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Gravitational Anomaly and Hawking Radiation of Apparent Horizon in FRW Universe
Eur. Phys. J. C (2009) 62: 455–458 DOI 10.1140/epjc/s10052-009-1081-4 Letter Gravitational anomaly and Hawking radiation of apparent horizon in FRW universe Ran Lia, Ji-Rong Renb, Shao-Wen Wei Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, Gansu, China Received: 27 February 2009 / Published online: 26 June 2009 © Springer-Verlag / Società Italiana di Fisica 2009 Abstract Motivated by the successful applications of the have been carried out [8–28]. In fact, the anomaly analy- anomaly cancellation method to derive Hawking radiation sis can be traced back to Christensen and Fulling’s early from various types of black hole spacetimes, we further work [29], in which they suggested that there exists a rela- extend the gravitational anomaly method to investigate the tion between the Hawking radiation and the anomalous trace Hawking radiation from the apparent horizon of a FRW uni- of the field under the condition that the covariant conser- verse by assuming that the gravitational anomaly also exists vation law is valid. Imposing boundary condition near the near the apparent horizon of the FRW universe. The result horizon, Wilczek et al. showed that Hawking radiation is shows that the radiation flux from the apparent horizon of just the cancel term of the gravitational anomaly of the co- the FRW universe measured by a Kodama observer is just variant conservation law and gauge invariance. Their basic the pure thermal flux. The result presented here will further idea is that, near the horizon, a quantum field in a black hole confirm the thermal properties of the apparent horizon in a background can be effectively described by an infinite col- FRW universe. -
Chiral Currents from Anomalies
ω2 α 2 ω1 α1 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 1 Chiral currents from Anomalies Michael Stone Institute for Condensed Matter Theory University of Illinois Santa-Fe July 18 2018 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 2 Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 3 Experiment aims to verify: k ρσαβ r T µν = F µνJ − p r [F Rνµ ]; µ µ 384π2 −g µ ρσ αβ k µνρσ k µνρσ r J µ = − p F F − p Rα Rβ ; µ 32π2 −g µν ρσ 768π2 −g βµν αρσ Here k is number of Weyl fermions, or the Berry flux for a single Weyl node. Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 4 What the experiment measures Contribution to energy current for 3d Weyl fermion with H^ = σ · k µ2 1 J = B + T 2 8π2 24 Simple explanation: The B field makes B=2π one-dimensional chiral fermions per unit area. These have = +k Energy current/density from each one-dimensional chiral fermion: Z 1 d 1 µ2 1 J = − θ(−) = 2π + T 2 β(−µ) 2 −∞ 2π 1 + e 8π 24 Could even have been worked out by Sommerfeld in 1928! Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 5 Are we really exploring anomaly physics? Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 6 Yes! Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 7 Energy-Momentum Anomaly k ρσαβ r T µν = F µνJ − p r [F Rνµ ]; µ µ 384π2 −g µ ρσ αβ Michael Stone (ICMT ) Chiral Currents Santa-Fe July 18 2018 8 Origin of Gravitational Anomaly in 2 dimensions Set z = x + iy and use conformal coordinates ds2 = expfφ(z; z¯)gdz¯ ⊗ dz Example: non-chiral scalar field '^ has central charge c = 1 and energy-momentum operator is T^(z) =:@z'@^ z'^: Actual energy-momentum tensor is c T = T^(z) + @2 φ − 1 (@ φ)2 zz 24π zz 2 z c T = T^(¯z) + @2 φ − 1 (@ φ)2 z¯z¯ 24π z¯z¯ 2 z¯ c T = − @2 φ zz¯ 24π zz¯ z z¯ Now Γzz = @zφ, Γz¯z¯ = @z¯φ, all others zero. -
Scientists Observe Gravitational Anomaly on Earth 21 July 2017
Scientists observe gravitational anomaly on Earth 21 July 2017 strange that they named this phenomenon an "anomaly." For most of their history, these quantum anomalies were confined to the world of elementary particle physics explored in huge accelerator laboratories such as Large Hadron Collider at CERN in Switzerland. Now however, a new type of materials, the so-called Weyl semimetals, similar to 3-D graphene, allow us to put the symmetry destructing quantum anomaly to work in everyday phenomena, such as the creation of electric current. In these exotic materials electrons effectively behave in the very same way as the elementary Prof. Dr. Karl Landsteiner, a string theorist at the particles studied in high energy accelerators. These Instituto de Fisica Teorica UAM/CSIC and co-author of particles have the strange property that they cannot the paper made this graphic to explain the gravitational be at rest—they have to move with a constant speed anomaly. Credit: IBM Research at all times. They also have another property called spin. It is like a tiny magnet attached to the particles and they come in two species. The spin can either point in the direction of motion or in the opposite Modern physics has accustomed us to strange and direction. counterintuitive notions of reality—especially quantum physics which is famous for leaving physical objects in strange states of superposition. For example, Schrödinger's cat, who finds itself unable to decide if it is dead or alive. Sometimes however quantum mechanics is more decisive and even destructive. Symmetries are the holy grail for physicists. -
On the Trace Anomaly of a Weyl Fermion in a Gauge Background
On the trace anomaly of a Weyl fermion in a gauge background Fiorenzo Bastianelli,a;b;c Matteo Broccoli,a;c aDipartimento di Fisica ed Astronomia, Universit`adi Bologna, via Irnerio 46, I-40126 Bologna, Italy bINFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy cMax-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut) Am M¨uhlenberg 1, D-14476 Golm, Germany E-mail: [email protected], [email protected] Abstract: We study the trace anomaly of a Weyl fermion in an abelian gauge background. Although the presence of the chiral anomaly implies a breakdown of gauge invariance, we find that the trace anomaly can be cast in a gauge invariant form. In particular, we find that it does not contain any odd-parity contribution proportional to the Chern-Pontryagin density, which would be allowed by the con- sistency conditions. We perform our calculations using Pauli-Villars regularization and heat kernel methods. The issue is analogous to the one recently discussed in the literature about the trace anomaly of a Weyl fermion in curved backgrounds. Keywords: Anomalies in Field and String Theories, Conformal Field Theory arXiv:1808.03489v2 [hep-th] 7 May 2019 Contents 1 Introduction1 2 Actions and symmetries3 2.1 The Weyl fermion3 2.1.1 Mass terms5 2.2 The Dirac fermion8 2.2.1 Mass terms9 3 Regulators and consistent anomalies 10 4 Anomalies 13 4.1 Chiral and trace anomalies of a Weyl fermion 13 4.1.1 PV regularization with Majorana mass 14 4.1.2 PV regularization with Dirac mass 15 4.2 Chiral and trace anomalies of a Dirac fermion 16 4.2.1 PV regularization with Dirac mass 16 4.2.2 PV regularization with Majorana mass 17 5 Conclusions 18 A Conventions 19 B The heat kernel 21 C Sample calculations 22 1 Introduction In this paper we study the trace anomaly of a chiral fermion coupled to an abelian gauge field in four dimensions. -
Chiral Gauge Theories Revisited
CERN-TH/2001-031 Chiral gauge theories revisited Lectures given at the International School of Subnuclear Physics Erice, 27 August { 5 September 2000 Martin L¨uscher ∗ CERN, Theory Division CH-1211 Geneva 23, Switzerland Contents 1. Introduction 2. Chiral gauge theories & the gauge anomaly 3. The regularization problem 4. Weyl fermions from 4+1 dimensions 5. The Ginsparg–Wilson relation 6. Gauge-invariant lattice regularization of anomaly-free theories 1. Introduction A characteristic feature of the electroweak interactions is that the left- and right- handed components of the fermion fields do not couple to the gauge fields in the same way. The term chiral gauge theory is reserved for field theories of this type, while all other gauge theories (such as QCD) are referred to as vector-like, since the gauge fields only couple to vector currents in this case. At first sight the difference appears to be mathematically insignificant, but it turns out that in many respects chiral ∗ On leave from Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany 1 νµ ν e µ W W γ e Fig. 1. Feynman diagram contributing to the muon decay at two-loop order of the electroweak interactions. The triangular subdiagram in this example is potentially anomalous and must be treated with care to ensure that gauge invariance is preserved. gauge theories are much more complicated. Their definition beyond the classical level, for example, is already highly non-trivial and it is in general extremely difficult to obtain any solid information about their non-perturbative properties. 1.1 Anomalies Most of the peculiarities in chiral gauge theories are related to the fact that the gauge symmetry tends to be violated by quantum effects. -
Global Anomalies in M-Theory
CERN-TH/97-277 hep-th/9710126 Global Anomalies in M -theory Mans Henningson Theory Division, CERN CH-1211 Geneva 23, Switzerland [email protected] Abstract We rst consider M -theory formulated on an op en eleven-dimensional spin-manifold. There is then a p otential anomaly under gauge transforma- tions on the E bundle that is de ned over the b oundary and also under 8 di eomorphisms of the b oundary. We then consider M -theory con gura- tions that include a ve-brane. In this case, di eomorphisms of the eleven- manifold induce di eomorphisms of the ve-brane world-volume and gauge transformations on its normal bundle. These transformations are also po- tentially anomalous. In b oth of these cases, it has previously b een shown that the p erturbative anomalies, i.e. the anomalies under transformations that can be continuously connected to the identity, cancel. We extend this analysis to global anomalies, i.e. anomalies under transformations in other comp onents of the group of gauge transformations and di eomorphisms. These anomalies are given by certain top ological invariants, that we explic- itly construct. CERN-TH/97-277 Octob er 1997 1. Intro duction The consistency of a theory with gauge- elds or dynamical gravity requires that the e ective action is invariant under gauge transformations and space-time di eomorphisms, usually referred to as cancelation of gauge and gravitational anomalies. The rst step towards establishing that a given theory is anomaly free is to consider transformations that are continuously connected to the identity. -
A Note on the Chiral Anomaly in the Ads/CFT Correspondence and 1/N 2 Correction
NEIP-99-012 hep-th/9907106 A Note on the Chiral Anomaly in the AdS/CFT Correspondence and 1=N 2 Correction Adel Bilal and Chong-Sun Chu Institute of Physics, University of Neuch^atel, CH-2000 Neuch^atel, Switzerland [email protected] [email protected] Abstract According to the AdS/CFT correspondence, the d =4, =4SU(N) SYM is dual to 5 N the Type IIB string theory compactified on AdS5 S . A mechanism was proposed previously that the chiral anomaly of the gauge theory× is accounted for to the leading order in N by the Chern-Simons action in the AdS5 SUGRA. In this paper, we consider the SUGRA string action at one loop and determine the quantum corrections to the Chern-Simons\ action. While gluon loops do not modify the coefficient of the Chern- Simons action, spinor loops shift the coefficient by an integer. We find that for finite N, the quantum corrections from the complete tower of Kaluza-Klein states reproduce exactly the desired shift N 2 N 2 1 of the Chern-Simons coefficient, suggesting that this coefficient does not receive→ corrections− from the other states of the string theory. We discuss why this is plausible. 1 Introduction According to the AdS/CFT correspondence [1, 2, 3, 4], the =4SU(N) supersymmetric 2 N gauge theory considered in the ‘t Hooft limit with λ gYMNfixed is dual to the IIB string 5 ≡ theory compactified on AdS5 S . The parameters of the two theories are identified as 2 4 × 4 gYM =gs, λ=(R=ls) and hence 1=N = gs(ls=R) . -
Thoughts About Quantum Field Theory Nathan Seiberg IAS
Thoughts About Quantum Field Theory Nathan Seiberg IAS Thank Edward Witten for many relevant discussions QFT is the language of physics It is everywhere • Particle physics: the language of the Standard Model • Enormous success, e.g. the electron magnetic dipole moment is theoretically 1.001 159 652 18 … experimentally 1.001 159 652 180... • Condensed matter • Description of the long distance properties of materials: phases and the transitions between them • Cosmology • Early Universe, inflation • … QFT is the language of physics It is everywhere • String theory/quantum gravity • On the string world-sheet • In the low-energy approximation (spacetime) • The whole theory (gauge/gravity duality) • Applications in mathematics especially in geometry and topology • Quantum field theory is the modern calculus • Natural language for describing diverse phenomena • Enormous progress over the past decades, still continuing 2011 Solvay meeting Comments on QFT 5 minutes, only one slide Should quantum field theory be reformulated? • Should we base the theory on a Lagrangian? • Examples with no semi-classical limit – no Lagrangian • Examples with several semi-classical limits – several Lagrangians • Many exact solutions of QFT do not rely on a Lagrangian formulation • Magic in amplitudes – beyond Feynman diagrams • Not mathematically rigorous • Extensions of traditional local QFT 5 How should we organize QFTs? QFT in High Energy Theory Start at high energies with a scale invariant theory, e.g. a free theory described by Lagrangian. Λ Deform it with • a finite set of coefficients of relevant (or marginally relevant) operators, e.g. masses • a finite set of coefficients of exactly marginal operators, e.g. in 4d = 4. -
Nonperturbative Definition of the Standard Models
PHYSICAL REVIEW RESEARCH 2, 023356 (2020) Nonperturbative definition of the standard models Juven Wang 1,2,* and Xiao-Gang Wen3 1School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08540, USA 2Center of Mathematical Sciences and Applications, Harvard University, Cambridge, Massachusetts 02138, USA 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 14 June 2019; accepted 21 May 2020; published 17 June 2020) The standard models contain chiral fermions coupled to gauge theories. It has been a longstanding problem to give such gauged chiral fermion theories a quantum nonperturbative definition. By classification of quantum anomalies (including perturbative local anomalies and nonperturbative global anomalies) and symmetric interacting invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and by the existence of a symmetric gapped boundary (designed for the mirror sector) on the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3 + 1D lattice without fermion doubling, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the standard models from the 16n-chiral fermion SO(10) grand unification can be defined nonperturbatively via a 3 + 1D local lattice model of bosons or qubits. Furthermore, we propose that standard models from the 15n-chiral fermion -
U(1) Gauge Extensions of the Standard Model
U(1) Gauge Extensions of the Standard Model Ernest Ma Physics and Astronomy Department University of California Riverside, CA 92521, USA U(1) Gauge Extensions of the Standard Model (int08) back to start 1 Contents • Anomaly Freedom of the Standard Model • B − L • Le − Lµ and B − 3Lτ • U(1)Σ • Supersymmetric U(1)X • Some Remarks U(1) Gauge Extensions of the Standard Model (int08) back to start 2 Anomaly Freedom of the Standard Model Gauge Group: SU(3)C × SU((2)L × U(1)Y . Consider the fermion multiplets: (u, d)L ∼ (3, 2, n1), uR ∼ (3, 1, n2), dR ∼ (3, 1, n3), (ν, e)L ∼ (1, 2, n4), eR ∼ (1, 1, n5). Bouchiat/Iliopolous/Meyer(1972): The SM with n1 = 1/6, n2 = 2/3, n3 = −1/3, n4 = −1/2, n5 = −1, is free of axial-vector anomalies, i.e. 2 [SU(3)] U(1)Y : 2n1 − n2 − n3 = 0. 2 [SU(2)] U(1)Y : 3n1 + n4 = 0. 3 3 3 3 3 3 [U(1)Y ] : 6n1 − 3n2 − 3n3 + 2n4 − n5 = 0. U(1) Gauge Extensions of the Standard Model (int08) back to start 3 It is also free of the mixed gravitational-gauge anomaly, U(1)Y : 6n1 − 3n2 − 3n3 + 2n4 − n5 = 0. Geng/Marshak(1989), Minahan/Ramond/Warner(1990) : Above 4 equations ⇒ n1(4n1 − n2)(2n1 + n2) = 0. n2 = 4n1 ⇒ SM; n2 = −2n1 ⇒ SM (uR ↔ dR); n1 = 0 ⇒ n4 = n5 = n2 + n3 = 0. Here eR ∼ (1, 1, 0) may be dropped. (u, d)L, (ν, e)L have charges (1/2, −1/2) and (uR, dR) have charges (n2, −n2). -
Green-Schwarz Anomaly Cancellation
Green-Schwarz anomaly cancellation Paolo Di Vecchia Niels Bohr Instituttet, Copenhagen and Nordita, Stockholm Collège de France, 05.03.10 Paolo Di Vecchia (NBI+NO) GS anomaly cancellation Collège de France, 05.03.10 1 / 30 Plan of the talk 1 Introduction 2 A quick look at the abelian axial anomaly 3 Few words on forms 4 Anomaly cancellation in type IIB superstring theory 5 Anomaly cancellation in type I superstring 6 Conclusions Paolo Di Vecchia (NBI+NO) GS anomaly cancellation Collège de France, 05.03.10 2 / 30 Introduction I The theory of general relativity for gravity was formulated by Einstein in 1915. I It is a four-dimensional theory that extends the theory of special relativity. I While special relativity is invariant under the transformations of the Lorentz group, general relativity is invariant under an arbitrary change of coordinates. I In the twenties it was proposed by Theodor Kaluza and Oskar Klein to unify electromagnetism with gravity by starting from general relativity in a five-dimensional space-time and compactify the extra-dimension on a small circle. I In this way one obtains general relativity in four dimensions, a vector gauge field satisfying the Maxwell equations and a scalar. I This idea of extra dimensions was not pursued in the years after. I In the sixties and seventies, when I started to work in the physics of the elementary particles, everybody was strictly working in four dimensions. Paolo Di Vecchia (NBI+NO) GS anomaly cancellation Collège de France, 05.03.10 3 / 30 I Also the dual resonance model, being a model for hadrons, was obviously formulated in four dimensions. -
Anomaly-Free Supergravities in Six Dimensions
Anomaly-Free Supergravities in Six Dimensions Ph.D. Thesis arXiv:hep-th/0611133v1 12 Nov 2006 Spyros D. Avramis National Technical University of Athens School of Applied Mathematics and Natural Sciences Department of Physics Spyros D. Avramis Anomaly-Free Supergravities in Six Dimensions Dissertation submitted to the Department of Physics of the National Technical University of Athens in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics. Thesis Advisor: Alex Kehagias Thesis Committee: Alex Kehagias Elias Kiritsis George Zoupanos K. Anagnostopoulos A.B. Lahanas E. Papantonopoulos N.D. Tracas Athens, February 2006 Abstract This thesis reviews minimal N = 2 chiral supergravities coupled to matter in six dimensions with emphasis on anomaly cancellation. In general, six-dimensional chiral supergravities suffer from gravitational, gauge and mixed anomalies which, being associated with the breakdown of local gauge symmetries, render the theories inconsistent at the quantum level. Consistency of the theory is restored if the anomalies of the theory cancel via the Green-Schwarz mechanism or generalizations thereof, in a similar manner as in the case of ten-dimensional N = 1 supergravi- ties. The anomaly cancellation conditions translate into a certain set of constraints for the gauge group of the theory as well as on its matter content. For the case of ungauged theories these constraints admit numerous solutions but, in the case of gauged theories, the allowed solutions are remarkably few. In this thesis, we examine these anomaly cancellation conditions in detail and we present all solutions to these conditions under certain restrictions on the allowed gauge groups and representations, imposed for practical reasons.