#1
Nov406, 03:18 PM

P: n/a

Googling around I recently ended up on a beautiful old spr thread
([1]) about QG, which I found extremely interesting both for its scientific content and for the hopeful mood it conveys (the overall mood in QG is not really hopeful these days, is it? My perception is that it changed quite suddenly around one year ago). The thread also contains a vivid exemplification of the semantic problems arising in GR ([2],cf.[3]), which I like to juxtappose to the following recent Zeilinger ([4]) quote: "Physics will in the future put less emphasis on equations and mathematics but more on verbal understanding". Scary, huh? IV [1] http://groups.google.com/group/sci.p...520204717c753c [2] http://groups.google.com/group/sci.p...a0bb6897aa8ccc "... there might be two U. C. Riversides with clock towers, or U. C. Riversides with dubious entities that we might or might not consider a clock tower." [3] http://groups.google.com/group/sci.p...ece53891b06e6d [4] http://xxx.lanl.gov/abs/quantph/0505187 , quoted by Peter Zoller. 


#2
Nov406, 03:18 PM

P: n/a

I.Vecchi wrote:
> Googling around I recently ended up on a beautiful old spr thread > ([1]) about QG, which I found extremely interesting both for its > scientific content and for the hopeful mood it conveys (the overall > mood in QG is not really hopeful these days, is it? My perception is > that it changed quite suddenly around one year ago). > > The thread also contains a vivid exemplification of the semantic > problems arising in GR ([2],cf.[3]), which I like to juxtappose to the > following recent Zeilinger ([4]) quote: > > "Physics will in the future put less emphasis on equations and > mathematics but more on verbal understanding". > > Scary, huh? > > IV > > [1] > http://groups.google.com/group/sci.p...520204717c753c > [2] > http://groups.google.com/group/sci.p...a0bb6897aa8ccc > "... there might be two U. C. Riversides with clock towers, or U. C. > Riversides with dubious entities that we might or might not consider a > clock tower." > [3] > http://groups.google.com/group/sci.p...ece53891b06e6d > [4] http://xxx.lanl.gov/abs/quantph/0505187 , quoted by Peter Zoller. > Thank you for the fascinating read. It strikes me how the idea of GR spacetime is similar to long forgotten ideas of the electromagnetic ether. In both cases we have some allpenetrating "substance" that affects physical objects, but cannot be detected itself. GRist will tell you that the "spacetime as rubber sheet" analogy popularized in documentaries is an oversimplification, but I have a feeling that this analogy plays a major role in their thinking process. They talk about spacetime being bent, curved, twisted, "foamed", even torn apart; they speak about including spacetime topologies in path integrals as if these were different states of some real object. John Baez talks (though jokingly) about applying Newton's third law to the spacetime (matter must act on spacetime just as spacetime is "acting" on matter). On the other hand, the spacetime idea is conspicuously absent from the vocabulary of quantum mechanics (or QFT). It seems that x and t arguments of quantum fields psi(x,t) have no direct relationship to the measurable time and positions of particles/events. After all, when we calculate something related to experiment, like the Hamiltonian or the Smatrix, these arguments get integrated out. In the Hilbert space of quantum mechanics we do have the observable of particle position R, but it is not distinguished among other observables, like momentum, energy, spin, etc. (Should I mention that there is no time operator in quantum mechanics?) The eigenvalues of R are triples of real numbers, but so are other eigenvalues of mutually commuting operators: momentum, or energy/spinsquared/spinz. There is no more (and no less) sense in combining triples (x,y,z) into some kind of manifold or continuum than in considering "momentum space" or "energyangular momentum space" or any number of other "spaces". Can we introduce the "curvature" of the "momentum space"? It seems to me that quantum mechanics rejects the idea of the preferred status of the position observable, and/or spacetime continuum. I find rather odd the attempts to quantize gravity by including "states" of the GR "rubber sheets" into QM Hilbert space. IMHO, space (or spacetime) is not an observable object, it can't be measured, there are no observables associated with it, so it has no place in the Hilbert space. If there is one lesson we can learn from quantum mechanics this lesson should be: "never ask questions about something you are not measuring" (e.g., whether the electron passed through the left slit or through the right slit?). Since spacetime curvature is something we cannot directly observe, I think our life would be much easier if we just abandon the attempts to describe how the spacetime "looks like", and focus instead on directly measurable observables of real particles or systems of particles. I anticipate your objections about how wonderful are predictions of GR regarding light bending or binary pulsars. However, successful predictions of QM and QFT are much broader in scope and few orders of magnitude more accurate. So, I would place my bet on QM rather than on GR. Eugene. P.S. I have zero credentials in quantum gravity research, so my thoughts can be safely ignored. 


#3
Nov406, 03:18 PM

P: n/a

In article <1131438756.304268.219710@g43g2000cwa.googlegroups.com>,
I.Vecchi <vecchi@weirdtech.com> wrote: >Googling around I recently ended up on a beautiful old spr thread >([1]) about QG, which I found extremely interesting both for its >scientific content and for the hopeful mood it conveys (the overall >mood in QG is not really hopeful these days, is it? My perception is >that it changed quite suddenly around one year ago). I remember the thread you mention: http://groups.google.com/group/sci.p...520204717c753c and it was indeed a lot of fun. It concerned a key conceptual problem of quantum gravity: the problem of describing quantum systems without using a fixed background metric on spacetime. This conceptual problem is largely ignored in string theory, but people working on other approaches to quantum gravity have thought about it for many years: A. Ashtekar and J. Stachel, Editors; conceptual Problems of Quantum Gravity. Proceedings of the 1988 Osgood Hill Conference (Birkhauser, N. Y., 1991), 602 pages. and eventually a good rough understanding was worked out. The problem was then to take this understanding and use it to come up with a specific theory of quantum gravity! All attempts to do this raise difficult "technical" challenges  "technical" in the sense that they involve a bit less verbal reasoning, and a lot more detailed math. Loop quantum gravity was the first theory to meet a bunch of these technical challenges. It did so starting in the late 1980s, and into the mid 1990s. This is perhaps responsible for the optimistic mood you detect. Loop quantum gravity moved on to the next stage: the problem of "getting the right semiclassical limit". This means showing the theory gives results that match those of general relativity at length scales far larger than the Planck scale. Doing this is a really crucial basic test for any theory of quantum gravity. It's preliminary to the really fun part: getting *new* predictions. This is where things bogged down and people became less optimistic: http://math.ucr.edu/home/baez/dynamics/ The story in string theory is very different: they made progress on different questions and became bogged down in different ways. The mood in string theory is also not very optimistic these days. But that's another story. People in modern urban societies tend to be very impatient. They think of 10 or 20 years as a long time. So, they find it upsetting to imagine that it might take a century or more to solve the problem of quantum gravity. I think it's good to relax a bit. The problem of quantum gravity is not going away; the only danger is that we'll destroy ourselves or lose interest in science before we solve this problem. If we manage to keep our society going and stay interested in science, I feel sure we'll make serious progress on quantum gravity  eventually. And even if *we* don't, *someone* probably will: there's lots of room out there. So, when I get myself in a sufficiently highlevel frame of mind, surveying the whole history of the universe and imagining its immensity, I'm very optimistic about the problem of quantum gravity. We (the people reading this now) probably won't be around to see it solved, but at least we can have fun trying. That should be enough. 


#4
Nov406, 03:19 PM

P: n/a

Quantum gravity confusion redux
Eugene Stefanovich wrote:
> If there is one lesson we can learn from quantum mechanics this lesson > should be: "never ask questions about something you are not measuring" > (e.g., whether the electron passed through the left slit or through the > right slit?). Since spacetime curvature is something we cannot directly > observe, I think our life would be much easier if we just abandon the > attempts to describe how the spacetime "looks like", and focus > instead on directly measurable observables of real particles or > systems of particles. > The mass of a particle can be taken as a direct measure of the "curvature of spacetime" due to that particle. Using the tired old rubber sheet analogy the deeper a dent a particle is in the harder it will be to move, it will have more inertia. Mass is a measure of inertia and a curvature tensor is a mathematical expression of inertia. > I anticipate your objections about how wonderful are predictions of > GR regarding light bending or binary pulsars. However, successful > predictions of QM and QFT are much broader in scope and few orders > of magnitude more accurate. So, I would place my bet on QM rather than > on GR. > GR is a classical theory of course it is going to be less precise. There is no argument that QM is more fundamental. That QM is more fundamental is the reason a theory of quantum gravity is so sought after. > Eugene. > > P.S. I have zero credentials in quantum gravity research, so my > thoughts can be safely ignored.  "...we advanced from the telegraph to telephones to email?" Mr ef'n conductor George Carlin. www.geocities.com/hontasfx 


#5
Nov406, 03:19 PM

P: n/a

I.Vecchi skrev: > Googling around I recently ended up on a beautiful old spr thread > ([1]) about QG, which I found extremely interesting both for its > scientific content and for the hopeful mood it conveys (the overall > mood in QG is not really hopeful these days, is it? My perception is > that it changed quite suddenly around one year ago). > My own heretical idea is that you can learn something about QG in 4D by trying to generalize the key aspects of QG in 2D, which are well understood. Actually, 2D QG proper is quite boring, because the Einstein action in 2D is a topological invariant  the Euler characteristic. However, we are really interested in QG coupled to other fields, e.g. the standard model. The Ddimensional free bosonic string in the Polyakov formulation is nothing but 2D gravity coupled to scalar fields, and thus it may be regarded as the simplest possible toy model of gravity. To me, the key lesson from 2D gravity is something that both string theory and LQG has missed: ______________________________________________________    Some gauge components of the metric become physical   after quantization.  ______________________________________________________ In classical 2D gravity, the metric is purely gauge, because it has 3 components and there are 3 gauge symmetries, 2 diffeomorphisms and 1 Weyl rescaling of the metric. But after quantization, the trace of the metric becomes physical. Depending on your formalism, this phenomenon manifests itself in different ways. In lightcone quantization, the physical Hilbert space becomes unexpectedly large, because you cannot factor out the conformal symmetry unless the number of scalar fields D = 26, due to the conformal anomaly. Provided that D < 26, the theory is consistent in spite of the anomaly, because the Hilbert space has a positivedefinite inner product (the spectrum is ghostfree). Alternatively, one can introduce the trace of the metric as a dynamical field already at the classical level; it is then known as the Liouville field. The parameters can be finetuned so that the Liouville field has central charge c = 26  D, cancelling the matter and ghost contributions, and proper conformal invariance is restored. However, the physical Hilbert space is still larger than expected classically, because it now includes the Liouville field. The moral is that a field which is classically a pure gauge becomes physical after quantization, due to a gauge (conformal) anomaly. This shows that, contrary to popular belief, some (not all) gauge anomalies are indeed consistent, and that the quantum theory as a result has more degrees of freedom than its classical limit. In particular, this does happen in 2D gravity, which strongly suggests that it will happen in 4D gravity as well. Gravity is not Weyl invariant except in 2D, so the conformal anomaly does not directly generalize to higher dimensions. However, if you start by gaugefixing Weyl symmetry rather than diffeomorphisms, the conformal anomaly reappears as a diffeomorphism anomaly instead, see e.g. http://www.arxiv.org/abs/hepth/9501016 . This immediately suggests what is missing from all approaches to QG: _________________________________________________________    Just as Weyl/diffeomorphism anomalies make the trace   of the metric physical in 2D gravity, diffeomorphism   anomalies make gauge components of the metric (beyond   the two graviton polarizations) physical in 4D gravity.  _________________________________________________________  In order to implement this program, I discovered how to generalize the Virasoro algebra beyond 1D and how to build quantum representations. Not alone, but as the only one who contributed to this discovery with a motivation from physics. Note that the existence of a multidimensional Virasoro algebra is not a priori obvious, in view of two nogo theorems: 1. The diffeomorphism algebra has no central extension except in 1D. 2. In field theory, there are no gravitational anomalies in 4D. However, it turns out that these theorems, although correct, depend on unnecessarily strong assumptions. The Virasoro extension is not central (commutes with everything) except in 1D, and one has to go slightly beyond field theory proper, by introducing and quantizing the observer's trajectory, in order to formulate the extension. Of course, the discovery of the multidimensional Virasoro algebra is by itself not enough to quantize gravity, just as the discovery of the usual Virasoro algebra was not enough to quantize the free string. But it is a quite nontrivial, and IMO crucial, step. Besides, it is undoubtedly the discovery of my lifetime. 2D gravity permits an unrelated but interesting observation. The Liouville field can be viewed as a Goldstone boson for the conformal symmetry. So what in lightcone quantization looks like anomalous symmetry breaking, looks like spontanous symmetry breaking in Liouville theory. Without having thought too much about it, I think that this suggests an interesting idea: _______________________________________________________    Anomalous and spontanous breaking of gauge symmetries   are closely related, perhaps different sides of the   same coin.  _______________________________________________________ In particular, the algebra of gauge transformations in 4D admits an extension which makes it into a higher dimensional analog of affine KacMoody algebra. This is a kind of gauge anomaly, but not a conventional one because it is proportional to the second rather than to the third Casimir operator. It is tempting to speculate that the symmetry breaking induced by these gauge anomalies is related to the Higgs effect, much like conformal anomalies are related to the Liouville field. Finally, let me point out that my key observation, that infinitedimensional constraint algebras generically acquire anomalies at the quantum level, has passed the most stringent test conceivable: it was repeated approvingly by Lubos Motl at http://motls.blogspot.com/2005/09/wh...nsteinii.html Alas, I suspect that Motl didn't realize from whom the formulation originated, see my comment at http://www.haloscan.com/comments/lum...0241393188648/ 


#6
Nov406, 03:19 PM

P: n/a

Eugene Stefanovich ha scritto:
> Thank you for the fascinating read. Well, its authors are a remarkable lot. Some of them still post here, some unfortunately don't do it anymore. > ... It strikes me how the > idea of GR spacetime is similar to long forgotten ideas of the > electromagnetic ether. In both cases we have some allpenetrating > "substance" that affects physical objects, but cannot be detected > itself. Other people seem to share your insight and beef it up with historical references ([1], #4). > GRist will tell you that the "spacetime as rubber sheet" analogy > popularized in documentaries is an oversimplification, but I have > a feeling that this analogy plays a major role in their thinking > process. They talk about spacetime being bent, curved, twisted, > "foamed", even torn apart; they speak about including spacetime > topologies in path integrals as if these were different states of some > real object. John Baez talks (though jokingly) about applying > Newton's third law to the spacetime (matter must act on spacetime just > as spacetime is "acting" on matter). > > On the other hand, the spacetime idea is conspicuously absent from the > vocabulary of quantum mechanics (or QFT). It seems that x and t > arguments of quantum fields psi(x,t) have no direct relationship to > the measurable time and positions of particles/events. After all, > when we calculate something related to experiment, like the Hamiltonian > or the Smatrix, these arguments get integrated out. In the Hilbert > space of quantum > mechanics we do have the observable of particle position R, > but it is not > distinguished among other observables, like momentum, energy, spin, etc. > (Should I mention that there is no time operator in quantum mechanics?) I would see that rather as a limitation of current QM, confronting us with the key question "what is a clock?" (cf. the references in [2]). ... > > It seems to me that quantum mechanics rejects the idea of the > preferred status of the position observable, and/or spacetime continuum. > I find rather odd the attempts to quantize gravity by including "states= " > of the GR "rubber sheets" into QM Hilbert space. IMHO, > space (or spacetime) > is not an observable object, it can't be measured, there are no > observables associated with it, so it has no place in the Hilbert space. Formulating a viable measurement theory for spacetime is indeed crucial. As Wigner writes under the heading "Quantum Limitations of the Concepts of General Relativity([3]), "In relativity theory, the state is described by a metric which consists of a network of points in spacetime, that is, a network of events, and the distances between these events. If we wish to translate these general statements into something concrete, we must decide what events are, and how we measure the distance between events" I think the last sentence is basic to any QG theory. In [3] Wigner then builds a relevant model, first noting that "the establishment of of a close network of points in spacetime requires a reasonable energy density, a dense forest of world lines wherever the network is to be established" and noting further that "it is desirable ... to reduce all measurements in spacetime by to measurements by clocks". Then "the simplest framework in spacetime ... is a set of clocks which ... tick off periods and these ticks form the network of events which we wanted to establish". Wigner then goes on to investigate the properties of such a network and the constraints that physical clocks impose on the theory, analysing the relationship between the mass of the clock and its accuracy and concludes that "the essentially nonmicroscopic nature of the general relativistic concepts seems ... inescapable" . Wigner then draws a parallel between the situation in GR and in QFT and writes that in the latter "if we analyse the way in which we 'get away' with the use of an absolute space concept , we simply find that we do not. In our experiments we surround the microscopic objects with a very macroscopic framework and observe *coincidences* between the particles emanating from the microscopic system and parts of the framework. ... There is therefore a boundary in our system ..." , indeed the infamous boundary. A similar line of thought glimmers in a very interesting post by Peter Peldan ([4]) about diffeormorphism invariance in QG, in the thread we are discussing. There was no reply, but it's never too late. ... > I have zero credentials in quantum gravity research, so my > thoughts can be safely ignored. ... or daringly heeded. Audaces fortuna iuvat. Cheers, IV [1] Dosch, Müller and Sieroka "Quantum Field Theory, its Concepts Viewed from a Semiotic Perspective" at http://philsciarchive.pitt.edu/archive/00001624 . [2] http://groups.google.com/group/sci.p...e33639e73f3acb [3] E. Wigner "Relativistic Invariance and Quantum Phenomena" Rev. Mod. Phys. 29, 255 (1957) [4] http://groups.google.com/group/sci.p...ddaf9b47103c1c  "It is the free in symbols acting spirit which constructs itself in physics a frame to which he refers the manifold of phenomena. He does not need for that imported means like space and time and particles of substance; he takes everything from himself." Hermann Weyl "Wissenschaft als symbolische Konstruktion der Menschen" 1949, quoted in [1], those were the days when people like Weyl confronted their ideas with those of people like Cassirer and Panovsky. The Hoelderlinian German original is also available at [1]. 


#7
Nov406, 03:19 PM

P: n/a

"Eugene Stefanovich" <eugenev@synopsys.com> schrieb > Thank you for the fascinating read. It strikes me how the > idea of GR spacetime is similar to long forgotten ideas of the > electromagnetic ether. In both cases we have some allpenetrating > "substance" that affects physical objects, but cannot be detected > itself. Moreover, it is quite easy to find an ether interpretation of the metric. All you need are harmonic coordinates and a choice of a timelike time coordinate between them. g^00 sqrt(g) = rho > 0 g^0i sqrt(g) = rho v^i g^ij sqrt(g) = rho v^i v^j  p^ij where rho is the ether density, v^i its velocity and p^ij its pressure/stress tensor. The harmonic condition becomes the continuity and Euler equation. To obtain the harmonic equation as a physical (EulerLagrange) equation we have to add two cosmological terms to GR. L = L_GR + (Y g^00 + X g^ii) sqrt(g) Even more, this Lagrangian may be derived from simple axioms, where the key axiom is that continuity and Euler equations appear as the Noether conservation laws in a special form of Noether's theorem. See grqc/0205035 for details. Thus, the Lagrangian and therefore the relativistic symmetry (EEP) may be explained. The unexplained "conspiracy" of the Lorentz ether no longer exists. > After all, > when we calculate something related to experiment, like the Hamiltonian > or the Smatrix, these arguments get integrated out. In the Hilbert > space of quantum > mechanics we do have the observable of particle position R, > but it is not > distinguished among other observables, like momentum, energy, spin, etc. > (Should I mention that there is no time operator in quantum mechanics?) A strong hint that the nonmeasurability of absolute time is not a strong argument against absolute time. > Since spacetime curvature is something we cannot directly > observe, At least in principle spacetime curvature is easily observable. We measure the radius and circumsphere of a circle, and if u = 2 pi r does not hold we have measured curvature. But we have to remember that "curvature" is just a mathematical word. "Curvature" may appear on very flat things, if we measure them with distorted rulers. Ilja 


#8
Nov406, 03:19 PM

P: n/a

"Ilja Schmelzer" <Ilja.Schmelzer@FernUniHagen.de> wrote in message
news:dl9k03$onq$1@tamarack.fernunihagen.de... > > Since spacetime curvature is something we cannot directly > > observe, > > At least in principle spacetime curvature is easily observable. > We measure the radius and circumsphere of a circle, and > if u = 2 pi r does not hold we have measured curvature. My point was that the "circle" we are measuring is made of some "real stuff", i.e., particles interacting (or not interacting) with each other. Quantum mechanically, our measurements refer to expectation values of the position operators of these particles in the Hilbert space. So, in order to describe these measurements you just need the Hilbert space of the system, the operators of observables (e.g., position) of particles belonging to the system, and the unitary representation of the Poincare group in the Hilbert space which expresses the properties of invariance for measurements made from different reference frames. This is sufficient for a complete description of the system (circle) and all possible measurements that we can make there, including the circumferencetoradius ratio. In this description, the spacetime continuum does not play any role at all. If we found, for example, that the ratio u/r is different from 2 pi, I would rather attribute it to some pecularities of interparticle interactions in the circle, than to "curvature" of some mysterious spacetime that evades direct observation. Eugene. 


#9
Nov406, 03:19 PM

P: n/a

Eugene Stefanovich wrote:
> "Ilja Schmelzer" <Ilja.Schmelzer@FernUniHagen.de> wrote in message > news:dl9k03$onq$1@tamarack.fernunihagen.de... > > At least in principle spacetime curvature is easily observable. > > We measure the radius and circumsphere of a circle, and > > if u = 2 pi r does not hold we have measured curvature. > > My point was that the "circle" we are measuring is made of some "real > stuff", i.e., particles interacting (or not interacting) with each > other. Quantum mechanically, our measurements refer to expectation > values of the position operators of these particles in the Hilbert > space. So, in order to describe these measurements you just need the > Hilbert space of the system, the operators of observables (e.g., > position) of particles belonging to the system, and the unitary > representation of the Poincare group in the Hilbert space which > expresses the properties of invariance for measurements made from > different reference frames. Spacetime is not Poincare symmetric. That is what our telescopes tell us. That is what our satelites tell us. That is what the cosmic microwave background tells us. > This is sufficient for a complete description of the system (circle) and > all possible measurements that we can make there, including the > circumferencetoradius ratio. In this description, the spacetime > continuum does not play any role at all. If we found, for example, that > the ratio u/r is different from 2 pi, I would rather attribute it to > some pecularities of interparticle interactions in the circle, than to > "curvature" of some mysterious spacetime that evades direct observation. You are free to choose an interpretation of experimental data, and so is everyone else. The tough part is predicting what the outcome of the next experiment will be, where the circle will be moved, tilted, rotated, or the particle will go faster, slower, be of different species, etc. Success in theoretical physics is not measured by interpretation. Igor 


#10
Nov406, 03:19 PM

P: n/a

"Eugene Stefanovich" <eugene_stefanovich@usa.net> schrieb
> "Ilja Schmelzer" <Ilja.Schmelzer@FernUniHagen.de> wrote > > > Since spacetime curvature is something we cannot directly > > > observe, > > > > At least in principle spacetime curvature is easily observable. > > We measure the radius and circumsphere of a circle, and > > if u = 2 pi r does not hold we have measured curvature. > > My point was that the "circle" we are measuring is made of some "real > stuff", i.e., particles interacting (or not interacting) with each > other. Quantum mechanically, our measurements refer to expectation > values of the position operators of these particles in the Hilbert > space. My point was about classical gravity. What is observable in quantum gravity is, of course, a quite different question. On the other hand, the correspondence principle requires that the classical measurement I have described appears to be the limit of some quantum measurement. > If we found, for example, that > the ratio u/r is different from 2 pi, I would rather attribute it to > some pecularities of interparticle interactions in the circle, than to > "curvature" of some mysterious spacetime that evades direct observation. I don't like the mystery related with words like "curved spacetime" too. It suggests not only some embedding in some higherdimensional space but also that the current measurement of distance and duration has fundamental, almost philosophical character. Therefore I fully agree that we can attribute with u/r =!= 2pi to something much less mysterious. In the ether interpretation I have described in my last posting spatial curvature is the same effect we know as "inner stress" in condensed matter theory. No mystery at all. (My only objection was that the thing which was given the mysterious name "spacetime curvature" is, nonetheless, something we can measure, as far as we have rulers and clocks.) Ilja 


#11
Nov406, 03:19 PM

P: n/a

Ilja Schmelzer wrote: >>My point was that the "circle" we are measuring is made of some "real >>stuff", i.e., particles interacting (or not interacting) with each >>other. Quantum mechanically, our measurements refer to expectation >>values of the position operators of these particles in the Hilbert >>space. > > > My point was about classical gravity. > > What is observable in quantum gravity is, of course, a quite different > question. On the other hand, the correspondence principle requires > that the classical measurement I have described appears to be the > limit of some quantum measurement. What I said is applicable to the classical case too. In order to translate, you should change "Hilbert space" > "phase space", "operator" > "real function on the phase space" "state vector" > "point in the phase space" "expectation value" > "value of the function at a point" Eugene. 


#12
Nov406, 03:20 PM

P: n/a

Igor Khavkine wrote:
> Spacetime is not Poincare symmetric. That is what our telescopes tell > us. That is what our satelites tell us. That is what the cosmic > microwave background tells us. These observations tell us that 1) there is gravity and 2) the force of gravity is different from the Newtonian 1/r^2. These two things you can take to the bank. Regarding "curved spacetime"... This hypothesis worked pretty well so far, but it also has a few defects. One of them is the fundamental incompatibility with the laws of quantum mechanics. Even if it will be definitively proven that gravity is a spacetime curvature (which I doubt), the Poincare group will be here to stay with us. If you consider an isolated system in empty space, i.e. two interacting masses (instead of a mass in an external field), this system is clearly invariant with respect to Poincare transformations: space and time translations, rotations, and boosts of the system as a whole. Eugene. 


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