DOCSLIB.ORG

Detecting the Rotating Quantum Vacuum

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Detecting the Rotating Quantum Vacuum PHYSICAL REVIEW D VOLUME 53, NUMBER 8 15 APRIL 1996 Detecting the rotating quantum vacuum Paul C. W. Davies* Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia 5005, Australia Tevian Dray† Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia 5005, Australia and Department of Mathematics, Oregon State University, Corvallis, Oregon 97331 Corinne A. Manogue‡ Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, South Australia 5005, Australia and Department of Physics, Oregon State University, Corvallis, Oregon 97331 ~Received 31 August 1995! We derive conditions for rotating particle detectors to respond in a variety of bounded spacetimes and compare the results with the folklore that particle detectors do not respond in the vacuum state appropriate to their motion. Applications involving possible violations of the second law of thermodynamics are briefly addressed. PACS number~s!: 04.62.1v I. INTRODUCTION and even to link this quantum vacuum with Machian effects @8#. We might envisage the quantum vacuum playing the role The nature of the quantum vacuum continues to be the of Newton’s substantival space. It is therefore of some inter- subject of surprising discoveries. One of the more curious est to investigate the effects of rotation in quantum field properties to be discussed in recent years is the prediction theory. that an observer who accelerates in the conventional quan- In what follows we attempt to progress this discussion by tum vacuum of Minkowski space will perceive a bath of investigating the properties of particle detectors moving in radiation, while an inertial observer of course perceives noth- circular paths. The results turn out to involve some remark- ing. In the case of linear acceleration, for which there exists able and subtle issues, especially with regard to the definition an extensive literature, the response of a model particle de- of a rotating quantum vacuum state. tector mimics the effects of its being immersed in a bath of The paper is organized as follows. In Sec. II, we first thermal radiation ~the so-called Unruh effect!. The investiga- review previous results about rotating particle detectors in tion of rigid rotation, as opposed to linear acceleration, unbounded Minkowski space. We then add a cylindrical seems, however, to have been somewhat limited @1–4#,in boundary in Sec. III, and consider compact spaces in Sec. IV. spite of the fact that experimental tests of the theory are more The possibility of violating the second law of thermodynam- feasible for the rotating case @5#. ics is discussed in Sec. V, and in Sec. VI we conclude by The problem of rotation is a very deep one in physics discussing the physical interpretation of our results. We use which goes back at least to Newton. Through his famous units with \5c51, our metric signature convention is bucket experiment, Newton sought to demonstrate that rota- (1222), and we treat massless scalar fields for simplicity. tion is an absolute effect taking place in a substantival space, later to be identified with the aether. By contrast, Mach ar- II. UNBOUNDED MINKOWSKI SPACE gued that rotation is purely relative to the distant matter in We first summarize the results of Letaw and Pfautsch @1#. the universe. Then with the inception of the theory of rela- Consider a DeWitt model particle detector @6# moving in a tivity the aether was abandoned, but the question of whether circular path of radius r at constant angular velocity V in the rotation is relative or absolute was not settled by the theory conventional Minkowski ~inertial! vacuum state. Here, and of relativity, and the status of Mach’s principle remains con- in all the cases we shall be investigating later, the response of tentious even today. the detector is independent of time. According to the stan- In recent years there has been an attempt to revive a type dard theory, the probability of excitation of a detector per of aether concept by appealing to the quantum vacuum @6,7#, unit ~detector! proper time is given by E! ` F ~ 2iEDt 1 *Electronic address: [email protected] 5 dDte G „x~t!,x~t8!…, ~1! T E2` †Permanent address: Oregon State University. Electronic address: [email protected] where G1 is the positive frequency Wightman function. The ‡Permanent address: Oregon State University. Electronic address: Minkowski space Wightman function in cylindrical polar co- [email protected] ordinates x5(t,r,u,z)is 0556-2821/96/53~8!/4382~6!/$10.0053 4382 © 1996 The American Physical Society 53 DETECTING THE ROTATING QUANTUM VACUUM 4383 1 1 G1~x,x8!5 . ~2! 4p2 ~t2t82ie!22@r21r8222rr8cos~u2u8!1~z2z8!2# The detector’s trajectory is given by r5const, z5const, zero response in the Rindler vacuum state, and will give a and u5Vt, so that G1 is a function only of the difference consistent thermal response to the Minkowski vacuum state Dt in proper time. The response function is then given by @6#. We might therefore expect a set of rotating detectors to the Fourier transform integral similarly reveal the state of a rotating quantum field. This is not the case here: Since the inertial and rotating vacuum ¯ F ~E¯ ! A12r2V2 ` e2iEDt states are identical, the response of a rotating particle detec- 5 2 dDt , tor to the corotating vacuum will be the same as ~3! above, T 4p E2` VDt ~Dt2ie!224r2sin2 and we have the curious result that a particle detector S 2 D adapted to the rotating vacuum apparently does not behave ~3! as if it is coupled to a vacuum state at all. Before accepting this odd result, however, we must con- ¯ 2 2 where E5EA12r V corresponds to the Lorentz rescaling front a serious problem. For a given angular velocity V, Dt5t2t85Dt/A12r2V2 arising from the transformation there will be a maximum radius rmax51/V~the light cylin- between proper time t and coordinate time t. This expres- der! beyond which a point at fixed r and u will be moving sion was evaluated numerically by Pfautsch. The fact that it faster than light, i.e., the rotating Killing vector ]t2V]u is nonzero is already an important result, analogous to the becomes spacelike. It is far from clear what to make of the famous linear acceleration case, where the detected spectrum rotating field modes, or even the notion of Bogolubov trans- of particles is thermal. Although Pfautsch’s spectrum is remi- formation, outside this limiting light circle. This complica- niscent of a Planck spectrum, the similarity with the thermal tion compromises any straightforward attempt to construct linear case is only superficial. rotating quantum states in terms of the mode solutions just The Minkowski vacuum state is defined with respect to described. the usual field modes based on mode solutions uqmk of the To circumvent this problem we eliminate the light cylin- wave equation in inertial coordinates: der altogether. We accomplish this in two ways: first, in Sec. III, by introducing a cylindrical boundary within the light 1 imu ikz 2ivt cylinder and confining the quantum field to the bounded re- uqmk5 Jm~qr!e e e , ~4! 2pA2v gion, and then, in Sec. IV, by making the space compact. where v25q21k2. III. ROTATING DETECTOR IN A BOUNDED DOMAIN The transformation from Minkowski to rotating coordi- The Wightman function ~2! inside a cylinder of radius nates appears at first sight to have only a trivial implication a,r , on which the field satisfies vanishing ~i.e., Dirich- for the modes of the quantum field: We may readily trans- max ˆ let! boundary conditions, can be written as a discrete mode form to a rotating coordinate system, by letting u5u2Vt. sum Mode solutions uˆ qmk of the rotating wave equation are then found to be identical to the conventional ~nonrotating! ` ` ` N2 j r j r8 Minkowski modes, and can be obtained, up to normalization, 1 mn mn G ~x,x8!5 ( ( dk Jm Jm by the replacement vˆ 5v2mV. In particular, the Bogol- m52` n51 E2` v S a D S a D ubov transformation between the rotating and nonrotating im~u2u8! ik~z2z8! 2iv~t2t8! modes is trivial, amounting to a simple relabeling of modes. 3e e e , ~5! Furthermore, since the two sets of modes are identical, they have identical norms, so that no mixing of positive and nega- where tive ‘‘frequencies’’ occurs irrespective of a possible change 1 of sign between v and vˆ . It is the norms, not the frequen- N5 , cies, of the rotating modes which determine the commutation 2pauJm11~jmn!u relations of the associated creation and annihilation opera- tors. jmn is the nth zero of the Bessel function Jm(x), and If these rotating modes are used to define a rotating 2 vacuum state, it coincides with the conventional Minkowski jmn v5 1k2. ~6! vacuum. This is in contrast to the case for linear acceleration A a2 @9#, where the so-called Rindler ~Fulling! vacuum contains Minkowski particles and vice versa. Furthermore, there is a As is appropriate for the positive frequency Wightman func- general expectation that a family of model particle detectors tion, we assume v.0. For a particle detector moving in a ‘‘adapted’’ to the coordinate system on which the quantized circular path of radius r about the axis of the cylinder with modes are based will reveal the particle content of those uniform angular velocity V, we set r5const, z5const, and modes. Thus, in the Rindler case, a set of uniformly acceler- u5Vt. Substituting from ~5! into ~1!, we obtain, for the ating particle detectors sharing common asymptotes will give detector response per unit time, 4384 DAVIES, DRAY, AND MANOGUE 53 ` ` F ~E! ` N2 evaluate ~11! numerically, but we expect on general grounds 5A12r2V2 dk for the spectrum to have a sawtooth form at low energies T ( ( E m52` n51 2` v @10#.
Load more

Recommended publications
  • Curiens Principle and Spontaneous Symmetry Breaking
    Curie’sPrinciple and spontaneous symmetry breaking John Earman Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh, PA, USA Abstract In 1894 Pierre Curie announced what has come to be known as Curie’s Principle: the asymmetry of e¤ects must be found in their causes. In the same publication Curie discussed a key feature of what later came to be known as spontaneous symmetry breaking: the phenomena generally do not exhibit the symmetries of the laws that govern them. Philosophers have long been interested in the meaning and status of Curie’s Principle. Only comparatively recently have they begun to delve into the mysteries of spontaneous symmetry breaking. The present paper aims to advance the dis- cussion of both of these twin topics by tracing their interaction in classical physics, ordinary quantum mechanics, and quantum …eld theory. The fea- tures of spontaneous symmetry that are peculiar to quantum …eld theory have received scant attention in the philosophical literature. These features are highlighted here, along with a explanation of why Curie’sPrinciple, though valid in quantum …eld theory, is nearly vacuous in that context. 1. Introduction The statement of what is now called Curie’sPrinciple was announced in 1894 by Pierre Curie: (CP) When certain e¤ects show a certain asymmetry, this asym- metry must be found in the causes which gave rise to it (Curie 1894, p. 401).1 This principle is vague enough to allow some commentators to see in it pro- found truth while others see only falsity (compare Chalmers 1970; Radicati 1987; van Fraassen 1991, pp.
    [Show full text]
  • A Discussion on Characteristics of the Quantum Vacuum
    A Discussion on Characteristics of the Quantum Vacuum Harold \Sonny" White∗ NASA/Johnson Space Center, 2101 NASA Pkwy M/C EP411, Houston, TX (Dated: September 17, 2015) This paper will begin by considering the quantum vacuum at the cosmological scale to show that the gravitational coupling constant may be viewed as an emergent phenomenon, or rather a long wavelength consequence of the quantum vacuum. This cosmological viewpoint will be reconsidered on a microscopic scale in the presence of concentrations of \ordinary" matter to determine the impact on the energy state of the quantum vacuum. The derived relationship will be used to predict a radius of the hydrogen atom which will be compared to the Bohr radius for validation. The ramifications of this equation will be explored in the context of the predicted electron mass, the electrostatic force, and the energy density of the electric field around the hydrogen nucleus. It will finally be shown that this perturbed energy state of the quan- tum vacuum can be successfully modeled as a virtual electron-positron plasma, or the Dirac vacuum. PACS numbers: 95.30.Sf, 04.60.Bc, 95.30.Qd, 95.30.Cq, 95.36.+x I. BACKGROUND ON STANDARD MODEL OF COSMOLOGY Prior to developing the central theme of the paper, it will be useful to present the reader with an executive summary of the characteristics and mathematical relationships central to what is now commonly referred to as the standard model of Big Bang cosmology, the Friedmann-Lema^ıtre-Robertson-Walker metric. The Friedmann equations are analytic solutions of the Einstein field equations using the FLRW metric, and Equation(s) (1) show some commonly used forms that include the cosmological constant[1], Λ.
    [Show full text]
  • 1 the LOCALIZED QUANTUM VACUUM FIELD D. Dragoman
    1 THE LOCALIZED QUANTUM VACUUM FIELD D. Dragoman – Univ. Bucharest, Physics Dept., P.O. Box MG-11, 077125 Bucharest, Romania, e-mail: [email protected] ABSTRACT A model for the localized quantum vacuum is proposed in which the zero-point energy of the quantum electromagnetic field originates in energy- and momentum-conserving transitions of material systems from their ground state to an unstable state with negative energy. These transitions are accompanied by emissions and re-absorptions of real photons, which generate a localized quantum vacuum in the neighborhood of material systems. The model could help resolve the cosmological paradox associated to the zero-point energy of electromagnetic fields, while reclaiming quantum effects associated with quantum vacuum such as the Casimir effect and the Lamb shift; it also offers a new insight into the Zitterbewegung of material particles. 2 INTRODUCTION The zero-point energy (ZPE) of the quantum electromagnetic field is at the same time an indispensable concept of quantum field theory and a controversial issue (see [1] for an excellent review of the subject). The need of the ZPE has been recognized from the beginning of quantum theory of radiation, since only the inclusion of this term assures no first-order temperature-independent correction to the average energy of an oscillator in thermal equilibrium with blackbody radiation in the classical limit of high temperatures. A more rigorous introduction of the ZPE stems from the treatment of the electromagnetic radiation as an ensemble of harmonic quantum oscillators. Then, the total energy of the quantum electromagnetic field is given by E = åk,s hwk (nks +1/ 2) , where nks is the number of quantum oscillators (photons) in the (k,s) mode that propagate with wavevector k and frequency wk =| k | c = kc , and are characterized by the polarization index s.
    [Show full text]
  • The Quantum Vacuum and the Cosmological Constant Problem
    The Quantum Vacuum and the Cosmological Constant Problem S.E. Rugh∗and H. Zinkernagel† Abstract - The cosmological constant problem arises at the intersection be- tween general relativity and quantum field theory, and is regarded as a fun- damental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically dis- cuss how the problem rests on the notion of physical real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. Introduction Is empty space really empty? In the quantum field theories (QFT’s) which underlie modern particle physics, the notion of empty space has been replaced with that of a vacuum state, defined to be the ground (lowest energy density) state of a collection of quantum fields. A peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise ‘empty’ (i.e. devoid of matter and radiation). These zero-point arXiv:hep-th/0012253v1 28 Dec 2000 fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρvac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ appearing in Einstein’s field equations from 1917, 1 8πG R g R Λg = T (1) µν − 2 µν − µν c4 µν where Rµν and R refer to the curvature of spacetime, gµν is the metric, Tµν the energy-momentum tensor, G the gravitational constant, and c the speed of light.
    [Show full text]
  • Arxiv:1508.01017V1 [Gr-Qc] 5 Aug 2015
    The Bohmian Approach to the Problems of Cosmological Quantum Fluctuations Sheldon Goldstein,∗ Ward Struyve,y Roderich Tumulkaz August 5, 2015 Abstract There are two kinds of quantum fluctuations relevant to cosmology that we focus on in this article: those that form the seeds for structure formation in the early universe and those giving rise to Boltzmann brains in the late universe. First, structure formation requires slight inhomogeneities in the density of matter in the early universe, which then get amplified by the effect of gravity, leading to clumping of matter into stars and galaxies. According to inflation theory, quantum fluctuations form the seeds of these inhomogeneities. However, these quantum fluctuations are described by a quantum state which is homogeneous and isotropic, and this raises a problem, connected to the foundations of quantum theory, as the unitary evolution alone cannot break the symmetry of the quantum state. Second, Boltzmann brains are random agglomerates of particles that, by extreme coincidence, form functioning brains. Unlikely as these coincidences are, they seem to be predicted to occur in a quantum universe as vacuum fluctuations if the universe continues to exist for an infinite (or just very long) time, in fact to occur over and over, even forming the majority of all brains in the history of the universe. We provide a brief introduction to the Bohmian version of quantum theory and explain why in this version, Boltzmann brains, an undesirable kind of fluctuation, do not occur (or at least not often), while inhomogeneous seeds for structure formation, a desirable kind of fluctuation, do. arXiv:1508.01017v1 [gr-qc] 5 Aug 2015 1 Introduction The notion of observation or measurement plays a fundamental role in standard quan- tum theory.
    [Show full text]
  • Quantum Vacuum Structure and Cosmology
    Quantum Vacuum Structure and Cosmology Johann Rafelski, Lance Labun, Yaron Hadad, Departments of Physics and Mathematics, The University of Arizona 85721 Tucson, AZ, USA, and Department f¨ur Physik, Ludwig-Maximillians-Universit¨at M¨unchen Am Coulombwall 1, 85748 Garching, Germany Pisin Chen, Leung Center for Cosmology and Particle Astrophysics and Graduate Institute of Astrophysics and Department of Physics, National Taiwan University, Taipei, Taiwan 10617, and Kavli Institute for Particle Astrophysics and Cosmology, SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305, U.S.A. Introductory Remarks Contemporary physics faces three great riddles that lie at the intersection of quantum theory, particle physics and cosmology. They are 1. The expansion of the universe is accelerating – an extra factor of two appears in the size. 2. Zero-point fluctuations do not gravitate – a matter of 120 orders of magnitude 3. The “True” quantum vacuum state does not gravitate. The latter two are explicitly problems related to the interpretation and the physi- cal role and relation of the quantum vacuum with and in general relativity. Their resolution may require a major advance in our formulation and understanding of a common unified approach to quantum physics and gravity. To achieve this goal we must develop an experimental basis and much of the discussion we present is devoted to this task. In the following, we examine the observations and the theory contributing to the current framework comprising these riddles. We consider an interpretation of the first riddle within the context of the universe’s quantum vacuum state, and propose an experimental concept to probe the vacuum state of the universe.
    [Show full text]
  • Quantum Mechanics Vacuum State
    Quantum Mechanics_vacuum state In quantum field theory, the vacuum state (also called the vacuum) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. Zero- point field is sometimes used[by whom?] as a synonym for the vacuum state of an individual quantized field. According to present-day understanding of what is called the vacuum state or the quantum vacuum, it is "by no means a simple empty space",[1] and again: "it is a mistake to think of any physical vacuum as some absolutely empty void."[2] According to quantum mechanics, the vacuum state is not truly empty but instead contains fleeting electromagnetic waves and particles that pop into and out of existence.[3][4][5] The QED vacuum of Quantum electrodynamics(or QED) was the first vacuum of quantum field theory to be developed. QED originated in the 1930s, and in the late 1940s and early 1950s it was reformulated by Feynman, Tomonaga andSchwinger, who jointly received the Nobel prize for this work in 1965.[6] Today theelectromagnetic interactions and the weak interactions are unified in the theory of theElectroweak interaction. The Standard Model is a generalization of the QED work to include all the known elementary particles and their interactions (except gravity).Quantum chromodynamics is the portion of the Standard Model that deals with strong interactions, and QCD vacuum is the vacuum ofQuantum chromodynamics. It is the object of study in the Large Hadron Collider and theRelativistic Heavy Ion Collider, and is related to the so-called vacuum structure of strong interactions.[7] Contents 1 Non-zero expectation value 2 Energy 3 Symmetry 4 Electrical permittivity 5 Notations 6 Virtual particles 7 Physical nature of the quantum vacuum 8 See also 9 References and notes 10 Further reading Non-zero expectation value Main article: Vacuum expectation value The video of an experiment showing vacuum fluctuations (in the red ring) amplified by spontaneous parametric down-conversion.
    [Show full text]
  • The Quantum Vacuum and the Cosmological Constant Problem
    The Quantum Vacuum and the Cosmological Constant Problem S.E. Rugh∗and H. Zinkernagely To appear in Studies in History and Philosophy of Modern Physics Abstract - The cosmological constant problem arises at the intersection be- tween general relativity and quantum field theory, and is regarded as a fun- damental problem in modern physics. In this paper we describe the historical and conceptual origin of the cosmological constant problem which is intimately connected to the vacuum concept in quantum field theory. We critically dis- cuss how the problem rests on the notion of physically real vacuum energy, and which relations between general relativity and quantum field theory are assumed in order to make the problem well-defined. 1. Introduction Is empty space really empty? In the quantum field theories (QFT’s) which underlie modern particle physics, the notion of empty space has been replaced with that of a vacuum state, defined to be the ground (lowest energy density) state of a collection of quantum fields. A peculiar and truly quantum mechanical feature of the quantum fields is that they exhibit zero-point fluctuations everywhere in space, even in regions which are otherwise ‘empty’ (i.e. devoid of matter and radiation). These zero-point fluctuations of the quantum fields, as well as other ‘vacuum phenomena’ of quantum field theory, give rise to an enormous vacuum energy density ρvac. As we shall see, this vacuum energy density is believed to act as a contribution to the cosmological constant Λ appearing in Einstein’s field equations from 1917, 1 8πG R g R Λg = T (1) µν − 2 µν − µν c4 µν where Rµν and R refer to the curvature of spacetime, gµν is the metric, Tµν the energy-momentum tensor, G the gravitational constant, and c the speed of light.
    [Show full text]
  • Attractive and Repulsive Casimir Vacuum Energy with General Boundary Conditions
    Available online at www.sciencedirect.com Nuclear Physics B 874 [FS] (2013) 852–876 www.elsevier.com/locate/nuclphysb Attractive and repulsive Casimir vacuum energy with general boundary conditions M. Asorey a,∗, J.M. Muñoz-Castañeda b a Departamento de Física Teórica, Facultad de Ciencias, Universidad de Zaragoza, E-50009 Zaragoza, Spain b Institut für Theoretische Physik, Universität Leipzig, Brüderstr. 16, D-04103 Leipzig, Germany Received 5 December 2012; received in revised form 20 May 2013; accepted 17 June 2013 Available online 25 June 2013 Abstract The infrared behaviour of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The anal- ysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth–Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials.
    [Show full text]
  • Abstract Hybrid Optomechanics and The
    ABSTRACT HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT by Robert A. McCutcheon We explore various aspects of hybrid optomechanics. A hybrid optomechanical model in- volving a cavity, atom, and moving mirror is introduced as the intersection between cavity quantum electrodynamics and cavity optomechanics. We describe how numerical simulations of such systems are performed and demonstrate a method that can reduce the computational scope of certain simulations. A method for calculating time-dependent spectra is also ex- plained. Finally, we explain how the dynamical Casimir effect can be achieved using an optical cavity with and without an atom and look at notable behavior of this system. HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT Thesis Submitted to the Faculty of Miami University in partial fulfillment of the requirements for the degree of Master of Science by Robert A. McCutcheon Miami University Oxford, Ohio 2017 Advisor: Perry Rice Reader: Samir Bali Reader: Karthik Vishwanath c 2017 Robert A. McCutcheon This thesis titled HYBRID OPTOMECHANICS AND THE DYNAMICAL CASIMIR EFFECT by Robert A. McCutcheon has been approved for publication by the College of Arts and Science and Department of Physics (Perry Rice) (Samir Bali) (Karthik Vishwanath) Contents 1 Introduction 1 1.1 Cavity . .2 1.2 Cavity and Atom . .5 1.3 Cavity and Mirror . .9 1.4 Cavity, Atom, and Mirror . 11 2 Theoretical Frameworks 13 2.1 Wavefunctions and Density Matrices . 13 2.2 Correlation Functions and Emission Spectrum . 16 2.3 Spectrum of Squeezing . 18 2.4 QuTiP Implementation . 19 3 Steady-State Value Computational Method 21 3.1 Procedure . 21 3.2 Results .
    [Show full text]
  • Instability of Quantum De Sitter Spacetime Open Access Article Funded by SCOAP Abstract: Chiu Man Ho and Stephen D.H
    Published for SISSA by Springer Received: January 21, 2015 Accepted: March 31, 2015 Published: April 20, 2015 JHEP04(2015)086 Instability of quantum de Sitter spacetime Chiu Man Ho and Stephen D.H. Hsu Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, U.S.A. E-mail: [email protected], [email protected] Abstract: Quantized fields (e.g., the graviton itself) in de Sitter (dS) spacetime lead to particle production: specifically, we consider a thermal spectrum resulting from the dS (horizon) temperature. The energy required to excite these particles reduces slightly the rate of expansion and eventually modifies the semiclassical spacetime geometry. The resulting manifold no longer has constant curvature nor time reversal invariance, and back- reaction renders the classical dS background unstable to perturbations. In the case of AdS, there exists a global static vacuum state; in this state there is no particle production and the analogous instability does not arise. Keywords: Models of Quantum Gravity, Classical Theories of Gravity ArXiv ePrint: 1501.00708 Open Access, c The Authors. 3 doi:10.1007/JHEP04(2015)086 Article funded by SCOAP . Contents 1 Introduction 1 2 Back-reaction from de Sitter thermal spectrum 2 3 Anti-de Sitter spacetime 3 4 Remarks 5 JHEP04(2015)086 1 Introduction In classical general relativity, a cosmological constant Λ has the special property that the equation of state must satisfy p w = = −1 (1.1) ρ exactly, where p is the associated pressure and ρ the energy density. This is equivalent to the energy-momentum tensor Tµν satisfying Tµν = Λ gµν , (1.2) where gµν is the metric tensor.
    [Show full text]
  • Bouncing Quantum Cosmology
    universe Review Bouncing Quantum Cosmology Nelson Pinto-Neto Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro 22290-180, RJ, Brazil; [email protected]; Tel.: +55-21-21417381 Abstract: The goal of this contribution is to present the properties of a class of quantum bounc- ing models in which the quantum bounce originates from the Dirac canonical quantization of a midi-superspace model composed of a homogeneous and isotropic background, together with small inhomogeneous perturbations. The resulting Wheeler-DeWitt equation is interpreted in the frame- work of the de Broglie-Bohm quantum theory, enormously simplifying the calculations, conceptually and technically. It is shown that the resulting models are stable and they never get to close to the Planck energy, where another more involved quantization scheme would have to be evoked, and they are compatible with present observations. Some physical effects around the bounce are discussed, like baryogenesis and magnetogenesis, and the crucial role of dark matter and dark energy is also studied. Keywords: bounce; quantum cosmology; inflation; cosmological perturbations; stability; dark energy 1. Introduction Cosmological observations (the cosmic microwave background radiation [1], Big Bang nucleosynthesis [2], large scale structure [3] and cosmological red-shift [4]) strongly indicate Citation: Pinto-Neto, N. Bouncing that the Universe is expanding from a very hot era, when the geometry of space was highly Quantum Cosmology. Universe 2021, homogeneous and isotropic (maximally symmetric space-like hyper-surfaces), with tiny 7, 110. https://orcid.org/10.3390/ deviations from this special symmetric state. universe7040110 Cosmology is a very peculiar part of Physics, because the physical system under investigation is the Universe as a whole, and one cannot control its initial conditions, as in Academic Editors: Yi-Fu Cai, other physical situations.
    [Show full text]