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Natural Science Forum / Physics / Research / September 2006If we had a Renormalized Theory of Gravitation, What Broad Features Would we Expect it to Have?
In QED and QCD, is it correct to hold the view that the upshot of renormalization after all the loop diagrams are calculated, cutoffs chosen, etc., is that the interaction couplings "run," and more to the point, that the measurable objects one observes must be specified -- not in the abstract -- but in relation to the probe energy mu used for their measurement? Would it then be fair to suppose that one feature of a renormalized gravitational theory, would a metric tensor g_uv(mu) and an invariant metric interval ds(mu) which must similarly be specified only in relation to the probe energy at which they are measured? Thanks, Jay. _____________________________ Jay R. Yablon Email: jyablon@nycap.rr.com Website: http://home.nycap.rr.com/jry/FermionMass.htm Dear, Jay. First, let me address your post's subject. One does not say that a theory is or is not "renormalized", but rather that it is or is not "renormalizable". The qualifier "renormalized" refers to measurable quantities like mass, charge, etc. This qualifier is used to distinguish them from similar "bare" quantities. As you are learning field theory at the moment, here's a pop quiz: as far as you know, what is the difference between "bare" and "renormalized" quatities? > In QED and QCD, is it correct to hold the view that the upshot of > renormalization after all the loop diagrams are calculated, cutoffs > chosen, etc., is that the interaction couplings "run," and more to the > point, that the measurable objects one observes must be specified -- not > in the abstract -- but in relation to the probe energy mu used for their > measurement? Since you seem to be thinking of "running couplings" a lot, it is best to be very clear about what it is. Another question: as far as you know, what is the definition of a "running coupling constant"? > Would it then be fair to suppose that one feature of a renormalized > gravitational theory, would a metric tensor g_uv(mu) and an invariant > metric interval ds(mu) which must similarly be specified only in > relation to the probe energy at which they are measured? I think that once you are well founded with the notion of what is a "running coupling", you'll be able to answer this question yourself. A last question for the quiz. Thinking about it will help you find the answer you are looking for. Here are some quantities from QED: A - EM vector potential, psi - electron field, alpha - fine structure constant, aka electromagnetic coupling. If you've got a theory of, say, pure gravity, then the metric tensor g is the gravitational field. Which of the listed QED quantities is g most like and least like? Igor > Dear, Jay. > [quoted text clipped - 3 lines] > quantities like mass, charge, etc. This qualifier is used to > distinguish them from similar "bare" quantities. Picky, picky, Igor. ;-) But thank you. I appreciate anything that helps me to communicate more clearly. I changed the subject to reflect this. > As you are learning field theory at the moment, here's a pop quiz: as > far as you know, what is the difference between "bare" and > "renormalized" quatities? OK, Igor, I'm game. The renormalized quantity is the quantity that is observed / measured, and includes all the effects of screening. For example, for an electron, the "bare" "charge" emits photons which annihilate into electron-positron pairs and the pairs polarize around the electron similarly to a dielectric so that when one approaches the electron with a second, "test charge," the strength of the attraction (+ test charge) or repulsion (- test charge) depends on how close the test charge is placed relative to the "bare" charge, i.e., how much of the dielectric polarization gets in the way of the measurement, and how much does not. The closer we get to the bare charge, the stronger the attraction or repulsion grows (in QED anyway, non-Abelian interactions work oppositely because, e.g., gluons themselves carry the pertinent (strong) charge and thus themselves affect the screening while photons do not). Because it takes more energy to bring the test charge close to the bare electron than to keep it further away, we characterize the strength of the observed interaction as a function of probe energy, or equivalently, probe distance. Particle physicists like to think in terms of energy mu, condensed matter physicists like to think in terms of distance. Now, because physics is all about what we can measure in the lab, the bare charge is something of a fiction, because we never really measure that. What we measure is the bare electron plus the sum total of all its screening effects, and that is what I think of as the "renormalized" charge e(mu). Because we can explain what we observe by presuming that "behind the scenes" there "exists" a "bare" charge surrounded by all of this "pair production" and its "polarization," that explanation becomes the theory used to explain what we have observed. The "renormalized" running coupling is then just the square of the charge, a(mu) = e(mu)^2 /4pi, and measures the strength of interaction between two charges, as a function of how close you get them together (condensed matter view) or how much energy you input to get them together (particle view). >> In QED and QCD, is it correct to hold the view that the upshot of >> renormalization after all the loop diagrams are calculated, cutoffs [quoted text clipped - 9 lines] > to be very clear about what it is. Another question: as far as you > know, what is the definition of a "running coupling constant"? Per above, the running coupling a(mu) = e(mu)^2 /4pi, is defined in relation to the running charge. If you add the word "constant" after running coupling, that seems to be an oxymoron because something which "runs" is not a "constant" expect at a fixed probe energy mu. I take this in one of several ways. First, it could be interpreted as an historical throwback to the days before we knew that couplings run and people talked about the "fine structure constant." Second, it could be interpreted as the "theoretical," but not observable, measure of the square/4pi of the "bare" charge e_0. Third, and probably most sensible in terms of what we know today and the primacy of measurement in physics, I would think of "running coupling *constant*" as the observable, asymptotic value of the running coupling. For EM interactions, that asymptotic value ~1/137.036 is observed at low probe; for QCD, it is observed at high probe. But, I would probably myself refrain from using the term "running coupling constant" because of the oxymoron and would try to make clear which of those three views I have in mind. Is there a fourth view I have neglected? >> Would it then be fair to suppose that one feature of a renormalized >> gravitational theory, would a metric tensor g_uv(mu) and an invariant [quoted text clipped - 13 lines] > quantities > is g most like and least like? I would say: g_uv is most like most like A^u, because g_uv = n_uv + kappa h_uv contains the gravitational field h_uv (n_uv is the Minkowski tensor) which in gravitational theory is the "potential" and plays an identical role as the A^u in EM theory. For example, the Laplacian of each is the source of fields, J^u for EM and T^uv for gravitation. Both A^u and h_uv are energy quantities (the ^0 and _00 components in particular), and so, would renormalize in a similar manner to, say, a rest mass. Next, I'd pick psi - electron field, because that is also a field. But, it is further from g_uv because the field quanta of psi are half-integer-spin sources and the field quanta of g_uv are integer-spin-2 interaction mediators just like the integer-spin-1 A^u mediators of EM. Alpha is least like g_uv, because it does not have any field quanta associated with it. It is just a parameter (the square/4pi of the running charge) which runs with energy. > Igor Now, in light of all that, let me pose my underlying question a bit more precisely: Recognizing that a running coupling is least like the metric tensor g_uv, I still have the question: we ordinarily take the metric tensor g_uv to be an "objective" quantity that one does not need to think about as a function of probe energy g_uv(mu). And, we use this to take measurements via the metric interval ds=g_uv dx^u dx^v. But that is because g_uv is used for macroscopic physics where quantum phenomenon are safely neglected. Masses beget gravitational energy which begets more gravitational energy which . . . non-linearly, ad inifinitum, and this begets curvature which is captured in the metric tensor g_uv and its second derivatives R^uv. When we don't probe too deeply, and are using gravitational theory macroscopically as we always do, then the question of penetrating this non-linear screening is a non-issue. It can be ignored. Whatever "bare" gravitational quantities exist behind the scenes, will be fully screened in relation to our observed measurements. What I am thinking about, however, is what would happen if we were to take a high-powered microscope/accelerator and look at the metric really closely, like at a Fermi or sub-Fermi probe energy, where we begin to penetrate some of that gravitational screening energy. As a result, being partly inside the gravitational screen rather than wholly outside, it would seem that the metric itself -- as we observe it -- must change. Thus, it seems that we would then need to consider g_uv(mu) and ds(mu), i.e., we would need to consider each of these to be a function of probe energy because when we get real close, we penetrate some of the gravitational screening, and therefore we have probed past some of the energy which generates the curvature observed from afar, and thereby *changed the observed curvature*. Where the running couplings come in, in my thinking, is insofar as the g_uv *might* be a function, in part, of a running coupling a(mu). In other words, might there be a g_uv(a(mu)), which effectively leads to a g_uv(mu)? Not that a running coupling is in any way *like* the g_uv. But, that if the g_uv should in some circumstance be a function of a(mu), then we have g_uv(mu) and ds(mu) which are defined in relation to how far we have penetrated the gravitational screen. These, in the parlance with which you introduce your reply, would be "renormalized" gravitational objects. Then, a "renormalizable" theory of gravitation would have to be capable of generating and accommodating such objects. Does that make sense? Jay. StrangeRep already pitched in with some excellent observations. I'll add some of my own. > > Dear, Jay. > > [quoted text clipped - 7 lines] > helps me to communicate more clearly. I changed the subject to reflect > this. Picky for a reason. Clarity of language implies clarity of thought. If one cannot yet achieve the latter, one must strive for the former. > > As you are learning field theory at the moment, here's a pop quiz: as > > far as you know, what is the difference between "bare" and > > "renormalized" quatities? > > OK, Igor, I'm game. The renormalized quantity is the quantity that is > observed / measured, and includes all the effects of screening. I'm going to have to stop you right away. The notions of what is renormalized and what is bare are strictly theoretical ones. The quantities that we observe and measure in the lab are independent of any theory that we may use to explain them. [...snip informal explanation of screening...] > The "renormalized" > running coupling is then just the square of the charge, a(mu) = e(mu)^2 > /4pi, and measures the strength of interaction between two charges, as a > function of how close you get them together (condensed matter view) or > how much energy you input to get them together (particle view). This is still not quite right. The crucial step in the renormalization procedure that allows us to talk of bare quantities at all is regularization. Here's a sketch of how it's done. The first step is to write down the continuum theory that we want to analyze. As is, no matter what values for the coupling constants like mass and charge we put in, we will get divergent answers. No good. The next step is to *change the theory*. We introduce a one parameter family of theories which tends to the continuum theory in a limit. What is important is that for any value of the parameter, each theory gives finite answers, but not necessarily so in the limit toward the continuum. There are many types of regularizations. Some that are in common use are lattice regularization, ultra-violet momentum cutoff, dimensional regularization. Each has its advantages and disadvantages. Next, we decide on how to compare theory and experiment. Here, it is necessary to introduce operational definitions for the coupling constants that we want to put into the theory. A description of how the electromagnetic coupling is defined operationally is what's contained in your last quoted paragraph. To be more precise, let the electromagnetic coupling alpha be defined by some kinematical factors multiplying a certain scattering amplitude of two electrons. A good question is: which scattering amplitude? Supposing for a moment that physicists can agree on the geometry of the scattering experiment, there is still the issue of which value, mu, of the center of mass momentum of the incoming electrons should be used. It's possible that this issue is a matter of some contention. So each experimentalist picks his or her value of mu and goes to measure alpha at that energy scale. In the end, we are left with several measurements at different mu. Lets collectively refer to them as Alpha(mu), note the capitalization. The final step is to make our theory agree with experiment. There are two requirements at this stage; (1) we must restore the continuum limit and (2) we must make sure that the calculated agreed-upon scattering amplitude reproduces the measured values of Alpha(mu). Here, some notation will be helpful: L - regularization parameter, (L -> oo gives the continuum limit), a_0 - coupling constant inserted into the regularized theory, alpha'(E,a_0,L) - the calculated value of the agreed-upon scattering implitude, with E denoting the energy scale, and finally alpha(E,mu,A) - renormalized coupling, whose nature I will explain below. Because the continuum limit of the theory is divergent, we find that, keeping E and a_0 fixed, alpha'(E,a_0,L) -> oo as L -> oo. Often the divergence is of the specific form alpha'(E,a_0,L) = a_0 + d(E,L), where d(E,L) -> oo as L -> oo. However, we find that there exists a function c(L) of the regulariztion parameter such that the calculated quantity alpha'(E,a_0+c(L),L) tends to a finite limit as L -> oo. Namely, c(L)+d(E,L) is finite as L -> oo. A famous example of a pair of functions of this form is c(L) = -log(L) and d(E,L) = log(L/E), then c(L)+d(E,L) -> -log(E) as L -> oo. Notice how we changed a_0 -> a_0+c(L). This change means that we added an extra term to the regularized Lagrangian that looks exactly like the electromagnetic coupling term, but with the coefficient c(L). This is what's know as "adding a counter term". Is this function c(L) unique? Unfortunately, no. For instance, we could use c(L) = -log(L/Q) or c(L) = -log(L) + 1/L or many others. Note that we could add any term to c(L) that goes to zero as L -> oo, so lets ignore this freedom. To avoid burdening the notation even further, lets write a_0(L) instead of a_0+c(L) and speak of choosing a_0(L) instead of c(L). What remains is a single free parameter that lets us choose the limit value of alpha'(E,a_0(L),L) through a choice of a_0(L). Notice that we can only choose the value of this limit for a single fixed value of E, say E = mu. The values of the limit for other E will henceforth be dictated to us by the calculation. Incidentally, this a_0(L) is what's known as the "bare coupling constant". And the important point to remember about it is that a bare coupling constant only appears at the regularized stage of the calculation. With that in mind, lets write a_0(L,mu,A) for the choice of bare coupling constant which yields the limit alpha'(mu, a_0(L,mu,A), L) -> A as L -> oo. As a final piece of notation and as the last step toward comparison with experiment, lets define alpha(E,mu,A) = limit alpha'(E, a_0(L,mu,A), L) , L -> oo where have kept E aribtrary compared to the previous equation. At the end of this, somewhat long winded procedure, we end up with a finite quantity associated to the continuum limit of our theory. This quantity, alpha(E,mu,A), is what's usually referred to as "the renormalized coupling". Once it's defined, agreement with experiment is encoded in the equation alpha(E,mu,Alpha(mu)) = Alpha(E). Notice that this is a very strong restriction, by choosing a *single* value of the renormalization scale mu and the *single* value Alpha(mu) that is measured at that scale, we make a prediction for the measured coupling strength at *all* values of E. This prediction is indeed confirmed to a limited extent (see endnote 4 in Chapter 18 of Weinberg's QFT, vol.2). Of course, it is very hard to calculate alpha(E,mu,A) to very high orders in perturbation theory. > > Since you seem to be thinking of "running couplings" a lot, it is best > > to be very clear about what it is. Another question: as far as you [quoted text clipped - 7 lines] > historical throwback to the days before we knew that couplings run and > people talked about the "fine structure constant." Point well taken, however, there's little we can do now that terminology involving the word "constant" is wide spread. > Second, it could be > interpreted as the "theoretical," but not observable, measure of the > square/4pi of the "bare" charge e_0. Hmm, this view should be revised with respect to the above explanation of what is "bare" charge. > Third, and probably most sensible > in terms of what we know today and the primacy of measurement in [quoted text clipped - 5 lines] > oxymoron and would try to make clear which of those three views I have > in mind. Is there a fourth view I have neglected? I guess I really threw you off by carelessly inserting the word "constant" into the question. :-) I should emphasize that the above description is only a sketch of the full procedure, which must be modified slightly to admit more than one bare coupling constant. The dynamical fields that appear in the regularized Lagrangians (referred to as "bare fields") are changed slightly, mostly by an overall scaling transformation. However, this scaling is renormalized by purely kinematic considerations without recourse to experimental data. This last statement can be summarized as follows: couplings run, but dynamical fields don't. That's what I tried to nudge you to think about with the previous question. Which brings me to the last one... > > Here are some quantities from QED: A - EM vector potential, psi - > > electron field, alpha - fine structure constant, aka [quoted text clipped - 10 lines] > particular), and so, would renormalize in a similar manner to, say, a > rest mass. Jumping to conclusions again? And how do you think will rest mass "renormalize"? Have you ever heard to A^u as "running" in QED? > Alpha is least like g_uv, because it does not have any field quanta > associated with it. It is just a parameter (the square/4pi of the > running charge) which runs with energy. Good. Now you just have to notice one more property that makes alpha and g_uv different. As a coupling, alpha may "run" with energy scale, but as a dynamical field, g_uv may not. I hope this finally answers your original question in this thread. Igor > Dear, Jay. > [quoted text clipped - 3 lines] > quantities like mass, charge, etc. This qualifier is used to > distinguish them from similar "bare" quantities. Well let me try this. I am taking QFT right now. We only just started a week ago. From reading the book. havent had any homework on this yet. A renormalized quantity is one that contained a divergence that was canceled out to a finite quantity by some other divergence that was of opposite sign. Renormalizeable theories often contain divergences that are not observeable therefore we do not worry about them. They are just unobserveable mathematical objects. The bare quantity is the computation before the process of renormalizeation. > As you are learning field theory at the moment, here's a pop quiz: as > far as you know, what is the difference between "bare" and [quoted text clipped - 10 lines] > to be very clear about what it is. Another question: as far as you > know, what is the definition of a "running coupling constant"? Let's see. A running coupling constant would have to be one that varies in space-time. Like saying that the permeiablity or permitivity of free space would vary over time. Which they would not. But then the scalar mass of a particle would be able to change inresponse to different chains of events. Such as capture of a neutron or radiation etc. I know this is not a formal definition but is it close. >> Would it then be fair to suppose that one feature of a renormalized >> gravitational theory, would a metric tensor g_uv(mu) and an invariant [quoted text clipped - 13 lines] > > Igor Well the metric tensor in general relativity is just that a tensor of rank 2. So I would rank these in order of most like g_{\mu\nu} as follows. The vector potential A (a tensor of rank 1), the fine structure constant, then the electron coupling field. The electron coupling field is least like the metric g because it is a quantum mechanical entity whiel the others are purely classical ideas. (if the word wrap on this is messed up somebody please tell me.)  Signature"....as long as you don't earn a living at it." A.E www.geocities.com/hontasfx/shortform.pdf www.geocities.com/hontasfx Igor Khavkine posed a pop quiz: >> [...] what is the difference between "bare" and >> "renormalized" quatities? I hesitate to put in my $0.02 here, since Igor knows a lot more QFT than I do. But I'll risk it anyway... Jay Yablon answered: > The renormalized quantity is the quantity > that is observed / measured, and includes all [quoted text clipped - 8 lines] > the test charge is placed relative to the "bare" > charge, [...] Although this type of explanation appears in many textbooks, it's important to understand that it's only an interpretation of the underlying math of renormalization... When one tries to solve an interacting QFT by performing a continuous perturbation around the free theory, one encounters infinities in the integrals. For a restricted class of theories it turns out that if we postulate some extra counter-terms in the Lagrangian to remove the infinities, these terms have the same form as some of the original terms, allowing the infinities to be absorbed into the "constants" of the original theory, and that this remains possible to all orders of perturbation. Such theories are called "renormalizable". In general, the new (renormalized) constants then depend on the energy scale and so are given the adjective "running". For other types of theories, this is not possible, as new parameters must be introduced at each successive order of perturbation. Such theories are called "non-renormalizable" because we cannot extract physical sense from them. The popular notion of virtual particles and screening is thus only an interpretation of the maths that allows us to squeeze sense out of our attempt to perturb around the free theory. > Because we can explain what we observe by > presuming that "behind the scenes" there > "exists" a "bare" charge surrounded by all of > this "pair production" and its "polarization," > that explanation becomes the theory used to > explain what we have observed "That explanation" (based on virtual particles and screening) certainly does not "become the theory". It's only an interpretation of the maths, used for explaining the theory to people who can't yet cope with the maths, or who object (naively) to the fact that the infinities appeared at all. The real theory remains in the gory detail of how renormalization actually works. This is another reason why people keep urging you to get deeper into QFT (pen-in-hand). For gravitation, there were attempts ages ago to quantize it by the canonical formalism. It was found that such a quantized theory of a spin-2 field is non-renormalizable. Since renormalizability of an interacting theory really just indicates whether we can extract physical sense out of a perturbation around the free theory, non-renormalizability just means that we can't solve the interacting theory by such means. OTOH, we know rather little about solving QFTs by other means. So I suspect your original question about "what a renormalizable theory of gravity would look like" is actually an ill-formed question, since we know that the QFT of such a spin-2 field is in fact non-renormalizable. I.e: it cannot be expressed in terms of physically-sensible running couplings. <snip> > For gravitation, there were attempts ages ago to > quantize it by the canonical formalism. It was [quoted text clipped - 8 lines] > such means. OTOH, we know rather little about > solving QFTs by other means. Even though any canonically quantized gravitational perturbation theory is non-renormalizable, do they have anything sensible to say? Or is it complete nonsense at the one-loop level? What prompts my question are Hawkings arguments for Black-Hole radiation, which I only understand at a layman's pictorial level. Even though gravity may may not be canonically renormalizable, can it still be quantized and regulated at some energy level much higher than needed for Hawking's arguments? If so, can such a quantized, regulated (non-renormalizable) theory be coupled with, say, the standard model or some renormalizable particle theory, and generate (at some low order of perturbation theory about the classical Black Hole solution) the Black Hole evaporation effect? My layman's picture of the evaporation effect resembles Feynman diagrams, which is another reason for the question. alan > Even though any canonically quantized gravitational > perturbation theory is non-renormalizable, do they have > anything sensible to say? Or is it complete nonsense at > at the one-loop level? If there are infinities, which cannot be renormalized away into a small number of parameters for all orders, then it's hard to see how any physical sense can be extracted from such a theory. > What prompts my question are Hawkings arguments for > Black-Hole radiation, which I only understand at > a layman's pictorial level. Even though gravity may > may not be canonically renormalizable, can it still be > quantized and regulated at some energy level much higher > than needed for Hawking's arguments? [...] Hawking radiation and Unruh radiation can be derived by considering quantum fields in curved spacetime and non-inertial frames, respectively. You don't need a full-on quantum theory of gravity. I don't know what level of math and QFT you're comfortable with, but a (comparatively) simplified derivation of Unruh radiation can be found in a paper by Alsing and Milonni as quant-ph/0401170. They show, when one transforms a quantum field from an inertial frame to a uniformly-accelerating frame, the accelerated observer will (apparently) perceive himself to be in a thermal bath (i.e: immersed in Unruh radiation). Their derivation just proceeds by taking some basic QFT concepts and applying the transformation. Hawking radiation arises in (sort-of) a related way by considering quantum fields in curved spacetime. The dubious point in all this is that there's no experimental evidence for such effects, and the calculations are done without first constructing a full (general-covariant) QFT. The notions of what "particle" and "vacuum" mean in non-inertial frames are thus controversial: they mean different things to different observers. But sensible physics is only expressed in terms of *invariant* quantities, so maybe all this stuff about Hawking-Unruh radiation is just a reminder that we haven't yet constructed our QFTs comprehensively in terms of general-covariant concepts. > > Even though any canonically quantized gravitational > > perturbation theory is non-renormalizable, do they have [quoted text clipped - 38 lines] > is just a reminder that we haven't yet constructed our QFTs > comprehensively in terms of general-covariant concepts. Shouldn't it be said that QFT in curved space time is fully covariant with respect to the fields, only not with our notions of particle? The field theory is fully covariant. The particle content is not and cannot be. But is that a problem with the theory or is it a problem with our hangup on particles? > Shouldn't it be said that QFT in curved space time is > fully covariant with respect to the fields, only not > with our notions of particle? Probably. (That's one reason I used the word "controversial" earlier.) > The field theory is fully covariant. The particle content > is not and cannot be. But is that a problem with the > theory or is it a problem with our hangup on particles? Indeed. But... first let me ask you... which formulation of QFT in curved spacetime do you have in mind, i.e: which textbook or whatever? > > Shouldn't it be said that QFT in curved space time is > > fully covariant with respect to the fields, only not [quoted text clipped - 10 lines] > QFT in curved spacetime do you have in mind, i.e: which > textbook or whatever? Birrell and Davies > Shouldn't it be said that QFT in curved space time is fully covariant > with respect to the fields, only not with our notions of particle? The > field theory is fully covariant. The particle content is not and cannot > be. But is that a problem with the theory or is it a problem with our > hangup on particles? Indeed, in semiclassical gravity only the notion of particles seems to be problematic. But what about the renormalization of the energy-momentum tensor? We need it to proceed from Hawking radiation to the evaporation of a black hole. And I see no covariant way of renormalization. Ilja > Even though any canonically quantized gravitational perturbation > theory is non-renormalizable, do they have anything sensible to say? > Or is it complete nonsense at the one-loop level? Yes, perturbative gravity does have something sensible to say. Non-renormaliziablity does not imply inutility. It just means that given a finite number of low energy observations, we won't be able to extrapolate our calculations to arbitrarily high-energy processes (which require arbitrarily high orders in perturbation theory). See for instance Cliff Burgess's article: Cliff P. Burgess, "Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory", Living Rev. Relativity 7, (2004), 5. http://www.livingreviews.org/lrr-2004-5 Sticking to the low energy regime (equivalently for spatial separations much larger than the Planck length), in Section 4.1 he gives corrections to the Newtoniam potential arising from quantized gravitational effects (one-loop gravition diagrams). Since, in that case, we can stick to the low-energy approximation, these corrections turn out to be unambiguious (from the renormalization point of view) and require only Newton's constant as input. Moreover, these effects turn out to be extremely tiny. So tiny, that they are unobservable even in the vicinity of the event horizon of large black holes. The latter estimate is not surprising, because for such black holes the curvature is small at the horizon. The real lesson here, and the reason this kind of calculation is expected to be valid, is that whenever we see a classical harmonic oscillator, we're pretty sure we'll find a quantum one, with all the accompanying quantum effects. It is not important that the degrees of freedom that we consider as individual oscillators are collective (say, elastic waves in solids) or effective (some believe that GR and its long wavelength modes are only an approximation to some more complicated theory), their behavior is well understood both classically and quantum mechanically. > What prompts my question are Hawkings arguments for Black-Hole > radiation, which I only understand at a layman's pictorial level. > Even though gravity may may not be canonically renormalizable, can it > still be quantized and regulated at some energy level much higher > than needed for Hawking's arguments? (Non-)renormalizability of gravity has no effect on Hawking's arguments, as he takes gravity as classically while applying quantum treatment to other matter. Even if perturbative gravity were included in a similar analysis, its effects would be extremely tiny, as per the estimation made in Burgess's paper cited above. Hope this helps. Igor >> Even though any canonically quantized gravitational perturbation >> theory is non-renormalizable, do they have anything sensible to say? [quoted text clipped - 12 lines] > Living Rev. Relativity 7, (2004), 5. > http://www.livingreviews.org/lrr-2004-5 Thanks, Igor, it's a fascinating paper, which I am working through slowly. I think I appreciated before that renormalizability + symmetry was a very strong guiding principle. The eye-opener for me with this paper is symmetry plus power counting with a cutoff is almost as strong a principle. If we could only do the measurements and take a few parameters from experiments, then quantum gravity would apparently be just as successful perturbatively as QED. I would assume that these perturbative results are non-controversial. If so, do modern QFT texts show these low loop orders for gravity as examples of how to merge gravity and quantum mechanics in a completely standard way? > Would it then be fair to suppose that one feature of a renormalized > gravitational theory, would a metric tensor g_uv(mu) and an invariant > metric interval ds(mu) which must similarly be specified only in > relation to the probe energy at which they are measured? To answer the question on the subject header: in a theory of quantum general relativity, one would expect (at the very least) a definition for the space of coherent states |g> and (as is generally the case for state-based, Berezin or coherent state quantization), a mapping from the family of classical states { W(g): g metric } to coherent states; W(g) |-> |g><g|, along with a reproducing kernel |g> = integral K(g,g') |g'> Dg' and a formula for the overlap transition probability T(g,g') = |<g|g'>|^2. There is no clear-cut barrier of a dynamic origin that prevents non-zero overlaps for |g> with a coherent state |g'> corresponding to a metric of different signature, a metric for a non-globally hyperbolic spacetime, or even a metric for a Galilean spacetime(!) or degenerate metric. So, no superselection principle <g|g'> = 0 for g, g' in different ones of these subsets, should be expected. The question therefore is focused all the more on handling the issue of quantization in achronal spacetimes and even accommodating the issue of signature-change -- notable weak points, for instance, with LQG. The underlying manifold M parametrizes the theory exactly as the manifold M = (-infinity, intinity) parametrizes the quantum "field" theory for particles (with t being the parameter). One should then expect that the classical property S = integral_{dV}(p^{mu}_a dq^a) ds_{mu} for actions in terms of the boundary state modes (q^a(x): x in dV) should be of prime importance (where V is a compact subset of M and dV its boundary), where q^a represents the components of the field under study within the quantum general relativistic setting. In a coherent state formalism, the parameter |g> in the coherent state represents an compendium of all the boundary modes. It may include other fields, in addition to the metric g. The state is in the Heisenberg picture and essentially provides a representation for an entire history. The expression for the reproducing kernel is therefore really a kind of "sum over histories" formalism, but one which doesn't quite use or require as huge a space as Feynman's "path space", nor requires the likes of Osterwalder-Schroeder rotations. But now, all of this gets is laid down to get to the final feature one should expect to see. The boundary modes are still infinite in number in the classical theory. In the quantum theory, a "3rd quantization" is in effect to implement the Bekenstein Bound, effectively reducing the number of modes to Area(dV)/(Planck Area/4). This is similar to the idea of quantizing cavity modes, only here the role of the "cavity" is played by the volume under consideration V, itself, the modes residing on the boundary dV. sr wrote: > Chris H. Fleming wrote: > [quoted text clipped - 12 lines] > QFT in curved spacetime do you have in mind, i.e: which > textbook or whatever? Why the infatuation with curved spacetime? Metric, affine, and teleparallel gravitations are broadly indistinguishable in prediction and all correspond to observed reality within experimental error. Spacetime torsion is as valid as spacetime curvature. In fact, it's better for having fewer failures and being disjointly testable: 1) Earth's day lengthens by 0.023 msec/year causing lunar recession of 3.84 cm/year. Symmetry of the Einstein curvature tensor and contingent energy-momentum tensor prohibit exchange of spin and orbital angular momenta in General Relativity. General Relativity is empirically wrong. http://en.wikipedia.org/wiki/Einstein-Cartan_theory 2) The patch is Einstein-Cartan theory with an affine connection to lift symmetry. This also creates a chiral pseudoscalar vacuum background. Space is a left foot if the Earth and moon are spin-orbit coupled. Live with it.. 3) Metaphoric left and right shoes of identical composition but inverse geometries will not fit identically into left-footed space. They will fit with different energies. The strongest constraint is Parity Violating Energy Difference experiments that allow more than 10 parts-per-trillion divergence, <http://www.mendcomm.org/pages/goodies/mc0303/pdf/mc1780.pdf> Mendeleev Commun. 13(3) 129 (2003)(pdf) Angew. Chem. Int. Ed. 41(24) 4618 (2003) If identical composition opposite shoes are melted into indistinguishable socks, the latent heat of melting/gram (enthalpy of fusion) must be different given a chiral pseudoscalar vacuum background, http://www.mazepath.com/uncleal/shoes.png For enantiomorphic space groups P3(1)21 and P3(2)21 benzil (mp = 95 C) and 10^(-13) divergence (limit of Eotvos experiment sensitivity) we expect an 8.99 J/g divergence or 8% relative divergence of enthalpy of fusion textbook value. Calorimeters have typical 0.1% precision. Cake walk. http://www.mazepath.com/uncleal/lajos.htm#b4 It is an undergraduate experiment conducted over two days in two differential scanning calorimeters and sensitive to 3x10(-18) Equivalence Principle violation: 300,000 times more sensitive than the best Eotvos balance in 1/45 the run time. http://www.mazepath.com/uncleal/lajos.htm#b2 4) Opposite shoes will vacuum freefall along non-parallel minimum action trajectories. They will violate the EP. Do they? The parity Eotvos experiment opposes single crystal solid spheres of enantiomorphic space groups P3(1)21 and P3(2)21 alpha-quartz. Adelbeger has a wonderfully updated Eot-Wash page, http://www.npl.washington.edu/eotwash/ and he remains staunchly committed to *never* investigating an EP parity violation - the only EP violation consistent with existing theory and the only EP violation *required* by lunar laser ranging measurement of lunar orbital recession. Curious. No composition Equivalence Principle violation experiment of any kind has ever had a quantitiative prediction of divergence. Equivalence Principle parity violation has a prediction of divergence. Unlike composition experiments, EP parity violation is strongly supported by predictive theory, http://arxiv.org/abs/gr-qc/0608090 http://arxiv.org/abs/gr-qc/9309027 String Theory demands the Equivalence Principle. A reproducible EP parity violation falsifies the achiral half of String Theory. How much fun is that? Somebody should look. It's only $100 in consummables, and $80 of that is for a jug of denatured alcohol to recrystallize the benzil. SignatureUncle Al http://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals) http://www.mazepath.com/uncleal/qz3.pdf |
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