random_variables" , but all we need here is that a Physical state and a random variable together give us an expected value associated with the random variable in that state). Two random variables, $$V_1$$ and $$V_2$$, say, are enough. A continuous random field effectively makes the index set be the points of space-time, so that we could talk about random variables $$\phi(x)$$ at every point of a (Minkowski) space-time, very similarly to a quantum field, but there are technical (and also notational) matters that make it mathematically much better to make the index set be a linear space of functions, which are called "test functions". It is possible to talk of a continuous random field $$\phi(x)$$ as a "random variable valued distribution", which is fairly directly comparable to the "operator valued distribution" way of talking about a quantum field. That continuous random fields are conceptually quite close to quantum fields, and can be presented in a Hilbert space and operator algebra formalism that is very close to quantum field theory, is a large part of why there is a small hope of casting some light on quantum mechanics. From here on, all the random fields are continuous.

With honorable exceptions, almost all attempts to understand quantum mechanics have been through the nonrelativistic quantum theory and classical particles, quantum field theory has been felt to be too complicated for any understanding of it to be possible until quantum mechanics has already been understood. Big mistake, I believe.

Of course, as classical models, random fields prima facie allow Bell inequalities to be derived, contradicting experiment, but in "Bell inequalities for random fields", http://dx.doi.org/10.1088/0305-4470/39/23/018" , I showed that Bell inequalities cannot be derived for random fields that have thermal or quantum fluctuations. Essentially, equilibrium states have non-local correlations at all times, and an absence of any pre-existing nonlocal correlations is required for Bell inequalities to be derived. Technically, this is usually called the conspiracy loophole. A lot of detail is required to get a paper on Bell inequalities into a good Physics journal; "Bell inequalities for random fields" takes several different approaches and gives a number of supporting arguments and references to papers that have previously argued that classical field theories allow the derivation of Bell inequalities. John Bell wrote the seminal paper, as so often, "The theory of local beables", which is in "Speakable and unspeakable in quantum mechanics", but it has a flaw, that it effectively works with particle properties and with random fields even-handedly, when the assumptions that are largely reasonable for one are not as reasonable for the other.

Note that I'm claiming that the voluminous literature that proves that classical particle property models cannot be used as physical models is just taking on straw man theories, and is irrelevant to classical random fields that have non-zero fluctuations. Personally, I think that a classical particle property model that I like is not possible, despite the claims of the detection loophole, de Broglie-Bohm models, Nelson models, etc.

My earlier papers do some mathematics, but they are largely non-constructive, so they have not so far provoked comment or rebuttal. "Lie fields revisited", in contrast, constructs a new class of interacting random field models that are not available to quantum field theory. "Lie fields" were introduced in the 1960s, but it became apparent after a few years that they are incompatible with the Wightman axioms, so they became moribund. Once we admit classical random fields as a possibility, and adopt an algebraic approach to constructing them that exploits the mathematics of quantum field theory, it turns out that we can sidestep renormalization completely, and we discover a mathematically well-defined concept of interacting particles that is quite closely analogous to the free quantum field's Fock space structure (without having to work with asymptotic states).

The approach I take attempts to be as empirical as possible, to the extent that it takes a view that is common in Physics, that correlations of "the field" are the observables for the purpose of theoretical physics. Renormalization, for example, takes as its empirical starting point that correlation functions must be finite. The class of models that I construct, however, generalizes the generalized free quantum field, with an arbitrary (complexification of a) Källén-Lehmann weight function required to specify a particular dynamics, so there is a "landscape" of possible models, as they say in string theory. From the point of view of empirical usefulness, this is good, but such a large class of models is essentially not very explanatory. Still, it seems good to have an intermediate explanation of the relationship between quantum (field) theory and classical modeling that I like better than anything I have seen elsewhere in the literature.

The most discursive account of my approach (but which pre-dates the mathematical developments in "Lie fields revisited") is probably in a Växjö Conference Proceedings, http://dx.doi.org/10.1063/1.2713457" .

I will post separately a couple of responses that I have received from main-line physicists, and my responses to them, which hopefully will give some indication of the sorts of interpretational and mathematical objections that can be raised about my approach.

 

[Peter Morgan]

Physicist A (whom I will not identify) wrote:

Dear Peter,
I looked at your paper but I could not find an answer that is of utmost importance for me. What is your proposal for the function $$\xi$$ in Eq.(5) for photons? I could not find an answer to this question in Section V where you discuss the electromagnetic field. Your proposal may become experimentally testable only if you propose a specific form of the deformation for the commutation relations and, of course, the photons are the best testing ground of your hypothesis.

My reply:

Dear A,
Thank you for your reply. Yes, there are open questions. My knowledge of modern mathematics is not good enough to be sure, but I believe some of the open questions are of a nature that has not previously been considered. I have hit my head against what the 3-point function looks like for a complex function $$G$$ that has squared modulus equal to the mass weight function with no success at all so far. That's the problem you picked out instantly, and I think it may be hard. It is also a major question for me whether there will even be fermions at all in this approach to modeling correlation functions, since fermion fields are both distributions and have to be multiplied at a point for them to be observables (however, fermion fields might still be present as non-observable fields, just as a way of implementing observable correlations between the electromagnetic field).

As well as this question of analysis, which I believe may be both hard and of interest, there are also combinatorial questions, including the proof of my conjecture in Appendix B, but I suspect these will be relatively easy for someone with the right mathematical tools. I hope that some mathematicians will be interested enough in these questions. I hope also that everyone will gradually be intrigued that this approach sidesteps renormalization and puts interactions on a solid footing instead of having to switch off the interaction at $$t = \pm\infty$$, informs our understanding of quantum (field) theory, and offers a progressive research program. As a new research program, I imagine there may be a slow start. I've been unable to find the quote, but didn't Dirac say of the late 1920s that even mediocre men did good work, and good men did brilliant work? Do you know where he says that?

As a direct model of correlation functions, experimental testability is closer, I hope, than the 30 years it has taken string theory to get nowhere. In contrast to the usual request for empirical verification, I think this class of models may be too empirical to give really satisfactory explanations. If empirically successful models can be constructed, but they require the function G to be specified to do so, that doesn't satisfy the current preoccupation with reducing the number of parameters, however useful engineers might find the models to be.
I doubt this reply will inspire you, but thank you again for your reply,
Peter.

As usual, I'm too long-winded. I haven't received a reply to this, nor did I expect to. Some of the readers of this post will be only too familiar with not getting replies from people whom they cold-call, so these few lines are gold-dust, to be carefully considered over the next year or two, but as usual I write in a way that guarantees no reply (actually, I have met this person, and he does remember me, but I'm pushing something conceptually new, which he doesn't think has much value. I have, of course, very sadly, made it just slightly more likely that he won't answer at all next time I venture to write to him).

Hopefully I will change my approach enough over time to address the issues raised here. Note that A is right about everything. Although my approach is moderately mathematically sophisticated, more than most crank papers, still the paper has holes a mile wide in it, which mean that I will probably have to write another, more detailed paper, if I can, to persuade anyone of the interest of my approach. A has picked on what I think probably is the biggest hole in the paper, and expressed his feeling that the hole is currently too big for him to spend any more time on it. Being married to an academic, who is currently head of the classics department at Yale, makes it easier to understand how busy A is, and to cut him some slack.

The request for experimental testability is of course harder to satisfy than the mathematical question. THAT has to wait, but I hope there's as much possibility of useful models for experiment being constructible within this approach as there is within the string theory approach. Reading between the lines, you may notice that A is not so robustly demanding of experimental testability as he might be. Cautiously, that makes me hopeful that he's recognized that this is not purely theoretical, although A wrote this e-mail in 42 seconds, so not too much should be taken from it. I reckon a thousand Physicists and Mathematicians working on this approach would have a fairly decent chance of getting first Physics in 10 years, but I haven't yet lined up the funding.

I'm frankly expecting no responses to this topic. Still, any response you do come up with will be just as welcome as the above.

 

[marcus]

Thanks for giving us this announcement of your work, and opening a thread where people can ask you about it. And welcome to PF's Beyond forum!

I have the impression that you have published quite a lot (half dozen or so?) in peer-review physics journals, and that before joining the Yale physics department you were at Oxford (philosophy of science/foundations of physics, I think). I could be wrong about any or all all these details.

It is interesting to see the correspondence---showing the effort involved in calling attention to ideas off the beaten track. I'll stop answering so i can get back and read your last post more carefully.
============

You know it is quite possible that you will get some reaction to your ideas here. I was struck by the coincidence that the mathematician John Baez who has sometimes visited here, and posted pedagogical notes on things that interest him, is also married to a classicist. I forget his wife's name, Lisa Raphals, I think. Her specialty is comparing classical Greek social and literary themes with their classical Chinese counterparts. Maybe that comes under the heading of Comp Lit, but at least it isn't modern Comp Lit.

============
I will isolate a few lines to look at more carefully (although other people here are specifically more expert.)

*Dear Peter,
I looked at your paper but I could not find an answer that is of utmost importance for me. What is your proposal for the function xi in Eq.(5) for photons? I could not find an answer to this question in Section V where you discuss the electromagnetic field. Your proposal may become experimentally testable only if you propose a specific form of the deformation for the commutation relations and, of course, the photons are the best testing ground of your hypothesis.*

It sounds like he is challenging you to propose a specific formulation, that could lead to testing.

*"Lie fields" were introduced in the 1960s, but it became apparent after a few years that they are incompatible with the Wightman axioms, so they became moribund. Once we admit classical random fields as a possibility, and adopt an algebraic approach to constructing them that exploits the mathematics of quantum field theory, it turns out that we can sidestep renormalization completely, and we discover a mathematically well-defined concept of interacting particles...*

This helps me understand your title "Lie fields revisited". I had never heard of Lie flelds. Just from the name I would assume it refers to a vector field taking values in a Lie algebra. that seems much too simple, what is it really? and what was the original motivation for studying Lie fields? (in basic entry-level terms please )

 

[Peter Morgan]

Physicist B wrote:

Dear Sir,
As you seem to anticipate in your mail, I am not able to understand what your paper is trying to say. The word "random" is used but not explained, and I fail to understand your introduction of creation an annihilation operators for a classical random field. It would have been nice to see an equation of motion or a field equation. The excessive use of test functions does not clarify the dynamics; in my experience test functions merely serve as security blankets that hide the locality properties of the fields one really wants to discuss, obscuring things sufficiently to make you believe that renormalization would not be necessary.

Without understanding the dynamical system you are discussing, I cannot judge the relevance of the paper.
Cordially,
B

That's more of a downer! Physicist B doesn't get the paper. Still, I write a reply, knowing that there's no chance of getting anything else back, but I need to formulate a reply for my own good. I had no intention of posting this or the previous e-mail when I sent it, so I have to edit this one a little to keep it anonymous. You'll notice that I naturally take the role of supplicant-who-can-be-ignored, further ensuring that I don't get a reply.

Dear B,
Many, many, many thanks for your reply. My approach is different enough that I do indeed expect not to be easily understood, but it is not so different that I cannot publish my work in good journals. I think my JMP paper is a little different from many other JMP papers. I won't expect you to go back to the paper or to reply or even to read this, you clearly didn't get the paper very much, but here are some responses to your comments. Formulating them is good for me. When did I ever write a paper that is clear? You will have seen that I'm not that good a mathematician, but I stumble on as best I can.

Dear Sir,
As you seem to anticipate in your mail, I am not able to understand what your paper is trying to say.
The word "random" is used but not explained,

The most straightforward construction of a (continuous) random field, I think, is as a quantum field that satisfies $$[\phi(x),\phi(y)]=0$$ for all positions $$x$$ and $$y$$ in Minkowski space. I'm pretty sure that's implicit in my paper to someone who reads it closely, but of course I cannot expect close readings by most people at this point. I explicitly define a random field in the other paper I mentioned in my previous e-mail, "Bell inequalities for random fields", in J. Phys. A. I also give references there to a few books on random fields. Amongst Physicists I have only ever heard astrophysicists express much familiarity with random fields. A "random field" is just an indexed set of random variables (even two random variables will do); taking a space of test functions as the index set is another way to construct a "continuous" random field (perhaps that's mathematically nicer, but not by very much in my rather practical view). [That last sentence is horrible, better ignore it!] A random field is more-or-less classical, with a classical measurement theory, but it's not a classical differentiable field theory. A random field considered at a point is not well-defined, but in loose terms we can think of a random field that has non-zero thermal or quantum fluctuations being $$\pm\infty$$ at almost all points, which averages out to finite values when smeared with a test function. Convolution with a test function (from a suitable test function space) yields a classical differentiable field.

[...] surely [we] should be taking a field approach. All the most successful physical theories, for almost 200 years, have been field theories, but with honorable exceptions like Shelly Goldstein [who has developed de Broglie-Bohm type trajectories for field theories perhaps as well as anyone can] all the classical realists in the world plug away at classical particle ways of thinking that it has been shown over and over have to be nonlocal and/or contextual and/or conspiratorial for them to work, and hence not acceptable to almost all Physicists. Some classical field theory, please![emphasis added] For a field approach to be at all close to a quantum field, it must be a random field, a classical continuous field just can't model either quantum and thermal fluctuations. Furthermore, the conspiracy approach that I understand you to take to the violation of Bell inequalities is entirely natural to a random field at finite temperature, but is a rather forced modification of classical particle or differentiable field physics.

and I fail to understand your introduction of creation an annihilation operators for a classical random field.

The creation and annihilation operators ensure that $$[\phi(x),\phi(y)]=0$$ because the inner product is not restricted to positive frequency. That's a little heterodox, and requires some careful conceptual thinking to make it work. Taking this algebraic and Hilbert space approach minimizes the distance from classical to quantum, making it necessary only to understand the difference between the measurement theories.

It would have been nice to see an equation of motion or a field equation.

For free fields, there is a correspondence between the classical dynamics, the free particle propagator of the quantum theory, and the inner product between test functions of the quantum theory. I believe that to be well-known, but I may perhaps have been thinking in more-or-less axiomatic terms for too long.
Then there is a question of how the deformation of the free field should be described. The standard way, in cartoon form, is to modify the classical dynamics, quantize it in one's favorite way, regularize it, renormalize it, et voilà. My way is to modify the algebraic structure of the free quantum field, et voilà. The renormalization process is conditioned by the requirement that the empirical content of the theory, the statistics of and correlations between the field observables, must be finite, whereas I more-or-less generate those statistics and correlations directly.
Precisely because my approach is so close to the empirical data, I take it to be essentially an empirical half-way house, it is not as explanatory as a dynamics. Having this approach as well as what we already have, however, allows the inventions of others, I hope, to be less straight-jacketed by the successes of particle physics.

The excessive use of test functions does not clarify the dynamics; in my experience test functions merely serve as security blankets that hide the locality properties of the fields one really wants to discuss, obscuring things sufficiently to make you believe that renormalization would not be necessary.
Without understanding the dynamical system you are discussing, I cannot judge the relevance of the paper.

There are of course a lot of details to understand. I have so far found the mathematics of the space-time behavior of 3-point correlation functions particularly difficult. A correspondence between perturbed classical dynamics and my algebraic deformations may be possible through the Hopf algebraic approach to renormalization (which, however, I don't yet understand enough).
For the time being, to my knowledge, I'm the only person working on anything at all like this approach, so progress will be slow unless I can find a small group of people who also find it worthwhile to push it forward. I have been working at this approach on my own for the last ten years, so my progress certainly has been slow so far. The approach may never take off, it may be shown to be obviously unusable or outright wrong in a few weeks or years, but I'm trying to find someone who sees the possibilities enough to work with me at least a little. If in a few months I fail, I will settle down to doing more mathematics on my own and writing a further paper.

I'm somewhat sad that you were not the referee at JMP. I asked for someone to be assigned as a referee who "has a moderate sensibility towards “Foundations of Physics”, without compromise, however, to a proper review of either the mathematical or foundational content of the paper", which could have been you if you ever referee for JMP these days. I guess you would have recommended that it be rejected, but the paper would have been better when I submitted it later to another journal. [...] Getting comments as substantive as you have just given me is very difficult, so I'm always very happy indeed when I do get some. Again, many, many, many thanks!
Best wishes,
Peter Morgan.

That's enough of a conceptual miss that I really may not get anyone interested at all, although B focuses on the fact that I conceived the mathematics of "Lie fields revisited" as particularly targeted for Mathematicians who work on what is known as "Local Quantum Physics" (which makes my mathematics look like 2+2=4 -- that is, I hope I'm not erroneously saying 2+2=3.7). Perhaps there's hope! You'll probably notice that's far too much to infer from B's reply. I also hope that B just read the paper very fast; I'm in no doubt that my approach is conceptually far enough from the ordinary that it can't be understood or any of the consequences appreciated in a moment.
I have met B, and talked to him for about a minute about Bell inequalities, perhaps two years before my J.Phys.A paper was published. The joys of living in Oxford, where almost everyone seemed to come through. He's not someone who just dismisses the hidden variables question instantly, but if he remembers me then I'm in a Bell inequalities slot in his head, not a serious physicist slot, so it would probably be better not to be remembered.

Note that I want my papers rejected. I'm not just puffing this guy. If my paper had been rejected by JMP, I would have had to try to find a way to improve it, perhaps even to rewrite it, for a submission to another good journal. It would have been better. "Bell inequalities for random fields" was accepted by the sixth journal I submitted it to, after six years of effort, by the skin of its teeth -- one referee thought it was OK, two thought it shouldn't be published, the editor, amazingly on that showing, sent the paper to a journal board member, who, it turned out, knows me and was willing to work on the paper enough with me to make it acceptable. It's really no wonder the paper has no citations yet, being published is absolutely only the first step, you've got to be read, and you've got to be understandable enough for someone to think that they can write a paper that cites your methods. I hope with this blitzkrieg that someone will read the paper, particularly in the context of "Lie fields revisited", and find it interesting.

That's it, at least for now. No-one else has replied. I'm sitting on my hands worrying that I won't get enough feedback to take a worthwhile next step. I might have to provide feedback myself! After 10 years, I'd almost welcome someone showing me an easy proof that it's all nonsense. I said almost, right? Where did outrageous over-self-confidence go for a second? A half-dozen Physicists have been kind enough to reply, saying that they will look at my paper, to my e-mails bringing "Lie fields revisited" to their attention, but none of them may reply substantively, of course. It may also be that Physics Forums will ask me to cease and desist from this form of posting, fair enough if they do.

 

[Peter Morgan]

Marcus, thanks for your reply. Very much. But I have to pick up my daughter from school now. She may want to play on her Webkinz account, or we may go out in the snow. So probably no reply to yours until later. I'm looking forward to trying to reply to the particular way you have phrased your questions, which looks productive.

 

[Haelfix]

This is best put in the quantum mechanics forum. Moreover, you really want to talk in person with constructive field theorists and measurement theorists about this. I am not aware of a significant presence of them on PF though.. This is sufficiently advanced and esoteric that you probably won't find much help here other than explaining confusion. Perhaps alt.sci.physics or something like that if you absolutely have to scrounge the internet wasteland.

 

[Peter Morgan]

This is best put in the quantum mechanics forum.

Maybe. If BYSM decides they want it elsewhere, they'll let me know or just move it.

Moreover, you really want to talk in person with constructive field theorists and measurement theorists about this. I am not aware of a significant presence of them on PF though..

That may well be true. I have e-mails to a few of them that are as yet unreplied to. The paper is posted on http://www.lqp.uni-goettingen.de/lqp/papers/".

This is sufficiently advanced and esoteric that you probably won't find much help here other than explaining confusion. Perhaps alt.sci.physics or something like that if you absolutely have to scrounge the internet wasteland.

If I can explain confusion, I guess that will reduce it from the esoteric. Einstein tells me I must teach my Grandmother, right? If what I'm doing comes across as advanced, I haven't explained properly. A random field is no harder in concept than a quantum field, which is easy enough. Isn't string theory harder? The only difference between random fields and quantum fields is that they commute at time-like as well as at space-like separation. Indeed, the presentation of random fields that "Lie fields revisited" proposes to make random fields a particular type of quantum field.

What I'm doing is beyond the standard model, but perhaps as yet too far, or else beyond in the sense of wrong.

 

[Peter Morgan]

It sounds like he is challenging you to propose a specific formulation, that could lead to testing.

I think that's right. I believe it may be a novel mathematical problem, however, so if that is what it takes it will take a while.

*"Lie fields" were introduced in the 1960s, but it became apparent after a few years that they are incompatible with the Wightman axioms, so they became moribund. Once we admit classical random fields as a possibility, and adopt an algebraic approach to constructing them that exploits the mathematics of quantum field theory, it turns out that we can sidestep renormalization completely, and we discover a mathematically well-defined concept of interacting particles...*

This helps me understand your title "Lie fields revisited". I had never heard of Lie flelds. Just from the name I would assume it refers to a vector field taking values in a Lie algebra. that seems much too simple, what is it really? and what was the original motivation for studying Lie fields? (in basic entry-level terms please )

I hadn't heard of Lie fields before a year ago either. I came across them while reading through a review paper (which is cited in "Lie fields revisited") by Streater, who was a mover and shaker in axiomatic quantum field theory. Coincidentally, Streater has a fairly well-known web-page on http://www.mth.kcl.ac.uk/~streater/lostcauses.html" that makes good cautionary reading.

The creation and annihilation operators of a free quantum field satisfy a commutation relation
$$[a_f,a_g^\dagger]=(g,f)$$,
where $$(g,f)$$ is a Lorentz invariant inner product on the test functions $$g$$ and $$f$$. The field itself is $$\hat\phi_f=a_f+a^\dagger_{f^*}$$, and the vacuum state is a zero eigenstate of the annihilation operators, $$a_f\left|0\right>=0$$. This may not be familiar to people used to the long-winded descriptions of quantum fields in Physics textbooks, but there it is.

A Lie field satisfies a commutation relation
$$[a_f,a_g^\dagger]=(g,f)+a_{\xi(g,f)}+a_{\xi(f,g)}^\dagger$$,
where $$\xi(g,f)$$ is a linear Lorentz covariant functional of $$g$$ and $$f$$. This equation is similar enough to the commutation relations satisfied by a Lie algebra to have earned the name. The inner product and $$\xi(g,f)$$ are effectively structure functions.

I was trying to construct just this structure on my own, because linearity of the quantum field as a functional of the test functions -- so that $$a_{\lambda f+\mu g}=\lambda a_f+\mu a_g$$ -- is almost enough to require only this structure, so I was primed to recognize the reference in Streater's paper when I saw it. The Lie field structure was tried with some enthusiasm in the early 60s because it was seen to be mathematically very simple and beautiful -- which it is -- but being incompatible with the Wightman axioms was seen as too much. It's only the developments across the whole literature since then that make it possible to think of classical random fields; I think it would have been unthinkable 10 years ago for JMP to publish a paper that relies on such a delicate discussion of the relationship between quantum and classical measurement theories.

 

[marcus]

Thanks for a brief clear explanation!

...it would have been unthinkable 10 years ago for JMP to publish a paper that relies on such a delicate discussion of the relationship between quantum and classical measurement theories.

Maybe the editor just happened to like your prose style

 

[Thomas Larsson]

The first time I heard of Lie fields was recently in http://arxiv.org/abs/0711.0627 . The following quote from p 2 may be relevant:

"After the first attempts to construct (4D) Poincar´e invariant Lie fields led to examples violating energy positivity, it was proven, that scalar Lie fields do not exist in three or more dimensions."

My own interest in that paper has to do with the immediately preceding lines

"The spectacular development of 2-dimensional (2D) conformal field theory in the 1980’s is based on the preceding study of infinite dimensional (Kac–Moody and Virasoro) Lie algebras and their representations. A straightforward generalization of this tool did not seem to apply in higher dimensions."

with which I completely disagree.

 

[Fra]

I just skimmed this and something makes me curious. I'm no matematician but I'm interested in the foundations of QM. As I understand it you are trying to communicate ideas on new perspective to a physical problem?

Every problems has to be analysed from some starting points, and I'm not matematician to mind and my starting points are different and to most I think philosophical.

I'm not sure if the questions make sense to you but...

1) When you consider random fields, which you say are basically indexed random variables. Then my first question is how do you _establish the indexing_ from an operational point when the variables are random? I've got a feeling that the indexing is implementing as a background structure? I see this related to the choice of microstructure to apply and ergodic hypothesis? Does this make sense?

2) Did you try to connect your thinking in a new way in the context of GR? Does your idea offer a supposed (yet to be seen of course) competitive advantage in your opinion? Or are you considerations considered in isolation from the problem of unifiying QM and GR?

/Fredrik

 

[Peter Morgan]

The first time I heard of Lie fields was recently in http://arxiv.org/abs/0711.0627 . The following quote from p 2 may be relevant:

"After the first attempts to construct (4D) Poincar´e invariant Lie fields led to examples violating energy positivity, it was proven, that scalar Lie fields do not exist in three or more dimensions."

My own interest in that paper has to do with the immediately preceding lines

"The spectacular development of 2-dimensional (2D) conformal field theory in the 1980’s is based on the preceding study of infinite dimensional (Kac–Moody and Virasoro) Lie algebras and their representations. A straightforward generalization of this tool did not seem to apply in higher dimensions."

with which I completely disagree.

Properly, what I have called Lie fields should be called Lie random fields; then we could say that Lie quantum fields do not exist in three or more dimensions.

http://arxiv.org/abs/0711.0627 is not an easy read. I can't say I agree or disagree with much of it at all.

I attempted to correspond a year ago with Seiler, Rehren (one of the authors of the above), and Fredenhagen over an aspect of the paper "Quantum Field Theory: Where We Are": I sought to point out that the principles they adopt there for quantum fields are remarkably weak:

• the superposition principle for quantum states, and the probabilistic interpretation of expectation values. These two principles together are implemented by the requirement that the state space is a Hilbert space, equipped with a positive definite inner product.
• the locality (or causality) principle. This principle expresses the absence of acausal influences. It requires the commutativity of quantum observables localized at acausal separation (and is expected to be warranted in the perturbative approach if the action functional is a local function of the fields).
• In addition, one may (and usually does) require
  covariance under spacetime symmetries (in particular, Lorentz invariance of the dynamics), and
• stability properties, such as the existence of a ground state (vacuum) or of thermal equilibrium states.

As written, but as it emerged not as intended, these principles admit certain kinds of classical models. They didn't like my point of view, which is that these principles are weak, and hence perhaps an interesting new departure; they preferred to see them as a non-binding summary of existing axiomatic systems. "Quantum Field Theory: Where We Are" has started to be cited in the literature.

 

[Peter Morgan]

I just skimmed this and something makes me curious. I'm no matematician but I'm interested in the foundations of QM. As I understand it you are trying to communicate ideas on new perspective to a physical problem?

Yes.

Every problems has to be analysed from some starting points, and I'm not matematician to mind and my starting points are different and to most I think philosophical.

I'm not sure if the questions make sense to you but...

1) When you consider random fields, which you say are basically indexed random variables. Then my first question is how do you _establish the indexing_ from an operational point when the variables are random? I've got a feeling that the indexing is implementing as a background structure? I see this related to the choice of microstructure to apply and ergodic hypothesis? Does this make sense?

2) Did you try to connect your thinking in a new way in the context of GR? Does your idea offer a supposed (yet to be seen of course) competitive advantage in your opinion? Or are you considerations considered in isolation from the problem of unifiying QM and GR?
/Fredrik

In answer to 1), there is a pre-existing background space-time in all my work. That's how QFT also proceeds. High Theory often takes correlation functions as the ground empirical input. Renormalization is conditioned, for example, by the requirement that correlation functions are finite at the end of the calculation. This is not what all theorists would take as ground empirical input, and I presume that experimentalists would never do so. Whether taking correlation functions to be a general enough empirical input to describe everything is OK from a Philosophy of Physics point of view is certainly open to discussion, but I have taken it as my unquestioned ground for now. Insofar as I want my audience to be Physicists, I don't want to have this discussion at this point in my research.

In answer to 2), Oh Yes. Most attempts at QG try to quantize GR, with all the difficulties that results in, whereas my approach, by classicizing quantum field theory, opens up a very different perspective. Loosely, variation of Planck's constant from place to place would look like curved space. From Appendix B of "Lie fields revisited" (which is quite verbose, and hopefully more generally readable):

In reverting to a classical random field model that explicitly includes experimental apparatuses and the effects of quantum fluctuations as well as the effects of thermal fluctuations in models, we treat thermal and quantum fluctuations in a more even-handed way. If we understand the Unruh effect to describe transformations between quantum and thermal fluctuations under non-inertial changes of coordinates, the principle of equivalence and a preference for covariant presentation of physical models suggest that we ought to prefer classical random field models to quantum field models. Making the expressions given in this paper for $$(...;...)$$ and $$\xi(...;...)$$ generally covariant and independent of or determining the underlying geometry, however, is a nontrivial problem.

Whether this is competitive with other approaches to QG is of course yet to be seen -- I really mean the "nontrivial" -- but it certainly opens up a new front. In fact, this is a front that is already open, in the book of http://ltl.tkk.fi/personnel/THEORY/volovik.html", for example, and the literature on "dumb holes" in general [I believe that literature somewhat needs the concept of quantum entropy, which I allude to in section 2 of "Lie fields revisited", as thermodynamic dual to Planck's constant, in analogy to entropy as the thermodynamic dual to temperature -- both Planck's constant and ordinary temperature being measures of fluctuations in a classical random field approach, but distinguished by different symmetry properties].

 

[Fra]

Thanks Peter for your clear and honest answers.

What you say motivates me to read and think about all this at least a second round.

I was hoping for another answer on the background thing because I saw a possibility that you considered random fields in the sense of not random state microstructure, but random state random microstructures, so as to also "randomize" the space of possibly microtructures (spacetime included). Of course that is probably horribly complicated mathematically but also conceptually. Or would you say this relates to your landscape note?

The other things you mention makes me curious. Maybe I'll get back with more questions when I've given it a second round of thinking.

/Fredrik

 

[Peter Morgan]

Thanks Peter for your clear and honest answers.

What you say motivates me to read and think about all this at least a second round.

I was hoping for another answer on the background thing because I saw a possibility that you considered random fields in the sense of not random state microstructure, but random state random microstructures, so as to also "randomize" the space of possibly microtructures (spacetime included). Of course that is probably horribly complicated mathematically but also conceptually. Or would you say this relates to your landscape note?

The other things you mention makes me curious. Maybe I'll get back with more questions when I've given it a second round of thinking.

/Fredrik

To move this to a dynamic space-time may require completely new thinking and mathematical structure, because everything currently depends on Fourier transforms, which are globally defined on the space-time. Deformation of Fourier transforms is possible, with what I find to be a significantly higher level of mathematical difficulty, to QFTs on a non-dynamic curved space-time, but if there's no background space-time then the Fourier transform has to be replaced by something else. There is also explicit use of the metric tensor at almost all points of my current approach.

The "landscape" note is purely about methodology of research: what are we looking for when we try to construct new models? Something that we think "explains" how things are is more highly prized than something that just describes. My approach, I'm pretty sure, is essentially descriptive (it may be useful as an effective model at some scales as a fairly general deformation of free field theories even if it turns out to be wrong as a model below scales of 10^9 meters, say). An alternative description opens up new ways to look for more explanatory theories, which may well be conceptually and mathematically very different from my approach, as is indicated by my note on dynamic space-time above.

 

[Fra]

Peter, I don't want to take focus of your thread. I hope others will give you the targeted feedback you wanted rather than the philosophical aspects you don't want to focus on, but here are some more comments on your last comment.

but if there's no background space-time then the Fourier transform has to be replaced by something else.

Relating to my own thinking - what if from the space of possible transforms, the Fourier transform will be emergent, and perhaps there is a logic as to why it is emergent. Start with random transformations of random fields, and the question is what is most likely to be observed, by most observers living in this "random world"?

It may seems like as soon as we are starting to talk about the space of possible transformations and so on, we immediately run into the well known problems of convergence issues because the options just grow fields of fields of fields etc.*But* if you add the constraint of limited information capacity of an observer, this puts a bound on the complexity on all relatable structures which means that even in the continuum limit the sets should as far as I see at least be countable. The more complex and iterated the structures get, the less "weight" do they get - given an obsever.

And if you see it from the view of self-organisation, the most random scenario seems to be when also the observers, clocks and rods are random. And if you start by considering low weight observers, the contraints of what is observable by an observable is very tight simply because there are only simple observers around.

I think this makes a lot of sense, but exactly what the choice of formalism will be is not so clear.

I hope to find an answer to the uniqueness of the Fourier transform in this emergent sense. But I don't know how to do it yet, but I'm trying to figure it out.

/Fredrik

 

[Peter Morgan]

Peter, I don't want to take focus of your thread. I hope others will give you the targeted feedback you wanted rather than the philosophical aspects you don't want to focus on, but here are some more comments on your last comment.



Relating to my own thinking - what if from the space of possible transforms, the Fourier transform will be emergent, and perhaps there is a logic as to why it is emergent. Start with random transformations of random fields, and the question is what is most likely to be observed, by most observers living in this "random world"?

It may seems like as soon as we are starting to talk about the space of possible transformations and so on, we immediately run into the well known problems of convergence issues because the options just grow fields of fields of fields etc.*But* if you add the constraint of limited information capacity of an observer, this puts a bound on the complexity on all relatable structures which means that even in the continuum limit the sets should as far as I see at least be countable. The more complex and iterated the structures get, the less "weight" do they get - given an obsever.

And if you see it from the view of self-organisation, the most random scenario seems to be when also the observers, clocks and rods are random. And if you start by considering low weight observers, the contraints of what is observable by an observable is very tight simply because there are only simple observers around.

I think this makes a lot of sense, but exactly what the choice of formalism will be is not so clear.

I hope to find an answer to the uniqueness of the Fourier transform in this emergent sense. But I don't know how to do it yet, but I'm trying to figure it out.

/Fredrik

Fredrik, what you're suggesting looks less like philosophy than it looks like mathematics to me, but I can't see a mathematization of your verbal presentation. Do you have a mathematical literature in mind when you speak of emergence in its relationship to Fourier transforms? Renormalization and effective field theory methods come to mind, but only weakly, since at the QFT level the mathematical methods very much depend on the metric and the Fourier transform.

The concept of "limited information capacity of an observer" seems to me to be a very flimsy theoretical assumption to rest much weight on. I can do some mathematics, I can run this fast, I can understand and appreciate poetry only somewhat vaguely, I can't play a musical instrument, etc., etc. These are only a finite beginning to an infinite characterization of my information processing capacity, which I take to be infinitely multi-dimensional, so it cannot be characterized well by a single number. I think that you get into describing the psychology of the observer if you introduce "limited information capacity of an observer", which I'm not even sure is open to any comprehensive mathematical model.

As far as my own approach is concerned, I think I believe, I expect, I hope, that there is no need to introduce anything except Fourier transforms or their non-dynamical space-time equivalents to achieve empirical adequacy for experiments that are characterized by length scales within, say, 10 orders of magnitude of human length scales. Claims that the observer is necessarily a part of physical models don't appeal to me, insofar as I haven't yet seen a good argument. Appeals to the quantum mechanical formalism certainly do not justify enthusiasm for the observer as a necessity.

It does hi-jack the thread as I see it, Fredrik, and I'd rather not do it here, but apparently no-one else sees the thread, so perhaps it doesn't matter. Still, you should more respond here to the mathematical modeling aspect of things -- although I am also interested in the cultural issue of how I'm ever going to get anyone to read and vigorously critique these papers.

 

[Fra]

> Fredrik, what you're suggesting looks less like philosophy than it looks like mathematics > to me, but I can't see a mathematization of your verbal presentation

You just formulated the quest. To identify the representation and exact formalism is the task from my point of view at least, I'm working on it. Since the mathematical formalism is in progress, common language is actually the most accurate so far. But as progress is made I'd expect a transition from philosophy to mathematics. Needless to say I think the QM and QFT formalism on static backgrounds is not the answer.

Since this is your thread I'll refrain from further expanding on my comments too much. It's not my intention to contaminate it. I did wait a couple of days to respond to your last message because I didn't want to prevent others from jumping in...

About the mathematical formalism applied to physical models that implements the fuzzy reasoning - this is exactly what I am looking for. Given that I'm no librarian, if there was a paper I was aware of that solved this to my satisfaction I would certainly have quoted it. That's is why I got curious on your work if there was a connection. You're right that information bounds are fuzzy, but I would say this is fuzzy simply because I'm still looking for the satisfactory formalism. Once it's nailed, it will not be fuzzy at all. To me the difference between philosophy and mathematics

> Still, you should more respond here to the mathematical modeling aspect of things --
> although I am also interested in the cultural issue of how I'm ever going to get anyone
> to read and vigorously critique these papers.

I can only speak for myself. The questions I asked where the first questions of choice that would determine my further motivation to analyse your paper ni more detail. If you had provided an model for the emergence of the indexing process I would have been excited! from the simple reason that I would be able to directly relate to it.

Given my current impression I don't think I personally can add anything of value as feedback at this point, because I feel that your posed question is formlated relative too much baggage so to speak. But so does all of QFT, so this isn't a specific critique of your paper of course, it's just me.

Edit:
> These are only a finite beginning to an infinite characterization of my > information processing capacity, which I take to be infinitely multi-
> dimensional, so it cannot be characterized well by a single number.

I understand the objection and I've thought of that too. Your enumeration of new combinations is a process in my view, and processing I think it related to changes (and time) and "learning". And information vs access times. When do we know what we know? Like the distinction between having the answer, or to be able computer the answer, and how long is the computing time?

I agree, there is no clear limit to what we can LEARN, and our information capacity can also change. Which I think is the key. What we know, and what we "could know" or will become to know, are different. But this is IMO similar to the problem of dynamical references.

/Fredrik

 

[Peter Morgan]

> I feel that your posed question is formulated relative too much baggage so to speak. But so does all of QFT, so this isn't a specific critique of your paper of course, it's just me.

I do agree with your assessment of this: you'll see a coded acknowledgment that I'm specifically working within QFT's acceptance that statistics and correlations of field observables, set against a background space-time, are sufficient as an empirical ground at the end of the introduction to my "Lie fields revisited". However my acceptance is certainly not without reservations.

When doing innovative research, it's important eventually to restrict to a context in which there can be a formulation that makes contact with other approaches well enough for people who use those approaches to be able to see the advantages of the new relative to the well-known (or at least for others to see the possibility of advantage). To make progress in Physics, that means you have to create and/or find mathematics that expresses and refines your conceptual ideas. There was a stage in my research when no-one could see that what I was doing could possibly come to anything mathematical, it's taken me ten years to get to the point of publishing one paper in J. Math. Phys. and a few other papers in other journals.
Your approach, I don't know. It looks to me as if you have at least five years before you reformulate, for the tenth time, what it is you think you're doing well enough that you'll be able to recognize applicable mathematics when you see it in the literature. I don't know how long you've been doing your research, but unless your experience is very different from mine you had better settle in for the long haul. Note also that I had to have random fields pushed in my face twice by friendly Physicists before I realized this was exactly the mathematics I wanted, when I'd already been working on what were effectively random fields for at least 5 years and I was very lucky that I was well-enough prepared immediately to recognize Lie fields as exactly what I had been trying to do when I saw a reference to them in a 1970s review article. Without that recognition, I doubt I could ever have published my work in J. Math. Phys.
I'm somewhat projecting my problems onto you, which may well not be appropriate, so you will have to decide how seriously to take my experience and the advice I'm sort of offering you here. Your posts look informed enough about the role of mathematics, however, that you may agree with much of what I'm saying, but you want a mathematical ground that's more the next thing after the sort of thing that mine is. Good Luck!
I will be on PF probably only for a few months. I will return again, it's useful, but I could not do research while posting heavily. It's a little addictive, so that I haven't yet found a way of posting only a little. I find posting here intellectually and emotionally demanding, and rewarding, but once I settle into reading and writing for The Next Paper, I don't think I will be able to keep this up as well.

 

[Fra]

Thanks for your responses Peter.

Everything you just said makes great sense to me and thanks for your advice.

I have no illusions (and even if I did, I fail to figure out how I could tell), and I'm fully aware of that this is a long process. I am not affiliated with anything or anyone, this is a sincere interest and hobby of time and my driver is curiousity. My impression is that the research politics and social constraints seemed like interfering too severly with creativity. I wouldn't stand it and I have the wrong type of personality to align.

I'm 34 and I will take the time it needs because what's the alternative? The choice is easy. I want the answer and I will keep looking until I find it, and in the meantime this is about the most intellectually rewarding hobby I can imagine, I ask for no morem and I have no reason to take shortcuts.

I mention my thinking in the connection to your thinking to relate, but I have no plans whatsoever to seriously "promote" my ideas at this stage. At minimum I have to blow myself off the chair before even trying to convince others and _this_ stage is at minimum a few years away I suspect, probably longer.

> what it is you think you're doing well enough that you'll be able to recognize applicable mathematics when you see it in the literature

Either I recognize it or I don't. Wether it's not there or I just can't see it - I don't see how I could ever distinguish between the options

Good luck to you as well.

/Fredrik

 

[Peter Morgan]

... the research politics and social constraints seemed like interfering too severly with creativity. I wouldn't stand it and I have the wrong type of personality to align.

I sympathize, but even from a sort-of-similar starting point, I find that trying to understand the relationship of my approach to other approaches well enough to explain helps my own understanding of what I'm doing. Once you've scooped up something out of almost nothing, it's demanding, fun, and frustrating to undertake the very different task of telling what it is. I consider that whether I can explain the benefits of my approach to others is a valid test of my understanding. You've noticed that I can't, at least not well enough, so I have more to do, right?

It's not clear to me whether you have access to an academic library. If you do not, and you more-or-less agree with my attitude to the Physics community, then you must get access somehow. I'm pretty sure that it's not possible to find enough of what is known in Physics from books.

I'm 34 and I will take the time it needs because what's the alternative? The choice is easy. I want the answer and I will keep looking until I find it, and in the meantime this is about the most intellectually rewarding hobby I can imagine, I ask for no more and I have no reason to take shortcuts.

Absolutely! It's there to climb, so up we go, hoping we don't fall off the face of analogy. Nothing else that I can do especially well? Check! That makes for the possibility of too much emotional attachment, however, so try not to let your commitment to it destroy real life. I perennially don't know whether I'm spending enough time, interest, and energy with my 8-year-old daughter, I hope you have a measure of your insanity that is as effective as she is for me.

 

[sooperdooper]

Forward your infomation to Stephan Wolfram re:NKS maybe he could get the quantum folks off his back :)
That is if I'm grasping your papers, seems a good fit.

 

[Peter Morgan]

Forward your infomation to Stephan Wolfram re:NKS maybe he could get the quantum folks off his back :)

Strange and wise that you should say that. I was invited to present at the NKS 2007 Wolfram Science conference in Burlington Vermont, as a result of communication with them earlier this year about my "Bell inequalities for beables" paper, which as you say is almost custom made for them. It seemed as if the argument went over most of their heads, I think including Wolfram's (the conversation I had with him was more concerned with his own approach to evading the strictures of the Bell inequalities), but then I never get much response to it, despite it changing the terms of the Bell inequalities debate wholesale, IMO.

As I said in an earlier post, a random field approach doesn't take off until there is a constructive way to do Physics, and it starts to make contact with existing quantum field theoretic approaches, which "Lie fields revisited" partially does. It seems likely, however, that "Lie fields revisited" is probably too abstract an approach for Wolfram.

There are plenty of other people who are developing classical/near-classical approaches; I've just read Caticha's paper http://arxiv.org/abs/gr-qc/0508108" , for example, with some interest as a result of a suggestion (privately) from Fra (who posted earlier on this Topic). Caticha's approach seems to need an understanding of Bell inequalities something like mine. 't Hooft, as you may know, is subjected to vigorous criticism by Physicists for defending a conspiracy loophole evasion of the Bell inequalities, but the very brief communications I have had with him suggest that he has not engaged with the details of the argument I make, despite its usefulness for his purposes, or else I have failed to understand his good reasons for not finding my approach helpful. Stephen Adler has an interest in the argument as well, and was involved in "Bell inequalities for random fields" being accepted by J. Phys. A.

 

[sooperdooper]

Any way that you could refactor or refit Bell's experiment in a way that would limit or reduce the size of his observed violations by introducing a piece of what you have in mind, in a way that incremental introductions of your theory displace his results. And with those observations then facilitate a full investagation with a new experiment of your design? Is making a hybrid of the actual experiment he performed with adaptations feasible? On a side note these experiments could be carried out as a "further investigation" into Bell's experiment, the results are what they are. Doubt you would get funded to beat up Bell, but "validate" or further research into something commonly accepted is not a big deal. Look at string theory. ;)

 

[Peter Morgan]

And with those observations then facilitate a full investagation with a new experiment of your design?

My http://arxiv.org/abs/cond-mat/0403692" demonstrates that no experiment of the Bell type can rule out classical random field models. Experiments that violate Bell inequalities do not determine in favor of QM, when compared with classical random field models, although they do determine against classical particle property models (up to the usual reservations). Such experiments are nowadays more being done for technological reasons, not to distinguish the Foundations of Physics more clearly.

Although the whole discussion in "Bell inequalities for random fields" is necessary, there is a very simple "in principle" argument, at least for the sort of model that Bell introduces in "The theory of local beables" (which can be found in "Speakable and unspeakable in quantum mechanics"), in which experimental results are presented as probability densities on a space-like hypersurface: such probability densities are precisely (partial) initial conditions for a classical field theory, which are unconstrained in classical Physics. We can say that the observed statistics seem unlikely, but the results of experiments are more important than our preconceived ideas of what is likely.

It should also be noted that there is a strand of discussion of the Bell inequalities that got going with Luigi Accardi and Arthur Fine in the early 80s, which explicitly derives Bell inequalities without any reference at all to locality/nonlocality. A few references and a brief discussion can be found in my http://dx.doi.org/10.1063/1.2713457" .

Before the post quoted above appeared, I had intended to post on this in https://www.physicsforums.com/showthread.php?t=204571", where TimH posted a basic question quite graphically:

I am trying to understand the very basics of entanglement. I've read the Mermin article with the machines the light up red and green, and the quantum cakes article http://chaos.swarthmore.edu/courses/phys6_2004/QM_PDF/AJP68_2000c.pdf"

I'm trying to figure out if nonlocality is a real, mind-blowing fact or does it only appear to exist when you allow yourself to believe that quantum systems "have properties without being measured."

The cakes article seems to me to suggest the latter, based on the first full sentence on p.35 of that article (the sentence beginning "In particular..."). This sentence talks about the 9% of cakes with quick-rising batter, but this group only exists if we do two rising measurings, so how can we talk about this group in other contexts?

My reply was going to confirm the latter. Not talking about particle properties goes a long way towards resolving the Bell inequalities, but of course leaves classical particle physics short of anything else to talk about. Hence we move to classical random fields.

I should add that I'm not going after quantum (field) theory to disprove it, I'm going after it to understand it.