Functional Determinants in Quantum Field Theory 2
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Functional Determinant of the Massive Laplace Operator and the Multiplicative Anomaly
Functional Determinant of the Massive Laplace Operator and the Multiplicative Anomaly a,b c a,b Guido Cognola ∗, Emilio Elizalde †, and Sergio Zerbini ‡ (a) Dipartimento di Fisica, Universit`adi Trento via Sommarive 14, 38123 Trento, Italia (b) TIFPA (INFN), Trento, Italia (c) Consejo Superior de Investigaciones Cientficas ICE (CSIC-IEEC), UAB Campus, 08193 Bellaterra, Barcelona, Spain Abstract After a brief survey of zeta function regularization issues and of the related multiplica- tive anomaly, illustrated with a couple of basic examples, namely the harmonic oscillator and quantum field theory at finite temperature, an application of these methods to the computa- tion of functional determinants corresponding to massive Laplacians on spheres in arbitrary dimensions is presented. Explicit formulas are provided for the Laplace operator on spheres in N = 1, 2, 3, 4 dimensions and for ‘vector’ and ‘tensor’ Laplacians on the unitary sphere S4. 1 Introduction In quantum field theory (QFT), the Euclidean partition function plays a very important role. The full propagator and all other n point correlation functions can be computed by means of it. − Moreover, this tool can be extended without problem to curved space-time [1]. As a formalism this is extremely beautiful but it must be noticed that in relativistic quantum field theories an infinite number of degrees of freedom is involved and, as a consequence, ultraviolet divergences will be present, thus rendering regularization and renormalization compulsory. In the one-loop approximation, and in the external field approximation too, one may describe (scalar) quantum fields by means of a (Euclidean) path integral and express the Euclidean parti- tion function as a function of functional determinants associated with the differential operators involved. -
Classical Chaos Described by a Density Matrix
Article Classical Chaos Described by a Density Matrix Andres Mauricio Kowalski , Angelo Plastino * and Gaspar Gonzalez IFLP, Universidad Nacional de La Plata, La Plata 1900, Argentina; kowalski@fisica.unlp.edu.ar (A.M.K.); [email protected] (G.G.) * Correspondence: plastino@fisica.unlp.edu.ar Abstract: In this paper, a reference to the semiclassical model, in which quantum degrees of freedom interact with classical ones, is considered. The classical limit of a maximum-entropy density matrix that describes the temporal evolution of such a system is analyzed. Here, it is analytically shown that, in the classical limit, it is possible to reproduce classical results. An example is classical chaos. This is done by means a pure-state density matrix, a rather unexpected result. It is shown that this is possible only if the quantum part of the system is in a special class of states. Keywords: classical limit; semiclassical system; semiclassical chaos; clasiccal chaos; maximum entropy density matrix 1. Introduction The quantum–classical transition is a frontier issue, a rather transcendental physics topic [1–5]. The use of semi-classical systems to describe problems in physics has a long history; see, e.g., [6–12]. A particularly important case to be highlighted is the one, in which the quantum features of one of the two components of a composite system are negligible in comparison with those of the other. Regarding the first component as a classical one simplifies the description and provides deep insight into the combined system Citation: Kowalski, A.M.; Plastino, dynamics [13]. This methodology is used for the interaction of matter with a field. -
Semiclassical Modeling of Multiparticle Quantum Mechanics
UNIVERSITY OF CALIFORNIA Los Angeles Semiclassical Modeling of Quantum-Mechanical Multiparticle Systems using Parallel Particle-In-Cell Methods A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Dean Edward Dauger 2001 © Copyright by Dean Edward Dauger 2001 To my parents, Alan and Marlene, who were always fully supportive of my work and my endeavors. And Charlie and Allegra, my cats. And to all those who have a vision, a dream of something new and inspiring, to express that Idea, limited only by its own conclusion if it has one. iii Contents I. Introduction A. Motivation ···················································································1 B. Existing Methods ···················································································6 C. Outline ···················································································8 D. Conventions ···················································································9 II. Theory A. The Approach ·················································································11 B. Feynman Path Integrals···············································································13 C. The Semiclassical Approximation·······························································18 D. Initial Position and Final Momentum·························································21 E. The Matrix ··················································································23 F. The Determinant -
Fact and Folklore in Quantum Gravity∗
Draft. Do not quote or distribute. Final version forthcoming in Philosophy of Science To Quantize or Not to Quantize: Fact and Folklore in Quantum Gravity∗ Christian W¨uthrich University of Pittsburgh 30 September 2004 Abstract Does the need to find a quantum theory of gravity imply that the gravitational field must be quantized? Physicists working in quantum gravity routinely assume an affirmative answer, often without being aware of the metaphysical commitments that tend to underlie this assumption. The ambition of this article is to probe these commitments and to analyze some recently adduced arguments pertinent to the issue of quantization. While there exist good reasons to quantize gravity, as this analysis will show, alternative approaches to gravity challenge the received wisdom. These renegade approaches do not regard gravity as a fundamental force, but rather as effective, i.e. as merely supervening on fundamental physics. I will urge that these alternative accounts at least prove the tenability of an opposition to quantization. 1 Introduction All major approaches to quantum gravity endorse, implicitly or explicitly, the view that gravity must be “quantized” in order to be amenable to a description at the fundamental level. To be sure, different approaches pro- pose to execute this task in radically different manners. But they agree ∗I wish to thank John Earman, Florian Girelli, John Moffat, Carlo Rovelli and Danny Terno for discussions. This work is supported by the Swiss National Science Foundation (grant PBSK1–102693). 1 in that without a quantization of gravity no consistent quantum theory of gravity will be forthcoming. What such a quantization involves will vary with the approach taken. -
Functional Determinants, Index Theorems, and Exact Quantum Black Hole Entropy
Prepared for submission to JHEP Functional determinants, index theorems, and exact quantum black hole entropy Sameer Murthy1 and Valentin Reys2 1Department of Mathematics, King’s College London The Strand, London WC2R 2LS, U.K. 2Nikhef theory group, Science Park 105, 1098 XG Amsterdam, The Netherlands Abstract: The exact quantum entropy of BPS black holes can be evaluated us- ing localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the Q operator). We compute the fluctuation determinant for V vector multiplets and hyper multiplets around Q-invariant off-shell configurations in four-dimensional = 2 supergravity with AdS S2 boundary conditions, using the N 2 × Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in = 2 supergravity. We explain cancel- 1 N lations concerning 8 -BPS black holes in = 8 supergravity that were observed in arXiv:1504.01400v3 [hep-th] 2 Dec 2015 N arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory. Keywords: Black hole entropy, Localization, Off-shell supergravity Contents 1 Introduction and summary: Quantum entropy of supersymmetric -
On Functional Determinants of Matrix Differential Operators with Multiple
On functional determinants of matrix differential operators with multiple zero modes G. M. Falco1, Andrei A. Fedorenko2 and Ilya A. Gruzberg3 1Amsterdam University of Applied Studies, Weesperzijde 190, 1097 DZ, Amsterdam, Netherlands 2 Laboratoire de Physique, ENS de Lyon, Univ Claude Bernard, Univ Lyon, CNRS, F-69342 Lyon, France 3 Ohio State University, Department of Physics, 191 W. Woodruff Ave, Columbus OH, 43210, USA E-mail: [email protected], [email protected], [email protected] Abstract. We generalize the method of computing functional determinants with a single excluded zero eigenvalue developed by McKane and Tarlie to differential operators with multiple zero eigenvalues. We derive general formulas for such functional determinants of r r matrix second order differential operators O with 0 < n 6 2r linearly × independent zero modes. We separately discuss the cases of the homogeneous Dirichlet boundary conditions, when the number of zero modes cannot exceed r, and the case of twisted boundary conditions, including the periodic and anti-periodic ones, when the number of zero modes is bounded above by 2r. In all cases the determinants with excluded zero eigenvalues can be expressed only in terms of the n zero modes and other r n or 2r n (depending on the boundary conditions) solutions of the homogeneous − − equation Oh = 0, in the spirit of Gel’fand-Yaglom approach. In instanton calculations, the contribution of the zero modes is taken into account by introducing the so-called collective coordinates. We show that there is a remarkable cancellation of a factor (involving scalar products of zero modes) between the Jacobian of the transformation to the collective coordinates and the functional fluctuation determinant with excluded zero eigenvalues. -
Heat Trace and Functional Determinant in One Dimension
New Mexico Tech (June 5, 2014) Heat Trace and Functional Determinant in One Dimension Ivan G. Avramidi Department of Mathematics New Mexico Institute of Mining and Technology Socorro, NM 87801, USA E-mail: [email protected] arXiv:1406.1499v2 [math-ph] 10 Nov 2014 We study the spectral properties of the Laplace type operator on the cir- cle. We discuss various approximations for the heat trace, the zeta function and the zeta-regularized determinant. We obtain a differential equation for the heat kernel diagonal and a recursive system for the diagonal heat ker- nel coefficients, which enables us to find closed approximate formulas for the heat trace and the functional determinant which become exact in the limit of infinite radius. The relation to the generalized KdV hierarchy is discussed as well. 1 Introduction The heat kernel of elliptic partial differential operators is one of the most powerful tools in mathematical physics (see, for example, [12, 13, 4, 5, 17, 22] and further references therein). Of special importance are the spectral functions such as the heat trace, the zeta function and the functional determinant that enable one to study the spectral properties of the corresponding operator. The one-dimensional case is a very special one which exhibits an underly- ing symmetry that has deep relations to such diverse areas as integrable systems, infinite-dimensional Hamiltonian systems, isospectrality etc (see [1, 20, 21, 9, 6, 10, 7, 14, 15, 16], for example). Moreover, it has been shown that one can ob- tain closed formulas which express the functional determinant in one dimension in terms of a solution to a particular initial value problem; see, e.g., [19, 11, 18]. -
Path Integrals with Μ1(X) and Μ2(X) the first and Second Moments of the Transition Probabilities
1 Introduction to path integrals with ¹1(x) and ¹2(x) the ¯rst and second moments of the transition probabilities. v. January 22, 2006 Another approach is to use path integrals. We will use Phys 719 - M. Hilke the example of a simple brownian motion (the random walk) to illustrate the concept of the path integral (or Wiener integral) in this context. CONTENT ² Classical stochastic dynamics DISCRETE RANDOM WALK ² Brownian motion (random walk) ² Quantum dynamics The discrete random walk describes a particle (or per- son) moving along ¯xed segments for ¯xed time intervals ² Free particle (of unit 1). The choice of direction for each new segment ² Particle in a potential is random. We will assume that only perpendicular di- rection are possible (4 possibilities in 2D and 6 in 3D). ² Driven harmonic oscillator The important quantity is the probability to reach x at 0 0 ² Semiclassical approximation time t, when the random walker started at x at time t : ² Statistical description (imaginary time) P (x; t; x0; t0): (3) ² Quantum dissipative systems For this probability the following normalization condi- tions apply: INTRODUCTION X 0 0 0 0 P (x; t; x ; t) = ±x;x0 and for (t > t ) P (x; t; x ; t ) = 1: Path integrals are used in a variety of ¯elds, including x stochastic dynamics, polymer physics, protein folding, (4) ¯eld theories, quantum mechanics, quantum ¯eld theo- This leads to ries, quantum gravity and string theory. The basic idea 1 X is to sum up all contributing paths. Here we will overview P (x; t + 1; x0; t0) = P (»; t; x0; t0); (5) the technique be starting on classical dynamics, in par- 2D h»i ticular, the random walk problem before we discuss the quantum case by looking at a particle in a potential, and where h»i represent the nearest neighbors of x. -
Observations of Hawking Radiation
Published for SISSA by Springer Received: December 1, 2020 Accepted: March 8, 2021 Published: April 28, 2021 Observations of Hawking radiation: the Page curve JHEP04(2021)272 and baby universes Donald Marolf and Henry Maxfield Department of Physics, University of California, Santa Barbara, CA 93106, U.S.A. E-mail: [email protected], [email protected] Abstract: We reformulate recent insights into black hole information in a manner empha- sizing operationally-defined notions of entropy, Lorentz-signature descriptions, and asymp- totically flat spacetimes. With the help of replica wormholes, we find that experiments of asymptotic observers are consistent with black holes as unitary quantum systems, with density of states given by the Bekenstein-Hawking formula. However, this comes at the cost of superselection sectors associated with the state of baby universes. Spacetimes stud- ied by Polchinski and Strominger in 1994 provide a simple illustration of the associated concepts and techniques, and we argue them to be a natural late-time extrapolation of replica wormholes. The work aims to be self-contained and, in particular, to be accessible to readers who have not yet mastered earlier formulations of the ideas above. Keywords: Black Holes, Models of Quantum Gravity ArXiv ePrint: 2010.06602 Open Access, c The Authors. https://doi.org/10.1007/JHEP04(2021)272 Article funded by SCOAP3. Contents 1 Introduction1 2 Hawking radiation and the path integral4 2.1 Hawking’s Heisenberg picture calculation4 2.2 Path integral version6 2.2.1 Path -
Sharp Inequalities, the Functional Determinant, and the Complementary Series
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 347, Number 10, October 1995 SHARP INEQUALITIES, THE FUNCTIONAL DETERMINANT, AND THE COMPLEMENTARY SERIES THOMAS P. BRANSON Abstract. Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequali- ties, are combined in a study of functional determinants of natural differential operators. The setting is that of compact Riemannian manifolds. We concen- trate especially on the conformally flat case, and obtain formulas in dimensions 2, 4, and 6 for the functional determinants of operators which are well be- haved under conformai change of metric. The two-dimensional formulas are due to Polyakov, and the four-dimensional formulas to Branson and Orsted; the method is sufficiently streamlined here that we are able to present the six- dimensional case for the first time. In particular, we solve the extremal problems for the functional determinants of the conformai Laplacian and of the square of the Dirac operator on S2 , and in the standard conformai classes on S4 and S6 . The S2 results are due to Onofri, and the S4 results to Branson, Chang, and Yang; the S6 results are presented for the first time here. Recent results of Graham, Jenne, Mason, and Sparling on conformally covariant dif- ferential operators, and of Beckner on sharp Sobolev and Moser-Trudinger type inequalities, are used in an essential way, as are a computation of the spectra of intertwining operators for the complementary series of SOo(m + 1, 1), and the precise dependence of all computations on the dimension. In the process of solving the extremal problem on S6 , we are forced to derive a new and delicate conformally covariant sharp inequality, essentially a covariant form of the Sobolev embedding L2(S6) «-» L3(S6) for section spaces of trace free symmetric two-tensors. -
Functional Determinants in Quantum Field Theory
Functional Determinants in Quantum Field Theory Gerald Dunne Department of Physics, University of Connecticut May 21, 2009 Abstract Lecture notes on Functional Determinants in Quantum Field Theory given by Gerald Dunne at the 14th WE Heraeus Saalburg summer school in Wolfersdorf, Thuringia, in September 2008. Lecture notes taken by Babette D¨obrich and exercises with solutions by Oliver Schlotterer. Contents 1 Introduction 3 1.1 Motivation . .3 1.2 Outline . .4 2 ζ-function Regularization 5 2.1 Riemann ζ-function . .6 2.2 Hurwitz ζ-function . .7 2.3 Epstein ζ-function . .8 3 Heat kernel and heat kernel expansion 10 3.1 Properties of the heat kernel and relation to the ζ-function . 10 3.2 Heat Kernel expansions . 11 3.3 Heat kernel expansion in gauge theories . 12 1 2 CONTENTS 4 Paradigm: The Euler-Heisenberg effective action 13 4.1 Borel summation of the EH perturbative series . 14 4.2 Non-alternating series . 17 4.3 Perturbative vs non-perturbative . 18 5 The Gel'fand-Yaglom formalism 19 5.1 Preliminaries . 19 5.2 One-dimensional Schr¨odingeroperators . 21 5.3 Sine-Gordon Solitons and zero modes of the determinant . 24 5.4 Gel'fand Yaglom with generalized boundary conditions . 26 6 Radial Gel'fand-Yaglom formalism in higher dimensions 27 6.1 d-dimensional radial operators . 27 6.2 Example: 2-dimensional Helmholtz problem on a disc . 29 6.3 Renormalization . 31 7 False vacuum decay 34 7.1 Preliminaries . 34 7.2 The classical bounce solution . 35 7.3 Computing the determinant factor with radial Gel'fand Yaglom . -
Hologram of a Pure State Black Hole
RI-HUJI xx/15 LMU-ASC 33/15 Hologram of a pure state black hole Shubho R. Roy1;2 and Debajyoti Sarkar3;4 1Racah Inst. of Physics Hebrew University of Jerusalem, Jerusalem 91904 Israel [email protected] 2Theory Division Saha Institute of Nuclear Physics, Calcutta 700064, West Bengal, India 3Arnold Sommerfeld Center Ludwig-Maximilians-University, Theresienstr. 37, 80333 Munchen, Germany [email protected] 4Max-Planck-Institut f¨urPhysik, F¨ohringerRing 6, D-80805 Munich, Germany Abstract In this paper we extend the HKLL holographic smearing function method to re- construct (quasi)local AdS bulk scalar observables in the background of a large AdS arXiv:1505.03895v3 [hep-th] 29 Sep 2015 black hole formed by null shell collapse (a \pure state" black hole), from the dual CFT which is undergoing a sudden quench. In particular, we probe the near horizon and sub-horizon bulk locality. First we construct local bulk operators from the CFT in the leading semiclassical limit, N ! 1. Then we look at effects due to the finiteness of N, where we propose a suitable coarse-graining prescription involving early and late time cut-offs to define semiclassical bulk observables which are approximately local; their departure from locality being non-perturbatively small in N. Our results have important implications on the black hole information problem. 1 Introduction The AdS/CFT duality [1,2,3,4] is, in principle, a fully non-perturbative definition of quan- tum gravity in asymptotically anti deSitter spacetimes (aAdSd+1) in terms of a large N factorizable conformal field theory supported on its conformal boundary (CFTd).