Essay Abstract

There have already been many attempts in the literature to modify the Wightman or Haag-Kastler axioms to be closer to the empirically successful Lagrangian approach to quantum field theory. It is suggested here that insofar as one fundamental difficulty of both Wightman and Lagrangian QFT is the postulate that the quantum field is an operator-valued distribution ---a linear map from a linear space of test functions to a linear space of operators---, we are motivated to consider taking a quantum field to be a *non*-linear map from a linear space of test functions to a linear space of operators, an approach that to my knowledge has not previously been proposed. Constructively, some of the non-linear possibilities for the scalar field case are introduced and discussed. The introduction of non-linearity widens the range of well-defined theories enough that they may provide worthwhile effective field models even if they cannot provide ultimately correct models. [I apologize that I cannot yet make the mathematical level be that of Nature or SciAm.]

Author Bio

I have been trying to "understand" quantum field theory for 20 years, more-or-less. I have focused on interacting QFT for approximately the last 5 years.

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"[I apologize that I cannot yet make the mathematical level be that of Nature or SciAm.]"

In my opinion, some of the essays in the contest could use a tad more mathematical links to mainstream assumptions but this reads very much like a specific journal article. Were you not concerned it might be too far outside the realm of the average Nature or Scientific American reader? I am sure that you are well qualified to write this, but could you explain in simpler terms what this means towards a foundational assumption? I would hate to skip over a great essay simply because I lacked some small concept.

    Dear Peter Warwick Morgan,

    Perturbative corrections, that proceeds with the notations of Wightman axioms in QFT is indicative of the unfathomable supersymmetry that arises from matter-antimatter asymmetry.

    Time paradox expressional with Minkowski spacetime that proceeds with the notations of Haag-Kastler axiomatic framework is indicative of ambiguous extra dimensions in that the emergence of time within systems is inexpressive in entirety.

    Thus we may have to redefine the dimensionality with string dynamics in that a different quantum mechanics may be applied with quark-gluon plasma.

    With best wishes,

    Jayakar

      Indeed, Community Rating may not work out well for this, but I think it's a curious essay for anyone who knows their QFT. I may put it on the arXiv as well, and I can publish it or not after the competition has run its course, so there is life after FQXi spits it out undigested. AFAICT, all the interesting essays so far are expounding ideas that are already in the literature in quite similar form, where the idea in this essay is not (although there may be some slight hints towards it in my previous work). Not having been previously published makes it inevitable that the paper must be more mathematically detailed to make any sense as Physics.

      To paraphrase, or to try to, there are specific kinds of non-linearity in axiomatic QFT already, but this essay introduces a kind of non-linearity that is a mathematically fairly natural consequence of the form of the deformations that are introduced by perturbative QFT (and that are empirically successful). I feel confident that it's not the only kind of non-linearity that could be introduced, and it's not necessary to introduce non-linearity because sophisticated approaches to distributions can be introduced, but it is fairly natural. On the other hand, it's a *somewhat* non-obvious kind of non-linearity, and I have not yet found straightforward ways to explicate the consequences of this introduction for the interpretation of QFT (which, as I say in my essay, I take to be a signal processing formalism).

      Perhaps significantly (at least it may clarify matters for some physicists), this construction is not akin to introducing nonlinear Schrödinger equations, and does not change the probability interpretation of QM in terms of the Born rule. Indeed, the Born rule is effectively taken to be fundamental, insofar as the GNS construction of a Hilbert space from a state over a *-algebra of observables can be taken to presuppose the Born rule. Another perspective is that a formalism of covariantly constructed states over *-algebras takes energy and/or action as derived rather than as fundamental quantities (in that sense only, I'm working in a Newtonian rather than in a Hamiltonian or Lagrangian mathematical formalism, although this is a choice that is congenial to me, not something that I regard as fundamental).

      Thanks for what I found to be a useful question to try to answer.

      Hi JJJ,

      Thanks for your comment. I suppose I take a different approach, in that I'd like to clarify our understanding of interacting QFT a little before trying to understand the even deeper mysteries and complexities of Mathematics such as QCD and string theory that I take to depend on QFT. I hope one can say something interesting about fundamental Physics without taking into account the finest details of hadronic matter and gravity, even if they are undeniably omnipresent, but it's good to have people on both sides of weighty dichotomies.

      Best wishes, Peter.

      Dear Peter,

      Problems like Haag's theorem seem indeed to be avoidable by allowing non-linearity. Taking the quantum fields to be non-linear maps from a linear space of test functions to a linear space of operators seem, after reading your essay, very promising in moving towards interactions. Do you think your approach can offer a better behavior for quantum fields on curved spacetime too? Could the desired non-linearity be obtained somehow from the general covariance?

      Best regards,

      Cristi Stoica

        Hi Christi, and thanks.

        I have thought about QFT-CST and about QG only a very little. So little, indeed, that I have a prejudice that I've so far been unable to shake from my head, in favor of working with torsion on an otherwise Minkowski space-time (something of the type of Einstein-Cartan, teleparallelism, or Poincaré gauge theory, but I know too little to know what formalism I would prefer), although perhaps only as an approach to local modeling, because Fourier transforms are so central to QFT, but difficult on a variable metric CST and impossible(?) if the background space-time is dynamical.

        Within such a scheme, my first attempt would be to take the torsion tensor to be the dynamical object of a quantum field theory, insofar as the non-linear approach I suggest in my essay may be less concerned with the renormalizable/non-renormalizable distinction than is standard QFT. I don't know how well that would work out, of course.

        That's not an answer to your question, but it's a vague answer to the question that your question suggests to me. Best wishes, Peter.

        22 days later

        Dear Peter,

        I also have an essay questioning the postulate of quantum theory - "Is there really no reality beneath quantum theory by Hou Ying Yau". Just like you, I spend many years trying to understand physics. I hope you will find the paper interesting.

        Sincerely,

        Hou Ying Yau

        Dear Peter Morgan,

        I have read what I can of your essay. You have chosen a problem to analyse that I am not sure is a wrong basic physical assumption but may be something to do with the best mathematical representation of nature. Not having studies QFT I was not even aware of the problem that you chose to write about.The level of mathematics that you included is a barrier to clear communication with those who do not have a background in mathematics and maybe not even physics.

        There are other entrants talking about mathematical approaches, such as Julian Barbour, who have non the less produced very accessible papers that can be understood by non specialists, are educational and enjoyable. Searching For New Mathematics by Ivars Peterson This article may help explain the problem for non mathematicians.

        Those who do have a good grounding in mathematics, (T H Ray, Lawrence Crowell and Joy Christian spring to mind), might be able to give give a valuable critique. Perhaps you could introduce yourself to them if you have not already and they might also be able to suggest other people who would find your essay accessible and something that they would be able to discuss with you.

        I am sorry I can not be more helpful. You no doubt feel very strongly about what you are doing as do many of the entrants, myself included. Good luck to you in getting the constructive feedback you are seeking.

        Thanks for reading through it, Georgina. I figured out weeks ago that I had misjudged the nature of the contest. I saw up front that it clearly asks for a level between Scientific American and Nature, but I'm not at a point in my research at which I can do that, so I did what I could. Of course if someone doesn't understand an essay, they can hardly give it a high community rating, and indeed it's reasonable for them to give it a low community rating for not hitting one of the contest specifications, of getting the right level. Now I can see that it was not worthwhile trying to do my research here; I suppose one can only present a much more complete idea.

        If one is arguing for something that has been in the literature as an idea for a while, as is the case for Julian Barbour, then a non-mathematical way of stating the issues has probably developed. An idea that hasn't yet become old enough to have a non-mathematical statement is probably wrong, as has always been true of my many previous ideas and approaches and as is probably true of my proposal here.

        To some extent there's a catch-22 here, which I suppose means that the winner is not likely to achieve the principle aim of the contest as I think of it, the pointing out of an assumption that genuinely hasn't been noticed and that can *usefully* be teased into a different form. AFAICT, the assumption I address is the only assumption questioned in all the essays here, mathematical or not, that I have not seen questioned many times before.

        I suppose everyone dies with less constructive criticism than they need. Best not to get too frustrated about it. It's perhaps ironic that I pretty much agree with everything that Peterson says in the interesting article that you link to (which definitely constitutes constructive feedback), but that does not mean that I ever succeed when I try to write lucidly.

        Thanks again. Peter.

        Peter,

        While I sympathize with your inability to stir up an audience, I have to agree with Georgina that the audience you require is extremely select. There is the unhelpful Catch 22 that such detail oriented work requires equally detail oriented observers to analyze it and nature contains far more detail than observers. If you notice some of the other threads, many are not even participating in discussions of their own ideas.

        I have something of the opposite problem with my entry. It is so elementally basic that no one takes it seriously. It is simply that we assume the effect of time, sequence of events, is fundamental, rather than the action causing them. That it is not the present moving from past to future, but the changing configuration of what exists, turning future into past. For example, the earth doesn't travel the fourth dimension from yesterday to tomorrow, but tomorrow becomes yesterday because the earth rotates. One effect of this you might consider is that it turns time from a linear dimension into a non-linear dynamic, since duration doesn't transcend the present moment, but is the state of the present between the occurrence of events.

        Hi Peter,

        I feel compelled to echo Georgina's opinion that it would be very difficult to find a referee for your work among the denizens here. (Don't look at me -- I've had Weinberg's 3-volume QFT text for years and every book is in pristine condition.)

        If it's worth anything to hear -- I don't think there will ever be a plain language version of QFT; the subject is clearly specialized for experts. Any continuous field theory that attempts quantization has already jumped the comprehensibility barrier of common language -- because we all understand "things," not fields. I suspect that's how string theory captured the popular imagination; the public at large doesn't have a clue of supersymmetry or the field excitations that the theorist derives from QFT, but they know what kind of thing a string is.

        I did see your request for someone to find an existing derivation of section 3 weeks ago, but I'm not up to that research, and I couldn't tell you who is.

        FWIW, I think your theme suffers from the difficulty of understanding in physics that probability theory has in mathematics -- i.e., not really much is known about it except in formal language. So doing a calculation ends up having a meaning of its own independent of any constructive meaning that the problem it purportedly solves, is trying to convey. That's the reason I switched from number theory to complex systems science, where the problems are clearly correspondent to the real world.

        Nevertheless, of the parts of your paper that I understand I am seeing some potentially valuable insights into quantum vaccum field solutions that could very well end up explaining how the field varies so radically from point to point yet sums to zero. That definitely brings things back to reality -- or rather, reality back to "things."

        All best wishes, Peter, in the contest and in your further research.

        Tom

        Tom, thanks also for your remarks. I want to address your "I don't think there will ever be a plain language version of QFT", because I think there is a possibility, and indeed that section III of my essay here /may/ be a starting point for at least a slightly more accessible account. The problem is not the basic formalism, insofar as I claim that QFT is at heart just a form of stochastic signal analysis ---notwithstanding the incompatibility of joint measurements at time-like separation---, the problem is renormalization, which makes the literature and textbook accounts almost entirely obscure. The difficulties of understanding QFT are notoriously extreme, but a lot of work has been done to make Feynman's dictum that they are insurmountable less true than it used to be.

        It's in the nature of the evolution of our understanding of mathematical formalisms that they are gradually reformulated, and that each reformulation leads to a wider understanding, until a popularizer at some level takes hold of the new ideas and brings them to wider audiences. Whether it's my work or someone else's that brings QFT into clearer focus is not of course of the essence.

        Looking at your FQXi essay this season, I see that you have detected at least something of this process in the Bell literature. If you haven't previously seen my "Bell inequalities for random fields - cond-mat/0403692 (v4, 24th May, 2006), J. Phys. A: Math. Gen. 39 (2006) 7441-7455", perhaps you might find my approach there to /those/ questions a little different from much of the Bell literature (though people also find that paper opaque). IMO, it's just a question of time, perhaps another 5-10 years (or 10-20, ...), before the penny drops for the Physics community that for a wide variety of reasons the violation of Bell inequalities proves almost nothing. At some point it will become possible to say that of course most Physicists always understood that was the case, which it will be possible to justify by pointing out that 't Hooft, Bohm, and other Serious Physicists were never outright refuted, and the derision nay-sayers were subjected to will be forgotten except by Historians. After the paper above I decided to stop worrying about Bell inequalities and move on to thinking much more single-mindedly about QFT.

        If you want to understand QFT, I'd recommend trying to engage with Haag's "Local Quantum Physics" for a few years, though it's definitely not easy going. Amongst the standard textbooks, I find the wildly old-fashioned (yet still modern enough) approach and notation of Itzykson & Zuber far more congenial than Weinberg. The path integration of modern textbooks, in particular, obscures the algebraic and combinatorial structure that is the only path to reconciling QFT with ordinary QM, IMO. That doesn't remove the ultimate need for at least some people to understand the relationships between such different approaches, of course.

        Although I operate at not much above the level of popular accounts, I believe that a lot of what can be shown about complex systems is probabilistic in character, right? I suspect the relationship between probability and statistics will be forever difficult.

        It's becoming more worthwhile having posted here. Thank you.

        Peter,

        This is in reply to your post on my thread. Since you sound more respectfully considerate, than genuinely interested, I repost it here:

        Peter,

        Thank you for the reply. I would first have to agree we are on opposite sides of a significant fence and I can understand why you might see my side as lacking necessary detail to be informative. My position is that while your side of the fence might be finely structured, it is still emergent from the underlaying dynamic. Which is to say I don't see the need for a platonic realm of fundamental laws governing nature. I see laws as patterns which emerge with the actions and relationships they define. Bottom up and top down are complementary functions that emerge as one. Yes, nature is exponentially complex, but the principles describing it are interactive and complementary. Knowledge and information must be static in order to maintain the very details of which they consist, but that doesn't mean reality is so fundamentally frozen. If reality were frozen, it would be a complete lack of thermodynamic activity and nothing would happen, or exist. A non-fluctuating vacuum. No nouns, or verbs. No factors, or functions.

        So for me, it's a matter of how to get from nothing to something. I would start with space as the aphysical infinite equilibrium. In this void, there is a cycle of expanding energy and contracting structure. Now if we were to relate that dichotomy to sentience and knowledge, the energy is the element of awareness and knowledge is the structure it conceives. Much as in my essay I point out that while our awareness is constantly moving onto new thoughts, these thoughts coalesce out of received information and then are replaced. So as awareness goes from past to future thoughts, the thoughts go from future to past. Just as energy is constantly inhabiting structure, then breaking it down and moving onto other forms.

        Now consider in your essay, the conceptual process which is going on. Much like a puzzle, modern physics consists of many static components that seem like they should fit together, but however it is done, there seem to be gaps and the solutions often create new problems, as they solve current ones. They are all obviously parts of some larger whole, but not a singular whole. So you find a connection that is "worthwhile," but not "ultimately correct." Possibly it is because there is no "ultimately correct model?" As I point out in the last line of my essay, "Neither academic or religious authority can turn an ideal into an absolute." There is no more a universal model than there is a universal god. Both models and perspective are inherently subjective. Oneness and one/unity and unit are not the same thing.

        I know this sounds philosophical, but if your ivory tower is built on sand, would you want to know, or would you prefer not to know?

          I think I have to say that I know it's built on sand. I'm no Platonist, to think that I build my models of anything else (though I've no way that I know of to gainsay someone who thinks Mathematics transcends our experience). You'll have to read between the lines of my various web presences to realize that there isn't much ivory in my tower.

          It looks as if your "there is a cycle of expanding energy and contracting structure" is your sand, or some part of it, where functional-algebraic models for the way we look and for what we see when we look are mine. I take almost everyone to need eventually to accommodate complexity in their models. I'm playing with this stuff that other people have built, as you are; it's good that we have chosen the work of different people as our starting points. Insofar as I prefer a more abstract way of thinking, as I think of it being, I just do what I prefer to do, for as long as I am given to do it and for as long as something shinier doesn't distract me. FWIW, I prefer to construct ways of thinking in which there are no "laws", there are just ways to guide our engineering of models for our experience and of edifices.

          If I construct stochastic 4-dimensional models, in which time is one of the four dimensions, that doesn't mean that I think this is the way the world is. It's just a model, not necessarily different in character from a spherical cow or from a map, except insofar as additional complexity makes a map different from a spherical cow.

          So in response to your "So for me, it's a matter of how to get from nothing to something", I suppose I would say that for me it's a matter of accepting the appearance that there is something, and looking for something that it seems worthwhile, to me, to do with what I see, given what I have to it with. And hoping that my judgement is not always as bad as it sometimes is.

          I've been trying to get back to some computations that I've been trying to get through, pursuing what I introduced in my FQXi essay, which are hard enough that posting here for a few days has been a welcome interlude. Peter.

          Peter,

          Thank you for the thoughtful and considerate response. I have to say I generally get a more emotional and negative reaction when I try to push the buttons of others in the field.

          For me, the concepts basic physics deals with; Energy, order, structure, expansion/contraction, attraction/repulsion, complexity/simplicity, etc. underlay and manifest all aspects of existence. Every person alive, from some kid kicking a soccer ball, to an astronaut, has to understand physics. We live what theorists analyze.

          Obviously there are levels of understanding, but there is no clear line between what people experience and theorists try to divine. Which is to say that I feel I have a stake in the field, even if my thoughts don't meet with approval by those more directly engaged in the various disciplines.

          While you are quite right to say our understanding starts with what is, my strongest impulse is that we stand over the abyss of mortality, so the "nothing" is as much a feature to explain, as complexity.

          For me, I accept there is an infinite amount of complexity I can never conceive, from quantum math, to biological and neurological functions and so the best I can do, is to sense those general patterns and how they interact and manifest in different ways. This gives me some foundation to deal with the unknown.

          I would say life is a game where the goal is to figure out the rules and the first rule is that many rules are subject to circumstance.

          Good luck with putting together your work and finding the right audience.

          Dear Peter,

          I read you message concerning the competition voting. You mentioned different categories for the different kinds of essays. I think it is a very good idea. I left a reply there.

          I understand when you say about research lacking fun, thats how I feel right now. I have spent far longer than is healthy thinking about, explaining and defending the things I write about over the last few years. It is disheartening when the enthusiasm is not reciprocated. It is very hard to justify the unpaid time spent on it, and the neglect of other important aspects of life, when the output is not valued and is mostly overlooked.

          The only reward is the tiny scraps of recognition. So though I don't like to ask as it is a bit impertinent and I'm trying hard not to offend anyone; I would greatly appreciate it if you would reciprocate and tell me what you think of my essay -especially concerning the explanatory framework set out in diagram 1. There is a high resolution version in my essay thread.