From: Charles Francis Subject: Questions about Constructing QED Date: Fri, 11 Aug 2000 04:55:29 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8c0ubm$mn6$1@pravda.ucr.edu>, thus spake John Baez >In article , >Charles Francis wrote: > >>[...] I do not >>necessarily agree that the question is correctly stated by you or that >>the Garding-Wightman axioms are even correct. Yes, I will modify the >>rules if I can see that the rules are not the rules of nature, and I am >>not interested in a mathematical expression which is not according to >>the rules of nature, except in so far as it helps research in finding a >>better expression of those rules. > >That's fine. This is probably the best attitude to take when >doing physics! All I was saying is that the problem of "whether >one can rigorously construct QED" is a famous and fairly well-defined >mathematical problem, a bit like Fermat's Last Theorem or Goldbach's >Conjecture. Lots of people have worked on it and anyone who solves >it will become famous. This does not mean the problem is important; >indeed, in a certain sense it's a completely useless and silly problem, >just like Fermat's Last Theorem and Goldbach's Conjecture. Glimme & Jaffe has finally arrived (thanks to all those who e-mailed me sources), and in so far as the Garding-Wightman axioms are concerned, I can say that the construction of field operators in discrete QED has nothing to do with them. (actually Glimme & Jaffe only mention Wightman axioms, I assume they are the same). The reason is that, as suspected, the Wightman axioms are phrased in terms of a construction of operators on an infinite dimensional hilbert space, and discrete QED constructs operators on a set of finite dimensional Hilbert spaces which are defined by observers each using a cubic lattice to represent possible results of measurement of position. Of more interest are the Haag-Kastler axioms, which do not specify the Hilbert space. At least superficially, they do appear to be satisfied. If I were to provide a formal treatment, would that be considered satisfactory as a construction of field theory? I have two other questions about quantities which I am now seeing regularly referred to, but never defined. What are the definitions of a) the mass gap b) the Landau pole -- Regards Charles Francis charles@clef.demon.co.uk ============================================================================== From: Jacques Distler Subject: Re: Questions about Constructing QED Date: 11 Aug 2000 18:24:34 GMT Newsgroups: sci.physics.research In article , Charles Francis wrote: >Of more interest are the Haag-Kastler axioms, which do not specify the >Hilbert space. At least superficially, they do appear to be satisfied. >If I were to provide a formal treatment, would that be considered >satisfactory as a construction of field theory? > >I have two other questions about quantities which I am now seeing >regularly referred to, but never defined. What are the definitions of > >a) the mass gap > >b) the Landau pole Ooops! These are the CENTRAL issues in constructing *any* QFT. If you haven't addressed them, it's as if you haven't really even *thought* about contructing the QFT. a) Is there a mass gap: does the spectrum of particle excitation have a continuum which extends all the way down to zero, or is there an energy gap between the vacuum and the lowest-lying particle excitation? Since you are *starting* with a finite lattice, of COURSE the spectrum is discrete. The *question* is whether in the infinite-volume, continuum limit, the spectrum becomes continuous near zero. b) The Landau pole: If you "naively" try to define QED via perturbation theory, you find that the coupling constant goes to *infinity* at a FINITE value of the cutoff. We can't define the theory for larger values of the cutoff and it is, therefore, impossible to take the cutoff to infinity and thereby define the continuum theory. The question is: does the Landau pole seen in perturbation theory persist in the full theory? Or, instead, does QED approach some (strongly-coupled) fixed point, which allows one to define a continuum theory? Despite many attempts, no evidence for such a strongly-coupled fixed point has been found in QED. Therefore, most people believe that the Landau pole is *real*, and the theory does not exist as a continuum QFT. Now that you know what the QUESTION is, you can set about attempting to answer it. Best of luck . . . -- PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc ============================================================================== From: Charles Francis Subject: Re: Questions about Constructing QED Date: Tue, 15 Aug 2000 10:51:20 +0100 Newsgroups: sci.physics.research In article <8n0apc$h3a$1@geraldo.cc.utexas.edu>, thus spake Jacques Distler >In article , Charles Francis > wrote: >>Of more interest are the Haag-Kastler axioms, which do not specify the >>Hilbert space. At least superficially, they do appear to be satisfied. >>If I were to provide a formal treatment, would that be considered >>satisfactory as a construction of field theory? >> >>I have two other questions about quantities which I am now seeing >>regularly referred to, but never defined. What are the definitions of >> >>a) the mass gap >> >>b) the Landau pole > >Ooops! These are the CENTRAL issues in constructing *any* QFT. If you >haven't addressed them, it's as if you haven't really even *thought* >about contructing the QFT. > No, what I have thought about is constructing a realist model of particle interactions in the absence of a background space-time, including photon absorption/emission. What I have found is a form of the equations of qed which applies in a family of finite dimensional Hilbert spaces. This includes a definition of "field" operators obeying the usual commutation relations. I am still trying to work out whether this constitutes a construction of qft in anyone else's language. As far as I can see the main difference between this model and the expected is that am using observer dependent Hilbert spaces (the field operators are covariant). >a) Is there a mass gap: does the spectrum of particle excitation have a >continuum which extends all the way down to zero, or is there an energy >gap between the vacuum and the lowest-lying particle excitation? > >Since you are *starting* with a finite lattice, of COURSE the spectrum >is discrete. The *question* is whether in the infinite-volume, continuum >limit, the spectrum becomes continuous near zero. I had thought this was the mass gap, but as you say in one sense the question is trivial. I have a finite number of types of particles as an assumption of the model, so the mass of the smallest massive particle is the mass gap. There is no limit as the number of types of particle goes to infinity. On a deeper level the question is either completely unanswered in my model, or dependent on far deeper and more complex arguments which I have not seen. I certainly have no obvious relationship between mass spectrum and the minimum length scale, and I have never thought or expected to answer this question. In fact I am puzzled as to why anyone expects it to be answered in the next major developments towards unification. >b) The Landau pole: If you "naively" try to define QED via perturbation >theory, you find that the coupling constant goes to *infinity* at a >FINITE value of the cutoff. We can't define the theory for larger values >of the cutoff and it is, therefore, impossible to take the cutoff to >infinity and thereby define the continuum theory. > >The question is: does the Landau pole seen in perturbation theory >persist in the full theory? Or, instead, does QED approach some >(strongly-coupled) fixed point, which allows one to define a continuum >theory? > >Despite many attempts, no evidence for such a strongly-coupled fixed >point has been found in QED. Therefore, most people believe that the >Landau pole is *real*, and the theory does not exist as a continuum QFT. > >Now that you know what the QUESTION is, you can set about attempting to >answer it. This does not tell me enough about the divergence to identify what the Landau pole actually is. Can you clarify? I am aware of a number of divergences of which I feel I know the origin, and none for which I do not think I know the origin. From what you say I surmise that the Landau pole is not a divergence in an individual diagram, but is perhaps due to the fact that the continuum perturbation series is asymptotic rather than convergent? Is that right? If so then I have a definite answer, which is that in the discrete model the perturbation series has a finite number of terms, and that the standard perturbation series is asymptotically close to it for low orders in qed. I expect no such approximation for qcd. In fact I believe this is one of my strongest practical arguments that there is no continuum qft and that discreteness is a fundamental property of matter. The same argument applies to infinite self energy, which I often read is regarded as being the sole important divergence problem. If discreteness is there as a property of matter then only a finite mass renormalisation is necessary. -- Regards Charles Francis charles@clef.demon.co.uk [Moderator's note: the Landau pole is a pole in the running coupling constant as a function of momentum. For a 1-loop calculation of the running coupling constant in QED which illustrates the existence of this pole, see any introduction to quantum field theory, e.g. Peskin and Schroeder's "An Introduction to Quantum Field Theory", where the result is equation (7.96). - jb] ============================================================================== From: Matthew Nobes Subject: Re: Questions about Constructing QED Date: Wed, 16 Aug 2000 11:17:19 -0700 Newsgroups: sci.physics.research On Tue, 15 Aug 2000, Charles Francis wrote: [snip] > This does not tell me enough about the divergence to identify what the > Landau pole actually is. Can you clarify? I am aware of a number of > divergences of which I feel I know the origin, and none for which I do > not think I know the origin. From what you say I surmise that the Landau > pole is not a divergence in an individual diagram, but is perhaps due to > the fact that the continuum perturbation series is asymptotic rather > than convergent? Is that right? Let me just jump in to say that in your approach (as I understand it) the QED landau pole would not be a problem. The divergence due to it shows up at some ridiculously short length scale (way lower than the Planck length) where I would presume that your discrete model would have long since taken over. > [Moderator's note: the Landau pole is a pole in the running > coupling constant as a function of momentum. For a 1-loop > calculation of the running coupling constant in QED which > illustrates the existence of this pole, see any introduction > to quantum field theory, e.g. Peskin and Schroeder's "An > Introduction to Quantum Field Theory", where the result is > equation (7.96). - jb] Three alternative references which treat these issues in more detail: K. Huang, Quarks, Leptons and Gauge Fields Discusses the Landau pole in QED and \phi^{4} theory. Montvay and Munster: Quantum Fields on a Lattice An extensive discussion of the Landau pole with a long discussion of the Luscher and Weisz investigation of the LP at strong coupling. The general conclusion is that the LP is present in \phi^{4} theory both perturbativly and non-perturbativly (this has possible implications for the consistentcy of the Higg's sector of the SM). Barton (I think) Introduction to Advanced Quantum Field Theory Discusses the LP in some detail, including the exact solution of the Lee model which exhibits a LP (i.e. non-perturbatively). There is also some discussion as to how this result relates to the LP problem in QED. (this is an old book and might be hard to find). (Note to Charles: Montvay and Munster spend a fair amount of time discussing the mathmatical consistency of lattice theories, you might want to have a look) ------------------------------------------------------------------------ "After the suffering of decades of violence |Matthew Nobes and oppression, the human soul longs for |c/o Physics Dept. things higher, warmer and purer than those |Simon Fraser University offered by todays mass living habits, |8888 University Drive introduced ... by the revolting invasion |Burnaby, B.C. of commercial advertising ..." |Canada Alexander Solzhenitsyn |http://hapiland.phys.sfu.ca ============================================================================== From: Jacques Distler Subject: Re: Questions about Constructing QED Date: 16 Aug 2000 22:01:34 GMT Newsgroups: sci.physics.research In article , Charles Francis wrote: >>a) Is there a mass gap: does the spectrum of particle excitation have a >>continuum which extends all the way down to zero, or is there an energy >>gap between the vacuum and the lowest-lying particle excitation? >> >>Since you are *starting* with a finite lattice, of COURSE the spectrum >>is discrete. The *question* is whether in the infinite-volume, continuum >>limit, the spectrum becomes continuous near zero. > >I had thought this was the mass gap, but as you say in one sense the >question is trivial. I have a finite number of types of particles as an >assumption of the model, so the mass of the smallest massive particle is >the mass gap. There is no limit as the number of types of particle goes >to infinity. On a deeper level the question is either completely >unanswered in my model, or dependent on far deeper and more complex >arguments which I have not seen. I certainly have no obvious >relationship between mass spectrum and the minimum length scale, and I >have never thought or expected to answer this question. In fact I am >puzzled as to why anyone expects it to be answered in the next major >developments towards unification. Why are people interested? Because the answer is important. QED (if it exists as a QFT, which it probably doesn't) should not have a mass gap. QCD (which probably DOES exist a a QFT) SHOULD have a mass gap. Recall what that means: even in the linit as you sent the lattice spacing to zero and the size of your box to infinity, QCD has a FINITE energy gap between the vacuum and the lowest-lying particle excitation. Saying that the lattice theory (with a finite lattice) has a gap is a statement of such UTTER triviality, that most people would not even bother making it. Saying that there is (or is not) a gap in the continuum limit is a nontrivial statement (which people are actually interested in). >>b) The Landau pole: If you "naively" try to define QED via perturbation >>theory, you find that the coupling constant goes to *infinity* at a >>FINITE value of the cutoff. We can't define the theory for larger values >>of the cutoff and it is, therefore, impossible to take the cutoff to >>infinity and thereby define the continuum theory. >> >>The question is: does the Landau pole seen in perturbation theory >>persist in the full theory? Or, instead, does QED approach some >>(strongly-coupled) fixed point, which allows one to define a continuum >>theory? >> >>Despite many attempts, no evidence for such a strongly-coupled fixed >>point has been found in QED. Therefore, most people believe that the >>Landau pole is *real*, and the theory does not exist as a continuum QFT. >> >>Now that you know what the QUESTION is, you can set about attempting to >>answer it. > >This does not tell me enough about the divergence to identify what the >Landau pole actually is. Can you clarify? OK. Let us place two static test charges one meter apart and fix the coupling constant of our lattice QED theory by demanding that the coulomb force between them agree with the experimental value. Now, let's say we change the lattice spacing, a, (say, making the lattice spacing half what it was before). If we want the force between our two test charges (still 1 meter apart) to be unchanged, we will, in general, have to adjust the coupling constant, e, of the lattice theory accordingly. In general, this proceedure will define a function e(a), which tells us how the coupling constant of the lattice theory varies as we vary the lattice spacing. It is easy to calculate e(a) using lattice perturbation theory. At tree level, nothing happens ( e(a) is a constant), but that ceases to be true once we include loop effects. What one finds is that e(a) INCREASES as the lattice spacing, a, decreases. Of course, once one reaches some small-enough value of a, e(a) has grown so large that we can no longer trust perturbation theory (I presume you have some other method of treating the lattice theory once perturbation theory is no longer valid). But, just for fun, let us extrapolate the function e(a), that we computed perturbatively, beyond the validity of perturbation theory. What one finds (actually, the credit goes to Landau) is that e(a) goes to INFINITY for some FINITE value of a, a=a_crit. This is called the Landau Pole. Now, as I said, it is possible that the full, nonperturbative lattice theory avoids the Landau Pole, but no one has succeeded in making this plausible. Most people believe that the Landau Pole is real (even if the value of a_crit computed in perturbation theory is wrong) and that, therefore, QED does not exist as a continuum QFT. >I am aware of a number of >divergences of which I feel I know the origin, and none for which I do >not think I know the origin. From what you say I surmise that the Landau >pole is not a divergence in an individual diagram, but is perhaps due to >the fact that the continuum perturbation series is asymptotic rather >than convergent? Is that right? No, it has nothing to do with that. Jacques -- PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc ============================================================================== From: Charles Francis Subject: Re: Questions about Constructing QED Date: 16 Aug 2000 22:06:05 GMT Newsgroups: sci.physics.research In article , thus spake Charles Francis >In article <8n0apc$h3a$1@geraldo.cc.utexas.edu>, thus spake Jacques >Distler > >>b) The Landau pole: If you "naively" try to define QED via perturbation >>theory, you find that the coupling constant goes to *infinity* at a >>FINITE value of the cutoff. We can't define the theory for larger values >>of the cutoff and it is, therefore, impossible to take the cutoff to >>infinity and thereby define the continuum theory. >> >>The question is: does the Landau pole seen in perturbation theory >>persist in the full theory? Or, instead, does QED approach some >>(strongly-coupled) fixed point, which allows one to define a continuum >>theory? >> >>Despite many attempts, no evidence for such a strongly-coupled fixed >>point has been found in QED. Therefore, most people believe that the >>Landau pole is *real*, and the theory does not exist as a continuum QFT. >> > >This does not tell me enough about the divergence to identify what the >Landau pole actually is. Can you clarify? > >[Moderator's note: the Landau pole is a pole in the running >coupling constant as a function of momentum. For a 1-loop >calculation of the running coupling constant in QED which >illustrates the existence of this pole, see any introduction >to quantum field theory, e.g. Peskin and Schroeder's "An >Introduction to Quantum Field Theory", where the result is >equation (7.96). - jb] > It is unfortunate that I have no convenient library and that "any" introduction does not appear to include Bjorken & Drell or Gasiorowicz. But if this is the Landau pole is it not removed by the method of Epstein & Glaser? This is equivalent to my rediscovery in discrete QED. In this case the Landau pole is not real, but is simply the result of a mistake coming about from the use of improper integrals without paying sufficient attention to the manner of taking limits. As for whether one should take a limit and define a continuum theory, as do Epstein & Glaser, or not, as do I, is a purely philosophical issue about the nature of mathematics and its relationship to physics, and will not be resolved by any mathematical or empirical consideration. -- Regards Charles Francis charles@clef.demon.co.uk ============================================================================== From: Jacques Distler Subject: Re: Questions about Constructing QED Date: 17 Aug 2000 16:50:32 GMT Newsgroups: sci.physics.research In article <8nf38d$gg7$1@Urvile.MSUS.EDU>, Charles Francis wrote: >>>Despite many attempts, no evidence for such a strongly-coupled fixed >>>point has been found in QED. Therefore, most people believe that the >>>Landau pole is *real*, and the theory does not exist as a continuum QFT. >> >>This does not tell me enough about the divergence to identify what the >>Landau pole actually is. Can you clarify? >> >>[Moderator's note: the Landau pole is a pole in the running >>coupling constant as a function of momentum. For a 1-loop >>calculation of the running coupling constant in QED which >>illustrates the existence of this pole, see any introduction >>to quantum field theory, e.g. Peskin and Schroeder's "An >>Introduction to Quantum Field Theory", where the result is >>equation (7.96). - jb] >> >It is unfortunate that I have no convenient library and that "any" >introduction does not appear to include Bjorken & Drell or Gasiorowicz. There's always amazon.com. A "modern" (ie, less than 35 years old) textbook in QFT would be useful to anyone interested pursuing this subject. Glimme and Jaffe is a bit of a leap if you have never even seen the Renormalization Group, as introduced in, eg, Peskin and Schroeder. >But if this is the Landau pole is it not removed by the method of >Epstein & Glaser? No, it is not. > This is equivalent to my rediscovery in discrete QED. >In this case the Landau pole is not real, but is simply the result of a >mistake coming about from the use of improper integrals without paying >sufficient attention to the manner of taking limits. No, it arises precisely from a CAREFUL attention to the taking of limits. >As for whether one should take a limit and define a continuum theory, as >do Epstein & Glaser, or not, as do I, is a purely philosophical issue >about the nature of mathematics and its relationship to physics, and >will not be resolved by any mathematical or empirical consideration. No, it is not a philosophical issue, as to whether the limit EXISTS. You BLITHELY assume that it does. Almost everyone else who has looked at lattice QED has come to the conclusion that the limit probably DOES NOT exist. Knowing that the limit exists, we can take it or not, as we please. But when people say "Does QED exist as a QFT?" they are talking about the existence of this limit (and that the resulting continuum theory satisfy their favourite set of axioms). -- PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc ============================================================================== From: Jacques Distler Subject: Re: Questions about Constructing QED Date: 18 Aug 2000 23:15:09 GMT Newsgroups: sci.physics.research In <8njjpv$kfa$1@Urvile.MSUS.EDU> John Baez wrote: >I guess I should have said any *modern* introduction. Bjorken and Drell's >book is great for what it does, but it's too old for this stuff - it was >written in 1965, and the Callan-Symanzik equation wasn't worked out >until 1970, so it doesn't talk at all about Landau poles, running >coupling constants, the renormalization group, and all that stuff >which forms the basis of our modern understanding of renormalization. NO ONE understood quantum field theory in 1965. Indeed, if you read Bjorken & Drell's preface, you can get some sense of the scepticism about QFT (in its then primitive state) that was widespread at the time. I've had many conversation with Steve Weinberg about those "Dark Ages". It's amazing that anyone kept on working on QFT in those days; the subject was held in such disrepute. It was only in the '70s that people started to understand quatum field theory: the renormalization group, the operator product expansion, asymptotic freedom, . . . Even Lattice Gauge Theory (which, unlike "Discrete QED", is a VERY big subject among quantum field theorists) was yet to be discovered when B&D was written. "All" B&D do is present the Feynman Rules, in the hope that they "may well outlive the elaborate mathematical structure of local canonical quantum field theory." Therefore, let us develop these rules first, independently of the field theory formalism which in time may come to be viewed more as superstructure than as foundation. (Quotations are from the preface to B&D.) One can't really learn field theory from B&D because the subject, as we understand it today, didn't really exist then. CF:>In this case the Landau pole is not real, but is simply the result of a CF:>mistake coming about from the use of improper integrals without paying CF:>sufficient attention to the manner of taking limits. > >No. I urge you to study a modern introduction to quantum field theory >and learn what the Landau pole is before making guesses like this. >Again, I recommend Peskin and Schroeder's book. You will not get >an adequate explanation here on sci.physics.research; quantum field >theory is too big of a subject. In <8ncl58$vp$1@geraldo.cc.utexas.edu>, I gave a simple outline of the calculation that Charles would have to do in order to see the Landau Pole in his theory. But, as you say, that's no substitute for actually learning ("modern") quantum field theory. Jacques -- PGP public key: http://golem.ph.utexas.edu/~distler/distler.asc ============================================================================== From: Aaron Chou Subject: Re: A 'simple' explanation required Date: 18 Sep 2000 12:29:04 GMT Newsgroups: sci.physics.research Milivoj Uroic wrote: > Hi, there ! > > I would like to discuss the following problem: > > Let's say we have a hydrogen atom with the electron in 2P state. The > electron would spontaneously decay within some 10e-6 seconds in the ground > state. WHY? > > -the first stuff you learn about it is Schroedinger equation. It states that > all electron levels are stable, and unless induced by radiation electron > should remain in initial state. > > -spontaneous decay rate is 'explained' by equilibrium conditions from, say, > Planck's radiation formula. I don't doubt the decay rate, but there's not a > clue on the cause > > -Fermi golden rule is the best way to calculate decay rates, but when you > trace it to the first principles, all the textbooks give some pseudo-classic > double-talk > > -I'm wondering if you absolutely need QED to explain it fully. I would > really like phrases like 'interaction with virtual field' either explained > through some principles, or omitted. > > -All you have is one proton and one stupid electron. How complicated can it > get? > > Explanations? Comments? What am I missing? > > Thanks a lot, > > Milivoj Uroic, Zagreb, Croatia I used to wonder about this too. To understand this, you need to look at things from a different perspective. I like to think about quantum field theory using a billiard ball analogy. You have a bunch of balls carrying energy, and an interaction which allows energy to be transferred from one ball to another. In fact, just view the billiard ball dynamics as a way in which a bunch of energy moves around and goes into different configurations. Each ball is just a container of energy. When two balls collide, some energy is transferred from one ball to the other and the energy goes into a new configuration. Similarly, in QFT, energy is stored both as kinetic energy and as rest energy. Every particle can be viewed as a ball or container of energy. Even the vacuum modes (states of zero particle number which have only the zero-point energy) can be viewed as balls. Because the vacuum modes are everywhere in space, these balls are constantly coming close to each other. Sometimes they hit each other and sometimes they miss. You can use quantum mechanics to calculate the probability of transferring a certain amount of energy per unit time (by calculating the overlap of the various wavefunctions weighted by the hypothesized interaction operator). Fermi's Golden Rule is just the statement that you have to sum over all of the possible ways that the energy might be transferred. So for example, you have a bunch of energy stored in an unstable particle, say your 2P hydrogen atom. The electron is flying around the atom with some kinetic energy. While doing so, it is constantly running into nearby vacuum modes of the photon. Eventually, one of these collisions will transfer the energy to the vacuum mode, and create one photon. Now you are left with a 2S atom, and a photon mode with photon number = 1. All of the electron's kinetic energy has been transferred to the photon mode, just as in a elastic collision between a moving ball and a ball at rest. Decays into massive particles can be thought of as inelastic collisions in which some of the energy becomes stored as internal energy. The only remaining question is how did the time scale arise. This I think is somewhat accidental. The fundamental dynamical scales are set by hbar and c, and somehow the universe evolved such that our everyday values of angular momentum and velocity are much different from these fundamental scales. Hypothetically, there could be other universes in which hbar and c are closer to everyday values for some alien lifeform. For them, quantum mechanics and relativity would just be common sense.