Fixed points of quantum gravity
On the Frame Fixing in Quantum Gravity
a r Xiv:g r-qc/41127v 13Jan24February 4,20081:40WSPC/Trim Size:9.75in x 6.5in for Proceedings main ON THE FRAME FIXING IN QUANTUM GRA VITY S.MERCURI ICRA —International Center for Relativistic Astrophysics G.MONTANI ICRA —International Center for Relativistic Astrophysics Dipartimento di Fisica,Universit`a di Roma “La Sapienza”,Piazzale Aldo Moro 5,I-00185,Roma,Italy We provide a discussion about the necessity to fix the reference frame before quantizing the gravitational field.Our presentation is based on stressing how the 3+1-slicing of the space time becomes an ambiguous procedure as referred to a quantum 4-metric.In the Wheeler-DeWitt (WDW)approach 3,the quantization of gravity is per-formed in the canonical way,starting from the Arnowitt-Deser-Misner (ADM)ac-tion.The use of the ADM formalism 1is justified by the necessity to obtain Hamil-tonian constraints,but the straightforward quantization of such (3+1)-picture con-tains some relevant ambiguities.In fact,the aim of the WDW approach is to quantize the gravitational field in a particular representation and its outcoming provides essentially information on the quantum dynamics of the 3-metric tensordefined on spatial hypersurfaces.To use the ADM splitting is equivalent to a kind of “gauge fixing”,because it is pre-served only under restricted coordinates transformations (time displacements and 3-diffeomorphisms);the point here is that the “gauge fixing”depends on the field we are quantizing and therefore the canonical approach seems to be an ambiguous procedure.Since in the ADM action the conjugate momenta,πand πi ,respectively to the lapse function N and to the shift vector N i are constrained to vanish,then,on a quantum level,the wave functional of the system does not depend on the lapse function and on the shift vector.The ambiguity relies on regarding as equivalent the fully covariant approach and the “gauge fixed”ADM one,in fact passing from g µνto ADM variables involves a metric dependent procedure,in the sense that we must be able to define a unit time-like normal field n µ(g µνn µn ν=−1),which ensures the space-like nature of h ij (in this respect we recall that h ij≡g µν∂i y µ∂j y νcorresponds to the spatial components of the 4-tensor h µν=g µν+n µn ν).Now the following question arises:how is it possible to speak of a unit time-like normal field1February4,20081:40WSPC/Trim Size:9.75in x6.5in for Proceedings main2for a quantum space-time?Indeed such a notion can be recognized,in quantum regime,at most in the sense of expectation values;therefore assuming the existence of nµbefore quantizing the system dynamics makes the WDW approach physically ill defined.Our point of view is that the canonical quantization of the gravitationalfield can be performed in a(3+1)-picture only if we add,to such a scheme,some information about the existence of the time-like normalfield,as shown in7,5,this result can be achieved by including in the dynamics the kinematical action4,already adopted to quantize“matter”fields on afixed background4.The physical interpretation of such new term either on a classical as well as on a quantum level leads to recognize the existence of a referencefluid and in this sense the analysis of7,5,6converges with the literature on the framefixing problem(see2and references therein).We observe that to include the kinematical action can be regarded as a consequence of fixing in the gravity action the lapse function and the shift vector and,therefore, to choose four independent components of the gravitationalfield,which is just the outcoming of the framefixing.A more physical manner to ensure the existence of a time-like vector consistsoffilling the space time with afluid which plays the role of real reference frame.Here we discuss on a phenomenological ground,the canonical quantization of the gravitationalfield plus a dust referencefluid,outlining some relevant differences between the classical and quantum behavior of this system.The Einstein equations and the conservation law,for the coupled gravity-fluid sys-tem,take the formGµν=χεuµuν,uν∇νuµ=0,∇ν(εuν)=0,(1) where Gµνandχdenotes respectively the Einstein tensor and constant.Remembering a well-known result,it is easy to show that the following relations take place8Gµνuµuν=−H(h ij,p ij)h=χε,Gµνuµhνi=H i(h ij,p ij)h=0.(2)Here h ij(ij=1,2,3)denotes the3-metric of the spatial hypersurfaces orthogo-nal to uµand p ij its conjugate momenta,while H and H i refer respectively to the super-Hamiltonian and to the super-momentum of the gravitationalfield.The above relations hold if we make reasonable assumption that the conjugate momen-tum p ij is not affected by the matter variables(i.e.thefluid term in ADM formalism should not contain the time derivative of the3-metric tensor).Only the Hamilto-nian constraints are relevant for the quantization procedure and,in the comoving frame,when the4-velocity becomes uµ={1,0}(N=1N i=0),we have to retain also the conservation lawε√February4,20081:40WSPC/Trim Size:9.75in x6.5in for Proceedings main3 Thus,when the coordinates system becomes a real physical frame,the Hamiltonianconstraints readH=ω(x i)H i=0.(3) Now,to assign a Cauchy problem for such a system,for which equations(3)play therole of constraints on the Cauchy data,corresponds to provide on a(non-singular)space-like hypersurface,sayΣ(0),the values{h(0)ij,p(0)ij,ε(0)};from these values ω(0)can be calculated by(3).It follows that,by specifying a suitable initial condition,the value ofω(0)can bemade arbitrarily small;from the constraints point of view,a very small value ofω(0)means,if h(0)is not so,that thefluid becomes a test one(beingωa constant ofthe motion);we emphasize that forfinite values ofω,h should not vanish to avoidunphysical diverging energy density of thefluid.The canonical quantization of this system is achieved as soon as we implementthe canonical variables into quantum operators and annihilate the state functionalΨvia the Hamiltonian operator constraints.Thus the quantum dynamics obeysthe following eigenvalue problem:HΨ({hij},ω)=ωΨ({h ij},ω),(4) where{h ij}refers to a whole class of3-geometries,so that the super-momentum constraint holds automatically.We stress how the above result is equivalent to the eigenvalues problem obtainedin7.In the above equation(4),the spatial functionωplays the role of the super-Hamiltonian eigenvalue;in this respect,we observe how its values can no longerbe assigned by the initial values,but they have to be determined via the spectrumof H.We conclude that,in the quantum regime,a real dust referencefluid never approaches a test system.Moreover the presence of non zero eigenvalues for the super-Hamiltonian removesthe so called“frozen formalism”of the WDW equation and confirms the idea thatintroducing a physical unit time like vector provides a consistent and evolutivecanonical quantum gravity dynamics.References1.R.Arnowitt,S.Deser,C.Misner,(1959),Phys.Rev.116,1322.2.J.Bicak,K.Kuchaˇr,(1997),Phys.Rev.D56,4878.3. B.S.DeWitt,(1967),Phys.Rev.160,1113.4.K.Kuchaˇr,Canonical Methods of Quantization,(1981),‘Quantum Gravity2:A Sec-ond Oxford Symposium’,Clarendon Press,Oxford,pp.329-374.5.S.Mercuri,G.Montani,(2003),to appear on Int.Jour.Mod.Phys.D,available ong r-qc/0310077.6.S.Mercuri,G.Montani,(2003),submitted to Class.Quant.Grav.,available on g r-qc/0312077.7.G.Montani,(2002),Nucl.Phys.B634,370.8.T.Thiemann,(2001),available gr-qc/0110034.。
Twist Boundary Conditions of Quantum Spin Chains near the Gaussian Fixed Points
a rXiv:c ond-ma t/967161v123J ul1996Twist Boundary Conditions of Quantum Spin Chains near the Gaussian Fixed Points A.Kitazawa Department of Physics,Tokyo Institute of Technology,Oh-okayama,Meguro-ku,Tokyo 152,Japan February 1,2008Abstract Duality transformation,which relates a high-temperature phase to a low-temperature one,is used exactly to determine the critical point for several models (2D Ising,Potts,Ashkin-Teller,8-vertex),as the self dual condition.By changing boundary condition,numerically we can determine the self-dual(critical)point of the Ashkin-Teller(or Gaus-sian)model.This is the first explicit application of the duality to the numerical calculation,with the use of boundary conditions.In this short note,we propose a new method to determine the 2D Gaus-sian critical point of quantum spin chains.Although the finite-size scaling method is a powerful tool to determine the critical point,difficulty may occur for some cases.This difficulty comes from the structure of scaling operators.By changing the boundary condition,we have the other struc-ture of operators.Therefore,selecting boundary conditions,we can use the preferable structure to determine the critical point.The obtained resultsare summarized in eq.(20)and Fig.2.As an effective theory of the 1D quantum spin systems,the following sine-Gordon model (in Euclidean space-time)has been studiedS =12πα2 dτdx cos √We make the identificationφ≡φ+√2π.There exists the U(1)symmetry for thefieldθbut the second term of eq.(1)violates the U(1)symmetry forφ.For the freefield theory,the scaling dimensions of the vertex operators exp(±in√2φ)are n2/2K and Km2/2, where the integer variables n and m are electric and magnetic charges in the Coulomb gas picture.After the scaling transformation a→e dl a,we have the following renor-malization group equationsdK−18y2,dy2)y.These are the famous recursion relations of Kosterlitz.Up to thefirst orderof y,wefind that y is an irrelevantfield for K>4and relevant for K<4.There is a separatrix32K−1−8ln K−1−y2=0which separates the infrared unstable region from the infrared stable region,and on this separatrix,theBerezinskii-Kosterlitz-Thouless transition occurs.The2D Gaussianfixedline lies on y=0.For K<4and y=0,yflows to infinity.For y>0, φis renormalized toπ/√L x n+2πλC n1n2πz h1αin which h1and¯h1are the conformal weights of O1(x1=h1+¯h1).From eq.(4),we have the following RG equationdλ2cos√2are not allowed.)Thus the OPE coefficient in eq.(4)is zero.This indicates that we cannot expect the simple behavior of thefinite size scaling method.In addition,for the irrelevant scalingfield(x>2),the system is in the massless phase,so thefinite size scaling method does not work to determine thefixed points.If we put artificially half magnetic charges m=±1/2in the system,theOPE relations areO1(z,¯z)O e1/2(0,0)=√2 α¯zK/4O e1/2(0,0)+···,O1(z,¯z)O o1/2(0,0)=−√2 α¯zK/4O o1/2(0,0)+···,(6)whereO1=√2φ,O e1/2=√√O o 1/2=√√2π.(8)Hence when the twist angle Φis π,halfodd integer magnetic charges appear.Recently Fukui and Kawakami[3]studied this model analytically and their results are consistent with eq.(8).However,since their studies were based on the integrability,the off-critical behaviors were not treated.To see what happens when the boundary condition is changed in the Coulomb gas picture,we review the case of the following action[4]S =1√2L ,then we can eliminate the second term of eq.(9)with the additional constant term Φ2K/2πL ,but the boundarycondition is changed as θ(τ,L )=θ(τ,0)−Φ/√2πK ∞−∞dτ L0dx (∂µφ)2+i √2π ∞−∞dτ∂τφ(τ,0).(10)4This shows that there exist magnetic charges∓Φ/2πatτ=±∞.Thus we obtain the ground state energy as[2]2π2 Φ2π 2K.(12) We denote the state corresponding to the vertex operator V n,m=e i√2mφas|n,m .Since there exists a magnetic chargeΦ/2πatτ=−∞,wefind the change of this state as|n,m Φ=|n,m+Φ/2π Φ=0,(13) and because there exists a magnetic charge−Φ/2πatτ=∞,the conjugate state isΦ n,m|=0 n,m+Φ/2π|.(14) Hence we obtain[2]E n,m(Φ)−E0(0)=2π2K+K2π 2,(15)orE n,m(Φ)−E0(Φ)=2π2K+K2π .(16)From this equation,wefind that the state|n,0 Φcorresponds to|n,Φ/2π 0 which has the excitation energy E n,0(Φ)−E0(Φ)=E n,0(0)−E0(0),and the momentum nΦ/L.Note that Dotsenko and Fateev[5]considered the similar situationS=12 Φ′asΦ n,m|=0 n,m+Φ′/2π|(which is consistent to eq.(16)),but in(9) the conjugate relation does not change.This may be only the difference of picture.In the case ofΦ=π,we have half odd integer magnetic charges effec-tively.In this case,|0,−1 π(=|0,−1/2 0)and|0 π(=|0,1/2 0)are degen-erate for freefield theory.Introducing the perturbation term of eq.(1)and using thefirst order perturbation theory,we obtain the hybridized states|ψ1 π=12(|0,−1 π+|0 π)(18) whose parity is even,and|ψ2 π=12i(|0,−1 π−|0 π)(19)whose parity is odd.(Note that only whenΦ=0andπ,parity is a good quantum number.)Using the OPE(6),we obtain thefinite size dependence of energy up to thefirst order perturbation asE1(π)−E0(0)=2π8+2πλ√2 2πL K2LK/2−2+··· .(20)Thus wefind that the energy eigenvalues of these states cross linearly at λ=0.In this stage we consider the symmetry of the states(18),(19).In the Ashkin-Teller language,the half magnetic charge operator√2 (√2)corresponds to the operator˜P=σ1µ2(˜P∗=µ1σ2)[6].The sine-Gordon model(1)is invariant under the transformationφ→φ+π2,θ→θ,and y→−y,(21)and the operators√2and√2are transformed as√2→−√2,√2→√2,(22) thus at the point y=0the system has the self-duality[7].6To verify the above things numerically,we study the following S=1 quantum spin chain,H=Lj=1(1−δ(−1)j)(S x j S x j+1+S y j S y j+1+∆S z j S z j+1).(23)The effective action of this model is described as eq.(1).The whole phasediagram was shown in ref.[8].The transition between the dimer and theHaldane phases is of the2D Gaussian ing the Lanczos method,wecalculate energy eigenvalues offinite systems(L=8,10,12,14).Figure1shows the scaled gap behavior of L=10,12,14systems with the periodicboundary condition for∆=0.5.We can see a minimum of the scaledgap.In Fig.2,we show two low lying energies of the subspace S z= 0with the boundary condition S x L+1=−S x1,S y L+1=−S y1,S z L+1=S z1,which correspond to E1(π)and E2(π).We see the expected behavior(20)for this twisted boundary condition.The obtained Gaussianfixed pointsagree with those obtained by the other method[9].The conformal anomalynumber is calculated as c=0.998for the periodic boundary condition andc(π)=−3.185for theΦ=πtwisted boundary condition.In table1,weshow some extrapolated scaling dimensions.These numerical values areconsistent with eqs.(11),(12),(15),(20).With this method,we can alsodetermine the Gaussianfixed line in the massless XY phase[10]and applyto the S=1spin chains with the single ion anisotropy[11,12].Lastly we remark the case of the following sine-Gordon model,S=12πα2 dτdx cos√2φ(z1)√8φ(z2)e±i√√References[1]J.L.Cardy,Nucl.Phys.B270[FS16],186(1986).[2]F.C.Alcaraz,M.N.Barber,and M.T.Batchelor,Phys.Rev.Lett.58,771(1987),and Ann.Phys.(N.Y.)182,280(1988).[3]T.Fukui and N.Kawakami,preprint cond-mat/9606190[4]H.W.J.Bl¨o te,J.L.Cardy,and M.P.Nightingale,Phys.Rev.Lett.56,742(1986).[5]Vl.S.Dotsenko and V.A.Fateev,Nucl.Phys.B240[FS12],312(1984),B251[FS13],691(1985).[6]L.P.Kadanoffand A.C.Brown,Ann.Phys.(N.Y.)121,318(1979).[7]M.Kohmoto,M.den Nijs,and L.P.Kadanoff,Phys.Rev.B24,5229(1981).[8]A.Kitazawa,K.Nomura,and K.Okamoto,Phys.Rev.Lett.76,4038(1996).[9]K.Nomura and A.Kitazawa,preprint cond-mat/9606127[10]A.Kitazawa and K.Nomura,in preparation[11]J.S´o lyom and T.A.L.Ziman,Phys.Rev.B30,3980(1984).[12]H.J.Schulz and T.A.L.Ziman,Phys.Rev.B33,6545(1986).[13]K.Nomura and K.Okamoto,J.Phys.A:Math.Gen.27,5773(1994).[14]T.Giamarchi and H.J.Schulz,Phys.Rev.B39,4620(1989).x1,0=1/2K x0,1/2(=x0(π))0.17860.34972.7992.798Figure captionsFig.1:The scaled gap behavior of L=10(△),L=12(2)and L=14(3) systems with the periodic boundary condition for∆=0.5.Fig.2:The low lying energies of the L=14system with theΦ=πboundary condition for∆=0.5.Parity even state(E1(π))is denoted as solid line and parity odd state(E2(π))is denoted as dashed line.The crossing point is the critical point and its estimated value isδc=0.2524.9。
量子飞船英文作文
量子飞船英文作文Paragraph 1: Wow, have you ever heard of quantum spaceships? They are like something out of a sci-fi movie, but they actually exist! These incredible machines use the principles of quantum mechanics to travel through space at unimaginable speeds. It's mind-blowing to think about the possibilities that quantum spaceships could bring to the exploration of the universe.Paragraph 2: Picture this: you're sitting in a comfortable seat inside a quantum spaceship, ready for the adventure of a lifetime. Suddenly, the ship disappears from its current location and reappears light-years away in a matter of seconds. The sensation of traveling through space in a quantum spaceship must be absolutely exhilarating. Just imagine the rush of adrenaline as you zip through the cosmos at speeds faster than the speed of light!Paragraph 3: One of the most fascinating aspects of quantum spaceships is the concept of entanglement. Inquantum mechanics, entanglement refers to the phenomenon where particles become connected in such a way that the state of one particle affects the state of another, regardless of the distance between them. This means that quantum spaceships could potentially harness the power of entanglement to communicate instantaneously across vast distances, revolutionizing interstellar communication.Paragraph 4: Another mind-boggling feature of quantum spaceships is the ability to exist in multiple places at once. According to the principles of quantum mechanics, particles can exist in a superposition of states until they are observed or measured. Applying this concept to a spaceship, it means that a quantum spaceship could potentially occupy multiple points in space simultaneously, allowing for instantaneous travel between different locations. It's like having the ability to be in two places at once!Paragraph 5: The technology behind quantum spaceshipsis still in its infancy, but scientists and researchers are working tirelessly to make this futuristic dream a reality.They are exploring ways to overcome the immense challenges posed by quantum mechanics and engineering a spaceship that can harness the power of quantum phenomena. It's anexciting time to be alive, as we witness the birth of a new era in space exploration.Paragraph 6: In conclusion, quantum spaceships are a testament to the incredible advancements humans have made in the field of science and technology. These extraordinary machines hold the potential to revolutionize space travel and our understanding of the universe. As we continue to push the boundaries of what is possible, it's only a matter of time before quantum spaceships become a common sight in our exploration of the cosmos. The future is trulylimitless!。
关于量子的英文作文
关于量子的英文作文英文：Quantum physics is a fascinating and complex field that has revolutionized our understanding of the universe. Atits core, quantum mechanics is the study of the behavior of matter and energy at the atomic and subatomic level. The principles of quantum mechanics are fundamentally different from those of classical physics, and they have led to some truly mind-bending discoveries.One of the most famous examples of quantum mechanics is the concept of superposition. This refers to the idea that a particle can exist in multiple states simultaneouslyuntil it is observed, at which point it collapses into a single state. This can be difficult to wrap your head around, but it has been experimentally verified time and time again.Another important concept in quantum mechanics isentanglement. This occurs when two particles become linkedin such a way that the state of one particle affects the state of the other, regardless of the distance between them. This has led to the development of technologies like quantum cryptography, which uses the principles of entanglement to create secure communication channels.Despite its many successes, quantum mechanics is still not fully understood. There are many mysteries and paradoxes that continue to baffle scientists, such as the famous Schrödinger's cat thought experiment. However, researchers are making progress every day, and thepotential applications of quantum mechanics are truly exciting.中文：量子物理学是一个令人着迷而复杂的领域，它彻底改变了我们对宇宙的理解。
Quantum Mechanics
Quantum MechanicsQuantum mechanics, also known as quantum physics, is a fundamental theory in physics that describes the behavior of matter and energy at the smallest scales—typically at the level of atoms and subatomic particles. It is a complex and intriguing field that has revolutionized our understanding of the universe, challenging our classical intuition and providing the foundation for many modern technologies. However, it also presents various conceptual and interpretational challenges that have sparked debates and discussions among physicists and philosophers. One of the central tenets of quantum mechanics is the principle of superposition, which states that particles can exist in multiple states simultaneously until they are measured or observed. This idea is famously illustrated by Schr?dinger's thought experiment involving a cat in a sealed box, which can be both alive and dead at the same time due to the uncertain state of a radioactive atom. While this concept has been supported by numerous experimental observations, it continues to defy our everyday experience and has led to profound philosophical questions about the nature of reality and the role of observation in shaping it. Another key aspect of quantum mechanics is the phenomenon of entanglement, where particles become interconnected in such a way that the state of one particle instantaneously influences the state of another, regardless of the distance between them. This "spooky action at a distance," as Einstein famously referred to it, has been experimentally verified and forms the basis of quantum technologies such as quantum cryptography and quantum computing. However, it challenges our classical notions of locality and raises questions about the nature of causality and the interconnectedness of the universe. The interpretation of quantum mechanics has been a subject of intense debate since its inception. The Copenhagen interpretation, proposed by Niels Bohr and Werner Heisenberg, asserts that the act of measurement collapses the wave function of a particle, determining its definite state. This view emphasizes the role of observation and measurement in defining reality, giving rise to the famous quote, "If you're not confused by quantum mechanics, you haven't really understood it." On the other hand, alternative interpretations such as the many-worlds interpretation and the pilot-wave theory offer different perspectives on the nature of quantum reality, leadingto ongoing discussions about the true nature of the quantum world. From a technological standpoint, quantum mechanics has led to groundbreaking advancements in areas such as computing, cryptography, and precision measurement. Quantum computers, which harness the principles of superposition and entanglement, havethe potential to solve complex problems exponentially faster than classical computers, offering transformative possibilities for fields such as drug discovery, materials science, and cryptography. Quantum cryptography leverages the principles of quantum mechanics to create secure communication channels that aretheoretically immune to eavesdropping, offering unprecedented levels of data security. Furthermore, quantum sensors and atomic clocks based on quantumprinciples have enabled unprecedented levels of precision in measurements, with applications ranging from GPS systems to gravitational wave detection. Despiteits remarkable successes, quantum mechanics continues to pose profound challenges and mysteries. The concept of wave-particle duality, where particles exhibit both wave-like and particle-like behavior, remains a fundamental enigma. The measurement problem, which addresses the nature of wave function collapse and the role of observation, continues to be a subject of philosophical and theoretical inquiry. Furthermore, the quest for a unified theory that reconciles quantum mechanics with general relativity, the other pillar of modern physics, remains a major unsolved problem in theoretical physics. In conclusion, quantum mechanics stands as one of the most profound and enigmatic theories in the history of science, reshaping our understanding of the fundamental nature of reality and opening up new frontiers in technology and philosophy. Its implications reach far beyond the realm of physics, challenging our intuitions and offering new ways of thinking about the nature of the universe. As we continue to explore and grapple with the intricacies of quantum mechanics, we are certain to encounter newinsights and challenges that will further deepen our appreciation of the quantum world.。
Quantum Mechanics
Quantum MechanicsQuantum Mechanics is a branch of physics that deals with the behavior of matter and energy at the atomic and subatomic level. It is a fascinating fieldthat has challenged our understanding of the universe and has led to the development of new technologies such as transistors, lasers, and computers. One of the most interesting aspects of quantum mechanics is the concept of superposition. Superposition is the idea that a particle can exist in multiple states simultaneously until it is observed or measured. This means that a particle can be in two places at the same time or have two different spins at once. This concept goes against our intuition and common sense, but it has been proven to be true through experiments. Another important concept in quantum mechanics is entanglement. Entanglement is the idea that two particles can become connected in such a way that the state of one particle is dependent on the state of the other, no matter how far apart they are. This concept has been demonstrated through experiments and has led to the development of quantum cryptography, which is a method of encrypting information that is theoretically unbreakable. Despite the many successes of quantum mechanics, there are still many unanswered questions and challenges in the field. One of the biggest challenges is the problem of measurement. When a particle is measured, its wave function collapses, and it is forced to take on a specific state. This raises the question of what causes the wave function to collapse and why it collapses in the way that it does. Another challenge in quantum mechanics is the problem of interpretation. There are several different interpretations of quantum mechanics, each with its own strengths and weaknesses. Some interpretations, such as the Copenhagen interpretation, emphasize the role of the observer in the measurement process, while others, such as the many-worlds interpretation, propose that every possible outcome of a measurement actually occurs in a separate universe. Despite these challenges, quantum mechanics has had a profound impact on our understanding of the universe and has led to the development of new technologies that have transformed our world. As we continue to explore the mysteries of the quantum world, we are sure to uncover new insights and challenges that will keep us fascinated for years to come.。
Unitarity and the Hilbert space of quantum gravity
a r X i v :0803.4212v 1 [h e p -t h ] 28 M a r 2008Unitarity and the Hilbert space of quantum gravity ∗Stephen D.H.Hsu †and David Reeb ‡Institute of Theoretical Science,University of Oregon,Eugene,OR 97403(Dated:March 26,2008)Under the premises that physics is unitary and black hole evaporation is complete (no remnants,no topology change),there must exist a one-to-one correspondence between states on future null andtimelike infinity and on any earlier spacelike Cauchy surface (e.g.,slices preceding the formation ofthe hole).We show that these requirements exclude a large set of semiclassical spacetime configura-tions from the Hilbert space of quantum gravity.In particular,the highest entropy configurations,which account for almost all of the volume of semiclassical phase space,would not have quantumcounterparts,i.e.would not correspond to allowed states in a quantum theory of gravity.I.Concepts All fundamental physical theories that we know,like quantum mechanics or general relativity,obey the principle of unitarity:knowledge about the state of a system at one instant is equivalent to knowledge about its state at any other instant.This is due to a one-to-one correspondence,induced by the evolution equations,between states at two different instants,and allows for prediction of the future and retrodiction of the past if the present state of the system is known.It is appealing to hypothesize that unitarity is itself a fundamental feature of Nature,especially in light of the AdS/CFT correspondence [1],although it remains a challenge to understand how it survives the unification of gravity and quantum mechanics,with the resulting phenomena of black hole evaporation [2].Statistical (microcanonical)entropy S is the logarithm of the number of distinct microstates ψof a system consistent with some imposed macroscopic properties.Thus,entropy S is proportional to the dimensionality of the Hilbert space of allowed ψ’s and measures the amount of information that is encoded in a particular microstate ψ.Unitarity forbids any change in the size of this Hilbert space during evolution of the system,so,in particular,entropy cannot decrease (but may increase as we coarse-grain macroscopic information so that more microstates are accommodated).By this logic,the second law of thermodynamics is a consequence of unitarity.Without a theory of quantum gravity,we do not know,so cannot count,the microstates of black holes (for results in string theory,see [3]).But it has been established semiclassically that a large black hole of mass M emits thermal radiation of temperature T ∼M −1[4],so the entropy in this Hawking radiation is of order the area of the hole:S = dQ/T ∼ dM M ∼A (we use Planck units =c =G =1throughout).Strictly speaking,the Hawking process applies only to the semiclassical part of the evaporation,but the final quantum part releases at most of order the Planck energy,which can be made negligible compared to the initial mass of the hole and is thus unlikely to change the scaling with M of the total amount of radiation entropy.A total black hole entropy of S BH =A/4,corresponding to an entropy density ∼1069bit /m 2on the horizon,is consistent with other evidence ranging from black hole thermodynamics [4,5]to string theory [3],although there are other interpretations of this area entropy as well,e.g.[6].By unitarity or the second law,the entropy carried away in Hawking radiation cannot be smaller than the total black hole entropy if evaporation is complete (as in Figure 1),i.e.,if there are no remnants left at the end of the evaporation process and if no topology change occurs,whereby information that hit the singularity could escape into parts of the universe other than that containing the external observer [7].The existence of Planck size remnantsthat contain significant entropy is aesthetically unpleasing since their number would have to be essentially infinite to account for the unradiated entropy;and in the AdS/CFT picture,topology change in a universe corresponds to an implausible scattering process in the field theory,in which large numbers of degrees of freedom somehow cease to interact (become causally disconnected).II.The PuzzleIn this essay we assume that physics,in particular physics describing black hole phenomena,is unitary and that black hole evaporation is cking a quantum theory of gravity,these assumptions are unjustified,though perhaps well motivated.FIG.1:Penrose diagram of a collapsing spherically symmetric object of mass M:as it contracts,a horizon forms behind which eventually all matter falls into the black hole singularity r=0.Initially(event A),an outside observer sees a shrinking star; after a while(B),her observations are consistent with the existence of a black hole,i.e.dim red-shifted light from the frozen surface and outgoing thermal Hawking radiation,which causes decrease in the central mass;finally(C),all Hawking radiation has passed the observer and the black hole has evaporated completely,leaving behind empty space with all information in the radiation at I+and i+.Also shown is a spacelike Cauchy sliceΣ0preceding the black hole.Under our assumptions of unitarity and complete evaporation there exists a one-to-one correspondence between statesΣ0(matter+gravity)and states on future infinity I+∪i+.Under these assumptions,all information(e.g.,on the spacelike sliceΣ0in Figure1)which falls into a black hole has to re-emerge during evaporation in the Hawking radiation,thereby preserving unitary evolution without remnants or topology change.So there has to exist a one-to-one correspondence between the future states of the universe on future infinity I+∪i+and the past states of the universe on any earlier Cauchy slice(e.g.,one,likeΣ0,prior to the black hole).This requires the future and past Hilbert spaces to have the same dimensionality.And it allows us to answer the question:how big is the Hilbert space of states that can form a black hole of mass M?It has the same dimension as the future Hilbert space on I+∪i+,i.e.the same as the space of Hawking radiation states.Thus,the entropy∼M2in the black hole radiation implies a bound on the entropy of the black hole precursor, or equivalently the dimension of its Hilbert space(we assume an initially pure state).But:there exist semiclassical configurationsΣ0(matter+gravity)that collapse to form a black hole of mass M and much larger than∼M2(i.e.,much larger than that of a black hole of equal ADM mass).We will have entropy SΣdescribe examples below.The assumptions of unitarity and complete evaporation therefore force the exclusion of highly entropic(S>M2) semiclassically allowed states from the Hilbert space of quantum gravity.The,perhaps surprising,result is that not all semiclassical configurations(atfixed ADM mass)have quantum counterparts,i.e.the vast majority do not correspond to allowed states in a quantum theory of gravity.Our argument is logically distinct from the black hole information problem as usually considered(evolution of an initially pure state into a mixed state after evaporation[2];see also[8])–we assume unitary evolution and do not attempt to address how the information manages to escape the black hole.Rather,our result is a statement about the Hilbert space of quantum gravity based on fundamental considerations.And although it can be cast in the form of an entropy bound–restricting the amount of entropy allowed in a black hole precursor–the specific high entropyconfigurations which are excluded do not violate Bousso’s covariant entropy bound[9,10].III.High Entropy ConfigurationsNow we present two examples of classes of configurationsΣ0(matter+gravity)that cause the puzzle outlined above. In both examples,the curvature of space onΣ0makes the ADM mass of the configuration(i.e.,the energy a distant external observer sees and that determines the black hole area and hence Hawking radiation entropy)much smaller than would be suspected from the proper internal volume,to which the initial entropy SΣis proportional.In the case of example(a)(“monsters”),this effect can be ascribed to large negative binding energy[11]which almost cancels the proper mass to yield a relatively small ADM mass.In(b),the Kruskal-FRW example,the reason is the non-monotonic behavior of the radius r of2-spheres across the outer Einstein-Rosen bridge.(a)MonstersOurfirst example is a ball of material which is on the verge of collapsing to form a black hole.Its energy density profile is arranged to produce a curved internal space with large proper volume(see Figure2(a)).The configuration is spherically symmetric,defined by initial data on a Cauchy sliceΣ0at a moment of time symmetry(i.e.,configuration initially“at rest”)without(marginally)trapped surfaces,so thatΣ0has geometryds2 Σ0=ǫ(r)−1dr2+r2dΩ2,K ab Σ0=0,(1) withǫ(r)>0.For given initial matter distributionρ(r),Einstein’s(constraint)equations determine(e.g.,[12])ǫ(r)=1−2M(r)r γ,r0<r<R,(4) with someγ>0and r0≪R(to avoid poles),so that the configuration comes increasingly closer to forming trapped surfaces as rրR(long“neck”in Figure2(a)).It has ADM massM=R2∼R(5)and energy densityρ(r)=M′(r)8πr2∼1rγ/2∼A3/2−β+γ/4,(7)with A∼M2the area of the black hole formed in collapse of this monster.It is now evident that,ifβis constant,one can alwaysfind configuration parametersγsuch that the entropy of the monster exceeds area scaling(hence,the name).This is the case,e.g.,if we model the matter(initially)as a perfect fluid with equation-of-state parameter w.Thenβ=1/(1+w),and we would just have to chooseγ>1for a photon gas(w=1/3)orγ>2for dust(w=0;we assume the dust particles carry some kind of label or have spin). Figure2(b)depicts the time evolution of a monster,which resembles ordinary gravitational collapse(Figure1). The main difference is that,due to our construction,the entropy SΣon the initial Cauchy slice can be much bigger than the entropy S+on future infinity.This monster is therefore a semiclassical configuration with no correspondingFIG.2:(a)Embedding of the monster configurationΣ0intoflat space with one angular dimension suppressed.The“neck”has proper length much bigger than(R−r0),due to the huge factorǫ(r)−1/2,and contains all of the initial entropy SΣ.For r>R the geometry is just that of a Schwarzschild slice with mass M=M ADM.(b)The monster’s future time evolution is similar to ordinary gravitational collapse(cf.Figure1):(almost)all matter and entropy,if it was not already initially,will fall behind a horizon(infall of outer layers soon creates trapped surfaces)and form a black hole which then evaporates,radiating away entropy S+∼M2<SΣpast the external observer to future infinity I+∪i+.microstates in a quantum theory of gravity.Note,if r0is chosen a few orders of magnitude above the Planck length, all involved densitiesρ(r)and s(r)are sub-Planckian,so that our semiclassical analysis naively applies.Furthermore, Bousso’s covariant entropy bound[9]holds in the semiclassical monster spacetime since it falls under the general class of spacetimes for which the theorem of Flanagan,Marolf and Wald[15]applies.(This assumes no large entropy gradients due to,e.g.,shockwaves during evolution,which seems plausible,but has not been proven.)(b)Kruskal-FR W gluingThe second example consists of slices of closed FRW universes which are glued together across Einstein-Rosen bridges,eventually connecting to a large asymptoticallyflat universe(Figure3(a)).Again,a larger proper volume can be accommodated atfixed ADM mass.The configuration is specified,as before,by initial data on a spherically symmetric and time symmetric(K ab|Σ=0)Cauchy sliceΣ0:we take the part of a constant-time slice of the Kruskal spacetime with mass M1(e.g.,part of the U+V=0slice,in usual Kruskal coordinates)that contains one asymptotic region with outside observer A,the Einstein-Rosen bridge at its maximal extent r=2M1and the piece r1l>r>2M1 of the other asymptotic region(right part in Figure3(a)).This is then glued onto the partχ<χ1l of the hypersurface ds2=a212 dχ2+sin2χdΩ2 representing a closed FRW universe at the instant of its maximal expansion a12.By cutting this3-sphere offatχ=χ2r,a second piece of Kruskal containing an Einstein-Rosen bridge can be joined,etc. In our notation the integer subscript n denotes the n-th Einstein-Rosen bridge,and l,r denote left,right(see Figure 3).Matching the geometry across the common boundary requires the transverse metric to be continuous and continu-ously differentiable(i.e.the extrinsic curvature K(3)ab has to be the same on either side);its second derivative can bediscontinuous,as is the energy densityρ,consistent with Einstein’s equation G ab=8πT ab.At the rightmost joining surface in Figure3(a),continuity of the transverse metric means equality of the areas of the spherical sectionsχ=χ1l and r=r1l,i.e.a12sinχ1l=r1l.(8)FIG.3:(a)Embedding of glued Kruskal-FRW initial sliceΣ0intoflat space with one angular dimension suppressed.R is the proper radial distance from the innermost point and r=r(R)gives the radius of the2-sphere labeled R.Additional or larger closed FRW pieces could be adjoined,and there could also be a second asymptotic Kruskal piece(even with mass parameter different from M1)if the far left were not closed offwith a3-sphere.(b)By considering the rightmost Einstein-Rosen bridge, standard energy conditions suffice to show that a singularity will form and that the external observer will see a black hole of mass M1whose Hawking radiation then contains potentially much less entropy than was present onΣ0.In the case of pressureless dust,the time evolved spacetime can be given analytically as Kruskal spacetimes and FRW universes appropriately sewn together(Oppenheimer-Snyder collapse[16,17]).And equality of extrinsic curvatures is,in the case of spherical symmetry,equivalent to continuous differentiability of the area A(R)of2-spheres with respect to proper radial distance R:d−(1−2M1/r)−1/2dr 4πr2 r=r1l,(9)which forcesχ1l∈[π/2,π)and,with(8),2M1=a12sin3χ1l.(10) Equations like(8)and(10)hold at every joining surface,with a modified constraintχr∈[0,π/2]if joining just right of an Einstein-Rosen bridge.From these formulae,a configuration like Figure3(a)can be constructed,e.g.,in the following way:first pick masses M1,M2,...describing the Kruskal pieces(M=0forces the construction to an end), then sizes a12,a23...of the FRW pieces subject to constraints a12≥2M1,2M2,etc.Σ0is then uniquely determined. Invoking Friedmann’s equation with vanishing instantaneous expansion,the FRW pieces have energy densityρ12= 3/8πa212.With s∼ρβ,the entropy of one piece becomesS12=4πa312s χ1lχ2r dχsin2χ∼a3−2β12 χ1l−χ2r−12sin2χ2r .(11)The bracket in(11)approachesπ=O(1)as a12becomes a few times bigger than2M1and2M2.In that case,the total entropy onΣ0isSΣ0=S12+S23+...∼a3−2β12+a3−2β23+...,(12)and so can be made arbitrarily big(for anyβ=1/(1+w)<3/2)by either taking the size of the FRW pieces or their number to be large.Evolved forward in time(Figure3(b)),the entropy in the Hawking radiation that passes the external observer and reaches future infinity is S+∼M21,so again is potentially much less than the entropy on the initial slice(12).As in the case of monsters,the Kruskal-FRW configurations are reasonable semiclassical initial data insofar as all involved densities are well sub-Planckian(if the FRW pieces are a few orders of magnitude bigger than the Planck length). The spacetimes do not violate the covariant entropy bound by the same arguments[15]as before(see also[9]for more specific discussion of entropy bounds in closed FRW universes).Both types of configurations have the pathological property that,under isolated evolution,they must have emerged from a past singularity(white hole).This can be seen via backward evolution of the time symmetric initial data, noting that forward evolution leads to a black hole and future singularity.Furthermore,relaxing the assumption of isolation,the configurations cannot be constructed“in the laboratory,”even via intervention by an arbitrarily advanced civilization[13,14].Despite their pathologies,these configurations represent valid semiclassical states of a matter-gravity system:they are all locally well behaved,in particular do not require large energy or entropy densities, and(if present in the Hilbert space)could be accessible via tunneling starting from an ordinary matter configuration with the same quantum numbers(ADM energy,angular momentum,charge).IV.ConclusionUnder the assumptions that physics is unitary and that black hole evaporation is complete,we have shown that the Hilbert space of quantum gravity is much smaller than expected based on semiclassical considerations.The overwhelming majority(as defined by phase space volume or entropy)of semiclassical matter-gravity configurations are not allowed states in the corresponding quantum theory of gravity.Presumably some new physical principle is required to exclude these high entropy states.This might be due to the very formulation of quantum gravity,e.g.,if it is formulated as a lower-dimensional theory on future infinity,due to holography.Note that in both of our high entropy examples one needs several disconnected holographic screens[18] on which to project all of the spacetime.Whatever the new principle is,it must be global in nature:the excluded configurations have no exceptional characteristics when considered locally.Acknowledgments—The authors thank Sean Carroll,Nick Evans,Ted Jacobson,Hirosi Ooguri,John Preskill and Mark Wise for useful discussions.The authors are supported by the Department of Energy under DE-FG02-96ER40969.[1]For a review,see O.Aharony,S.S.Gubser,J.M.Maldacena,H.Ooguri and Y.Oz,“Large Nfield theories,string theoryand gravity,”Phys.Rept.323,183(2000)[hep-th/9905111].[2]S.W.Hawking,“Breakdown of predictability in gravitational collapse,”Phys.Rev.D14,2460(1976).[3]A.Strominger and C.Vafa,“Microscopic origin of the Bekenstein-Hawking entropy,”Phys.Lett.B379,99(1996)[hep-th/9601029];J.M.Maldacena,A.Strominger and E.Witten,“Black hole entropy in M theory,”JHEP12,002 (1997)[hep-th/9711053].[4]S.W.Hawking,“Particle creation by black holes,”Commun.Math.Phys.43,199(1975).[5]J.D.Bekenstein,“Black holes and entropy,”Phys.Rev.D7,2333(1973).[6]T.Jacobson,“On the nature of black hole entropy,”gr-qc/9908031.[7]S.D.H.Hsu,“Spacetime topology change and black hole information,”Phys.Lett.B644,67(2007)[hep-th/0608175].[8]L.Susskind and J.Lindesay,An introduction to black holes,information and the string theory revolution:The holographicuniverse,World Scientific,Hackensack,USA(2005).[9]R.Bousso,“A covariant entropy conjecture,”JHEP07,004(1999)[hep-th/9905177].[10]R.Bousso,“The holographic principle,”Rev.Mod.Phys.74,825(2002)[hep-th/0203101].[11]S.D.H.Hsu,“Zero energy configurations in general relativity,”gr-qc/9801106.[12]R.M.Wald,General Relativity,The University of Chicago Press,Chicago,USA(1984),chapters6.2and10.2.[13]S.D.H.Hsu and D.Reeb,“Black hole entropy,curved space and monsters,”Phys.Lett.B658,244(2008)[arXiv:0706.3239[hep-th]].[14]R.D.Sorkin,R.M.Wald and Z.J.Zhang,“Entropy of self-gravitating radiation,”Gen.Rel.Grav.13,1127(1981).[15]´E.´E.Flanagan,D.Marolf and R.M.Wald,“Proof of classical versions of the Bousso entropy bound and of the generalizedsecond law,”Phys.Rev.D62,084035(2000)[hep-th/9908070].[16]E.Poisson,A Relativist’s Toolkit–The Mathematics of Black-Hole Mechanics,Cambridge University Press,Cambridge,UK(2004),chapter3.8.[17]C.W.Misner,K.S.Thorne and J.A.Wheeler,Gravitation,W.H.Freeman and Company,San Francisco(1973),chapter32.4.[18]R.Bousso,“Holography in general space-times,”JHEP06,028(1999)[hep-th/9906022].。
Two-dimensional Quantum Field Theory, examples and applications
Abstract The main principles of two-dimensional quantum field theories, in particular two-dimensional QCD and gravity are reviewed. We study non-perturbative aspects of these theories which make them particularly valuable for testing ideas of four-dimensional quantum field theory. The dynamics of confinement and theta vacuum are explained by using the non-perturbative methods developed in two dimensions. We describe in detail how the effective action of string theory in non-critical dimensions can be represented by Liouville gravity. By comparing the helicity amplitudes in four-dimensional QCD to those of integrable self-dual Yang-Mills theory, we extract a four dimensional version of two dimensional integrability.
2 48 49 52 54 56
5 Four-dimensional analogies and consequences 6 Conclusions and Final Remarks
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a r X i v :h e p -t h /0312114v 2 16 M a r 2004CERN-TH-2003-299,SHEP 03-32Fixed points of quantum gravityDaniel F.LitimTheory Division,CERN,CH-1211Geneva 23and School of Physics and Astronomy,University of Southampton,Southampton SO171BJ,U.K.(Daniel.Litim@cern.ch)Euclidean quantum gravity is studied with renormalisation group methods.Analytical results for a non-trivial ultraviolet fixed point are found for arbitrary dimensions and gauge fixing parameter in the Einstein-Hilbert truncation.Implications for quantum gravity in four dimensions are discussed.PACS numbers:11.10.Hi,04.60.-m,,04.50.+hClassical general relativity is acknowledged as the theory of gravitational interactions for distances suffi-ciently large compared to the Planck length.At smaller length scales,quantum effects are expected to become important.The quantisation of general relativity,how-ever,still poses problems.It is known since long that quantum gravity is perturbatively non-renormalisable,meaning that an infinite number of parameters have to be fixed to renormalise standard perturbation the-ory.It has been suggested that Einstein gravity may be non-perturbatively renormalisable,a scenario known as asymptotic safety [1].This requires the existence of a non-trivial ultraviolet fixed point with at most a fi-nite number of unstable directions.Then it would suf-fice to adjust a finite number of parameters,ideally taken from experiment,to make the theory asymptotically safe.Non-perturbative renormalisability has already been es-tablished for a number of field theories [2].The search for fixed points in quantum field theory calls for a renormalisation group study.A particularly useful approach is given by the Exact Renormalisation Group,based on the integrating-out of infinitesimal mo-mentum shells within a path integral representation of the theory [3].The strength of the method is its flexibility when it comes to approximations.Furthermore,general optimisation procedures are available [4],increasing the domain of validity and the convergence of the flow.Con-cequently,the reliability of results based on optimised flows is enhanced [5].For Euclidean quantum gravity the Exact Renormal-isation Group has been made accessible by Reuter [6].Explicit flow equations have been constructed using back-ground field techniques [6,7,8].Diffeomorphism invari-ance under local coordinate transformations is controlled by modified Ward identities,similar to those known for non-Abelian gauge theories [9].In general,methods orig-inally developed for gauge theories [10],with minor mod-ifications,can now be applied to quantum gravity.So far most studies have been concerned with theEinstein-Hilbert truncation based on the operators√g R (g )in the effective action,where g is the de-terminant of the metric tensor g µνand R (g )the Ricci scalar.In four dimensions,the high energy behaviour of quantum gravity is dominated by a non-trivial ultraviolet fixed point [7,11,12],which is stable under the inclusionof R 2(g )interactions [8]or non-interacting matter fields [13].Further indications for the existence of a fixed point are based on dimensionally reduced theories [14]and on numerical studies within simplicial gravity [15].For phe-nomenological applications,see [16].In this Letter,we study fixed points of quantum grav-ity in the approach put forward in [6],amended by an adequate optimisation [4,5].The main new result is the existence of a non-trivial ultraviolet fixed point in the Einstein-Hilbert truncation in dimensions higher than four,a region which previously has not been accessible.Analytical results for the flow and its fixed points are given for arbitrary dimension.The optimisation ensures the maximal reliability of the result in the present trun-cation,thereby strengthening earlier findings in four di-mensions.Implications of these results are discussed.The Exact Renormalisation Group is based on a mo-mentum cutofffor the propagating degrees of freedom and describes the change of the scale-dependent effective action Γk under an infinitesimal change of the cutoffscale k .Thereby it interpolates between a microscopic action in the ultraviolet and the full quantum effective action in the infrared,where the cutoffis removed.In its modern formulation,the renormalisation group flow of Γk with the logarithmic scale parameter t =ln k is given by∂t Γk =116πG kd d x√2dependent functions.This is a consequence of the mo-mentum cutoff,expressed in (1).We introduce dimen-sionlessrenormalised gravitationalandcosmological con-stants gk and λk asg k =k d −2G k ≡k d −2Z −1N,k ¯Gλk =k −2¯λk (3)¯Gand ¯λdenote the unrenormalised Newtonian coupling and cosmological constant at some reference scale k =Λ,and Z N,k denotes the wave function renormalisation factor for the Newtonian coupling.Their flows follow from (1)by an appropriate projection onto the operators in (2).A scaling solution of the flow (1)in the truncation (2)corresponds to fixed points for the couplings (3).Explicit momentum cutoffs have been provided for the fluctuations in the metric field in Feynman gauge [6],and for its component fields in a traceless transverse de-composition in a harmonic background field gauge with gauge fixing parameter α[7].In either case the ten-sor structure of the regulator is fixed,while the scalar part is left free.Here,we employ the optimised cutoffR opt =(k 2−q 2)θ(k 2−q 2)for the scalar part [4,5].In the setup (1)–(3),we have computed the flow equa-tion for arbitrary αand arbitrary dimension.To simplify the notation,a factor 1/αis absorbed into the definition (3).Below we present explicit formulæonly for the limit α→∞where the results take their simplest form.The general case is discussed elsewhere.The β-functions areβλ≡∂t λ=P 1P 2+4(d +2)g(4)with polynomials P 1,2(λ,g ;d )P 1=−16λ3+4λ2(4−10d g −3d 2g +d 3g )+4λ(10d g +d 2g −d 3g −1)+d (2+d )(d −16g +8d g −3)g P 2=8(λ2−λ−d g )+2.A numerical factor c d =Γ(d(d −2)g −2(λ−11+2(2−d )g.(6)The flow (6)displays two fixed points,the Gaussian one at g =0and a non-Gaussian one at g ∗=1/(4d ).In the vicinity of the non-trivial fixed point for large k ,the gravitational coupling behaves as G k ≈g ∗/k d −2.This behaviour is similar to asymptotic freedom in Yang-Mills theories.The anomalous dimension of the graviton (5)remains finite and ≤0for all g between the infrared and the ultraviolet fixed point,because g k ≤g ∗<g bound (0)for all d >−2and g Λ≤g ∗.At the fixed point,universal observables are the eigenvalues of the stability matrix at criticality,i.e.the critical exponents.The universal eigenvalue θis given by ∂βg /∂g |∗=−θ.We findθG =2−d ,θNG =2dd −2g Λ−1/θGg ∗−g kΛ.(8)and g ∗,θG and θNG as given above.Fig.1shows the crossover from the infrared to the ultraviolet fixed point in the analytic solution (8)in four dimensions (and with c 4in g reinserted).The corresponding crossover momentum scale is associated with the Planck mass,M Pl =(G )−1/2.More generally,(8)is a solution at λ=0for arbitrary gauge fixing parameter and regu-lator.In these cases,the exponents θand the fixed point g ∗turn into functions of the latter.It is reassuring that the eigenvalues only display a mild dependence on these parameters.3ln(k/M Pl )ηgIRUV-4-224-20246Fig.1:Running of couplings according to(6),(8).Nowweproceed to the non-trivial fixed points implied by the simultaneous vanishing of βg and βλ,given in (3).A first consequence of P 2=0is g ∗=(λ∗−121±√2.The branch with λ∗≥12d (d 2−d −4)2+2)(4π)d/2−1167)/3,orθ′=1.667,θ′′=4.308.(10)More generally,the eigenvalues remain complex for alldimensions 2.56<d <21.4.In d =4,and for gen-eral gauge fixing parameter,the eigenvalues vary between θ′≈1.5−2and θ′′≈2.5−4.3.The range of variation serves as an indicator for the self-consistency of (2).The result (10)and the variation with αagree well with ear-lier findings based on other regulators [7,11].In the approximation (6),which does not admit complex eigen-values,the ultraviolet eigenvalue reads θNG =8/3in 4d .Both θNG and θ′agree reasonably well with the criti-cal eigenvalue θ≈3detected within a numerical study of 4d simplicial gravity with fixed cosmological constant [15].In view of the conceptual differences between the numerical analysis of [15]and the present approach,the precise relationship between these findings requires fur-ther clarification.Still,the qualitative agreement is very encouraging.g λFig.2:The separatrix in four dimensions.Next,we discuss the main characteristics of the phase portrait defined through (4).Finiteness of the flow im-plies that the line 1/η=0cannot be crossed.Slowness of the flow implies that the line η=0can neither be crossed:in its vicinity,the running of g is βg =(d −2)g +O (g 2),and the gravitational coupling approaches g =0without ever reaching (nor crossing)it for any scale k .Thus,dis-connected regions of renormalisation group trajectories are characterised by whether g is larger or smaller g bound and by the sign of g .Since ηchanges sign only across the lines η=0or 1/η=0,we conclude that the graviton anomalous dimension has the same sign along any tra-jectory.In the physical domain which includes the ultra-violet and the infrared fixed point,the gravitational cou-pling is positive and the anomalous dimension negative.In turn,the cosmological constant may change sign on trajectories emmenating from the ultraviolet fixed point.Some trajectories terminate at the boundary g bound (λ),linked to the present truncation (cf.Fig.2).The two fixed points are connected by a separatrix.In Fig.2,it has been given explicitly in four dimensions (with the factor c 4in g reinstalled).Integrating the flow starting in the vicinity of the ultraviolet fixed point and fine-tuning the initial condition leads to the trajectory which runs into the Gaussian fixed point.The rotation of the separa-trix about the ultraviolet fixed point reflects the complex nature of the eigenvalues (10).At k ≈M Pl ,the flow dis-plays a crossover from ultraviolet dominated running to infrared dominated running.For the running couplings and ηthis behaviour is displayed in Fig.3.The non-vanishing cosmological constant modifies the flow mainly in the crossover region rather than in the ultraviolet.A similar behaviour is expected for operators beyond the truncation (2).This is supported by the stability of the fixed point under R 2(g )corrections [8],and by the weak dependence on the gauge fixing parameter.In the in-frared limit,the separatrix leads to a vanishing cosmo-logical constant ¯λk =λk k 2→0.Therefore,it is inter-preted as a phase transition boundary between cosmolo-gies with positive or negative cosmological constant at large distances.This picture agrees very well with nu-merical results for a sharp cut-offflow [12],except for the location of the line 1/η=0which is non-universal.4ln(k/M Pl )ηλg IR UV-4-224-4-3-2-1012Fig.3:Running of couplings along the separatrix.Finally,we note that the qualitative picture detailed above persists in dimensions higher than four.Therefore quantum gravity in higher dimensions may well be formu-lated as a fundamental theory.This consideration is also of interest for recent phenomenological scenarios based on gravity in extra dimensions.In higher dimensions,higher dimensional operators beyond the truncation (2)are likely to be more relevant than in lower ones.Conse-quently,the projection of the full flow onto the Einstein-Hilbert truncation (2)and the respective domain of va-lidity are more sensitive to the cutoff.A first analysis for specific cutoffs in d >4has revealed that the fixed point exists up to some finite dimension,where λ∗reached the boundary of the domain of validity [12].No definite con-clusion could be drawn for larger dimensions.Based on an optimised flow,the main new result here is that a non-trivial ultraviolet fixed point exists within the domain of validity of (4)for arbitrary dimension.Furthermore,thefixed point is smoothly connected to its 4d counterpart,and shows only a weak dependence on the gauge fixing parameter.In the limit of arbitrarily large dimensions,this leads to λ∗→1/2and g ∗→c d /(2d 2)in the ex-plicit solution (9).Similar results are obtained for arbi-trary gauge fixing parameter.Note that λ∗approaches its boundary value very slowly,increasing from 1/4to 0.4for d ranging from 4to ing (9)and the defi-nition for g bound ,we derive g bound (λ∗)/g ∗=(2d )/(d −2)at the fixed point.Hence,g ∗stays clear by at least a factor of two from the boundary where 1/ηvanishes,and the ultraviolet fixed point resides within the domain of validity of (4)even in higher dimensions.This stabil-ity of the fixed point also strengthens the result in lower dimensions,including four.In summary,we have found analytic results for the flow and a non-trivial ultraviolet attractive fixed point of quantum gravity.Maximal reliability of the present trun-cation is guaranteed by the underlying optimisation.The fixed point is remarkably stable 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