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 My posts to the newsgroup sci.physics.research - Part I Full threads can be found at the URL http://groups.google.com/groups?hl=en&group=sci.physics.research and also http://www.lns.cornell.edu/spr/ From: ark (arkatcassiopaea.com) Subject: Re: W*-Category Newsgroups: sci.physics.research Date: 2001-10-08 15:10:06 PST  On Wed, 26 Sep 2001 06:19:10 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >This is also an excellent book in itself! - but if it's too hard, >just read the introduction and then jump to the bibliography. >The introduction will help answer your question about why this >stuff is important in physics. The one-line answer is: applications >of categories to quantum theory often use C*-categories. I would add to thi that W* category became especially important with the discovery of Tomita-Takesaki theorem - it deals with von Neumann algebras with a cyclic and separating vector. In physics it was first related to the Kubo-martin-Schwinge condition for thermal equilibrium, then it has found more potential applications. BTW it also relates to the geometry of quantum states, including Berry phase. There are many textbooks available, some are less known than other, and yet very good. Among those less known I would recommend S. Stratila "Lectures on von Neumann algebras" and Li Bing-Ren "Introduction to operator algebras" ark From: ark (arkatcassiopaea.com) Subject: Re: Clifford's geometry algebra Newsgroups: sci.physics.research Date: 2001-10-08 15:16:12 PST On Sun, 7 Oct 2001 15:58:24 GMT, "David Corfield" wrote: >One of my interests in this is that a PhD student I'm supervising is working >on the relationship between Clifford's math and his philosophy. It would >make his thesis all the more striking the greater the latent power in >Clifford's ideas. In any case, interesting philosophies deserve good ideas. It seems to me that, as in many other areas, we are dealing here with a particular lobby. It is tru that Clifford algebras are useful, also in physics. It is also true that there are amny results that you can get using Clifford algebras, which can be also obtained, at a much smaller cost, by using other algebraic and geometric techniques. Let me give an example: W. Rodrigues writes his papers using Clifford algebras - so that most of the volume of the paper deals with the formal structures that are needed to present the result in this particular language. But, he often adds a sentence telling the reader that the particular result can be as well obtained and expressed in a standard language of differential forms (with co-differential, Laplace-Beltarmi etc.). In theories of gravity, especially those where gravity results from some more fundamental structures, we need to tacle the problem of "degenerate metric" or "degenerate soldering form." These theories can hardly be based on a Clifford algebra. Although there are generalization of Clifford algebra techniques to the case where inner product can be degenerate, these do not belong to the standard courses. Clifford algebra is a particular structure and it offers a particular computational technique. What is perhaps interesting from a philosophical point of view is the famous modulo 8 periodicity - which may somehow mirror something very deep and fundamental in the structure of the universe.... ark From: ark (arkatcassiopaea.com) Subject: Re: Clifford's geometry algebra Newsgroups: sci.physics.research Date: 2001-10-08 18:33:31 PST On Sun, 7 Oct 2001 15:58:24 GMT, "David Corfield" wrote: >Let's see if this one works. > >Does anyone have any thoughts on the claims of the group at >www.mrao.cam.ac.uk/~clifford that a huge opportunity has been overlooked >for decades. (snip) Adding to comments from other people: Clifford algebras are also of importance in quantum theory. Algebra of canonical anticommutation relations (CAR) generated by fermionic creation-annihilation operators is a Clifford algebra. In quantum theory we hace to deal also with fermionic systems with infinitely many degrees of freedom - thus we need infinite dimensional Clifford algebras. To deal with this subject we have to invoke the theory of C* and their representations, thus also W* algebras - which connects to another thread on this group. When dealing with systems that need indefinite metric (usually we try to avoid it in quantum theory) - we have Clifford algebras with indefinite scalar product. We move to more general category or B* algebras (B for Banach). ark From: ark (arkatcassiopaea.com) Subject: Re: string theory & Berry phase; quantum bundles Newsgroups: sci.physics.research Date: 2001-10-09 17:57:22 PST On 8 Oct 2001 06:38:17 GMT, abergmanatPrinceton.EDU (Aaron J. Bergman) wrote: >>A quaternionic geometric phase could e.g. be associated with a rotation >>subgroup of the Lorentz group (as opposed to the complex phase on >>projective space), but I am not sure what this would mean. > >I'm not sure why it would be useful. The same with me. Read Adler's book on QQM and also old papers by Jauch, Emch, .... Quaternionic Quantum Mechanics serves a different purpose than complex one. We are not sure which purpose exactly. It is a good scheme from the point of view of quantum logic and noncommutative probability. It is not so good from the point of view of dynamics. In complex QM there is one "i", and it stands in front of one Planck constant. It serves as a crucial connection between group generators (which are anti-Hermitian) and observables (which are Hermitian). A generator of time translation, for instance, is anti-Hermitian, and it "happens to be" iH/hbar, where H is energey observable. Thus "energy conservation" etc. In Quaternionic QM we do not really know what to do in similar cases. Should time be thre-dimensional (see Tifft and Lehto) so as to connect to three imaginary units? If so, then what? There is also another problem in QQM - which stems from the fact that quaternions do not commute: we ca not easily build composed systems out of simple ones - because of the problem with tensor product of quaternionic vector spaces. I am not saying the there is no future for QQM. I am saying that I do not know yet what this future can possibly be - because of the problems sketched above. One more comment: it is possible to take another approach to Berry phaes (originally due to Armin Uhlmann). In this other approach you consider parallel transport along density matrices rather than along pure states. In this approach Berry phase has values in unitary operators, not just a complex phase. ark From: ark (arkatcassiopaea.com) Subject: Re: People in the Group/Backgrounds Newsgroups: sci.physics.research Date: 2001-10-11 10:45:52 PST On 10 Oct 2001 01:04:00 GMT, baezatgalaxy.ucr.edu (John Baez) wrote: >On the other hand, studying at a university also has its problems! >Besides costing a lot, the main problem is that you can easily fall >into the trap of believing that your teachers know everything you >need to learn, and that all you need to do is go to class and pass >tests, and everything will be fine. > >In short: there are problems with getting degrees from universities >and problems with studying on your own, so the only solution is to >do BOTH. I would add to the above one more thing: as soon as possible, while studying, do all you can so as to be able to go to as many conferences and seminars with guest scientists as possible. Do not be shy!!! You must overcome your shyness. Very important. When you have some idea or some question - approach speakers during a break, or a lunch, or a party, and ask your questions. Do you homework during the night, and next day ask the followup.. This makes a real differerence. Sometimes much of your future work depends on a single exchange with just one bright scientist! ark From: ark (arkatcassiopaea.com) Subject: Re: Kaluza-Klein theories Newsgroups: sci.physics.research Date: 2001-10-12 14:33:41 PST On 10 Oct 2001 07:17:24 -0700, mcartoajatmat.ulaval.ca (Mihai Cartoaje) wrote: >I would like to know what the current state of non-quantum >Kaluza-Klein theories are. I know that using them one can predict the >four forces. Have there been other fields predicted like the >Klein-Gordon or Dirac fields? Part of the mathematical formalism related to this question is in Chapter seven of "Riemannian geometry, fiber bundles and Kaluza-Klein theories." (World Scientific, I am one of the authors). This chapter deals with "Harmonic analysis and dimensional reduction." However, the starting assumptions in this book are much too restrictive (and thus not realistic as far as physics is concerned) For an alternative approach, not quite in the line of mainstream phsyics, and yet, in my opinion, interesting, search the net for papers by P.S. Wesson. ark From: ark (arkatcassiopaea.com) Subject: Re: Kaluza-Klein theories Newsgroups: sci.physics.research Date: 2001-10-15 21:20:04 PST On 14 Oct 2001 20:51:41 GMT, "alon" wrote: >why isnt P.S Wesson approach in the mainstream of physics? Perhaps I will withdraw this statement. The definition of "mainstream physics" will always be debatable. Instead let me quote from P.S. Wesson paper: "String theory is 'promising ...,' one worker said, '... and promising, and promising' [116]. M-theory which (unlike superstring theory) is not perturbative, is even more opaque (...)". Another quote: "(...) we do not assume compactification of the extra dimension or assume it space-like." ark From: Arkadiusz Jadczyk (arkatcassiopaea.com) Subject: Re: Kaluza-Klein theories Newsgroups: sci.physics.research Date: 2001-11-30 16:57:57 PST On Thu, 29 Nov 2001 05:15:22 +0000 (UTC), "Danny Ross Lunsford" wrote: >> 2) Through each point in 5d space passes only one geodesic > >this should read > >> 2) Through each point in 5d space passes only one geodesic >> that returns to the same point in the same direction as it left it > >This is part of the "projective" assumption. > >-ross Sorry, I posted my message before reading your correction. In 1938 Einstein and Bermann generalized KK assumptions: "we ascribe physical reality to the fifth dimension whereas in Kaluza's theory this fifth dimension was introduced only in order to obtain new components of the metric tensor representing the electromagnetic field." "We have, therefore to take the fifth dimension seriously although we are not encouraged to do so by plain experience." "The most essential point of our theory is the replacing of the hypothesis 2, of rigorous cylindricity by the assumption that space is closed (or periodic) in the x0- direction ." x0 is the fifth dimension in their notation. See Ref. 43 in http://xxx.lanl.gov/abs/gr-qc/9805018 ark From: Arkadiusz Jadczyk (arkatcassiopaea.com) Subject: Re: Kaluza-Klein theories Newsgroups: sci.physics.research Date: 2001-11-30 16:57:28 PST On Thu, 29 Nov 2001 01:55:31 GMT, Danny Ross Lunsford wrote: >For this reason Klein sought to abandon the cylinder condition. Because >the correct transformation properties of gm5 ~ Am (they have to look >like 4d gauge transformations) depends to some extent on the cylinder >condition, it turns out that the best you can do to preserve this >essential feature is to make the assumptions > >1) The dependence on x5 is angular (i.e. x5 is compactified) >2) Through each point in 5d space passes only one geodesic Are you sure 2), as written above, experesses what you have in mind? Through each point in 5d space passes infinitely many geodesics. Perhaps what you tried to say is that 2) the fibers are closed geodesics??? ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac theory on the KG kernel (?) Newsgroups: sci.physics.research Date: 2001-10-09 21:24:22 PST On Mon, 8 Oct 2001 20:59:48 +0200, "Urs Schreiber" wrote: >Given an n-dimensional pseudo-Riemannian manifold (M,g). > >Let > > K |psi> = 0 > >be a Klein-Gordon-type equation for a particle propagating on (M,g) >and let > > D |psi> = 0 > >be the corresponding Dirac equation, where the generalized Dirac >operator D satisfies > > D o D = K I would be careful here. On a curved space square of the Dirac operator is the "spinor Laplacian", not the ordinary one. You get an extra 1(1/4) scalar curvature term. It makes real difference when your manifold is not of constant curvature. ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac theory on the KG kernel (?) Newsgroups: sci.physics.research Date: 2001-10-10 19:54:12 PST On Wed, 10 Oct 2001 09:46:43 +0200, "Urs Schreiber" wrote: >Not if one uses e.g. an "N=2" Dirac operator where the curvature >terms of both "handednesses" cancel mutually. The simplest example is >the Dirac operator on the exterior bundle given by > > D = d + d* > >where d and d* are the exterior and coexterior derivative, >respectively. When trying to compare kernels of two operators, both operators must act on the same space. Which space? 0-forms? On zero-forms d+d* acts as d alone. Thus its kernel consists of constant function - whatever the metric is. Not very interesting. If not zero forms, then what else? Square of d+d* gives Laplace-Beltrami operator, not the Laplace operator - the two differ by curvature terms on forms of higher order. Therefore, I would insist that one should try, first, to specify the question: which exactly operators? Which exactly space they are supposed act upon. ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac theory on the KG kernel (?) Newsgroups: sci.physics.research Date: 2001-10-12 14:31:35 PST On Thu, 11 Oct 2001 16:34:49 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >>Which space? > >The space of all differential forms. That's what Schreiber >meant by "on the exterior bundle". All right. My fault. When he wrote "operator of Klein-Gordon type", I did not get that what he really means is the Laplace-Beltrami operator. (If that is what he really meant?) Now, space of all differential form, when the metric is pseudo-Riemannian, is an indefinite metric space, not a Hilbert space. One needs to be careful with spectral decomposition here indeed. Years ago I wrote a paper dealing with the peculiarities of spectral decomposition of selfadjoint operators on Krein spaces - but it was about bounded operators. So it will not be useful here. ark From: ark (arkatcassiopaea.com) Subject: Re: General covariance, background independence Newsgroups: sci.physics.research Date: 2001-10-15 21:24:19 PST On 12 Oct 2001 21:38:56 GMT, baezatgalaxy.ucr.edu (John Baez) wrote: >In article <%Esw7.53277\$3d2.2474513atbgtnsc06-news.ops.worldnet.att.net>, >AG wrote: >>Does "general covariance" mean "expressible as tensor equations?" >Did I just hear someone open a can of worms? > >The term "general covariance" has a long and convoluted history, >since it goes back at least to Einstein's early work, where he >had strong intuitions about what made general relativity different >from all previous theories, but said some misleading stuff, because >he hadn't yet figured everything out. > >Taken literally, "general covariance" means covariance under all >coordinate transformations. A modern translation would be "diffeomorphism >covariance". I would like to add to the above just few comments. First is that it can also mean: "use only NATURAL operations." Of course the question is which are natural operations? For this I recommend the book by Kolar, Michor and Slovak "Natural operations in differntial geometry." Second, when you peruse the book, you will notice that there is no chapter on spinors. Are not spinors natural objects too? Is not Dirac operator a natural operator? And here one needs to be careful. Diffeomorphism invariance of spinors is trcky - to say the least. But I am not going to open the can of worms even more.... ark From: ark (arkatcassiopaea.com) Subject: Re: D vs. E in vacuum Newsgroups: sci.physics.research Date: 2001-10-19 13:32:49 PST On 18 Oct 2001 00:56:34 GMT, "Eric Forgy" wrote: >After all that mess, the short answer to the question "Is D a pseudo >vector?" is apparent. Although D is a 2-form, that does not mean that D is a >pseudo vector because D is the Hodge dual of a 1-form E and the HODGE STAR >PICKS UP AN ADDITIONAL SIGN UNDER A PARITY TRANSFORMATION (in this case, but >not in general). > >Thanks for a nice question! It made me think. I hope my answer makes sense. One observation that is not strictly related to the original equation, but, somehow, is relevant: The relation between D and E, as well as the relation between H and B, in vacuum, involves Hodge operator on 2-forms. Hodge operators on 2-forms in 4d (as well as hodge operators on n/w forms in nd) is conformally invariant. Thsi is an easy exercise. But it less known that Hodge * operator on 2-forms in (-+++) signature , with the properies *^2=-1 and Hermitian (with respect to the natural inner product) is sufficient to define uniquely the light cone (= conformal) structure of space time. ark From: Charles Torre (TORREatcc.usu.edu) Subject: Re: D vs. E in vacuum Newsgroups: sci.physics.research Date: 2001-10-21 21:58:44 PST ark writes: > Also earlier: "Electromagnetic permeability of the vacuum and the > light cone structure" > A. Jadczyk, Bull. Acad. Pol. Sci. 27 (1979) 91-94. > Online at http://www.quantumfuture.net/quantum_future/emp.htm with extra > info. > Nice. I will keep your result showing how Hodge * on 2-forms determines conformal structure in mind next time the issue comes up, which it does from time to time. As long as I am writing, I might mention (changing the subject entirely) that your series of papers with Coquereaux and the corresponding book (all from long ago) on Kaluza-Klein reductions have been very useful to us here at USU since we are working on various aspects of symmetry reduction in gravitational theories. I would highly recommend them to anyone who is interested in such things. -charlie From: ark (arkatcassiopaea.com) Subject: Re: position operator Newsgroups: sci.physics.research Date: 2001-10-23 22:46:28 PST On Tue, 23 Oct 2001 18:54:05 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >In article , >pollux wrote: >>In page 24 of Peskin/Schroeder, they say that the quantized Klein- >>Gordon field \phi(x) (here x and p denote the spatial part of the >>corresponding 4-vectors in a fixed frame) acting on the vacuum >>state creates a particle at position x. >Really? That's a sort of odd thing to say - if this were true, >it would be very hard to understand why the inner product of phi(x)|0> >and phi(y)|0> is nonzero [....] (snip) >People have studied other candidates as well. The whole exercise >is of very limited use in practical particle physics, but it's sort >of fun to figure this stuff out. I would add to the above one comment. Operationally, there is a difference between "creating a particle" or "preparing a particle state" at (x,t) and "detecting a particle at (x,t)". The "measurement" issue is a complex one. If one studies the issues, one finds that standard prescriptions (or axioms) of QM (Born's interpretation, transition probabilities") are valid only as approximations. As for Newton-Wigner analysis - yes, it is really fun to follow their argument, however their assumptions are too strong. They assume projection valued measure, while there is no reason for it. It could be POV (positive-operator-valued) as well, and probably it is, and then their reasoning does not apply. But in old days POV measure were not that popular as they are today. P.S. There is a rather popular myth that all "observables" in quantum theory are necessarily represented by Hermitian linear operators. To see that it isn't necessarily so check the discussion of "time of arrival" in Helv. Phys. Acta 69 (1996) 613-636, available also online: http://www.quantumfuture.net/quantum_future/papers/time/time.html That is how I learned my lesson.... ark From: ark (arkatcassiopaea.com) Subject: Re: EF theory Newsgroups: sci.physics.research Date: 2001-11-01 19:43:12 PST On Thu, 1 Nov 2001 20:31:28 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >Consider EF theory on a G-bundle P -> M over an orientable >smooth manifold M of dimension n, without cosmological constant >term. The Lagrangian is > >tr(E ^ F) > >where E is an ad(P)-valued (n-2)-form and F is the curvature >of the connection A on P. > >1) What are the classical equations of motion? Perhaps you guys know it already, but this question can be answered using Souriau's method. I did similar calculations for point particles and strings with charges valued in Lie(G)* (Higgs charge), I could try it with EF setting. It should be specified whether we are looking for point particles or for strings. For strings, of course, classical equations of motion can not be specified until we postulate the necessary internal structure ("tension" tensor). ark From: ark (arkatcassiopaea.com) Subject: Re: Time travel Newsgroups: sci.physics.research Date: 2001-11-06 14:42:09 PST On 6 Nov 2001 02:55:26 GMT, Moataz Emam wrote: >Physics looks at both space and time as one inseparable structure, a geometric >manifold. So, in that sense, the past and the future do exist, because the >manifold exists as a whole. Probably you meant "Einstein theory of relativity" rather than "physics". Einstein theory of relativity is just one particular theory, and will probablu be superceeded by another theory. Quantum theory, for instance, looks at space and time as separate structures. Some physicists keep it in mind that Lorentz group symmetry may be violated. Only experiment can tell. ark ubject: Re: Unification of Electromagnetism and Gravity Newsgroups: sci.physics.research Date: 2001-11-07 19:04:02 PST On Tue, 6 Nov 2001 20:35:34 +0000 (UTC), Grace Shellac wrote: >I've been under the impression that gravity and electromagnetism >actually hasn't been unified. I asked Jerrold Marsden down at CalTech >if this was true. > >He responded: "...there is a well defined set of equations called the >Einstein-Maxwell equations that already unifies E and M with >gravity. They are discussed in standard books on general relativity." > >"Thus, I would say that most researchers believe that these theories >are already unified. The big question is how to unify quantum theory >with gravity." > >So, what is the deal here. Am I misinformed? No, you are not misinformed. You got a direct information about one particular point of view. To have a full picture you need to read what other physicists think of this problem. There are at least three other ways. > Or is Prof. Marsden >mistaken? Prof Marsden represents here a conservative trend, I would say, and a short description of one vesrsion of "already unified" theory (originated by Rainich) you can find in "Gravitation" by Misner et al. > Doing a google.com search didn't seem to help as there >doesn't seem to be a consensus that gravity and electromagnetism are or >have been satisfactorily unified. I see web pages by people like >Sweetster and others but no definitive comment that indicates that >anyone actually thinks gravity and electromagnetism have been unified. See above. Alternatives are: a) try to describe gravitation as a function of electromagnetic phenomena b) try to describe electromagnetism as a function of gravitational phenomena c) try to derive both from more fundamental fields d) try to derive both by making use of alternative geometries (Finsler, Einstein-Cartan, gauge theories, more than four space-time dimensions, noncommutative geometry etc e) try to derive Theory of Everything, and EM+G will come in a big bag of all kinds of goodies. >Help! What's the latest opinion? Different physicists will have different opinions. There is no consensus. Marsden is certainly right that we have also problems with adding quantum theory - and here you will find not so many physicists that will tell you that there are no problems there, The above is simply understanding of the present situation, based on my own research experience, not an autoritative answer. ark From: ark (arkatcassiopaea.com) Subject: Re: Time operator Newsgroups: sci.physics.research Date: 2001-11-07 23:59:46 PST On Wed, 7 Nov 2001 22:32:39 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >>The loop quantum >>gravity approach seems to take both views into account also, although I >>understand that they have a problem with how to define a time quantum >>operator. > >What's a "time quantum operator"? You mean an operator that >corresponds to the observable we hope we're measuring when we >look at a clock? More precisely, an operator T canonically >conjugate to the energy H: > >[H, T] = i hbar ? > >The lack of such an operator is a quite general feature of >quantum theory, nothing special about loop quantum gravity. >String theory doesn't have such an operator either. It's not >a problem. When discussing quantum-mechanical observables it is useful to make a distinction between Scroedinger and Heisenberg picture. Relativistic local quantum filed theory teaches us that it usually less confusing to use Heisenberg picture. Having this in mind, there is no such thing as a "position operator x of a particle". But, for each time t, there is a position operator x(t) for a particle at time t. Usually operators x(t1) and x(t2) do not commute! Similarly, even if only conceptually, it makes no sense to speak about "time operator in general". But we can think of "time operator t(x) for a particle to arrive at x". It makes no sense to think of "time in itself, or "time of a clock" as observable, because each observable needs a prescription of how to "measure it". Ther is no prescription as to how to measure time. Time in the morning, or time in the evening? Time at 6AM or time at 5PM? It makes no even sense. Nevertheless, without much thinking, just for curiosity, people were looking for an operator "T" satisfying [H, T] = i hbar and indeed soon they found that it can't exist as a regular operator, mainly because energy is usually bounded from one side, and also because energy spectrum is not continuously homogeneus. Clever formal substitutes for [H, T] = i hbar have been invented in the past - just to find a solution From the physical point of view a well defined observable for a single particle is "time of arrival at a point" or "time of crossing of a surface". The subject and different versions of these observables, and their relation to measurements of "tunnelling time" are discussed in the current literature. The complication here is that many particles will never arrive at x, thus we need to "normalize the probability", the result being that such a time of arrival is described by a nonlinear operator. Additional complications arise when quantum mechanics is treated relativistically or on a curved background. Indeed, as John explained, all that has nothing to do with loop quantum gravity - rather it has to do with clearly defining meaning of observables. One more comment is important. Quantum mechanics is usually about "particles". But it need not be so. Suppose instead of thinking about quantum mechanics of particles, we are conteplating quantum mechanics of extended objects - for instance wave fronts. In THAT case it makes sense to ask: "at which time a given wave front, will arrive at x. We will get answer to this experimental question almost always. Thus, in this case, we expect that for each each time-like geodesic gamma there will be a linear time operator t(gamma) describing time at which observer described by gamma will register arrival of the wave front. ark From: ark (arkatcassiopaea.com) Subject: Re: Lorentz invariace of inner products Newsgroups: sci.physics.research Date: 2001-10-29 17:36:06 PST [Moderator's note: I screwed up, so it seemed that this post was written by me, when in fact it was by "ark" - jb] On 21 Oct 2001 02:38:57 GMT, Iuval clejan wrote: >The thing that bothers me is >that under a passive boost (change of reference frames) this spinor >does not transform in a unitary manner, for example it does not >preserve its norm One thing I would like to add to what John already wrote: "who sets the norm?" You write about "norm" - but this norm is an artifficial construct in the framework of finite dimensional representations of the Lorentz group. On the other hand, you will usually find that there IS a "norm" that conserved - which comes from indefinite metric (i.e. scalar product that is invariant but not positive definite). Perhaps this "norm" has some physical meaning? Like for instance in Minkowski space..... Normally we think that there is a little if any relation between the fact that Minkowski space scalar product is indefinite, and the fact that quantum theory is noncommutative. But, note, that because of the indefinite scalar product, the double cones in Minkowski space (with natural intersection and union operations) form a complete orthomodular lattice, a "quantum logic" that is non-Boolean, it is not even modular. ark From: ark (arkatcassiopaea.com) Subject: Re: Lorentz invariace of inner products Newsgroups: sci.physics.research Date: 2001-11-08 11:34:20 PST On Tue, 06 Nov 2001 20:32:56 -0500, Iuval clejan wrote: >Now here we have a finite dimensional irreducible unitary >representation of a non compact group (Lorentz group plus parity)???? >Can someone please explain this? > >-Iuval This has already been discussed. John Baez addressed this problem already. You can find answers in previous posts on the subject. But let me say it again: standard probabilistic intepretation of quantum mechanics needs positive definite scalar product, not just "inner product". What you call "unitary representation" is indeed unitary but with respect to "indefinite metric" scalar product : there are vectors with =-1. When your space can be decomposed into a direct sum of positive and negative definite subspaces, it is called a Krein space. Studying of operators, group representations, spectral theory on Krein spaces is a separate branch of mathematics. Probabilistic interpretation of quantum mechanics needs Hilbert space, not a Krein space. This being said, I need to add a note: the so called Gupta-Bleuler quantization of EM field uses indefinite metric. Then a "physical subspace" is selected, which is semi-definite, by imposing gauge condition. Then a Hilbert space is cosnstructed by taking a quotient with respect to the "zero-norm" subspace. But this use of indefinite metric, although having the same source as in your example (Minkowski space signature) is not the same what you are talking about when discussing finite dimensional "unitary" representations. Second note: as the number of papers is growing on the subject of "negative probability" - one day physicists may be back to indefinite metric as in your example, and quantum mechanical interpretation may be given to finite dimensional "unitary" representations of the Lorentz group (possibly in terms of pre-geometry, ur-geometry or whatever), but this is a phantasy right now. ark From: ark (arkatcassiopaea.com) Subject: Re: Gauss and Euclidean Space Newsgroups: sci.physics.research Date: 2001-11-07 23:19:52 PST On Mon, 5 Nov 2001 14:49:30 GMT, "AG" wrote: >As Henri Poincare said, any attempt to discover by experiment the geometry >of space is futile: "Measurement is never of space itself, but always of >empirically given physical objects in space, whether rigid rods or light >rays. Regarding the structure of space as such, experiment can tell us >nothing; it can tell us only of the relations that hold among material >objects." [quoted in M. Jammer/Concepts of space] Interpretation of every experiment whatsoever is always through some underlying theory. The very fact that what we are doing constitutes and experiment is already based on prior assumptions. If the underlying theory of space and its geometry is such that it allows experiments to determine some aspects of space geoemetry - then it is so. In my opinion Poincare, in the above statement, was exhibiting the way of thinking of a philosopher-mathematician. Physicists and engineers have somewhat different mentality. For many years I was collaborating with Rudolf Haag (one of the pioneers of algebraic quantum field theory). His first education was of an engineer. If you read his recent papers, for instance "Objects, events and localization", you will realize that he proposes to use experiments to detect "a granular structure of space." Reference: R. Haag, "Objects, events and localization", in "Quantum Future. From Wolta and Como to the Present and Beyond." Springer, 1999 ark From: ark (arkatcassiopaea.com) Subject: Re: Gauss and Euclidean Space Newsgroups: sci.physics.research Date: 2001-11-12 13:49:51 PST On Sat, 10 Nov 2001 22:37:55 GMT, "AG" wrote: >I was unable to interpret this as a proposal to detect the granular >structure of "space." Is yours a literal quote from the paper? Because he >only writes "there is some granular structure," he does not say "...granular >structure of space." You are right. He does not say "granular structure of space". A short summary of his position is contained in his other paper "On the lessons of quantum physics" [Helv. Phys. Acta. 66(1996), 679-681] "An event is a fact and irreversibility is inherent in this concept. According to widespread belief the fundamental laws are time-reversal invariant and irreversibility appears only on the macroscopic level dues to coarse grained description. This should be seriously reconsidered in Quantum Physics.(...) we can assert that as soon as we accept the existence of discrete facts independent of the cognition by an observer we are forced to an evolutionary picture where in the progress of time new events are realized and the events are connected by directed causal ties which in the simplest case, are particles and when ultimately space-time is related to the pattern of events." Thus, it seems, he considers space-time as emerging in a limit, not as a priori given with its geometry. Perhaps I should also add that in "objects, Events and Localization" Haag is quoting papers by Ph. an myself on EEQT as "In a series of papers (Blanchard and Jadczyk 1995) Ph. Blanchard and A. Jadczyk have described a formalism which generates real event in the interaction of an atomic object with a macroscopic measuring device.(...) The scheme, called 'event enhanced quantum theory' introduces irreversible decisions into the interaction process and yields a good phenomenological description of the quantum measurement process. It is related in spirit to points 1 and 5 of the strategy outlined above but allocates events only to the interaction between small and a large system retaining the ad hoc distinction between a classical and a quantum level of the world." I should explain here that I consider classical-quantum and small-large as two different divisions. Large can be quantum, and small can be classical. In our formalism we never use limit of "large quantum systems." We simply admit a part that is classical, we allow for the algebra to have a nontrivial center. Back to Haag and Poincare and whether measurements tell us something about geometry of space and time and other structures of interest: While discussing algebraic quantum field theory in 1964 or so, Haag stressed the importance of weak topology where neighborhoods are determined by finite sets of measurements. ark From: ark (arkatcassiopaea.com) Subject: Re: Phase space of all possible universes Newsgroups: sci.physics.research Date: 2001-11-05 18:19:26 PST On Fri, 26 Oct 2001 17:55:39 +0000 (UTC), p.kinsleratic.ac.uk wrote: >> Yes, exactly: because the *process* of observation forces the system >> into an eigenstate. Nobody talks about probabilities in the context of >> QM when the system under consideration is in an eigenstate. After it is >> observed, there's no "probability" anywhere. It's in that state and >> that is all. > >An observation forces the system into an eigenstate of the measurement >operator. A subsequent measurement using a _different_ measurement >will push the system into an eigenstate of that different operator. Although this is one of the standard axioms of quantum mechanics, there is no compelling reason that it must be so. In fact there are good reasons to believe that this statement is an oversimplification. First of all there is no reason to associate every observation with a self-adjoint operator. Second, to even discuss this question within physics we need a framework within which the very concept of observation is part of the dynamics. Moreover, since we are talking here about early stage of the universe, it is clear that "observation" should not imply an "observer". Wheeler's concept of a Big Eye observing the early cosmos from the Future is a nice metaphor, but is not yet part of any formalism that I know about. So, how can we define an "observation", so that it works also when there is no "observer?" Following Bohr and Bell (and early Stapp) we can define it as a coupling between the quantum and the classical part of the system (the Universe in our case). If we apply now Lindblad's theory, generalized by Christensen to arbitrary C* algebras, and if we apply the theory of piecewise deterministic processes (a typical process governing stock market and its crashes), we find out that an observation results in "events", and these events involve "quantum jump". Typically a quantum jump maps |psi> into A|psi>, but the operator A need not be Hermitean and, moreover, the resulting state needs not be an eigenstate of some particular "observable". This happens only when A is a projection operator on a one-dimensional eigenvector of some "observable" that defines the observation. It is a very particular, very special case. ark From: ark (arkatcassiopaea.com) Subject: Re: Phase space of all possible universes Newsgroups: sci.physics.research Date: 2001-11-10 18:41:05 PST On Thu, 8 Nov 2001 03:00:04 GMT, "Urs Schreiber" wrote: >Following Zeh et al., however, we can define it as the coupling >between a quantum system with few degrees of freedom and a "bath" of >a large number of (also quantum) degrees of freedom, which leads to >decoherence of the "small" system, mimicking "state reduction". This >is more than a metaphor and indeed part of a formalism (actually of >*the* formalism). Yes, I know. But I disagree. If you couple a quantum system with a few degrees of freedom with a large number of degrees of freedom - you still have a quantum system and unitary linear evolution. Nothing will ever "happen", there will be no wave packet reduction. Things will happen "in the limit of infinite time", but nothing will ever happen in finite time. Of course you can try to argue that time large enough is infinite FAPP. But what is FAPP for one person, in one circumstances, will not necessarily be FAPP for another person, in different circumstances. FAPP is subjective, and we are striving at making physics objective - as much as it can be. Hepp and Araki, also Namichi, were discussing thse questions long ago. My point of view here is that the problem above is telling us something important. Decoherence formalism is interesting, sure .... FAPP. Quting from Kiefer: "The most importante feature of decoherence besides its ubiquity is its irreversibility". [in "Quantum Future", Ph. Blanchard and A. Jadczyk ed] You will not get irreversibility in a quantum system, however large, from a unitary dynamics! You can get it by replacing your system with another system, obtained in some clever limit. But thsi will not change the fact that YOUR original system, the one that obeys unitary evolution, is reversible. So, what to do? One way is to accept FAPP methods. Another way is to accept the facts and relax axioms of QM that are due to Bohr, von Neumann and others. 1) Istead of assuming that the algebra of observables is L(H) (all bounded operators on H), let us assume that it is a general C* algbenra, not necessarily simple, possibly with a nontrivial center (superselection observables) 2) Instead of assumint that time evolution is given by a one parameter group of aotomorphism, let us assume that it is given by a semigroup of positive-definite maps (positivity has a physical meaning, multiplication of quantum operators does not have such a meaning) So, we have these two options. Most physicists follow the first options. But Prigogine, Haag and some other propose to examine the possibility of irreversibility on the fundamental level. ark From: ark (arkatcassiopaea.com) Subject: Re: Phase space of all possible universes Newsgroups: sci.physics.research Date: 2001-11-14 09:52:10 PST On 13 Nov 2001 04:38:29 GMT, Urs Schreiber wrote: >> But I disagree. If you couple a quantum system with a few >> degrees of freedom with a large number of degrees of freedom - you >> still have a quantum system and unitary linear evolution. Nothing will >> ever "happen", there will be no wave packet reduction. Things will >> happen "in the limit of infinite time", but nothing will ever happen >> in finite time. > >Since we were talking about cosmology: Do we even need a >notion of time evolution for decoherence to be of interest and >useful? > Doesn't it all boil down to the statement that: "When >looking at a generic, highly entangled system and ignoring >most of its degrees of freedom, the remaining degrees of >freedom will look like a mixture, not like a superposition."? You see, I disagree here. Once we are talking about cosmology, it does not matter whether. Nobody is "looking", and it does not matter whether we ignore something or not. Once it is there - it is there. They may "look" like a mixture, but this is subjective (objectively they are not a mixture), and we aim at an objective description. >In my current understanding the role of any time evolution in >decoherence is merely to evolve a special state into a generic >state to which the above sentence applies, much like time >evolution in statistical mechanics is merely there to >thermalize everything and bring the theory into position. Yes, and it is subjective. There is no objective decoherence, except after taking some kind of a limit, or after taking some kind of subjective coarse-graining. > >> 1) Istead of assuming that the algebra of observables >> is L(H) (all bounded operators on H), let us assume that >> it is a general C* algbenra, not necessarily simple, possibly >> with a nontrivial center (superselection observables) >> 2) Instead of assumint that time evolution is given by a >> one parameter group of aotomorphism, let us assume >> that it is given by a semigroup of positive-definite >> maps (positivity has a physical meaning, multiplication >> of quantum operators does not have such a meaning) > >What would replace the usual path-integral formalism in this >setting? Mensky wrote a series of papers on path inegral formalism for systems under continuous observation. He was, in fact, deriving a form of disiipation from a modification of path inegral. I do not think that he has the final answer, but he has at least a good starting point. Check his "Continuously Measured Systems,Path Integrals and Information", quant-ph/0007089, as well as other papers. The path inegral formalism comes from Trotter's formula, and it applies to an arbitrary evolution kernel, but in QM the tricky thing is using space, or phase space variables in the propagator. Every Abelian C* algebra is essentially algebra of functions on some set. There is also "central decomposition" - probably it should be applied here, but how exactly? I do not know. ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac equation under Lorentz transformations Newsgroups: sci.physics.research Date: 2001-11-10 18:39:27 PST On Fri, 9 Nov 2001 00:01:22 +0000 (UTC), mcartoajatmat.ulaval.ca (Mihai Cartoaje) wrote: >I believe I am beginning to understand how the Dirac equation >transforms under Lorentz transformations. The Dirac equation is, > >(i gamma^mu del_mu - m) Psi(x) = 0. > >The gamma matrices transform as, > >gamma^mu ' = S gamma^mu S^(-1) = Lambda^mu_nu gamma^nu, I would be careful with wording here, because it may lead to a wrong understanding of what is going on. I wouldn't say "gamma matrices transform". I would say gamm matrices have the property. What transforms is \Psi and \del. Then, because gammas have this property, we find that the equation is invariant. Here is a schematic sequence (I will skip the mass) gamma del Psi = 0 gamma del S^{-1}S Psi =0 gamma S^{-1} del S Psi =0 S^{-1} S gamma S^{-1} del S Psi =0 thus (multiply with S from the left) S gamma S^{-1} del S Psi =0 thus (properties of gammas) gamma L del S Psi = 0 now L del = del^prime S Psi = Psi^prime thus gamma del^prime Psi^prime = 0 The deeper understanding being that we change the spinorial frame by a constant Lorentz transformation, and that del_mu are derivations along vectors of this frame. Gamma matrices are constants here (they keep their standard form). What is not constant are gamma^mu del_mu. ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac equation under Lorentz transformations Newsgroups: sci.physics.research Date: 2001-11-14 09:52:49 PST On Tue, 13 Nov 2001 21:45:53 +0000 (UTC), ultraman2002athotmail.com (mandro) wrote: >the electron is a section q of a vector bundle B on a manifold >M, the two "lorenz coordinate systems on M" in your case minkowsky space >are f, g : M --> R^4 >since--and only since, they determine orthonormal basis fields >E_i , E'_i i = 1,,,4 on M >we can say they determine trivializations t, t': B --> M x C^4 > (t, t' are like coordinates on B) Not exactly. You need more than an orthonormal basic field, you need a spinorial frame, and there are two spinorial frames over each orthonormal frames. You need a spinorial frame to trivialize the particular B. But you may call it a nuisance here. > >now f o g^{-1} = L is a lorenz transformation > > t o t'^{-1} = p(L) is a SL(2, C) transformation > > >then the precise content of that statement is that > > t o q o f^{-1} "is the wave fn in one coord system" > and t' o q o g^{-1} " is the wave fn in the other coord system" > >and, the dirac equation would have work in both (rouhly speaking) Yes, but the above says nothing about the Dirac operator. The fact that the Dirac operator is invariant follows from its definition, and need to be checked. Of course, in a sense, it follows from the fact the Dirac operator is a "natural" operator, but to see how natural it is, you need to contemplate its definition. >By the way did you know that you dot really need two coordinate systems >to show this all you really need is two vector basis fields >just consider > t o q and t' o q > enjoy Yes, but still, there is a work that need to be done to show that the Dirac operator is a well defined operator defined on sections of B. This work is equivalent to showing the invariance of the Dirac equation. ark From: ark (arkatcassiopaea.com) Subject: Re: Dirac equation under Lorentz transformations Newsgroups: sci.physics.research Date: 2001-11-19 19:23:54 PST On Fri, 16 Nov 2001 14:46:45 GMT, Mihai Cartoaje wrote: >> Psi itself does not know how to tranform to a frame that >> is not an orthonormal frame. > >Have you used the assumption that the gamma matrices are invariant to >reach such a conclusion? On sci.physics, I have posted a LaTeX document >showing how to transform the wavefunction so as to keep the Dirac >equation form invariant assuming the gamma matrices form a contravariant >system. It is only one page of easy reading. Shall you tell me where >I have made a mistake or made the assumption that the coordinate systems >are orthonormal? If you want to read it it is under the thread "Dirac >equation under coordinate tranformations". You can use groups.google.com >to find it if it has been removed from the server. Hi, It seem you propose to use Psi transforming as a scalar, rather than as a spinor. So you want to have a multiplet (as your Psi has values in C^4) of spin zero particles, instead of a doublet of spin 1/2 particles (as it is with the standard Dirac equation). Now, the good thing about the standard Dirac equation is that it has a conserved probablity current, with positive definite probability density. You should also attempt to propose your definition of the probability current with your method. You should be cateful though, because in the standard approach gamma_0 is being used. But in your approach there is no one gamma_0, you have different gammas at each point and for each coordinate system. So, what would be your formula for the probability current that a) defines a conserved vector field b) probability density is non-negative? ark From: ark (arkatcassiopaea.com) Subject: Re: Lorentz invariance of inner products Newsgroups: sci.physics.research Date: 2001-10-31 09:27:29 PST On Tue, 30 Oct 2001 01:37:05 +0000 (UTC), baezatgalaxy.ucr.edu (John Baez) wrote: >In particular, this fact just means that you cannot make up a >Lorentz-invariant quantum theory where the Hilbert space of >states is finite-dimensional. > >(Unless you use a trivial representation, which would be boring.) > >This is one reason why people needed to invent quantum field theory! >In quantum field theory, the space of states is an *infinite-dimensional* >*unitary* representation of the Lorentz group (or its double cover). A little correction: You do not need quantum field theory for that. The Hilbert space of solutions of Dirac equation carries an infinite-dimensional projective unitary representation of the Lorentz group. Thus the above can not be thought of as a reason for inventing quantum field theory. People invented quantum field theory because of Klein paradox and other problems of relativistic quantum mechanics in external fields. I am making this note just in case some kid is reading this newsgroup .... ark From: ark (arkatcassiopaea.com) Subject: Re: Lorentz invariance of inner products Newsgroups: sci.physics.research Date: 2001-11-02 19:19:35 PST On 2 Nov 2001 07:33:16 GMT, Iuval clejan wrote: >ohn Baez wrote: > >> >> >> We all agree that it's good to have a norm, or more precisely an inner >> product, to interpret a vector space as a space of "state vectors" >> in quantum theory. However, the spin-1/2 representation does not >> have a Lorentz-invariant norm. What this means is that in the context >> of special relativity, it is quite artificial to pick any particular >> norm on this space - whatever you pick, a Lorentz boost will change it! >> >> >> And this is consistent with the fact that we should not attempt to >> use this space as a space of state vectors in the context of special >> relativity + quantum theory. > >So what space should we use? The infinite dimensional unitary reps of the >Lorentz group have all possible spin values, so they are not what we want when >we talk about spin 1/2 particles. Let's concentrated on these things that can be handled mathematically, that is free fields. A free relativisttic quantum field has a particle number operator which splits the Fock space into n-particle subspaces. For n=0 we have vacuum which carries a trivial representation of SL(2,C). For n=1 we have usually a direct sum of few irreducible representations of the same spin. (why "few"? because we have particles and antitiparticles, and possibly some other "internal symmetries"). Apart of possible subtleties the n=1 (one-particle) space is isomorphic to the space of solutions of the appropriate wave equation. The subtleties involve the fact that solutions of classical wave quation split into positive and negative energies. Also, the classical wave equation can be real, while in quantum field theory we need complex Hilbert space. > The infinite dimensional (time and light-like) >unitary reps of the Poincare group are probably OK though (I am looking at my >group theory book), but since the Lorentz group is just the Poincare group >without translations, these should be equivalent, so I am still confused. > >> >There is absolutely no relation between the double cones in Minkowski >> >and quantum logic except a mathematical one. There is no physics there >> >at all. And why so? Here is an example, When we look for a relativistic position operator, we need into account causal structure of the Minkowski space. The natural regions for localization are double cones (diamonds). We want states to be orthogonal when localized in spacelike diamonds. These diamonds form a non-Boolean lattice. We want to represent this non-Boolean lattice, together with the symmetry group. The natural lattice to represent a non-Boolean lattice is the lattice of subspaces of a Hilbert space. This way you can rediscover Dirac equation. >> This relation probably *is* irrelevant to your question, so I don't >> know why ark mentioned it. However, it's *not* irrelevant to physics! I mention it, becuase there are finite dimensional representation of SO(3,1), but with indefinite metric. Thus they can not have a standard quantum mechanical probabilistic interpretation. However more and more people are looking into a possibility of using signed measures, thus "negative probabilities" (Feynman wrote a paper on this subject, recently Noyes wrote a paper too). This is NOT a standard theory to be told kids, but this is a valid research avenue. >> The point is this: Minkowski spacetime and the space of 2x2 hermitian >> matrices are equivalent representations of the Lorentz group. The >> causal ordering in Minkowski spacetime corresponds to the usual >> ordering on the space of 2x2 hermitian matrices, where A < B if B - A >> has positive eigenvalues. And if we restrict attention to >> projection operators, which correspond to "propositions" in quantum >> logic, this ordering corresponds to "implication": we obtain a >> complete orthomodular lattice of propositions. I am not talking about representing points. I am talking about generalization of Mackey imprimitivity theorem, when spectral measure is replaced by the lattice of diamond sets in the Minkowski space, with its natural orthogonal complement operation. There is no such a theory in fact. But there are partial results. ark From: ark (arkatcassiopaea.com) Subject: Re: Lorentz invariance of inner products Newsgroups: sci.physics.research Date: 2001-11-15 19:28:52 PST On Thu, 15 Nov 2001 05:16:25 GMT, George Jones wrote: >Bottom line: unitarity in quantum theory is defined with respect to a >positive definite inner product. > >Regards, >George This would be true if you formulate is as "usually unitarity in quantum theory is defined with respect to a positive definite inner product." The word "usually" is important here. If you do not use this word - than what you said is incorrect. There are books about "indefinite metric." For a recent paper on indefinite metric in C* algebraic formulation see e.g. http://www.lqp.uni-goettingen.de/lqp/papers/98/12/98122300.html It is important to undertand that there is no reason for the scalar product to be positive definite. The only thing that needs to be positive definite are probabilities of events coming from physical measurements. To see how can it be achieved see for intance Ph. Blanchard and A, Jadczyk Found. Phys. 26 (1996) 1669-1681: quant-ph/9610028 We write there: "We extend the ideas of L.P. Horwitz and C. Piron and we propose a relativistic version of Event Enhanced Quantum Theory, with an event generating algorithm for spin one-half particle detectors. The algorithm is based on proper time formulation of the relativistic quantum theory. Although we use indefinite metric, all the probabilities controlling the random process of the detector clicks are non--negative. " Thus it is only ""usually unitarity in quantum theory is defined with respect to a positive definite inner product." Usually, not always. ark From: ark (arkatcassiopaea.com) Subject: Re: unitary reps of Lorentz group Newsgroups: sci.physics.research Date: 2001-11-16 18:48:43 PST On Tue, 13 Nov 2001 07:56:10 +0000 (UTC), Toby Bartels wrote: >>George Jones wrote: >>>As Ark, John Baez, and Pertti Lounesto have pointed out, it is >>>possible to define a Lorentz-invariant scalar product on the space of >>>Dirac spinors, but this scalar product is not positive definite, and >>>so is not a true inner product. >>You mean the norm has to be positive definite (the inner product of >>course can be negative or zero) >What else could "positive definite" mean when speaking of a scalar product? >(The way that you've interpreted is useless, as you essentially note.) >By definition, a bilinear form is _positive_ or _definite_ >(the 2 properties are logically independent, so I separate them) >iff its associated quadratic form is respectively positive or definite. >I can't say with certainty that I've ever seen this before, >but it's how I would naturally use and interpret the terms. One needs to be careful with terminology. Inner product means often the same as "nondegenerate sesquilinear form". We can then specialize: "definite inner product" which means positive or negative definite or "indefinite inner product" which means is positive for some vectors and negative for some other vectors. For instance in "On Noncommutative and semi-Riemannian Geometry" http://xxx.lanl.gov/abs/math-ph/0110001 the Author writes uses explicitly the term "indefinite inner product" (in relation to the Dirac operator). He is also introducing the concept of a Krein space - worth reading. ark
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