Notes on "quantum gravity" and non-commutative geometry
NNotes on “quantum gravity” andnoncommutative geometry
Jos´e M. Gracia-Bond´ıa
Abstract
I hesitated for a long time before giving shape to these notes, originallyintended for preliminary reading by the attendees to the Summer School “New pathstowards quantum gravity” (Holbaek Bay, Denmark, May 2008). At the end, I decideagainst just selling my mathematical wares, and for a survey, necessarily very se-lective, but taking a global phenomenological approach to its subject matter. Afterall, noncommutative geometry does not purport yet to solve the riddle of quantumgravity; it is more of an insurance policy against the probable failure of the otherapproaches. The plan is as follows: the introduction invites students to the fruitfuldoubts and conundrums besetting the application of even classical gravity. Next, thefirst experiments detecting quantum gravitational states inoculate us a healthy doseof scepticism on some of the current ideologies. In Section 3 we look at the actionfor general relativity as a consequence of gauge theory for quantum tensor fields.Section 4 briefly deals with the unimodular variants. Section 5 arrives at noncom-mutative geometry. I am convinced that, if this is to play a role in quantum gravity,commutative and noncommutative manifolds must be treated on the same footing;which justifies the place granted to the reconstruction theorem. Together with Sec-tion 3, this part constitutes the main body of the notes. Only very summarily atthe end of this section we point to some approaches to gravity within the noncom-mutative realm. The last section delivers a last dose of scepticism. My efforts willhave been rewarded if someone from the young generation learns to mistrust currentmindsets.
Departamento de F´ısica Te´orica, Universidad de Zaragoza, Zaragoza 50009, SpainDepartamento de F´ısica, Universidad de Costa Rica, San Pedro 2060, Costa Rica 1 a r X i v : . [ h e p - t h ] M a y Jos´e M. Gracia-Bond´ıa “Quantum gravity” denotes a problem, not a theory. There is no theory of quantumgravity. There exist several competing schemes, as mathematically sophisticated andfecund, as a rule, as undeveloped in the face of experimental evidence and of thepurported aim of unifying gravity with other fundamental interactions.My account of the subject is unabashedly low-road . The concept was coined byGlashow in his thought-provoking book [1]. The low road: . . . is the path from the laboratory to the blackboard, from experiment to theory, from hard-won empirical observations to the mathematical framework in which they are described,explained and ultimately understood. This is the traditional path that science has so suc-cessfully followed since the Renaissance. . . In each of these cases, scientists built theirtheories upon a scaffold of experimental data. The Standard Model could not have beeninvented by theorists, however brilliant, just sitting around and thinking.Sometimes scientists have followed a different road. The high road tries to avoid themorass of mundane experimental data. . .
Glashow goes on portraying the invention by Einstein of classical general rela-tivity as the single example of successful pursuit of the high road; and exemplifyingmodern high-roaders with superstring theorists.However, we ought to say, string theory in general is a very reasonable bet com-pared with most “quantum gravity” schemes. What motivates them? From a text-book [2, p. 24] we quote Bergmann:
Today’s theoretical physics is largely built on two giant conceptual structures: quantum the-ory and general relativity. As the former governs primarily the atomic and subatomic worlds,whereas the latter’s principal applications so far have been in astronomy and cosmology, ourfailure to harmonize quanta and gravitation has not yet stifled progress in either front. Nev-ertheless, the possibility that there might be some deep dissonance has caused physicists anesthetic unease, and it has caused a number of people to explore avenues that might lead toa quantum theory of gravitation, no matter how many decades away the observations. . .
Dissonance, we claim, there is not: trees electromagnetically keep growing onthe third planet from the Sun, bound by gravity since as far as we can tell. Thereis theoretical ignorance about a vast region of possible experience unconstrainedby evidence. Be that as it may, “esthetic unease” is about the worst guide for sci-ence. Ugliness is in the eye of the beholder. Nobody claims the standard model ofparticle physics to be beautiful. However, it has survived more than 35 years of de-termined theoretical and —much more important— empirical assault. It possessesnow the beauty of staying power: any scheme whatsoever aiming to replace it needsto manage the Standard Model disguise.History is a better guide. The clash between classical mechanics and electromag-netism, seemingly leading to catastrophic atomic collapse, was overcome by moreprofound experiments and the quantum theories designed to explain them. There-fore we do little of the “dissonance” of the underpinnings of quantum theory andclassical gravity, since in all likelihood at least one of those is doomed to perish. otes on “quantum gravity” and noncommutative geometry 3
Glashow concludes:
History is on our side (i.e., of the low-roaders). Every few years there has been a world-shaking new discovery in fundamental physics or cosmology. . . Can anyone really believethat nature’s bag of tricks has run out? Have we finally reached the point where there is nolonger. . . a bewildering new phenomenon to observe? Of course not.
Fortunately, even classical gravity is in deep crisis. This opens a number of op-portunities. The crisis concerns almost every aspect. • Cosmic acceleration. In a nutshell, the expansion of the universe seems to be accelerating when it should be braking . This is the “cosmological constant” or“dark energy” problem. The question is obviously: why now? We shall comeback to this. • Galaxy clustering and cosmology. As it turns out, some think the previous to bea pseudo-problem. Wiltshire and coworkers [3, 4, 5, 6] have argued that:
Cosmic acceleration can be understood as an apparent effect, and dark energy as amisidentification of those aspects of cosmological gravitational energy that by virtueof the strong equivalence principle cannot be localized. . .
Wiltshire’s proposal is of the “radically conservative” kind. The implication isthat we truly do not know how to solve the Einstein equations.In a similar vein, current orthodoxy regarding gravitational collapse towardsblack holes and the “information loss” problem has been also called into ques-tion [7]. • The best-tested aspects of the theory are challenged by the Solar System anoma-lies. To begin with, at least since the eighties it has been known that the trajecto-ries of the
Pioneer
10 and
Pioneer
11 past the outer planets’ orbits deviate fromthe predictions, as though some extra force is tugging at them from the directionof the Sun [8, 9, 10].The unmodeled blue shift appearing in the
Pioneer missions data amounts to10 − cm s − ; it may not seem much, but it adds now to many thousands of kilo-metres behind the projected paths. A “covariant” solution to the anomaly seemsruled out —see for instance [11]. In desperation, some bold proposals are be-ing made. For instance that, because of the influence of background gravitationalsources in the universe on the evolving quantum vacuum [12, 13], astronomicaltime and time as nowadays measured by atomic clocks might not coincide. • To this, add the even more surprising and now apparently verified fact (spokenabout in hushed ones since 1990, when first noticed in the flight of probe
Galileo by Earth), that the slingshot manoeuvre of spacecraft delivers (or takes away)more energy than the current theory allow us to expect [14]. A simple empiricalformula describes rather accurately the deviations, which translate into a fewmillimitres a second of extra velocity.Both solar system anomalies belong in the category of “unexpected experi-ments”. • The existence of (non-baryonic) dark matter is better established than that of darkenergy, since several lines of evidence point to a relatively low baryon content ofthe universe.
Jos´e M. Gracia-Bond´ıa
However, models do exist that attribute the relatively high acceleration of stars ina typical galaxy, thus the appearance of dark matter, to mysterious deviationsfrom standard gravity. Particularly, Milgrom’s MOND (modified Newton dy-namics) model —see [15] and references therein, as well as the discussion inthe popularization book [16]. MOND postulates that Newton’s law is modifiedin very weak acceleration regimes. There is no “respectable” theory behind itas yet. However, as it happens, Milgrom’s hypothesis implies predictions on thesurface densities of galaxies and more; these have been pretty much verified tillnow. The Milgrom acceleration is pretty close to the cosmic acceleration. It isnot very different in order of magnitude from the “acceleration” of the
Pioneers .On the other hand, interacion with dark matter might explain the Pioneers’ blueshift. • Taken together, dark matter and energy signal the transition to a new cosmologi-cal paradigm. Whether they will emerge as modified gravity (massive graviton orother), new energy components, or pointers to strings and other noncommutativesubstructures, remains to be seen. • Among the questions of principle that periodically erupt into controversy, is thequestion of the speed of transmission of the gravitational interaction, or, if youwish, the lack of aberration of gravity [17].
Perhaps the most fundamental question of principle, for our purposes, concerns therole, if any, of the principle of equivalence in the interface of gravity with the quan-tum world. We begin by that in earnest. Now, there is little in the way of quantumgravity that we can probe in laboratory benches at present. The universe was cre-ated with a quarantine: gravity is so weak an interaction that it can only producemeasurable effects in the presence of big masses, and this very fact militates againstdetecting radiative corrections to it. To see quantum effects in pure gravity is farbeyond our power. What we can do with some confidence is to envisage quantumsystems in classical background gravitational fields, with back-reaction neglected,or approximately treated. In fact, only the interface of nonrelativistic quantum me-chanics with Newtonian gravity has been experimentally tested.Some wisdom is gained, however, by not discarding a priori such humble be-ginnings. For this writer, the alpha of quantum gravity is the Colella–Overhauser–Werner (COW for short, from now on) experiment [18]. It tests the equivalenceprinciple. The latter appears in textbooks in slightly different formulations. Forsome, the “strong” principle says that accelerative and gravitational effects are lo-cally equivalent; the “weak” principle states that inertial masses and gravitationalcharges are the same (up to a universal constant). Some others use the nomenclaturethe other way around. In both cases we refer to systems placed in external fields,such that the complicating effects of the gravitational pull by the system itself canbe neglected. From the second form it plainly follows that all classical masses fall otes on “quantum gravity” and noncommutative geometry 5
Fig. 1 (a) In the most com-mon interferometer three“ears” are cut from a perfectcrystal, ensuring coherenceover it (about 10 cm long).The incident beam is split(by Bragg scattering) at A into two, I and II. These areredirected at B and C andrecombine in the last ear.The relative phase at D de-termines the counting rate atthe detectors. (b) Top viewof the interferometer. Therelative phase can be changedin a known way by insert-ing a wedge in one beamat E , which thickness can bechanged by displacement. Theexperiment is performed at F . with the same acceleration in a gravity field. Thus, if the initial conditions for thosemasses coincide, their trajectories will coincide as well: Galielo’s uniqueness offree fall. In other words, mass is superfluous to describe particle motions in classi-cal gravity; it all belongs to the realm of kinematics. From this to the assertion [19,p. 334] that . . . geometry and gravitation were one and the same thing. is there but a near-vanishing step.So, what does the COW experiment mean for humanity? It and its follow-upslend support to the equivalence principle. It would have been earth-shaking if theydid not; but it is indispensable to reflect on which aspects of current orthodoxy areconfirmed, and which ones actually disproved by it.The COW tool is neutron (and neutral atom) interferometry. A typical neutroninterferometer —see Fig. 1, taken from [20]— is a silicon crystal of length L .The incident beam is split with half-angle θ in the first ear of the apparatus atone extreme, redirected halfway through it, and recombines in the third ear atthe other extreme. The neutron wavelength λ N and the atom spacing in the crys-tal need to be of the same order, about 10 − cm. Thus the momentum is in theballpark of ( ¯ h / λ N ) ∼ − erg. The neutron is relatively cold: with an inertialmass m i ∼ − g, this implies a velocity v ∼ cm/s; thus a nonrelativistic cal-culation will do.A gravitational phase shift is obtained simply by rotating the apparatus about theincident beam, say an angle α , so the acceleration is g sin α , with g the standardacceleration on Earth. The phase shift over one period is of the order of the quotientbetween the (difference in) potential energy and the kinetic energy of the beam;even with the small velocities involved, this is of the order ∼ − . Under suchconditions, it is not hard to see that the phase difference is given approximately by Jos´e M. Gracia-Bond´ıa
Fig. 2
Gravitational perturbation of the beam. (a) The interferometer is rotated around the incidentbeam by an angle α ; the beams will be at a different height (equal to 2 x sin θ between equivalentpoints along the paths), with an effective gravitational field g α = g sin α in the interferometer plane.(b) In the free-fall system, the neutrons beam are unaccelerated, but the interferometer scatteringplanes appear to be accelerating upwards. (cid:82) V dt ¯ h , where V denotes the difference in potential between the higher and the lower unper-turbed neutron paths and t is the time.Now, let x be a rectilinear coordinate along the long diagonal of the rhomb con-stituted by the two beam’s paths. Then the difference of height between the paths isas indicated in Fig. 2. The difference in potential is 2 mg sin α x sin θ . Thus we have: (cid:82) V dt ¯ h = mg sin α sin θ ¯ hv cos θ (cid:90) L x dx = mgA sin α ¯ hv , (1)with v the mean velocity of the neutrons and A the area of the rhomb, given by halfthe diagonals’ product: A = L tan θ . Actually the mass appearing in (1) is the gravitational charge; the inertial mass m i is hidden in the relation between v and the de Broglie wavelength. The shift (1) isaround 100 rad, and the resulting fringe pattern easily visible and measurable. (Wehave neglected the effect of the Earth’s rotation, which amounts to less of 2% ofthe total shift.) It turned out that the neutrons do fall in the Earth’s gravity field aspredicted by the Schr¨odinger equation, with m and m i identified.The experiment appears to confirm both versions of the equivalence principle,since the possibility of describing the problem in the neutron beam reference systemas an upward acceleration of the interferometer holds in the Schr¨odinger equation.This is discussed exhaustively in [21]. Use of the Dirac equation instead makes no otes on “quantum gravity” and noncommutative geometry 7 practical difference. Anyway, the experiment was repeated in “actually accelerated”interferometers, with the expected result [22].However, as soon as we try to translate the “weak” principle in geometrical termsin the quantum context, we run into trouble. The fact that “trajectories” have notmuch quantum-mechanical meaning is enough to make us suspicious. Nevertheless,let us for simplicity explore the situation in terms of circular Bohr orbits. (That theseare still pertinent concepts is plain to anybody who has done atomic physics with theWigner phase-space function [23, 24].) Assume a very large mass M bounds a smallone m gravitationally into a Bohr atom. For circular orbits with angular velocity ω ,Kepler’s laws give ω = GMr , with r restricted by mr ω = n ¯ h . Thus E n = − m ω r = − G M m h n . Therefore in quantum mechanics one can tell the mass of a gravitational boundparticle. The explanation for this lies in the very quantization rule [ x , p ] = i ¯ h , which is formulated in phase space. If we define velocity by p / m , we obtain thecommutator [ x , v ] = i ¯ h / m . This means that kinematical quantities are functions of ¯ h / m . In general, it is enoughto look at the Schr¨odinger equation to see that energy eigenvalues go like m f ( ¯ h / m ) ,or more accurately, m f ( ¯ h / mm i ) for some function f .Now, if we admit the previous, how does the dependence of the mass disappearin the classical limit? The only possibility is that the quantum number scales with m .This of course makes sense in the semiclassical limit: if particle 1 is heavier thanparticle 2, we expect its energy levels to be accordingly higher. But for low-lyingstates geometrical equivalence inevitably breaks down. We have here the curiouscase of a symmetry generated (rather than broken) by “dequantization”. The pointwas made in [20].In summary, lofty gravity is treated by quantum mechanics as lightly as lowlyelectrodynamics. In the classical motion of charged particles, only the parameter e / m appears. This is not interpreted geometrically, since e / m varies from systemto system, so nobody thinks it has fundamental significance. When the system isquantized, ¯ h comes along in both cases, and in gravity experiments, like the onesdescribed above with states in the continuum, we can tell the mass. Alas, for somethis destroys the beauty of the theory. So much that they never mention the fact. Jos´e M. Gracia-Bond´ıa
Before examining the consequences of the failure of the geometrical principle, letus see if we can find a way out. To preserve weak equivalence as an exact quantumsymmetry, we must take the canonical velocity as a dynamical quantity v . Then theHamiltonian is rewritten H = m ( v / + V ( x )) = m H ( x , v ) , with V the gravitational potential. If now we quantize the theory in terms of x and v ,we obtain a “quantum gravity” theory respecting the geometrical equivalence prin-ciple (although, of course, this flies in the face of the workings of ordinary quanti-zation for other interactions).Through existence of the constant c of nature, such a quantization method in-volves the introduction of a fundamental length [ x , v ] = icl . This is not quite “noncommutative geometry” in the superficial way it is mostlypractised nowadays (the present author is not innocent of such a sin), but resemblesit more than a bit. The point we are able to make is twofold: (i) of need the geomet-rical approach to quantum gravity will be noncommutative or will not be; (ii) it isnot at all required that l be of the order of Planck’s length scale. It has been arguedmany times, invoking mini-black holes in relation with the incertitude principle andsuch, that something must happen at that length scale —see [25] for example. Butnothing forbids that the critical length be bigger (a string length, for instance), pro-vided it could have escaped detection so far. If and how such fundamental lengthintervenes is a matter only for experiment to decide.We return to noncommutative geometry in Section 5. The understanding that geometry and gravitation are not to be one and the samething should be confirmed by some experiment checking (low-lying) states of aquantum system bound by gravity.Such an experiment —the first ever to observe gravitational quanta— has alreadytaken place [26].Ultracold neutrons ( v ∼
10 m/s) are stored in a horizontal vacuum chamber; amirror is placed below and a non-specular scatterer above. Thus the neutrons findthemselves in a sort of gravitational potential well, with a “soft wall” on one side.The Bohr–Sommerfeld formula is good enough to calculate its energy levels asso-ciated to vertical motion: otes on “quantum gravity” and noncommutative geometry 9
Fig. 3
Quantum states are formed in the “potential well” between the Earth’s gravity field and thehorizontal mirror on bottom. The vertical axis z is intended to give an idea about the spatial scalefor the phenomenon. E n = ( m N / ) / (cid:0) π ¯ hg [ n − ] (cid:1) / . We obtain E (cid:39) . (cid:39) − Ry . (2)A first remarkable thing is the minuteness of (2). In spite of being so small, quan-tum effects of gravity have been detected on a table-top! However, the main ques-tion here is that the difference between masses becomes of a yes/no nature. Supposethat the height of the “slit” formed by the upper and lower walls of the chamber issmaller than 10 − cm. If instead of neutrons one were trying to send through (say)aluminium atoms, they would be observed at the exit. However, that same slit onEarth is opaque to neutrons. The following rule of thumb is useful: the energy re-quired to lift a neutron by 10 − cm is classically 1 peV with a good approximation.Accordingly the width of the state (2) can be estimated: the height of the chambershould be bigger than 1 . × − cm for neutrons to be observed at the exit. Fig. 3illustrates this. The phenomenon has nothing to do with diffraction, since the wave-length of neutrons remains much smaller than the height of the slit; visible light,with a wavelength much bigger than those neutrons, is transmitted.Bingo! A slit has become a wall, impenetrable. Uniqueness of free fall fails.Gravitation is not just geometry.The point is even more forcefully brought home in Fig. 4, which describes theactual experimental situation. Put in a different way, at least for interaction with mat-ter, the (geometrical form of the) equivalence principle and the incertitude principleclash. No prizes to guess which must give way.Surprisingly, our viewpoint is found controversial by some. To put matters intoperspective, it is helpful to keep in mind that the equivalence principle is classicallyexpressed by the statements (1) Gravitational mass equals inertial mass or (2) The Fig. 4
Dependency of the particle flux on the slit size. The circles indicate the experimental re-sults [26] for a beam with an average value of 6.5 m/s for the horizontal velocity component. Thestars show the analogous measurement with 4.9 m/s. The solid lines correspond to the classicalexpectation values for these two experiments. The horizontal lines indicate the incertitude in thedetector background. motion of particles in a gravity field is indifferent to their mass. While the COWexperiment confirms (1), the second is untrue in the quantum world. Since pointparticles, paths and clocks play an apparently essential role in the foundations ofgeneral relativity (see the remarks further below), and since it is hard to see howgeometry could have come to such a preponderance in dynamics without (2), itwould seem the latter is bound to diminish. However, one can argue for an importantresidual role of geometry in quantum physics, as in the very readable article [27].(In the current experimental situation, there is not much more than can be donedirecly to measure quantum jumps in a gravitational field. Present hopes to improveon accuracy of measurement of the quantum states parameters rest on use of stor-age sources of ultra-cold neutrons and magnetic field gradients to resonate with thefrequency defined by the energy difference of two states [28].)Among the numerous works on “quantum gravity” that make much of the clas-sical geometry aspects of gravitation, a good representative is the homonymousbook [29]. Its philosophical position is staked out at the outset: . . . the question we have to ask is: what we have learned about the world from quantummechanics and from general relativity?. . . What we need is a conceptual scheme in whichthe insights obtained with general relativity and quantum mechanics fit together.This view is not the majority view in theoretical physics, at present. There is consensusthat quantum mechanics has been a conceptual revolution, but many do not view generalrelativity in the same way. . . According to this opinion, general relativity should not betaken too seriously as a guidance for theoretical developments.I think that this opinion derives from a confusion: the confusion between the specificform of the Einstein–Hilbert (EH) action and the modification of the notions of space andtime engendered by general relativity.otes on “quantum gravity” and noncommutative geometry 11
We are pleased to vote with the bread-and butter majority here. The trouble is thenon-geometrical cast of quantum dynamics . Since we know not the shape of thingsto come, the task is not so much to “fit general relativity with quantum mechanicstogether” as to —slowly and painstakingly— extend our knowledge to quantum andgravitational phenomena simultaneously taking place. It is somewhat saddening thatthe COW experiment and its successors are not found in the reference list of [29];nor are they mentioned in the history of quantum gravity given as an appendix inthat book —which is more in the “history of ideas” mold. In fact the sphere of ideasaround the proper interpretation of the COW experiment hails back to Wigner, who,long ago, had explained keenly the quantum limitations of the concepts of generalrelativity [30], concluding: . . . the essentially non-microscopic nature of the general relativistic concepts seems to usinescapable.
In otherwise mathematically subtle and full of gems [29], as in the works ofother practitioners of quantum gravity, the warning goes unmentioned, as well as un-heeded.To summarize, a generous dose of salt is in order when dealing with “quantumgravity” claims. Without necessarily enjoying the quarantine, we should go mostcarefully about breaking it. Not only “large fragments of the physics community”,but also thoughtful mathematicians like Yuri Manin, advise a useful skepticism, inthe respect of taking as physical what is just product of mathematical skill:
Well-founded applied mathematics generates prestige which is inappropriately generalizedto support quite different applications. The clarity and precision of mathematical deriva-tions here are in sharp contrast to the uncertainty of the underlying relations assumed. Infact, similarity of the mathematical formalism involved tends to mask the differences in thedifferences in the scientific extra-mathematical status. . . mathematization cannot introducerationality in a system where it is absent. . . or compensate for a deficit of knowledge.
This as very timely quoted in [31].
From our standpoint, the action for gravitational interactions is more important thanspeculative “background independency” in a “final unified theory”. Moreover, thepure gravity EH action can be rigorously derived from the theory of quantum fields:a simple lesson, often forgotten. We proceed to that in this section. (As a historicalnote, for once the Einstein–Hilbert surname is right on the mark: independentlyHilbert and Einstein gave the new equations of gravitation in the dying days ofNovember 1915.)
The book [32], containing lectures by Feynman on gravitation given at Caltech in1962-63, deals with the perturbative approach to classical gravity; to wit, with theself-consistent theory of a massless spin-2 field (we may call it graviton). The fore-word of this book (by John Preskill and Kip S. Thorne) is recommended reading.There the unfolding of (earlier) variants of the same idea by Kraichnan and Guptais narrated as well, with references to the original literature. The main aspect inKraichnan–Gupta–Feynman arguments is that a geometrical theory is obtained fromflat-spacetime physics by using consistency requirements. Later work by Deser andOgivetsky and Polubarinov in the same spirit is also remarkable.The distinctively non-geometrical flavour is welcome here, where we regard thegeometrical approach as suspect. An excellent review with references of the clas-sical path from the action for such field to the EH action is found in the recentbook [33, Chap. 3].Weinberg’s viewpoint in 1964 [34] is also very instructive and deserves mention.On the basis of properties of the S -matrix, he proves that gravitons must couple to allforms of energy in the same way. He moreover shows that any particle with inertialmass m i and energy E has, apart from Newton’s constant, an effective gravitationalcharge 2 E − m i / E . For E = m i , one recovers the usual equivalence result. While for m i = E , which gives the correct result for the deflection of light. (Also, a graviton mustrespond to an external gravity field with the same charge.)In this section we perform a parallel exercise to Feynman’s: assuming ignoranceof Einstein’s general relativity, we arrive again at the EH action by successive ap-proximation. Our method has little to do with the “effective Lagrangians” approachand differs from traditional ones mentioned above in at least one of several respects: • We consider only pure gravity. Coupling to matter is sketched after the fact, justfor completeness. • It is fully quantum field theoretical, in that recruits the canonical formalism onFock space and quantum gauge invariance. Our main tool is BRS technology,and ghost fields are introduced from the outset. In other words, we treat gravityas any other gauge theory in the quantum regime; we obtain a quantum theory ofthe gravitational field, in which at some point we put ¯ h = • We use the causal (or Epstein–Glaser) renormalization scheme [35], relying onthe (perturbative expansion in the coupling parameter of the) S -matrix. This en-tails a slight change of interpretation, in regard to renormalization, with respectto standard thinking; we briefly discuss the matter at the end of subsection 3.6.Epstein–Glaser renormalization is specially appropriate for gravity issues sinceit does not rely on translation invariance. • We never invoke the stress-energy tensor. otes on “quantum gravity” and noncommutative geometry 13
In some sense we close a circle opened as well by Feynman in the early six-ties [36], where he first realized that unitarity at (one-)loop graph calculations de-manded ghost fields, for gravity as well as for Yang–Mills theory. Through well-known work by DeWitt, Slavnov, Taylor, Fadeev and Popov, and Lee and Zinn-Justin, this would eventually lead to BRS symmetry by the mid-seventies.We mainly follow [37, 38]. The remote precedent for the last paper is an out-standing old article by Kugo and Ojima [39].
In order to make clear the strategy, we briefly recall here the similar treatment for(massive and massless) electrodynamics. Suppose we wish to effect the quantizationof spin-1 particles by means of real vector fields. The question is how to eliminatethe unphysical degrees of freedom, since a vector field has four independent com-ponents, while a spin-1 particle has three helicity states, or two if it is massless.A standard procedure is to impose the constraint ∂ µ A µ = : ( ∂ · A ) =
0. How-ever, this is known to lead to the Proca Lagrangian (density), which has very badproperties. Also, under quantization, use of Proca fields entails giving up covariantcommutators of the disarmingly simple form found for neutral scalar fields: [ A µ ( x ) , A ν ( y )] = i η µν D ( x − y ) , ( A µ ) + = A µ ; (3)with η the Minkowski metric and D the Jordan–Pauli propagator. We would like tokeep them instead. The Klein–Gordon equations ( (cid:3) + m ) A µ = + we understand the ordinary involution. However, it is possibleto do it through the introduction of a distinguished symmetry η (that is, an operatorboth selfadjoint and unitary), called the Krein operator. Whenever such a Kreinoperator is considered, the η -conjugate O + of an operator O with adjoint O † is: O + : = η O † η . Let ( · , · ) denote the positive definite scalar product in H . Then (cid:104)· , ·(cid:105) : = ( · , η · ) yields an “indefinite scalar product”, and the definition of O + is just that of theadjoint with respect to (cid:104)· , ·(cid:105) . Then A will be self-conjugate.The massive vector field model is known to be a gauge theory [40] if we introducethe auxiliary (scalar) St¨uckelberg field B (say with the same mass m ), and gaugetransformations of the form: δ A µ ( x ) = η µν ∂ ν θ ( x ) = ∂ µ θ ( x ) ; δ B ( x ) = m θ ( x ) . The trick now is to use the unphysical parts ∂ · A , B plus the ghosts u and anti-ghost ˜ u to construct the BRS operator Q = (cid:90) x = const d x ( ∂ · A + mB ) ←→ ∂ u , whose action should reproduce the gauge variations (where commutators [ ., . ] − oranticommutators [ ., . ] + are taken according to whether the ghost number of the var-ied field is even or odd): sA µ ( x ) = [ Q , A µ ( x )] ± = i ∂ µ u ( x ) ; sB ( x ) = [ Q , B ( x )] ± = imu ( x ) ; su ( x ) = [ Q , u ( x )] ± = s ˜ u ( x ) = [ Q , ˜ u ( x )] ± = − i (cid:0) ∂ · A ( x ) + mB ( x ) (cid:1) . (5)With these relations one easily proves 2-nilpotency modulo the field equation:2 Q = i (cid:90) x = const d x (cid:3) u ←→ ∂ u + im (cid:90) x = const d x u ←→ ∂ u = . Thus the right hand side of (5) are coboundary fields. With the help of nilpotency,the finite gauge variations for the same fields of (5) are easily computed. The su-percharge Q is conserved. The massless limit is not singular in this formalism: forphotons, we just put m =
0, and B drops out of the picture. A rank 2 tensor field under the Lorentz group decomposes into the direct sum of fourirreducible representations, corresponding to traceless symmetric tensors, a scalarfield, and self-dual and anti-self-dual tensors. We group the first two into a symmet-ric tensor field h ≡ { h µν } with arbitrary trace. Let us introduce as well ϕ : = h ρρ ; H ≡ (cid:8) H µν } : = (cid:8) h µν − η µν ϕ (cid:9) ; thus H ρρ = . (We wish to keep h to denote the whole tensor, and so we do not use the standardnotation for its trace.) Again the question is how to eliminate the superfluous degreesof freedom in the description of a spin-2 relativistic particle, which possesses onlytwo helicity states. A fortiori we do not want to follow for the graviton the path ofenforcing constraints, that was discarded for photons.For a free graviton one may settle on the Lagrangian otes on “quantum gravity” and noncommutative geometry 15 L ( ) = ( ∂ ρ h αβ )( ∂ ρ h αβ ) − ( ∂ ρ h αβ )( ∂ β h ρα ) − ( ∂ ρ ϕ )( ∂ ρ ϕ ) . (6)Of course this choice is not unique. The more general Lorentz-invariant action qua-dratic in the derivatives of h is of the form (cid:90) d x [ a ( ∂ ρ h αβ )( ∂ ρ h αβ ) + b ( ∂ ρ h αβ )( ∂ β h ρα ) + c ( ∂ ρ ϕ )( ∂ σ h ρσ ) + d ( ∂ ρ ϕ )( ∂ ρ ϕ )] . The frequently invoked Fierz–Pauli Lagrangian [41] is of this type, with a = , b = − , c = , d = − . The signs are conventionally chosen in both cases so that thefirst term has a positive coefficient. The Euler–Lagrange equations correspondingto (6): ∂ γ ∂ L ( ) ∂ ( ∂ γ h αβ ) = (cid:3) h αβ − ∂ γ ∂ β h αγ − ∂ γ ∂ α h βγ − η αβ (cid:3) ϕ = . (7)This form is essentially equivalent to the Fierz–Pauli equation, but more convenienthere. (For a critique of the Fierz–Pauli framework, consult [42].) A crucial point is the invariance of the Lagrangian L ( ) —thus of equation (7)—under gauge transformations δ h αβ = λ ( ∂ α f β + ∂ β f α − η αβ ( ∂ · f )) = λ b αβρτ ∂ ρ f τ , (8)where b αβρτ : = η αρ δ βτ + η βρ δ ατ − η αβ δ ρτ , for arbitrary f = ( f α ) . This entails δ ϕ = − λ ( ∂ · f ) . (9)To verify this invariance, with an obvious notation, and up to total derivatives, δ L ( ) I = − δ h αβ (cid:3) h αβ ; δ L ( ) II = δ h αβ ∂ ρ ( ∂ α h βρ + ∂ β h αρ ) ; δ L ( ) III = δ ϕ (cid:3) ϕ . One finishes the argument by use of (8) and (9).That tensor b will reappear often. Classically, one could specify here the trans-verse gauge condition: ∂ β ( h αβ + δ h αβ ) = . (10) (In the gravity literature a so-called de Donder gauge condition is more frequentlyused.) The last equation is obtained at once if f α solves λ (cid:3) f α = − ∂ β h αβ = : − ( ∂ · h ) α ;then (7) reduces to (cid:3) h = . As advertised, we refrain from quotient by imposing gauge conditions. In our BRS-like treatment, the elimination of the many extra degrees of freedom takes placecohomologically, rather than by use of constraints. The fields are promoted to (bynow still free) normally ordered quantum fields. Clearly, in this approach we needto add to L ( ) the gauge-fixing and free ghost terms: L free = L ( ) + ( ∂ · h ) · ( ∂ · h ) − ( ∂ µ ˜ u ν + ∂ ν ˜ u µ )( ∂ µ u ν + ∂ ν u µ − η µν ( ∂ · u )) . (11)One quantizes h in the most natural way [ h αβ ( x ) , h µν ( y )] = ib αβ µν D ( x − y ) ; (12)and therefore the propagators for H , ϕ are given by: [ H αβ ( x ) , H µν ( y )] = i (cid:0) η αµ η βν + η αν η β µ − η αβ η µν (cid:1) D ( x − y ) , [ ϕ ( x ) , ϕ ( y )] = − iD ( x − y ) , [ ϕ ( x ) , H µν ( y )] = . Also, for the fermionic ghosts we have the anticommutation relations [ u α ( x ) , u β ( y )] = ig αβ D ( x − y ) (13)All other anticommutators vanish. The new Euler–Lagrange equations give rise nowto the simplest possible, ordinary wave equations for all fields considered. (cid:3) h = (cid:3) u = (cid:3) ˜ u = . We can prove directly consistency of rules (12) and (13), analogous to (3) and (4),by constructing a explicit representation in a Fock–Krein space. The reader will seethis in a later subsection.Let us now introduce the BRS operator Q = (cid:90) x = const d x ( ∂ · h ) α ←→ ∂ u α = (cid:90) x = const d x (cid:0) ( ∂ · H ) α + ∂ α ϕ (cid:1) ←→ ∂ u α ; (14)where ( ∂ · h ) α denotes the divergence ∂ β h αβ , which in view of (10) is unphysical,and u α is the fermionic (vector) ghost field. The associated gauge variations are: otes on “quantum gravity” and noncommutative geometry 17 sh µν = [ Q , h µν ] = ib µνρτ ∂ ρ u τ = i ( ∂ µ u ν + ∂ ν u µ − η µν ( ∂ · u )) ; su = [ Q , u ] + = s ˜ u = [ Q , ˜ u ] + = − i ( ∂ · h ) µ . (15)Note that the action of the coboundary operator is dictated by the variation (8). Otherimportant coboundaries like s ϕ = i ( ∂ · u ) ; s ( ∂ · h ) µ = Q is 2-nilpotent and conserved. We make a temporary halt to examine whether, with our choices in subsection 3.3we are on the right track, after all. Let g : = ( g αβ ) denote the metric tensor and R the Ricci curvature. As hinted above, for this writer the EH action (with c =1, andwithout the “cosmological constant”) S EH = − π G (cid:90) d x (cid:112) − det g R = − π G (cid:90) d x g µν R µν . constitutes the alpha and omega of gravitation theory. Here G is Newton’s constant,equal to ¯ h / m . We recall Γ αβγ = g αµ ( ∂ γ g β µ + ∂ β g γµ − ∂ µ g βγ ) ; thus ∂ α g µν = − Γ µγα g γν − Γ νγα g γµ (vanishing covariant derivative); R µν = ∂ α Γ αµν − ∂ ν Γ αµα + Γ βµν Γ αβα − Γ βµα Γ αβν ; R = g αβ R αβ . (16)It is convenient to have a special notation for Γ µ : = Γ αµα = g αγ ∂ µ g αγ = ∂ µ ( det g ) g = ∂ µ (cid:16) log (cid:112) − det g (cid:17) . We have employed that the minors of g αβ in det g are equal to det g g αβ . Finally, theGoldberg tensor 1-density g αβ : = (cid:112) − det gg αβ is —quite canonically, according to [43, Sect. 2.1]— a hero of our story.Let us define λ = √ π G (essentially the inverse of Planck’s mass, in naturalunits). Since our approach to S EH is perturbative, we need to rewrite the correspond-ing Lagrangian L EH as a series in the coupling constant λ . An old trick in classicalgravity —see for instance [44, Sect. 93]— is to split off a divergence from L EH by using g µν ∂ α Γ αµν = ∂ α ( g µν Γ αµν ) − Γ αµν ∂ α ( g µν ) ; g µν ∂ ν Γ µ = ∂ ν ( g µν Γ µ ) − Γ µ ∂ ν ( g µν ) . With the help of previous equations, one finds g αβ R αβ = H − ∂ γ ( g µγ Γ µ − g µν Γ γµν ) = : H − ∂ γ D γ , (17)where H = g αβ ( Γ γαρ Γ ρβγ − Γ ραβ Γ ρ ) . The key step in our identification comes now: to make the contact between quan-tum field theory and general relativity, we postulate g µν = η µν + λ h µν . (18)Remark that do not assume h to be small in any sense. In (17) above we separate thepart of the vector D containing negative powers of λ : D γ = λ ( ∂ γ ϕ + ∂ ρ h γρ ) + D ( ) γ . (19)The inverse matrix g µν with g µρ g ρν = δ µν formally becomes a series g µν = η µν − λ h µν + λ h µγ h γν − λ h µγ h γτ h τν + · · · (20)Substituting this expression in the new form of the action ( / λ ) (cid:82) d x H , we obtaina series as well: L = ∞ ∑ λ n L ( n ) . (21)(Actually, the Neumann series (20) is somewhat suspect, in view of convergenceproblems and other technical difficulties. One could se the Cayley–Hamilton theo-rem to obtain an exact expression for ( g µν ) .) The lowest order, at any rate, is indeedof order λ in view of the two derivatives inside H ; and it is seen to coincide withthe free model of subsection 3.3. For completeness and use later on, we also reportthe three-graviton and four-graviton couplings: L ( ) = (cid:0) − ∂ ρ ϕ∂ σ ϕ + ∂ ρ h αβ ∂ σ h αβ + ∂ γ h αρ ∂ α h γσ (cid:1) h ρσ ; L ( ) = − h αβ h ρβ ( ∂ ν h αµ )( ∂ µ h βν ) − h ρσ h ρβ ( ∂ α h ρβ )( ∂ α ϕ ) − h νµ ( ∂ α h νµ ) h σρ ( ∂ α h σρ ) + h νµ ( ∂ α h νµ ) h a β ( ∂ β ϕ )+ h βρ h βσ ( ∂ µ h ρα )( ∂ µ h σα ) − h αρ ( ∂ µ h ρσ )( ∂ ν h αρ ) h µν + h αρ h βσ ( ∂ µ h ασ )( ∂ µ h βρ ) . (22) otes on “quantum gravity” and noncommutative geometry 19 Interacting fields in Epstein–Glaser formalism are made out of free fields. The start-ing point for the analysis is the functional S -matrix in the Dyson representationunder the form of a power series: S ( g ) = + T = + ∞ ∑ n = n ! (cid:90) dx . . . dx n T n ( x , . . . , x n ) g ( x ) · · · g ( x n ) . (23)The theory is constructed basically by using causality and Poincar´e invariance ofthe scattering matrix to determine the form of the time-ordered products T n . Onlythose fields should appear in T n that already are present in T . The adiabatic limit onthe “coupling functions” g ( x ) ↑ sT n = [ Q , T n ] ± mustbe a divergence, keeping in mind that T n and T (cid:48) n are equivalent if they differ bycoboundaries.In particular, first-order CGI means sT ( x ) = i ( ∂ · T / )( x ) . For T , let us try a general Ansatz containing cubic terms in the fields and leadingto a renormalizable theory. At our disposal there are three field sets: h , u , ˜ u . The mostgeneral coupling with vanishing ghost number without derivatives is of the form a ϕ + b ϕ h νµ h νµ + c h µν h νγ h γµ + ( u · ˜ u ) ϕ + e h νµ u ν ˜ u µ . Correspondingly, with ghost number one since the action of the BRS operator in-creases ghost number by one, we can have (with an obvious simplified notation) T µ / = a (cid:48) u µ ϕ + b (cid:48) u µ h · h + c (cid:48) ( u · h ) µ ϕ + d (cid:48) u α h αβ h β µ + e (cid:48) u ( u · ˜ u ) . Forlorn hope. It must be: s ( ∂ · T / ) = . This condition has only the trivial solution T / = T in-volving only h with two derivatives, and 21 combinations in T involving h , u , ˜ u ,with two derivatives and zero total ghost-number. At the end of the day, one obtains T = T h + T u , with T h uniquely proportional to L ( ) (modulo physically irrelevantdivergences), and T u = a (cid:0) − u α ( ∂ β ˜ u ρ ) ∂ α h βρ + ( ∂ β u α ∂ α ˜ u ρ − ∂ α u α ∂ β ˜ u ρ + ∂ ρ u α ∂ β ˜ u α ) h βρ (cid:1) . The calculations are excruciatingly long, and of little interest. They, as well as theexplicit expression of T / , can be found in [37], to which we remit. By the way, had we tried to use g µν = η µν + λ h µν instead of (18), then T h turns out much more complicated —even after eliminationof a host of divergence couplings.More intrinsically interesting are the calculations of CGI at second order, alsodone in [37], which indeed reproduce L ( ) . For the higher-order analysis, one needssome (rather minimal) familiarity with the Epstein–Glaser method to inductivelyrenormalize (i.e., to define) the time-ordered products T n , based on splitting of dis-tributions. This requires use of antichronological products, corresponding to the ex-pansion of the inverse S -matrix. If we write the inverse power series: S − ( g ) = + ∞ ∑ n ! (cid:90) d x . . . (cid:90) d x n T n ( x , . . . , x n ) g ( x ) . . . g ( x n ) , then we have T | N | ( N ) = ∑ nk = ( − ) k ∑ (cid:93) kj = I j = N T | I | ( I ) , . . . , T | I k | ( I k ) , where the disjointunion is over (non-empty) blocks I j . For instance, the second order term T ( x , x ) in the expansion of S − ( g ) is given by T ( x , x ) = − T ( x , x ) + T ( x ) T ( x ) + T ( x ) T ( x ) . The inductive step is performed using the totally advanced and totally retarded prod-ucts. For instance, at the lower orders: A ( x , x ) = T ( x ) T ( x ) + T ( x , x ) = T ( x , x ) − T ( x ) T ( x ) ; R ( x , x ) = T ( x ) T ( x ) + T ( x , x ) = T ( x , x ) − T ( x ) T ( x ) ; A ( x , x , x ) = T ( x ) T ( x , x ) + T ( x ) T ( x , x ) + T ( x , x ) T ( x )+ T ( x , x , x ) ; R ( x , x , x ) = T ( x ) T ( x , x ) + T ( x , x ) T ( x ) + T ( x , x ) T ( x )+ T ( x , x , x ) . (24)By the induction hypothesis D n + : = R n + − A n + depends only on known quanti-ties. Moreover D n + has causal support. If we can find a way to extract its retardedor the advanced part, that is, to split D n + , then we can calculate T n + ( x , . . . , x n + ) .Consider then D ( x , y ) = [ T ( x ) , T ( y )] , the first causal distribution to be split. Wehave thus sD ( x , y ) = [ sT ( x ) , T ( y )] + [ T ( x ) , sT ( y )]= i ∂ x µ [ T µ / ( x ) , T ( y )] + i ∂ y µ [ T ( x ) , T µ / ( y )] ; (25)so that D is gauge-invariant; and the issue is how to preserve gauge invariance in therenormalization or distribution splitting. That is, we must split D and the commu-tators —without the derivatives— in the previous equation; then gauge invariance: otes on “quantum gravity” and noncommutative geometry 21 sR ( x , y ) = i ∂ x µ R µ / ( x ) + i ∂ y µ R µ / ( y ) can only be (and is) violated for x = y , that is, by derivative terms in δ ( x − y ) . Thatis to say, if local renormalization terms N , N µ / , N µ / can be found in such a waythat s ( R ( x , y ) + N ( x , y )) = i ∂ x µ ( R µ / + N µ / ) + i ∂ y µ ( R µ / + N µ / ) , with an obvious notation, then CGI to second order holds.When computing in practice, one is liable to find identities in distribution theorylike ∂ x µ [ A ( x ) B ( y ) δ ( x − y )] + ∂ y µ [ A ( y ) B ( x ) δ ( x − y )]= ∂ µ A ( x ) B ( x ) δ ( x − y ) + A ( x ) ∂ µ B ( x ) δ ( x − y ) (26)and A ( x ) B ( y ) ∂ x µ δ ( x − y ) + A ( y ) B ( x ) ∂ y µ δ ( x − y )= A ( x ) ∂ µ B ( x ) δ ( x − y ) − ∂ µ A ( x ) B ( x ) δ ( x − y ) . (27)We make the following observation: since A ( x ) B ( y ) δ ( x − y ) = A ( x ) B ( x ) δ ( x − y ) , it must be ∂ x µ (cid:0) A ( x ) B ( y ) δ ( x − y ) (cid:1) = ∂ x µ (cid:0) A ( x ) B ( x ) δ ( x − y ) (cid:1) ;which forces B ( y ) ∂ x µ δ ( x − y ) = B ( x ) ∂ x µ δ ( x − y ) + ∂ µ B ( x ) δ ( x − y ) . (28)We are able to prove both (26) and (27) from (28). ∂ x µ [ A ( x ) B ( y ) δ ( x − y )] + ∂ y µ [ A ( y ) B ( x ) δ ( x − y )]= ∂ µ A ( x ) B ( x ) δ ( x − y ) + A ( x ) B ( y ) ∂ x µ δ ( x − y )+ ∂ µ A ( x ) B ( x ) δ ( x − y ) − A ( y ) B ( x ) ∂ x µ δ ( x − y )= ∂ µ A ( x ) B ( x ) δ ( x − y ) + A ( x ) B ( y ) ∂ x µ δ ( x − y ) − A ( x ) B ( x ) ∂ x µ δ ( x − y ) = ∂ µ A ( x ) B ( x ) δ ( x − y ) + A ( x ) ∂ µ B ( x ) δ ( x − y ) ;where we have used (28) twice. Analogously, A ( x ) B ( y ) ∂ x µ δ ( x − y ) + A ( y ) B ( x ) ∂ y µ δ ( x − y ) = A ( x ) B ( x ) ∂ x µ δ ( x − y )+ A ( x ) ∂ µ B ( x ) δ ( x − y ) − A ( y ) B ( x ) ∂ x µ δ ( x − y )= A ( x ) ∂ µ B ( x ) δ ( x − y ) − ∂ µ A ( x ) B ( x ) δ ( x − y ) , using (28) twice again.Again after excruciatingly long calculations, by the sketched method one recov-ers the four-graviton couplings (22), plus terms with ghosts that we omit. Nev- ertheless, the road seems barred in that, in order to rederive the EH Lagrangian,one would have to perform an infinite number of calculations. Put in another way,we could not never finish ascertaining that the EH Lagrangian fulfils CGI. (In a(re)normalizable theory it would be enough to verify CGI till third order, but thisis not the case here.) For the latter, a better way can be contrived, though. Leavingaside the question of uniqueness (in spite of “folk theorems”, uniqueness there isnot: see Section 4), one can jump to the conclusion that L EH does satisfy CGI. Inthe next subsection, we describe a simple, short and rigorous argument for this.Before pursuing, we take stock: a classical Lagrangian is extracted from a quan-tum theory because, for all computations, naturally starting at T , only tree diagrams are considered. Par ce biais-ci the limit ¯ h ↓ S EH are obtained; although this is not for the fainthearted.See [45] for the graviton self-energy; discrepancies between the coefficients of thosecorrections are still found in the literature. Anomalies are lurking there as well.A last comment is in order: we have not tackled the matter of (re)normalizabilityof the theory, which is in terms of the T n is a bit involved. Suffice here to say thatthe conclusion is similar to that of standard arguments (on the basis of the dimen-sionality of G , for instance). It is true that in causal (re)normalization, there are noultraviolet divergences as such. There is a problem of correct definition of distribu-tions involved in the perturbative expansion of the S -matrix. The price of a “non-normalizable” theory like Einstein’s is an infinite number of normalization constantsin the process of that definition. This is not automatically so damning (also in re-gard of the discussion in the previous section), since perhaps they could be fixedby experiments, or have unobservable consequences. At any rate, the famous one-loop finiteness result by ’t Hooft and Veltman —consult for instance the discussionin [46, Sect. III]— means that, at next order in pure gravity, no (new normalizationconstants and thus no) new geometrical invariants are introduced: another rule ofthe godly quarantine. We rely in the following on a theorem by D¨utsch [38]: BRS invariance of a La-grangian, depending only on the fields and their first derivatives and carrying non-negative powers of the couplings, implies local conservation of the BRS current. Thelatter implies CGI in the Heisenberg representation for tree graphs; and this result iskept in passing to time-ordered products. BRS invariance means precisely that theaction of the BRS operator on the Lagrangian is a divergence, without use of thefield equations. This admitted, the proof of CGI for the EH Lagrangian —modifiedlike in formula (17)— by means of the BRS formulation of gravity by Kugo andOjima [39] is simplicity itself.In (our version of) that formulation, one keeps (18) and uses new gauge varia-tions. The coboundary operator now is of the form otes on “quantum gravity” and noncommutative geometry 23 s = s + λ s . Here s acts exactly like s of (5) and s h µν = i ( h µρ δ ντ + h νρ δ µτ ) ∂ ρ u τ − i ∂ τ ( h µν ) u τ ; s u = − i ( u · ∂ ) u ; s ˜ u = . (29) Sotto voce we are introducing here the Lie derivative of ( g µν ) with respect to theghost vector field, thus diffeomorphism invariance. The new Lagrangian, completewith gauge-fixing and ghost terms, is: L total = − H + L gf + L ghost = − H + ( ∂ · h ) · ( ∂ · h ) + i ( ∂ ν ˜ u µ + ∂ µ ˜ u ν ) sh µν . Of course L total is not diffeomorphism-invariant. Compare (11). Note that L gf = L ( ) gf = − ( s ˜ u ) , while L ghost has terms of order λ . From this, s h = s u = s ˜ u µ = − δ S total δ ˜ u µ , vanishing on-shell. It is known that [39] that s L EH = − i λ ∂ · ( u L EH ) , and since, with F α : = ( ∂ ρ h βρ ) sh αβ we have s ( L gf + L ghost ) = i ∂ · F , it would seem that BRS invariance is checked, and we are done. Actually L EH doesnot fulfil the conditions of D¨utsch’s theorem. However, we can use (17) and (19) toconclude. Indeed − sH = − i λ ∂ · ( u L EH ) − i ∂ · ( sD ) + i λ ∂ · ( (cid:3) u − ∂ ( ∂ · u )) . The last vector is conserved, but the point is that it cancels the term of the form1 λ s (cid:0) ∂ γ ϕ + ∂ ρ h γρ (cid:1) , in sD . Then s ( L total ) = s ( − H + L gf + L ghost ) = − i ∂ · (cid:16) λ u L EH + sD − F − (cid:3) u λ + ∂ ( ∂ · u ) λ (cid:17) ;that is s ( L total ) = ∂ · I with I of the form I = ∞ ∑ k = λ k I ( k ) , and all is well. (The funny and revealing thing in all this is that the parts in 1 / λ and 1 / λ in the EH Lagrangian do not contribute to the equations of motion.)It is instructive to compare the tensor and vector cases. In order to see the parallel,one ought to replace (the massless version of) formulae (5) by sA µ a ( x ) = iD µ ab u b ( x ) ; su a ( x ) = − i g f abc u b u c ; s ˜ u a ( x ) = − i (cid:0) ∂ · A a ( x ) (cid:1) . Like there, it is plain that the action of the BRS operator increases ghost number byone. Here f abc denotes the structure constants of a Yang–Mills model, and D is thecorresponding covariant derivative. The reader might be curious to know how the physical degrees of freedom emergeunder our canonical recipe.Let us treat ghosts first. Consider a family of absorption and emission operators c α a ( k ) with a = , [ c α a ( k ) , c β b ( k (cid:48) )] + = δ ab δ αβ δ ( k − k (cid:48) ) , defining a bona fide Fock space; with the definitions u α ( x ) = ( π ) − / (cid:90) d µ ( k ) ( e − ikx c α ( k ) − g αα e ikx c α ( k ) † ) , ˜ u α ( x ) = − ( π ) − / (cid:90) d µ ( k ) ( e − ikx c α ( k ) + g αα e ikx c α ( k ) † ) , (30)where d µ ( k ) is the usual Lorentz-invariant volume over the lightcone. There is aKrein operator on the ghost Fock space that allows for u being self-conjugate and ˜ u being skew-conjugate. This can be achieved by c i ( k ) + = c i ( k ) † ; c i ( k ) + = c i ( k ) † ; c ( k ) + = − c ( k ) † ; c ( k ) + = − c ( k ) † , with i = , ,
3. Then formulae (30) are rewritten u α ( x ) = ( π ) − / (cid:90) d µ ( k ) ( e − ikx c α ( k ) + e ikx c α ( k ) + ) = u α ( x ) + , ˜ u α ( x ) = ( π ) − / (cid:90) d µ ( k ) ( − e − ikx c α ( k ) + e ikx c α ( k ) † ) = − ˜ u α ( x ) + , (31) otes on “quantum gravity” and noncommutative geometry 25 From (30) or (31) we obtain for u , ˜ u the wave equations. Covariant anticommutationrelations (13) also follow.Note now t αβ µν : = (cid:0) η αµ η βν + η αν η β µ − η αβ η µν (cid:1) = t µναβ . That is, (cid:0) t µναβ (cid:1) = / / / (cid:18) / − / − / / (cid:19) / / on a ( , ) , ( j , j ) , ( , j ) , ( j , l ) block basis, with j , l = , , , j (cid:54) = l ; and in particular T ≡ (cid:0) t µµαα (cid:1) = / / / / / − / − / / − / / − / / − / − / / on the ( , ) , ( , ) , ( , ) , ( , ) basis. Next we note that T = MM † , with M = / / / − / / /
20 1 / − / /
20 1 / / − / . Next we invoke operators defining a Fock space: [ b αβ ( k ) , b µν ( k (cid:48) )] = ( δ αµ δ βν + δ αµ δ β µ ) δ ( k − k (cid:48) ) , with b αβ = b βα . Define now operators a αβ , with a αβ = a βα as well, by a αβ = b αβ for α (cid:54) = β and a αα = ∑ β M αβ b ββ . The rule [ a αβ ( k ) , a † µν ( k (cid:48) )] = g αα g ββ t αβ µν δ ( k − k (cid:48) ) follows.The scalar field is now constructed in a way close to the standard one: ϕ ( x ) = ( π ) − / (cid:90) d µ ( k ) ( e − ikx a ( k ) − e ikx a † ( k )) , (32)where the (not Lorentz-covariant) operators a satisfy [ a ( k ) , a † ( k )] = δ ( k − k (cid:48) ) . The traceless sector is represented H αβ ( x ) = ( π ) − / (cid:90) d µ ( k ) ( e − ikx a αβ ( k ) + g αα g ββ t αβ µν e ikx a † αβ ( k )) . Now one can verify (12) painstakingly.The last task in this subsection is to identify finally the physical degrees of free-dom. For that, let us choose and fix k µ = ( ω , , , ω ) . One can verify that the onlystates not present in Q (that is, belonging to the kernel of [ Q , Q † ] + ) are ( b − b ) † | (cid:105) and b †12 | (cid:105) = b †21 | (cid:105) . They correspond to linear polarization states. Their complex combinations (circularpolarization states) may be represented by matrices ε ± : = ± i ± i − , which transform like ε (cid:48)± = e ± i φ ε ± under a rotation of angle φ about the direction of propagation. The reader can verifythis by using the generator of rotations − . The two ± ε µν ± k ν = . (33)These conditions are not Lorentz-invariant. Notice the associated gauge freedom ε µν ± → ε µν ± + k µ f ν + f µ k ν − η µν ( k · f ) . We may add ε ν ± ν = . (34)This five conditions (33) and (34) are also possible for a massive graviton —say k = ( m , , , ) . Thus they characterize the spin two case in general, with up to fivedegrees of freedom. Now, for k lightlike as above, let e , e denote two spacelikevectors orthogonal to k and mutually orthogonal, say ( , , , ) , ( , , , ) . The ten-sors otes on “quantum gravity” and noncommutative geometry 27 ( k µ k ν ) = ; ( k µ e ν + e µ k ν ) = ; ( k µ e ν + e µ k ν ) = verify (33) and (34) as well. They represent the three helicity states that disappearin the massless case. • The geometrical form of general relativity, due to Einstein, is supremely elegantfor some. However, the accompanying interpretation clashes with the one advo-cated here, based in the identification of the quanta of the gravitational field andmore-or-less standard quantum field theory procedures; not to speak of table-topexperiments. Since experiments probe gravity theory to very low orders in G , ¯ h ,one should keep an open mind, and welcome any consistent quantum theory per-turbatively compatible with general relativity. As string theory promises to be. • Coupling to matter. The graviton naturally couples to another symmetric tensorfield: T matter1 = i λ A αβ µν h αβ T µν with sT = . Consideration that sT matter1 must be a divergence leads at once to ∂ µ T µν = • Infrared freedom: in the Epstein–Glaser dispensation, vacuum diagrams, as anyothers, are ultraviolet-finite. Because of their high degree of singular order, how-ever, we are assured that they are infrared finite. Therefore the vacuum is stable(no colour confinement or anything of the sort): a bonus for quantum gravity. • The CGI formalism allows one can deal with massive gravity as well [47], al-though the shortcut in subsection 3.7 apparently is not available. At the price ofintroducing St¨uckelberg-like vector Bose ghosts, the massless limit of massivegravity is relatively smooth. Suggestively, a cosmological constant Λ = m / m the graviton mass, ensues; one is reminded of Mach’s principle, as well.Note that the Fadeev–Popov approach to ghosts in quantum gravity is linked toexistence of quasi-invariant measures on diffeomorphism groups [48]. • Path-integral quantization faces the stark difficulty (rather, the impossibility) of“counting” four-dimensional manifolds [49]. A way around it may be “dynamicaltriangulation” —see [50] and in the same vein the recent [51]. • We cannot close the section without mentioning the promise of “asymptoticsafety” in quantum gravity, developed by Reuter and coworkers. Consult [52],and references therein. There are intriguing results within this approach, point-ing out to effective 2-dimensionality of spacetime at the Planck scale —whichhas been used by Connes, somewhat dubiously, to justify that the finite noncom-mutative geometry part in his reinterpretation of the standard model Lagrangianbe of KO -dimension 6 [53]. While, at the other end of the scale, exceptionallygood infrared behaviour could mimic both “dark matter” and “dark energy” be-haviour. • In relation with the discussion at the end of subsection 3.6, support for the ideathat UV divergences in gravity are not so intractable has come recently fromwork by Kreimer [54].
A recent edition of a standard text about cosmology by a well-respected author [55]ends with a chapter on “Twenty controversies in cosmology today”. In the first one,about general relativity, he declares:
In fact it is theories without effective rivals that require the most vigilant testing.
Without contradicting this wisdom, let me point out that general relativity hassome rivals which are too close for comfort. In order to grapple with them, let us goback to the fundamentals. We did omit the proof of that, for suitable variations ofthe metric ( g αβ ) , the Einstein field equations in vacuum G αβ + Λ g αβ : = R αβ − Rg αβ + Λ g αβ = . (35)are equivalent to δ S EH δ g αβ = . It is worthwhile to go through that routine here. Now S EH = − π G (cid:90) d x (cid:112) − det g ( R − Λ ) . Clearly otes on “quantum gravity” and noncommutative geometry 29 δ S EH = π G (cid:90) d x (cid:104) − ( R − Λ ) δ √− det g δ g αβ (cid:105) + (cid:112) − det g (cid:2) R αβ δ g αβ + g αβ δ R αβ (cid:3) , where we take into account R αβ δ g αβ = − R αβ δ g αβ , since δ g ρσ g σε + g ρσ δ g σε = . Now, δ (cid:112) − det g = − √− det g ∂ ( − det g ) ∂ g αβ δ g αβ = (cid:112) − det g g αβ δ g αβ . It is easy to s how that the last term in δ S EH does not contribute to the variation ofthe action. Therefore δ S EH δ g αβ = √− det g π G (cid:0) R αβ − Rg αβ + Λ g αβ (cid:1) ;hence (35).It is apparent that life would be much simpler if √− g where not a dynamicalquantity. This is suggested by Weinberg in his well-known review [56], in relationwith the discussion in Section 6; the idea basically goes back to Einstein. Let us seewhat happens. First of all Λ seems to vanish from the picture. Second, since nowthe action has to be stationary only with respect to variations keeping det g invariant,that is g αβ δ g αβ =
0, one gathers the elegant R αβ trace − free = R αβ − g αβ R = . As it turns out, these are the Einstein equations again! The reason is that the con-tracted Bianchi identities ∇ β R αβ = ∇ α R , that is ∇ β G αβ = , are still valid. They can be derived from R µν = g σρ R σµρν and the uncontractedBianchi identities: ∂ τ R µνρσ + ∂ σ R µντρ + ∂ ρ R µνστ = . Therefore, by integration, − R = G αα = − κ ; and then G αβ + κ g αβ = , which is but (35) with κ replacing Λ . However, the interpretation has changed. Theterm in Λ in the action does not contribute anything (so the Minkowski space is asolution of the field equations even in the presence of such a term); and κ arises asan initial condition. Remark 1.
The discussion in this section is mainly pertinent in the presence of mat-ter. If we define here the matter stress-energy tensor T ≡ ( T αβ ) by δ S matter = : (cid:90) d x (cid:112) − det g T αβ δ g αβ , then varying S matter + S EH while keeping the determinant fixed results in R αβ trace − free = π G T αβ trace − free . Since the conservation law ∇ · T = R − π GT αα = κ , and finally R αβ − Rg αβ + κ g αβ = π G T αβ , exactly the usual Einstein equations in the presence of a cosmological constant termplus matter, with the mentioned replacement of Λ by κ , and the attending change ofinterpretation.It should be remarked that we are not implying that the classical action for gravi-tational physics is invariant only under coordinate transformations (“transverse dif-feomorphisms”) that preserve the volume element. This is a stronger claim. Elegantjustification for it is found in [57]. In accordance with the above, all known testsof general relativity probe equally the (several) unimodular theories. It has been ar-gued that the matter-graviton coupling gives rise to inconsistencies when “strong”unimodularity holds [58]; but this objection we know not in relation with weak uni-modularity. Only quantum effects would in principle allow tell it and general relativ-ity apart [59] —after all the “measure” of the quantum functional integral for gravityis changed. Meanwhile, the interest of the unimodular theory is twofold: as indicatedby Weinberg, it alleviates the cosmological constant problem (Section 6); moreover,it is natural from the current formulation of noncommutative manifold theory (sub-section 5.9.2). From the viewpoint of the preceding section, the key question is howthe unimodular theory is arrived at the ¯ h ↓ There is no general theory of noncommutative spaces. The practitioners’ tactics hasbeen that of multiplying the examples, whereas trying to anchor the generalizationson the more solid ground of ordinary (measurable / topological / differentiable / Rie- otes on “quantum gravity” and noncommutative geometry 31 mannian. . . ) spaces. This is what we try to do here, within the limitations imposedby the knowledge of the speaker.The first task is to learn to think of ordinary spaces in noncommutative terms.Arguably, this goes back to the Gelfand–Na˘ımark theorem (1943), establishing thatthe information on any locally compact Hausdorff topological space X is fully storedin the commutative algebra C ( X ) of continuous function over it, vanishing at ∞ . Thisis a way to recognize the importance of C ∗ -algebras, and to think of them as locallycompact Hausdorff noncommutative spaces. If we had just asked for the functionsto be measurable and bounded, we would had been led to von Neumann algebras.Vector bundles are identified through their spaces of sections, which algebraicallyare projective modules of finite type over the algebra of functions associated to thebase space —this is the Serre–Swan theorem (1962). In this way, we come to thinkof noncommutative vector bundles.Under the influence of quantum physics, the general idea is then to forget aboutsets of points and obtain all information from classes of functions; e.g. open setsin X are replaced by ideals. The rules of the game would then seem to be: (1) finda way to express a mathematical category through algebraic conditions, and then:(2) relinquish commutativity. This works wonders in group theory, which is replacedby bialgebra theory, relinquishing (co)commutativity. However, that kind of gener-alization quickly runs into sands, for two reasons: (i) Some mathematical objects,like differentiable manifolds, and de Rham cohomology, are reluctant to direct non-commutative generalization. The same is true of Riemannian geometry; after all, allsmooth manifolds are Riemann. (ii) Genuinely new “noncommutative phenomena”are missed.For instance, in the second respect, in many geometrical situations the associatedset is very pathological, and a direct examination yields no useful information. Theset of orbits of a group action, such as the rotation of a circle by multiples of anirrational angle θ , is generally of this type. In such cases, when we examine thematter from the algebraic point of view, we are sometimes able to obtain a perfectlygood operator algebra that holds the information we need; however, this algebra isgenerally not commutative.One can situate the beginning of noncommutative geometry (NCG) in the 1980paper by Connes, where the ‘noncommutative torus’ T θ was studied [60]. Not onlyis this algebra able to answer the question mentioned above, but one can decidewhat are the smooth functions on this noncommutative space, what vector bundlesand connections on T θ are and, decisively, how to construct a Dirac operator on it.Even now, the importance of this early example in the development of the the-ory can hardly be underestimated. The noncommutative torus provides a simple butnontrivial example of spectral triple ( A , H , D ) —see further on for the notation— or‘noncommutative spin manifold’, the algebraic apparatus with which Connes even-tually managed to push aside the obstacles to the definition of noncommutative Rie-mannian manifolds. The Dirac equation naturally lives on spin manifolds, and theseconstitute the crucial paradigm, too, for Connes program of research (and unifica-tion) of mathematics. The more advanced rules of the game would now seem to be: (1) Escape the diffi-culties “from above” by finding the algebraic means of describing a richer structure.If we reformulate algebraically what a spin manifold is, we can describe its de Rhamcohomology, its Riemannian distance and like geometrical concepts, algebraicallyas well. Choice of a Dirac operator D means imposing a metric. However, there isthe risk that the link to the commutative world is obscured. Therefore: (2) Makesure that the link is kept. In other words, prove that a noncommutative spin mani-fold is in fact a spin manifold in the everyday sense (!) when the underlying algebrais commutative. In point of fact, the second desideratum only received a definitive,satisfactory answer a few weeks ago. The following quotation of a popular book [61] provides a convenient rallying point.
When physicists talk about the importance of beauty and elegance in their theories, the Diracequation is often what they have in mind. Its combination of great simplicity and surprisingnew ideas, together with its ability both to explain previously mysterious phenomena andpredict new ones [spin], make it a paradigm for any mathematically inclined theorist.
Thus the irony is in that, first and foremost [61],
Mathematicians were much slower to appreciate the Dirac equation and it had little impacton mathematics at the time of its discovery. Unlike the case with the physicists, the equationdid not immediately answer any questions that mathematicians had been thinking about.
The situation changed only forty years later, with the Atiyah–Singer theory ofthe index.A second and minor irony is that, now that spin manifold theory is an establishedand respectable line of mathematical business, its community of practitioners seemsmostly oblivious to the fact it underpins a whole new branch/paradigm/method ofdoing mathematics (although something is being done to fill up this gap).Now come the informal rules for noncommutative geometers —rules which inany society insiders recognize as the most binding. These seem to be: (1) Keep closeto physics, and in particular to quantum field theory. There is no doubt that Connescame to his ‘axioms’ for noncommutative manifolds by thinking of the StandardModel of particle physics as a noncommutative space. (2) Try to interpret and solvemost problems conceivably related to noncommutative geometry by use of spec-tral triple theory. This of course is not to everyone’s taste, and a cynic could say:“Whoever is good with the hammer, thinks everything is a nail”; moreover it is ofcourse literally impossible, as the mathematical world teems with virtual objects forwhich complete taxonomy is an impossible task. It has proved surprisingly reward-ing, however.A caveat about (2): there is an underlying layer of index theory and K -theory,which is a deep way of addressing quantization. But even there, when you need to otes on “quantum gravity” and noncommutative geometry 33 compute K -theoretic invariants, you are led back to smoother structures where youhave more tools, like ( A , H , D ) . Let us we imagine a star, with NCG in the centre, of subjects intimately related toit. This will include: • Operator algebra theory • K -theory and index theory • Hochschild and cyclic homology • Bialgebras and Hopf algebras, including quantum groups • Foliations, groupoids • Singular spaces • Deformation and quantization theory • Topics in physics: quantum field theory, including noncommutative field theoryand renormalization; gauge theories, including the Standard Model; condensedmatter; gravity; strings
The root of the importance of spectral triples in NCG is found in algebraic topology .Noncommutative topology brings techniques of operator algebra to algebraic topo-logy —and vice versa. As indicated earlier, the method of rephrasing concepts andresults from topology using Gelfand–Na˘ımark and Serre–Swan equivalence, andextending them to some category of noncommutative algebras, recurs for a while.Moreover, deeper proofs of some properties of objects in the commutative world areto be found in their noncommutative counterparts, with Bott periodicity providingan outstanding example.Now, to extend the standard (co)homology functors (not to speak of homotopy)is rather difficult. On the other hand, Atiyah’s K -functor generalizes very smoothly.Given a unital algebra A , its algebraic K -group is defined as the Grothendieckgroup of the (direct sum) semigroup of isomorphism classes of finitely generatedprojective right (or left) modules over A . Then in view of the Serre–Swan theorem K ( C ( X )) = K ( X ) .Given an ordinary space X , the real K -group KO ( X ) —actually, it is a ring, withproduct given by pullback by the diagonal map of the tensor product— for X isobtained as the Grothendieck group for real vector bundles. Higher order groupsare defined by suspension. If X is Hausdorff and compact, we have KO i ( X ) (cid:39) KO i + ( X ) ; this is real Bott periodicity. Recall that we have: KO ( ∗ ) = Z , KO ( ∗ ) = KO ( ∗ ) = Z , KO ( ∗ ) = , KO ( ∗ ) = Z , KO ( ∗ ) = KO ( ∗ ) = KO ( ∗ ) =
0. There isan isomorphism of the spin cobordism classes of a manifold X onto KO • ( X ) [62]. The K - homology of topological spaces can been developed as a functorial theorywhose cycles pair with vector bundles in the same way that currents pair with differ-ential forms in the de Rham theory. Such cycles are given, interestingly enough, byspin c structures. On the other hand, the index theorem shows that the right partnersfor vector bundles are elliptic pseudodifferential operators (with the pairing given bythe index map). We can think of abstract K -cycles as of phases of Dirac operators.In NCG we want to generalize both this and the line element (entering the realm ofRiemannian geometry). Note the result: Proposition 1.
On a spin manifold the geodesic distance between two points obeysthe formula d ( p , q ) = sup { | f ( x ) − f ( y ) | : f ∈ C ( X ) , | [ D , f ] | ≤ } . (36)This is actually trivial, since | [ D , f ] | is the Lipschitz norm of f .The foregoing motivates: Definition 1.
A noncommutative geometry (spectral triple) is a triple ( A , H , D ) ,where A is a ∗ -algebra represented faithfully by bounded operators on the Hilbertspace H and D is a self-adjoint operator D : Dom D → H , with Dom D = H , suchthat [ D , a ] extends to a bounded operator and a ( + D ) − / is a compact operator,for any a ∈ A ; plus a postulate set of conditions given below.We do not explicitly indicate the representation in the notation. A spectral tripleis even when there exists on H a symmetry Γ such that A is even and D odd withrespect to the associated grading. Otherwise, it is odd. A spectral triple is compact when A is unital; it is then enough to require that ( + D ) − / be compact.One should think of A as of an algebra of ‘smooth’, not ‘continuous’ elements.Of course, it is important that K ( A ) = K ( ¯ A ) , with ¯ A the C ∗ -algebra completionof A . Sufficient conditions are known for this.In the compact case the maximal set of postulates includes:1. Summability or Dimension : for a fixed positive integer p , we have ( + D ) − / ∈ L p , + ( H ) , implying Tr ω (( + D ) − p / ) ≥ , for all generalized limits ω ; and moreover Tr ω (( + D ) − p / ) (cid:54) = A : a (cid:55)→ Tr ω ( a ( + D ) − p / ) is a hypertrace.2. Regularity : with δ a : = [ | D | , a ] , one has A ∪ [ D , A ] ⊆ ∞ (cid:92) m = Dom δ m . otes on “quantum gravity” and noncommutative geometry 35 Finiteness: the dense subspace of H which is the smooth domain of D , H ∞ : = (cid:92) m ≥ Dom D m is a finitely generated projective (left) A -module, which carries an A -valuedHermitian pairing ( · | · ) A satisfying (cid:104) ξ | a η (cid:105) = Tr ω (cid:0) a ( ξ | η ) A ( + D ) − p / (cid:1) when ξ , η ∈ H ∞ and a ∈ A . This also implies the absolute continuity propertyof the hypertrace:Tr ω ( a ( + D ) − p / ) > , whenever a > A .4. First-order condition : as well as the defining representation we require a com-muting representation of the opposite algebra A ◦ . Now H ∞ can be regarded as aright A -module. Then we furthermore ask for [[ D , a ] , b ] = a ∈ A , b ∈ A ◦ .(When A is commutative, we could still have different left and right actionson H . If they are equal, the postulate entails that the subalgebra C D A of B ( H ) generated by A and [ D , A ] belongs in End A ( H ∞ ) .)5. Orientation : let p be the metric dimension of ( A , H , D ) . We require that thespectral triple be even if and only if p is even. For convenience, we take Γ = p is odd. We say the spectral triple ( A , H , D ) is orientable if there exists aHochschild p -cycle c = n ∑ α = ( a α ⊗ b α ) ⊗ a α ⊗ · · · ⊗ a p α ∈ Z p ( A , A ⊗ A ◦ ) whose Hochschild class may be called the “orientation” of ( A , H , D ) , such that π D ( c ) : = ∑ α a α b α [ D , a α ] . . . [ D , a p α ] = Γ . (37)6. Reality : there is an antiunitary operator C : H → H such that Ca ∗ C − = a forall a ∈ A ; and moreover, C = ± , CDC − = ± D and also C Γ C − = ± Γ in theeven case, according to the following table of signs depending only on p mod 8: p mod 8 0 2 4 6 C = ± + − − + CDC − = ± D + + + + C Γ C − = ± Γ + − + − p mod 8 1 3 5 7 C = ± + − − + CDC − = ± D − + − + For the origin of this sign table in KR -homology, we refer to [63]. (This postulateis optional, but important in practice. It makes the difference between spin c andspin manifolds.)7. Poincar´e duality : the C ∗ -module completion of H ∞ is a Morita equivalence bi-module between ¯ A and the norm completion of C D A .With the exception of the last, they are essentially in the form given to them byConnes.What good are these terms? We have the following: Proposition 2.
Let M be a compact Riemannian manifold without boundary withRiemannian volume form ν g , and assume there exists a spinor bundle S over it, withconjugation C. Define the Dirac spectral triple associated with it as ( C ∞ ( M ) , L ( M , S ) , D / ) , where L ( M , S ) is the spinor space obtained by completing the spinor module Γ ∞ ( M , S ) with respect to the natural scalar product (using | ν g | ) and D / : = − i ( ˆ c ◦ ∇ S ) is the Dirac operator (for the notation: if c is the action of the Clifford algebra bun-dle over M, then ˆ c ( α , s ) = c ( α ) s, for α in that bundle and s a spinor). Also Γ = c ( γ ) ,where γ is the chirality element of the Clifford bundle, either the identity operatoror the standard grading operator on L ( M , S ) , according as dim M is odd or even.Then the Dirac spectral triple is a commutative noncommutative spin geometry. (Sorry for the bad joke!)The proof is routine. We can relax postulate 6 and obtain just a spin c geometry.The most important thing is to think of the spinor bundle as an algebraic object: thiscomes from Plymen’s characterization [64], suggested by Connes, of spin c struc-tures as Morita equivalence bimodules for the Clifford action induced by the met-ric. The existence of that equivalence is tantamount to the vanishing of the usualtopological obstruction to the existence of spin c structures. A precedent for this al-gebraization is Karrer’s [65]. A recent article by Trautman [66] contains interestinghistorical asides. So far, so good, but there will be a point to the precedent exercise only if we canprove that the algebraic terms of the previous section lead in an essentially uniqueway to a spin manifold. That is, assuming conditions 1 to 7, excluding 6 for the timebeing, and furthermore that A is commutative (this of course entails some simpli-fication in the orientation axiom), is there a spin c manifold M —with dim M = p —such that A (cid:39) C ∞ ( M ) and similarly all of the original spectral triple is reproducedby its Dirac geometry?Proof of this on the assumption that A (cid:39) C ∞ ( M ) for some M is found alreadyin [63]. An attempt to prove it without that strong assumption was announced in otes on “quantum gravity” and noncommutative geometry 37 October 2006 by A. Rennie and J. C. V´arilly [67]. However, this work had someflaws, recently corrected by Connes [68, 69].Some extra technical assumptions are needed for the proof. Rennie and V´arillyassume that the spectral triple ( A , H , D ) is irreducible , that is, the only operators in B ( H ) commuting (strongly) with D and with all a ∈ A are the scalars in C
1. (Thisensures the connectedness of the underlying topological space M .) Moreover, theypostulate the following closedness condition: for any p -tuple of elements ( a , .., a p ) in A , the operator Γ [ D , a ] . . . [ D , a p ]( + D ) − p / has vanishing Dixmier trace;thus, for any ω , Tr ω (cid:0) Γ [ D , a ] . . . [ D , a p ] ( + D ) − p / (cid:1) = . This is is an algebraic analogue of Stokes’ theorem.Their argument to show that the Gelfand–Na˘ımark spectrum M of A is a differ-ential manifold may be conceptually broken into two stages. The first is to constructa vector bundle over the spectrum which will play the role of the cotangent bundle.For that, one identifies local trivializations and bases of this bundle in terms of the‘1-forms’ [ D , a j α ] given by the orientability condition. The aim is then to show thatthe maps a α = ( a α , . . . , a p α ) : M → R p provide coordinates on suitable open subsetsof M ; for that, one must prove that the maps a α are open and locally one-to-one.At this stage one needs to deploy, besides the technical conditions, postulates 1to 5 on our spectral triple. A basic tool is a multivariate C ∞ functional calculus forregular spectral triples, that enables to construct partitions of unity and local inverseswithin the algebra A .However, the strategy of [67] failed to ensure that the maps a α are local homeo-morphisms. Instead, Connes [69] resorted to the inverse function theorem [70], byshowing that regularity and finiteness provide enough smooth derivations of A tobuild nonvanishing Jacobians where needed. This requires delicate arguments withunbounded derivations of C ∗ -algebras, and two other technical assumptions, replac-ing those of [67]: • Skewsymmetry of the Hochschild cycle c under permutations of a α , . . . , a p α . Thisenables one to bypass the cotangent bundle construction and omit the closednessproperty, but is arguably a stronger assumption. • Strong regularity : all elements of End A ( H ∞ ) , not merely those in C D A , lie in (cid:84) ∞ m = Dom δ m .The local injectivity of the maps a α is established by first showing that theirmultiplicity (as maps into R p ) is bounded: this needs delicate estimates in orderto invoke the measure theoretic results of Voiculescu [71]. The smooth functionalcalculus can then be used to construct local charts at all points of M by small shiftsof the original maps a α .Poincar´e duality in K -theory plays no role in the reconstruction of a manifoldas a compact space M with charts and smooth transition functions. However, oncethat has been achieved, it is needed to show that M carries a spin c structure and toidentify the class of ( A , H , D ) as the fundamental class of the spin c manifold. This is done by showing that in this case End A ( H ∞ ) coincides with C D A –see [67, 69]—and in particular strong regularity is moot. The Dirac operator is shown to differfrom D by at most an endomorphism of the corresponding spinor bundle. When M is spin, the latter can be eliminated by a variational argument —as shown byKastler, and by Kalau and Walze, the Wodzicki residue of ( + D ) − p / + gives theEH action; see [63, Sect. 11.4].Once one has at one’s disposal a spin c structure, axiom 6 (Reality) allows torefine it to a spin structure. For that, we refer to [64] —or consult [63]— wherein itis shown that the spinor module for a spin structure is just the spinor module for aspin c structure equipped with compatible change conjugation, which is none otherthan the real structure operator C (acting on H ∞ ); the spin structure is extracted,using C , from a representation of the real Clifford algebra of T ∗ M .It is unlikely [72] that the reconstruction theorem holds under the more stringentconditions set out originally by Connes [73]. Possible redundancy of the system ofpostulates has not been much investigated; but certainly there are indications thatthe ones related with dimension are independent. This was the early paradigm for nc manifolds, where everything works smoothly.For a fixed irrational real number θ , let A θ be the unital C ∗ -algebra generated bytwo elements u , v subject only to the relations uu ∗ = u ∗ u = vv ∗ = v ∗ v =
1, and vu = λ uv where λ : = e π i θ . (38)Let S ( Z ) denote the double sequences a = { a rs } that are rapidly decreasing in thesense that sup r , s ∈ Z ( + r + s ) k | a rs | < ∞ for all k ∈ N . The irrational rotation algebra or noncommutative torus algebra T θ is defined asT θ : = (cid:8) a = ∑ r , s a rs u r v s : a ∈ S ( Z ) (cid:9) . It is a pre-C ∗ -algebra that is dense in A θ . The product and involution in T θ arecomputable from (38): ab = ∑ r , s a r − n , m λ mn b n , s − m u r v s , a ∗ = ∑ r , s λ rs ¯ a − r , − s u r v s . The irrational rotation algebra gets its name from another representation, on L ( T ) :the multiplication operator U and the rotation operator V given by ( U ψ )( z ) : = z ψ ( z ) and ( V ψ )( z ) : = ψ ( λ z ) satisfy (38). In the C ∗ -algebraic framework, U generates the C ∗ -algebra C ( T ) and conjugation by V gives an automorphism α of C ( T ) . Under otes on “quantum gravity” and noncommutative geometry 39 such circumstances, the C ∗ -algebra generated by C ( T ) and the unitary operator V iscalled the crossed product of C ( T ) by the automorphism group { α n : n ∈ Z } ). Insymbols, A θ (cid:39) C ( T ) × α Z . The corresponding action by the rotation angle 2 πθ on the circle is ergodic andminimal (all orbits are dense); it is known that the C ∗ -algebra A θ is therefore simple.Using the abstract presentation by (38), certain isomorphisms become evident.First of all, T θ (cid:39) T θ + n for any n ∈ Z , since λ is the same for both. Next, T θ (cid:39) T − θ via the isomorphism determined by u (cid:55)→ v , v (cid:55)→ u . There are no more isomorphismsamong the T θ .The linear functional τ : T θ → C given by τ ( a ) : = a is positive definitesince τ ( a ∗ a ) = ∑ r , s | a rs | > a (cid:54) =
0; it satisfies τ ( ) = τ ( ab ) = τ ( ba ) . Also, it can be shown that τ extends to a faithful continuous traceon the C ∗ -algebra A θ ; and, in fact, this normalized trace on A θ is unique. The GNSrepresentation space H = L ( T θ , τ ) may be described as the completion of thevector space T θ in the Hilbert norm (cid:107) a (cid:107) : = (cid:112) τ ( a ∗ a ) . Since τ is faithful, the ob-vious map T θ → H is injective; to keep the bookkeeping straight, in this section weshall denote by a the image in H of a ∈ T θ . The GNS representation of T θ is just b (cid:55)→ ab . The vector 1 is obviously cyclic and separating, and the Tomita involutionis given by J ( a ) : = a ∗ , thus J = J † . The commuting representation is then given by b (cid:55)→ J π ( a ∗ ) J † b = J a ∗ b ∗ = ba . To build a two-dimensional geometry, we need to have a Z -graded Hilbert spaceon which there is an antilinear involution C that anticommutes with the grading andsatisfies C = −
1. There is a simple device that solves all of these requirements: wesimply double the GNS Hilbert space by taking H : = H ⊕ H and define C : = (cid:18) − JJ (cid:19) . In order to have a spectral triple, it remains to introduce the operator D . For D tobe selfadjoint and anticommute with Γ , it must be of the form D = − i (cid:18) ∂ † ∂ (cid:19) , for a suitable closed operator ∂ on L ( T θ , τ ) . The order-one axiom, together withthe regularity axiom and the finiteness property lead to ∂ , ∂ † being derivations of T θ .The reality condition CDC † = D is equivalent to the condition that J ∂ J = − ∂ † on L ( T θ , τ ) . Consider the derivations δ ( a rs u r v s ) : = π ir a rs u r v s ; δ ( a rs u r v s ) : = π is a rs u r v s . For concreteness, take ∂ to be a linear combination of the basic derivations basicderivations δ , δ . Apart from a scale factor, the most general such derivation is ∂ = ∂ τ : = δ + τδ with τ ∈ C . In fact, real values of τ must be excluded. Now, D − τ has discrete spectrum of eigenvalues ( π ) − | m + n τ | − , each with multiplicity 2.The Eisenstein series ∑ m , n (cid:54) = , ( m + n τ ) diverges logarithmically, thereby establishingthe two-dimensionality of the geometry. The orientation cycle is given by14 π ( τ − ¯ τ ) ( v − u − ⊗ u ⊗ v − u − v − ⊗ v ⊗ u ) . This makes sense only if τ − ¯ τ (cid:54) =
0, i.e., τ / ∈ R . Thus ( ℑ τ ) − is a scale factor in themetric determined by D τ . (Note a difference with the commutative volume form:since v − u − = λ u − v − , there is also a phase factor λ = e π i θ in the orientationcycle.)We conclude by indicating that the noncommutative torus can be regarded aswell as a deformation, as it corresponds to the Moyal product of periodic functions.There are of course nc tori of all dimensions greater than 2. Real noncompact spectral triples (also called nonunital spectral triples) have im-plicitly been already defined. In practice the data are of the form ( A , (cid:102) A , H , D ; C , Γ ) , where now A is a nonunital algebra and the new element (cid:102) A is a preferred unitiza-tion of A , acting on the same Hilbert space.To get an idea of the difficulties involved in the choice of A , consider the sim-plest commutative case, say of the manifold R p . Depending on the fall-off condi-tions deemed suitable, the smooth nonunital algebras that can represent the manifoldare numerous as the stars in the sky. The problem is compounded in the noncommu-tative case, say when A is a deformation of an algebra of functions. To be on thesafe side, one should take a relatively small algebra at the start of any investigationof examples.Postulates 2, 4 and 6 need no changes with respect to the compact case formula-tion.Now, we ponder: • Dimension of the geometry: for p a positive integer a ( + D ) − / belongs tothe generalized Schatten class L p , + for each a ∈ A , and moreover Tr ω ( a ( + | D | ) − p ) is finite and not identically zero. • Finiteness: the algebra A and its preferred unitization (cid:102) A are pre- C ∗ -algebras.There exists an ideal A of (cid:102) A , including A , which is also a pre- C ∗ -algebra withthe same C ∗ -completion as A , such that the space of smooth vectors is an A -pullback of a finitely generated projective (cid:102) A -module. Moreover, an A -valued otes on “quantum gravity” and noncommutative geometry 41 hermitian structure is defined on H ∞ with the noncommutative integral; this is anabsolute continuity condition. • Orientation: there is a
Hochschild p-cycle c on (cid:102) A , with values in (cid:102) A ⊗ (cid:102) A ◦ . Sucha p -cycle is a finite sum of terms like ( a ⊗ b ) ⊗ a ⊗ · · · ⊗ a p , whose naturalrepresentative by operators on H is given by π D ( c ) in formula (37); the volumeform π D ( c ) must solve the equation π D ( c ) = Γ (even case), or π D ( c ) = . The need for some preferred unitization is plain, as finiteness requires the pres-ence both of a nonunital and a unital algebra. Then examples show the need fora further subtlety, to wit, the nonunital algebra for which summability works is smaller than the nonunital algebra required for finiteness. Also, orientation is de-fined directly on the preferred unitization.The commutative examples were worked out in [74, 75]; there summabilityworks in view of asymptotic spectral analysis for the Dirac operator. In [76] —tosome surprise of Alain Connes— it was shown that Moyal algebras are noncompactspectral triples.It is worthwhile to point out that the NCG versions of the Standard Model arenoncompact spectral triples, too; while there is no end of algebraic intricacies forthe finite dimensional representation [77] required to reproduce the quirks of particlephysics, analytically the problem is to be tackled by the methods of the mentionedpapers [74, 75, 76].
How does one recover the metric geometry of the Riemann sphere S from spectraltriple data? If A is a dense subalgebra of a some C ∗ -algebra containing elements x , y , z and if the matrix p = (cid:18) + z x + iyx − iy − z (cid:19) is a projector, it is easy to see from the projector relations that x , y , z commute andthat x + y + z =
1. Thus A = C ( X ) where X ⊂ S is closed. The condition π D (cid:16) tr (cid:0) ( p − ) ⊗ p ⊗ p (cid:1)(cid:17) = Γ can only hold if X = S . In the same way, Connes sought to obtain the sphere S with its round metric by starting with an analogous projector in M ( A ) : p = (cid:18) ( + z ) qq ( − z ) (cid:19) , with q the quaternion q = (cid:18) a b − b ∗ a (cid:19) , imposing conditions so that π D (cid:16) tr (cid:0) ( p − ) ⊗ p ⊗ p ⊗ p ⊗ p (cid:1)(cid:17) = Γ . Again A is commutative and the 4-sphere relation holds. But then Landi surprisedeveryone by pointing out that one could substitute − λ b ∗ for the entry − b ∗ . With λ = e π i θ , this works into a spectral triple. It was called an isospectral deformation because the Dirac operator remains untouched [78].Again, this generalizes into a θ -deformation of any Riemannian manifold M thatadmits T as a subgroup of its group of isometries. And again, this is essentially aMoyal deformation: if M = G / K , with G compact of rank at least two, then C ∞ ( G ) can be deformed in such a way that C ∞ ( M θ ) is a homogenous space of the compactquantum group C ∞ ( G θ ) [79].The procedure can be generalized to a large family of noncompact Riemannianspin manifolds (with ‘bounded geometry’) that admit an action of T l , for l ≥ R l , for l ≥ S hids suprises, too, if one allows for relaxing the notion of what aDirac operator is [81].) So far, we have played it very safe, and we have said little on how to handle wilderexamples of nc manifolds. Connes himself recommends the following steps [82]:1. Given an algebra A (putative ‘of smooth functions on a nc manifold’), try firstof finding a resolution of it as an A -bimodule, with a view to compute itsHochschild cohomology, and eventually its cyclic homology and cohomology.This is not an easy task in general; it has been performed in the commutativecase and for foliations.2. Many nc spaces arise as ‘bad quotients’. Consider Y : = X / ∼ . If one tries to study C ( Y ) = { f ∈ C ( X ) : f ( a ) = f ( b ) , ∀ a ∼ b } , one often ends up with only constant functions. (It is true that, for proper actionsof Lie groups, even if M / G is not a manifold, there is however an interestingfunctional structure [83, 84], that can be usefully studied by a mixture of “com-mutative” and “noncommutative” methods.) It beckons to drop the commutativ-ity requirement by considering complex functions of two variables defined on otes on “quantum gravity” and noncommutative geometry 43 the graph of the equivalence relation. They will act as bounded operators on theHilbert space of the equivalence class, and they multiply with the convolutionproduct: ( f g ) ab = ∑ a ∼ c ∼ b f ac g cb . (39)Of course, when the quotient space is ‘nice’, one can do that, too; as a rule inthis case, the commutative and noncommutative algebras are Morita equivalent.But in a case as simple as X = [ , ] × Z with ∼ given by ( x , +) ∼ ( x , − ) for x ∈ ] , [ , we obtain for the convolution algebra the “dumbbell” algebra: { f ∈ C ([ , ]) ⊗ M C : f ( ) , f ( ) diagonal } ;and there is no such equivalence. The idea is then to compute the K -theory, inorder to learn as much as possible on the space. Ideally, one would also like tohave ‘vector bundles’, Chern character (using connections and curvature) andeven moduli spaces for Yang-Mills connections —this works wonderfully for nctori, which after all are quotient spaces.Incidentally, families of maps that are semigroups in the commutative word nat-urally become C ∗ -bialgebras in the noncommutative context. We may refer to therecent beautiful paper by Soltan [85], where the quantum family of maps from C to C is identified to the dumbbell algebra.Let us add as well that Connes contends that the foundational step of QuantumMechanics (by Heisenberg in 1925) amounts to replacing an abelian group lawby a groupoid law like (39), in order to make sense of the combination principlesof spectral lines.3. Then come the spectral triples. They respond for K -homology classes, smoothstructure, and metric. There is a surprisingly vast class of spaces that can be de-scribed in this way, under conditions in general less strict than the ones requiredfor the reconstruction theorem.4. The time evolution and thermodynamic aspects.That said, we can prepare our catalogue (leaving aside subjects related to physics,for a moment): • Spaces of leaves of foliations. This was an early, successful application of ncgeometry. By elaborating on the construction of point 2 above, Connes was ableto apply methods of operator theory to foliation theory. • Tilings (periodic and aperiodic). Also under point 2. • Dynamical systems. Also point 2. • Cantor sets and fractals. One can associate spectral triples (Dirac operators) tothem! The algebra of continuous functions on a Cantor set is AF commutative.We omit the details on the construction of ( H , D ) . It is then very interesting toinvestigate the dimension spectrum of the spectral triple. For the classical middle-third Cantor set: Tr ( | D | − s ) = ∑ k l sk = ∑ k ≥ k − sk = − s − − s , given that l k = − k with multiplicities 2 k − . This yields as dimension spectrumlog 2log 3 + π in log 3 , for n ∈ Z . For compact fractal subsets of R n . Christensen and Ivan recently haveconstructed spectral triples not satisfying Weyl’s asymptotic formula —there isno constant c so that the number of eigenvalues N ( Λ ) bounded by Λ fulfils N ( Λ ) − c Λ ∼ lower order in Λ . • Algebraic deformations. Of this the Moyal-like spaces are the outstanding exam-ple. More on that below. • Spherical manifolds which are not isospectral deformations. I refer to [86] andsubsequent papers by Connes and Dubois-Violette. • Nc spaces related to arithmetic problems (including some that have been used byConnes to try to prove the Riemann hypothesis). On this I claim zero expertise. • Quantum Hall effect, related to nc tori. This is due to Bellissard. • Nc spaces from axiomatic QFT. For instance, the local algebras in a supersym-metric model, together with the supercharge as a Dirac operator, constitute aspectral triple. • Nc spaces from renormalization, via dimensional regularization. This is has beenonly hinted at. • The mentioned Standard Model reconstruction from NCG. • Nc spaces from strings. If one goes to the physics archives and asks for “noncom-mutative geometry” or “noncommutative field theory”, what one finds is some-thing as puzzling as particular, that is, perturbative quantum field theory overMoyal hyperplanes. This was popularized by Seiberg and Witten [87] as a cer-tain limit of string theory, but has acquired a life of its own. Nevertheless [76]and subsequent papers [88, 89] tried to make a bridge between this and Connes’paradigm. • Lie algebroids, Lie–Rinehart algebras and the like. It is a little mystery why,while groupoids play a central role in NCG, their infinitesimal version does notseem to play any role. All the more so because the algebraic version of Lie alge-broids, the theory of Lie–Rinehart(–Gerstenhaber) algebras, which seems to bethe good framework for BRS theory, has very much the flavour of NCG, and isquite able to deal with many singular spaces [90]. otes on “quantum gravity” and noncommutative geometry 45
Lie–Rinehart algebras are usually commutative; but some of the results pertain-ing to them can be extended to “softly noncommutative” cases. Most impor-tantly, the theory of Adams operations, that plays such an important role in theHochschild and cyclic cohomology of commutative algebras, can be extended tothe realm of noncommutative spaces [91]. This connects the local index formulaby Connes and Moscovici [92] with combinatorial aspects (the Dynkin operatorof free Lie algebra theory and noncommutative symmetric functions) that havenot been fully explored. • Rota–Baxter operators and skewderivations. A poor man’s path to the nc world(akin to the one taken by some quantum group theorists) is to try to generalize theusual derivative/integral pair. This is elementary stuff with many ramifications. Askewderivation of weight θ ∈ R is a linear map δ : A → A fulfilling the condition δ ( ab ) = a δ ( b ) + δ ( a ) b − θ δ ( a ) δ ( b ) . (40)We may call skewdifferential algebra a double ( A , δ ; θ ) consisting of an alge-bra A and a skewderivation δ of weight θ . A Rota–Baxter map R of weight θ ∈ R on a not necessarily associative algebra A , commutative or not, is a linear map R : A → A fulfilling the condition R ( a ) R ( b ) = R ( R ( a ) b ) + R ( aR ( b )) − θ R ( ab ) , a , b ∈ A . (41)When θ = ( A , δ , R ; θ ) willdenote an algebra A endowed with a skewderivation δ and a corresponding Rota–Baxter map R , both of weight θ , such that R δ a = a for any a ∈ A such that δ a (cid:54) =
0, as well as δ Ra = a for any a ∈ A , Ra (cid:54) =
0. We can check consistency ofconditions (40) and (41) imposed on δ , R : θ δ R ( ab ) = R ( a ) b + aR ( b ) − δ ( R ( a ) R ( b ))= R ( a ) b + aR ( b ) − R ( a ) b − aR ( b ) + θ ab = θ ab ; R δ ( ab ) = R ( a δ ( b )) + R ( δ ( a ) b ) − θ R ( δ ( a ) δ ( b )) = R ( a δ ( b )) + R ( δ ( a ) b ) − R ( a δ ( b )) − R ( δ ( a ) b ) + ab = ab . Rota–Baxter operators have proved their worth in probability theory and combi-natorics, and in the Connes-Kreimer approach to renormalization; but their rangeof applications is much wider. • What is the natural noncommutative algebra structure than one should imposeon ordinary, well behaved manifolds? The author has long contended that theanswer, at least in the equivariant case, is: general Moyal theory. Given the nat-uralness of ordinary Moyal quantization on hyperplanes, the high number of ncspaces that turn out to be related to Moyal quantization, plus the usefulness ofMoyal quantization in proofs (for instance of Bott periodicity in the algebraiccontext), it is surprising that few nc geometers seem interested in general Moyaltheory.
But how to define the latter? It would run as follows. Let X be a phase space, µ a Liouville measure on X , and H the Hilbert space associated to ( X , µ ) . AMoyal or Stratonovich–Weyl quantizer for ( X , µ , H ) is a mapping Ω of X intothe space of selfadjoint operators on H , such that Ω ( X ) is weakly dense in B ( H ) ,and verifying Tr Ω ( u ) = , Tr (cid:2) Ω ( u ) Ω ( v ) (cid:3) = δ ( u − v ) , in the distributional sense. (Here δ ( u − v ) denotes the reproducing kernel forthe measure µ .) Moyal quantizers, if they exist, are unique, and ownership of aMoyal quantizer solves in principle all quantization problems: quantization of a(sufficiently regular) function or “symbol” a on X is effected by a (cid:55)→ (cid:90) X a ( u ) Ω ( u ) d µ ( u ) = : Q ( a ) , and dequantization of an operator A ∈ B ( H ) is achieved by A (cid:55)→ Tr A Ω ( · ) = : W A ( · ) . Indeed, it follows that 1 H (cid:55)→ Q ( a ) = (cid:90) X a ( u ) d µ ( u ) . Moreover, using the weak density of Ω ( X ) , it is clear that: W Q ( a ) ( u ) = Tr (cid:20)(cid:18) (cid:90) X a ( v ) Ω ( v ) d µ ( v ) (cid:19) Ω ( u ) (cid:21) = a ( u ) , so Q and W are inverses. In particular, W Q ( ) = (cid:55)→ H by quantiza-tion, and this amounts to the reproducing property (cid:82) X Ω ( u ) d µ ( u ) = H . Finally,we also have Tr [ Q ( a ) Q ( b )] = (cid:90) X a ( u ) b ( u ) d µ ( u ) . This is the key property. Most interesting cases occur in an equivariant context ;that is to say, there is a (Lie) group G for which X is a symplectic homoge-neous G -space, with µ then being a G -invariant measure on X , and G acts by aprojective unitary irreducible representation U on the Hilbert space H . A Moyalquantizer for the combo ( X , µ , H , G , U ) is a map Ω taking X to selfadjoint ope-rators on H that satisfies the previous defining equations and the equivarianceproperty U ( g ) Ω ( u ) U ( g ) − = Ω ( g · u ) , for all g ∈ G , u ∈ X . otes on “quantum gravity” and noncommutative geometry 47 The question is: how to find the quantizers? The fact that the solution in flatspaces leads to (bounded) parity operators points out to the framework of sym-metric spaces as the natural one to find Moyal quantizers by interpolation. Thisheuristic parity rule was found to work for orbits of the Poincar´e group [93].Noncompact symmetric spaces should provide a wealth of noncompact spec-tral triples (the compact case is somewhat pathological). Recently the author,together with V. Gayral and J. C. V´arilly, has given the Moyal quantization of thesurface of constant negative curvature [94]; a new special function plays therethe main role in framing a subtler version of the parity rule. • Algebraic K -theory, noncommutative geometry and field theory. The role of thetwo first functors of algebraic K -theory in QFT with external fields is ’“well-known”; Connes has dabbled on this, but he has not pursued the subject. To thiswriter, also in relation with [92], it seems extremely promising. This subsection is intended as a taunt. We just lift a corner of the veil.
Direct connection between noncommutative field theory and quantum gravity hasbeen sought in several papers. The basic idea is due to Rivelles [95]. In noncommu-tative gauge theories, translations are equivalent to gauge transformations. This atonce reminds one of gravitation (the case can be made that translations necessarilyinvolve gauge transformations in Yang–Mills theories, too [96]; but this is a weakerstatement). In general, the distinction between internal and geometrical degrees offreedom fades in noncommutative geometry [97]. Indeed in [95] it is shown, usingSeiberg-Witten maps [87], how the field action can be regarded as a coupling toa gravitational background. The idea has been further developed in [98]. In someother papers suggesting a noncommutative geometry formalism for pure classicalgravity, the apparatus is so heavy as to make it difficult to see the forest for thetrees [99]. A different approach is to look for noncommutative corrections to par-ticular classes of spacetimes. This is found in [100]. The “barriers to entry” in thisfield being relatively modest, we cut our remarks short.
There seems to be no good reason to exclude noncommutative manifolds in thesense of Connes from the approaches to quantum gravity based on “sum over geo-metries”. Already, in an important paper [101], Yang has showed that the Eguchiand Hanson gravitational instantons [102] give rise by isospectral deformation to noncommutative noncompact manifolds in the sense of [76]. Now, isospectral de-formation leaves the orientation condition unchanged. The general paradigm is afollows: any
Dirac operator, describing a K -homology class, corresponding to acommutative manifold (thus, for any Riemannian geometry over it) or noncommu-tative one, solves equally well, and on the same footing, the “topological equation”that defines the manifold itself. With the proviso that the volume form remains thesame. The advantages indicated in [57] should apply in this context, too.The punch line: in its present form at least, noncommutative geometry favoursthe unimodular theory. Notice that both terminologies “cosmological constant” and “dark energy” betraytheoretical prejudices.The first name, that we can deal with the observations pointing to an accelerationof the expansion rate of the universe by just including the so-called cosmologicalterm in the Einstein equations. In fact, we do not know the equation of state, notto speak of the evolution laws, of whatever exotic “substance” that might be in-volved [103].The second is related to the belief that the acceleration be caused by fluctuations,or “zero-point energies” of the quantum vacuum, somehow. Alas, this notion herewas entertained by nobody less than Weinberg, whose already mentioned [56] threwboth light and obscurity on the subject.The whole review hangs on the thread that there must be a problem, since: . . . the energy density of the vacuum acts just like a cosmological constant.
However, the effective cosmological constant is quite small (we wouldn’t be hereotherwise). On the face of it, zero-point energies are infinite (well, this is not the casein Epstein–Glaser renormalization, but let us go with the argument). If we take asa sensible cut-off the Planck scale, the amount of “fine-tuning” necessary to canceltheir contribution is mind-boggling. Thus,
Perhaps surprisingly, it was a long time before particle physicists began seriously to worryabout this problem, despite the demonstration in the Casimir effect of the reality of zero-point energies.
The trouble is, that “demonstration” is another urban legend. The negative weightof zero-point fluctuations is unobserved in any laboratory experiment, including theCasimir effect . The latter is measured nowadays well enough. However, the usualderivation in terms of differences of zero-point energies, and its neat result, whereonly c , ¯ h and the geometry of the plates enter, inviting us to think of it as a “propertyof the vacuum”, is misleading. The point has been made recently by Jaffe [104].In truth, the Casimir effect distinguishes itself from other quantum electrodynamics otes on “quantum gravity” and noncommutative geometry 49 only in that (for some geometrical configurations, not for all) it reaches a finite limitas the fine structure constant α ↑ ∞ ; this limit is the usually quoted result. In thatderivation, the plates are treated as perfect conductors. However, a perfect conductorat all frequencies is a physical impossibility. The plasma frequency ω pl = e (cid:114) π nm indicates the frequency above which the conductivity goes to zero; here n is thedensity of conduction band electrons and m their effective mass. So the perfect con-ductor approximation is good if c / d (cid:28) ω pl , with d the distance between plates; thatis for materials and plate distances such that1137 ∼ α (cid:29) mc π ¯ hnd . Still, it remains an approximation. Casimir forces can be and have been calculatedwithout reference to the vacuum. Whether there can be experimental evidence forzero point energies, apart from gravity, is an open question, which may be answeredin the negative for all we know. The lesson is that their putative contribution to thecosmological constant must be in doubt. As Jaffe puts it [104]:
Caution is in order when an effect, for which there is no direct experimental evidence, is thesource of a huge discrepancy between theory and experiment.
Indeed.We might add: nowadays there is a “vacuum fluctuations” branch of mathemat-ics, conductors which are always perfectly so and plates of vanishing thickness etsidaretur . This is to the good, and may be helpful, provided we keep the origins inmind and do not start to draw unwarranted physical inferences! We are remindedof Manin’s dicta. A mathematically rigorous and physically sound account of theCasimir effect without invoking “zero-point energies”, particularly unveiling the un-physical nature of Dirichlet boundary conditions, has been given by Herdegen [105].Parenthetically, one finds in the work of Vachapasti and coworkers on “blackstars” mentioned in the first section [7] a comendable retreat to consideration of physical black holes —collapsing bodies suspended above their Schwarzschild ra-dius forever from a remote observer viewpoint— rather than mathematical blackholes —vacuum solutions of the general relativity equations. While the mathemat-ical study of black holes remains a useful and fascinating subject, the former isrequired to explain astrophysical observations.On the other hand, it is hard to dispute that the energy density of the vacuumitself should act like a cosmological constant. Thus it is rather less clear why theflavourdynamics scale —whereby we are talking not of phantom fluctuations, but ofthe vacuum expected value of the energy itself— does not play a role. Even if one(as this writer) does not trust the Higgs mechanism, there is reason to worry aboutthe contribution of chiral symmetry breaking in the quark condensate, still twelve orders of magnitude above the “observed” range for the cosmological constant. Forthis reason unimodularity as discussed in Section 4 should be taken seriously.A recommended review on the cosmological constant is [106]. Its author dis-misses “fine-tuning” out of hand. Suggestive thinking on the dark energy problemis found in [107].We cannot conclude without mentioning the “LHC connection”. After all, fun-damental scalar fields, hitherto unseen, are assumedly involved in inflation, darkenergy and other cosmological scenarios. It is widely believed that the Higgs parti-cle will be observed after the few first stages of the LHC’s proper operation.Some skepticism is also warranted on that. The reason is that “minimality” of thescalar sector of the Standard Model of partcile physics is just a theoretical preju-dice. This has been particularly emphasized by Strassler [108]. (Yes, entes non suntmultiplicanda praeter necessitatem . But Nature does not care for Ockham’s razor:who ordered the muon?)There is the distinct possibility that something was overlooked at LEP and thatthe Higgs sector be considerably more complicated that in standard lore. Tension hasbeen growing for a while between precision results and direct Higgs searches. Thebasic trouble was laid down by Chanowitz a few years ago [109]: if one eliminatesfrom the precision electroweak data the (outlier) value of the forward-backwardasymmetry into b -quarks, then the expected value for the Higgs mass drops to lessthan 50 GeV or so; with the mentioned outlier attributable to new physics. Other-wise, the overall fit is poor. This leds us to take cum grano salis the exclusion resultsat LEP. For instance, mixing with “hidden world” scalars leads to reduction to thestandard Higgs couplings (consult [110] and references therein), in particular the ZZh coupling; and this could not be, and was not, ruled out by LEP for those rela-tively low energies. Other Higgs sector scenarios shielding the Higgs particle fromdetection have been discussed in [111, 112].Recent experiment has made the situation even murkier: on Halloween night of2008, ghostly (albeit rather abundant) multi-muon events at Fermilab were reportedby (a majority segment of) the CDF collaboration [113]. A possible explanationfor them invokes “new” light Higgs-like particles coupling relatively strongly to the“old” ones, and much less so to the SM fermions and MVB [114, 115]. There isalso the possibility that the visible Higgs boson be rather heavier than expected, thediscrepancy with the precision results being (somewhat brazenly) atributed to newphysics [116, 117]. Then the inert
Higgs boson would be a prime candidate for darkmatter.
Acknowledgements
Help of J. C. V´arilly in preparing these notes is most gratefully acknowl-edged. Thanks are due as well to Michael D¨utsch, who helped me with fine points of the argumentin subsection 3.7. I am also indebeted to the referee, whose detailed comments and remarks helpedto improve the final version.otes on “quantum gravity” and noncommutative geometry 51
References
1. S. L. Glashow,
Interactions , Warner Books, New York, 1988.2. C. Kiefer,
Quantum Gravity , Oxford University Press, Oxford, 2007.3. D. L. Wiltshire, “Exact solution to the averaging problem in cosmology”, Phys. Rev. Lett. (2007) 251101.4. D. L. Wiltshire, “Dark energy without dark energy”, astro-ph/0712.3984.5. B. M. Leith, S. C. Cindy Ng and D. L. Wiltshire, “Gravitational energy as dark energy:concordance of cosmological tests”, Astrophys. J. (2008) L91.6. D. L. Wiltshire, “Cosmological equivalence principle and the weak field limit”, Phys. Rev.D (2008) 084032.7. T. Vachaspati, D. Stojkovic and L. M. Krauss, “Observation of incipient black holes and theinformation loss problem”, Phys. Rev. D (2007) 024005.8. J. D. Anderson, P. A. Laing, E. L. Lau, A. S. Liu, M. M. Nieto and S. G. Turyshev, “Studyof the anomalous acceleration of Pioneer 10 and 11”, Phys. Rev. D (2002) 082004.9. M. M. Nieto and J. D. Anderson, “Search for a solution of the Pioneer anomaly”, Contemp.Phys. (2007) 41.10. M. M. Nieto, “New horizons and the onset of the Pioneer anomaly”, Phys. Lett. B (2008) 483.11. Q. Exirifard, “Constraints on f (cid:0) R ijklijkl (cid:1) gravity: an evidence against the covariant resolutionof the Pioneer anomaly”, Class. Quant. Grav. (2009) 025001.12. A. F. Ra˜nada and A. Tiemblo, “Time, clocks and parametric invariance”, Found. Phys. (2008) 458.13. A. F. Ra˜nada and A. Tiemblo, “The Pioneer anomaly as a quantum cosmological effect”,gr-qc/0804.1904.14. J. D. Anderson, J. K. Campbell, J. E. Ekelund, J. Ellis and J. F. Jordan, “Anomalousorbital-energy changes observed during spacecraft flybys of Earth”, Phys. Rev. Lett. (2008) 091102.15. M. Milgrom, “A modication of the Newtonian dynamics as a possible alternative to thehidden mass hypothesis”, Astrophys. J. (1983) 365.16. L. Smolin, The trouble with physics , Houghton Mifflin, New York, 2006.17. “The speed of gravity —what the experiments say”, T. Van Flandern, Phys. Lett. A (1998) 1.18. R. Colella, A. W. Overhauser and S. A. Werner, “Observation of gravitationally inducedquantum interference”, Phys. Rev. Lett. (1975) 1472.19. J. L. Anderson, Principles of relativity physics , Academic Press, New York, 1967.20. D. M. Greenberger, “The neutron interferometer as a device for illustrating the strange be-havior of quantum systems”, Rev. Mod. Phys. (1983) 875.21. D. M. Greenberger and A. W. Overhauser, “Coherence effects in neutron diffraction andgravity experiments”, Rev. Mod. Phys. (1979) 43.22. U. Bonse and T. Wroblewski, “Measurement of neutron quantum interference in noninertialframes”, Phys. Rev. Lett. (1983) 1401.23. M. Springborg and J. P. Dahl, “Wigner’s phase-space function and atomic structure. II.Ground states for closed shell atoms”, Phys. Rev. A (1987) 1050.24. Ph. Blanchard and Jos´e M. Gracia-Bond´ıa, “Density functional theory on phase space”,2010, in preparation.25. S. Doplicher, K. Fredenhagen and J. E. Roberts, “The quantum structure of spacetime at thePlanck scale and quantum fields”, Commun. Math. Phys. (1995) 187.26. V. V. Nesvizhevsky, A. K. Petukhov, H. G. B¨orner, T. A. Baranova, A. M. Gagarski, G. A.Petrov, K. V. Protasov, A. Yu. Voronin, S. Baeßler, H. Abele, A. Westphal and L. Lucovac,“Study of the neutron quantum states in the gravity field”, Eur. Phys. J. C (2005) 479.27. A. Herdegen and J. Wawrzycki, “Is Einstein equivalence principle valid for a quantum par-ticle?”, Phys. Rev. D (2002) 044007.2 Jos´e M. Gracia-Bond´ıa28. M. Kreuz, V. V. Nesvizhevsky, P. Schmidt-Wellenburg, T. Soldner, M. Thomas, H.G. B¨orner, F. Naraghi, G. Pignol, K. V. Protasov, D. Rebreyend, F. Vezzu, R. Flaminio,C. Michel, L. Pinard, A. Remillieux, S. Baessler, A. M. Gagarski, L. A. Grigorieva, T.M. Kuzmina, A. E. Meyerovich, L. P. Mezhov-Deglin, G. A. Petrov, A. V. Strelkov andA. Yu. Voronin, “A method to measure the resonance transitions between the gravita-tionally bound quantum states of neutrons in the GRANIT spectrometer”, physics.ins-det/0902.0156.29. C. Rovelli, Quantum Gravity , Cambridge University Press, Cambridge, 2004.30. E. P. Wigner, “Relativistic invariance and quantum phenomena”, Rev. Mod. Phys. (1957) 255.31. B. Booß-Bavnbek, G. Esposito and M. Lesch, “Quantum gravity: unification of principlesand interactions, and promises of spectral geometry”, hep-th/0708.1705.32. R. P. Feynman, F. B. Morinigo and W. G. Wagner, Feynman lectures on gravitation ,Addison-Wesley, Reading, MA, 1995.33. T. Ort´ın,
Gravity and strings , Cambridge University Press, Cambridge, 2004.34. S. Weinberg, “Photons and gravitons in S -matrix theory: derivation of charge conservationand equality of gravitational and inertial mass”, Phys. Rev.
135 B (1964) 1049.35. G. Scharf,
Quantum Gauge Theories: A True Ghost Story , Wiley, New York, 2001.36. R. P. Feynman, “Quantum theory of gravity”, Acta Phys. Polon. (1963) 697.37. G. Scharf and M. Wellmann, “Spin-2 gauge theories and perturbative gauge invariance”,Gen. Rel. Grav. (2001) 553.38. M. D¨utsch, “Proof of perturbative gauge invariance for tree diagramas to all orders”, Ann.Phys. (Leipzig) (2005) 438.39. T. Kugo and I. Ojima, “Subsidiary conditions and physical S -matrix unitarity in indefinite-metric quantum gravitation theory”, Nucl. Phys. B (1978) 234.40. J. M. Gracia-Bond´ıa, “BRS invariance for massive boson fields”, to appear in the proceed-ings of the Summer School “Geometrical and topological methods for quantum field the-ory”, Cambridge University Press, 2009; hep-th/0808.2853.41. M. Fierz and W. Pauli, “On relativistic wave equations for particles of arbitrary spin in anelectromagnetic field”, Proc. Roy. Soc. (London) A (1939) 211.42. S. G´omez-Avila, M. Napsuciale, J. A. Nieto and M. Kirchbach, “High integer spins beyondthe Fierz–Pauli framework”, Guanajuato preprint, 2005.43. N. Berline, E. Getzler, M. Vergne, Heat kernels and Dirac operators , Springer, Berlin, 1996.44. L. D. Landau and E. M. Lifshitz,
The classical theory of fields , Pergamon Press, Ox-ford, 1971.45. N. Grillo, “Scalar matter coupled to quantum gravity in the causal approach: finite one-loopcalculations and perturbative gauge invariance”, Ann. Phys. (New York) (2001) 153.46. E. Alvarez, “Quantum gravity: an introduction to some recent results”, Rev. Mod. Phys. (1989) 561.47. D. R. Grigore and G. Scharf, “Massive gravity as a quantum gauge theory”, Gen. Rel. Grav. (2005) 1075.48. M. U. Khafizov, “A quasi-invariant smooth measure on the diffeomorphism group of a do-main”, Math. Notes (Matematicheskie Zametkie) (1990) 134.49. E. Alvarez, “Some general problems of quantum gravity”, in Quantum gravity and cosmol-ogy , J. P´erez-Mercader, J. Sol`a and E. Verdaguer, eds., World Scientific, Singapore, 1992.50. J. Ambjørn and R. Loll, “Non-perturbative Lorentzian quantum gravity, causality and topo-logy change”, Nucl. Phys. B (1998) 407.51. J. Ambjørn, R. Janik, W. Westra and S. Zohren, “The emergence of background geometryfrom quantum fluctuations”, gr-qc/0607013.52. O. Lauscher and M. Reuter, “Asymptotic safety in quantum Einstein gravity: nonperturbativerenormalizability and fractal spacetime structure”, hep-th/0511260.53. A. Connes, “Noncommutative Geometry and the Standard Model with neutrino mixing”,J. High Energy Phys. (2006) 081.54. D. Kreimer, “Not so non-renormalizable gravity”, in
Quantum field theory. Competitivemodels , B. Fauser J. Tolksdorf and E. Zeidler, eds., Birk¨auser, Basel, 2009.otes on “quantum gravity” and noncommutative geometry 5355. M. Rowan-Robinson,
Cosmology , Oxford University Press, Oxford, 2004.56. S. Weinberg, “The cosmological constant problem”, Rev. Mod. Phys. (1989) 1.57. J. J. van der Bij, H. van Dam and Y. J. Ng, “The exchange of massless spin-two particles”.Physica A (1982) 307.58. E. Alvarez and A. F. Faedo, “A comment on the matter-graviton coupling”, Phys. Rev. D (2007) 124016.59. E. Alvarez, “Can one tell Einstein’s unimodular theory from Einstein’s general relativity?J. High Energy Phys. (2005) 002.60. A. Connes, “ C ∗ -algebras and differential geometry”, C. R. Acad Sci. Paris A (1980) 599.61. P. Woit, Not even wrong , Jonathan Cape, London, 2006.62. H. B. Lawson, jr. and M.-L. Michelsohn,
Spin geometry , Princeton University Press, Prince-ton University, New Jersey, 1989.63. J. M. Gracia-Bond´ıa, J. C. V´arilly and H. Figueroa,
Elements of Noncommutative Geometry ,Birkh¨auser, Boston, 2001.64. R. J. Plymen, “Strong Morita equivalence, spinors and symplectic spinors”, J. Oper. Theory (1986) 305.65. G. Karrer, “Einf¨uhrung von Spinoren auf Riemannschen Mannigfaltigkeiten”, Ann. Acad.Sci. Fennicae Ser. A I Math. (1963) 3.66. A. Trautman, “Connections and the Dirac operator on spinor bundles”, J. Geom. Phys. (2008) 238.67. A. Rennie and J. C. V´arilly, “Reconstruction of manifolds in noncommutative geometry”,math.OA/0610418.68. A. Connes, course given at the Coll`ege de France, January–March 2008.69. A. Connes, “On the spectral characterization of manifolds”, math.OA/0810.2088.70. R. S. Hamilton, “The inverse function theorem of Nash and Moser”, Bull. Amer. Math. Soc. (1982), 65.71. D. Voiculescu, “Some results on norm-ideal perturbations of Hilbert-space operators, I & II”,J. Oper. Theory (1979) 3; J. Oper. Theory (1981) 77.72. H. Moscovici, private communication.73. A. Connes, “Gravity coupled with matter and foundation of non-commutative geometry”,Commun. Math. Phys. (1996) 155.74. A. Rennie, “Smoothness and locality for nonunital spectral triples”, K -Theory (2003) 127.75. A. Rennie, “Summability for nonunital spectral triples”, K -Theory (2004) 71.76. V. Gayral, J. M. Gracia-Bond´ıa, B. Iochum and J. C. V´arilly, “Moyal planes are spectraltriples”, Commun. Math. Phys. (2004) 569.77. A. Chamseddine, A. Connes and M. Marcolli, “Gravity and the Standard Model with neu-trino mixing”, Adv. Theor. Math. Phys. (2007) 991.78. A. Connes and G. Landi, “Noncommutative manifolds, the instanton algebra and isospectraldeformations”, Commun. Math. Phys. (2001) 141.79. J. C. V´arilly, “Quantum symmetry groups of noncommutative spheres”, Commun. Math.Phys. (2001) 511.80. V. Gayral, B. Iochum and J. C. V´arilly, “Dixmier traces on noncompact isospectral defor-mations”, J. Funct. Anal. (2006) 507.81. A. Sitarz, “Quasi-Dirac operators and quasi-fermions”, math-ph/0602030.82. A. Connes and M. Marcolli, “A walk in the noncommutative garden”, math.OA/060154.83. G. W. Schwarz, “Smooth functions invariant under the action of a compact Lie group”,Topology (1975) 63.84. P. W. Michor, “Isometric actions of Lie groups and invariants”, lecture notes, Vienna, 1997.85. P. M. Soltan, “Quantum families of maps and quantum semigroups on finite quantumspaces”, math.OA/0610922.86. A. Connes and M. Dubois-Violette, “Noncommutative finite-dimensional manifolds. I.Spherical manifolds and related examples”, Commun. Math. Phys. (2002) 539.87. N. Seiberg and E. Witten, “String theory and Noncommutative Geometry”, J. High EnergyPhys. (1999) 032.4 Jos´e M. Gracia-Bond´ıa88. V. Gayral, “Heat-Kernel approach to UV/IR mixing on ssospectral deformation manifolds”,Ann. Henri Poincar´e (2005) 991.89. V. Gayral, J. M. Gracia-Bond´ıa and F. Ruiz Ruiz, “Position-dependent noncommutativeproducts: classical construction and field theory”, Nucl. Phys. B727 (2005) 513.90. J. Huebschmann, “Poisson cohomology and quantization”, J. reine angew. Math. (1990) 57.91. F. Patras, “Lambda-rings”, in
Handbook of Algebra , vol. 3, M. Hazewinkel, ed., Elsevier,Dordrecht, 2003.92. N. Higson, “The residue index theorem of Connes and Moscovici”, in
Surveys in Noncom-mutative Geometry , vol. 6, N. Higson and J. Roe, eds., American Mathematical Society,Providence, RI, 2006.93. J. F. Cari˜nena, J. M. Gracia-Bond´ıa and J. C. V´arilly, “Relativistic quantum kinematics inthe Moyal representation”, J. Phys. A (1990) 901.94. V. Gayral, J. M. Gracia-Bond´ıa and J. C. V´arilly, “Fourier analysis on the affine group,quantization and noncompact Connes geometries”, J. Noncommutative Geom. (2008) 215.95. V. O. Rivelles, “Noncommutative field theories and gravity”, Phys. Lett. B (2003) 191.96. R. Jackiw and N. S. Manton, “Symmetries and conservation laws in gauge theories”, Ann.Phys. (1980) 257.97. F. Lizzi, R. Szabo and A. Zampini, “Geometry of the gauge algebra in noncommutativeYang–Mills theory”, J. High Energy Phys. (2001) 032.98. H. Steinacker, “Emergent gravity from noncommutative gauge theory”, J. High Energy Phys. (2007) 049.99. P. Aschieri, M. Dimitrijevic, F. Meyer and J. Wess, “Noncommutative Geometry and grav-ity”, Class. Quant. Grav. (2006) 1883.100. S. Marculescu and F. Ruiz Ruiz, “Noncommutative Einstein-Maxwell pp-waves”, Phys. Rev.D (2006) 105004.101. C. Yang, “Isospectral deformations of Eguchi–Hanson spaces as nonunital spectral triples”,math.OA/0804.2114.102. T. Eguchi and A. J. Hanson, “Asymptotically at self-dual solutions to euclidean gravity”,Phys. Lett. B (1978) 249.103. R. Miquel, “Dark energy: an observational primer”, Acta Phys. Polon. B (2008) 2765.104. R. L. Jaffe, “The Casimir effect and the quantum vacuum”, Phys. Rev. D (2005) 021301.105. A. Herdegen, “No-nonsense Casimir force”, Acta Phys. Polon. B (2001) 55.106. S. Nobbenhuis, “Categorizing different approaches to the cosmological constant problem”,Found. Phys. (2006) 613.107. T. Padmanabhan, “Emergent gravity and dark energy”, gr-qc/0802.1798.108. M. J. Strassler, “New signatures and challenges for the LHC”, in Proceedings of the 38thInternational Symposium on Multiparticle Dynamics, hep-ph/0902.0377.109. M. S. Chanowitz, “ Z → b ¯ b decay asymmetry: lose-lose for the Standard Model”, Phys. Rev.Lett. (2001) 231802.110. S. Gopalakrishna, S. Jung and J. D. Wells, “Higgs boson decays to four fermions through anabelian hidden sector”, Phys. Rev. D (2008) 055002.111. J. J. Van der Bij, “No Higgs at the LHC”, hep-ph/0804.3534.112. R. Jora, S. Moussa, S. Nasri, J. Schechter and M. Naeem Shahid, “Simple two Higgs doubletmodel”, Int. J. Mod. Phys. A (2008) 5159.113. T. Aaltonen et al (of the CDF collaboration), “Study of multi-muon events produced in p ¯ p collisions at √ s = .
96 TeV”, hep-ex/0810.5357.114. P. Giromini et al , “Phenomenological interpretation of the the multi-muon events reportedby the CDF collaboration”, hep-ph/0810.5730.115. M. J. Strassler, “Flesh and blood, or merely ghosts? Some comments on the multi-muonstudy at CDF”, hep-ph/0811.1560.116. R. Barbieri, L. J. Hall and V. S. Rychkov, “Improved naturalness with a heavy Higgs boson:an alternative road to CERN LHC physics”, Phys. Rev. D (2006) 015007.117. P. Bin´etruy, “The LHC and the universe at large”, Int. J. Mod. Phys. A24