Quantum and Classical Fluctuations
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Observations of Wavefunction Collapse and the Retrospective Application of the Born Rule
Observations of wavefunction collapse and the retrospective application of the Born rule Sivapalan Chelvaniththilan Department of Physics, University of Jaffna, Sri Lanka [email protected] Abstract In this paper I present a thought experiment that gives different results depending on whether or not the wavefunction collapses. Since the wavefunction does not obey the Schrodinger equation during the collapse, conservation laws are violated. This is the reason why the results are different – quantities that are conserved if the wavefunction does not collapse might change if it does. I also show that using the Born Rule to derive probabilities of states before a measurement given the state after it (rather than the other way round as it is usually used) leads to the conclusion that the memories that an observer has about making measurements of quantum systems have a significant probability of being false memories. 1. Introduction The Born Rule [1] is used to determine the probabilities of different outcomes of a measurement of a quantum system in both the Everett and Copenhagen interpretations of Quantum Mechanics. Suppose that an observer is in the state before making a measurement on a two-state particle. Let and be the two states of the particle in the basis in which the observer makes the measurement. Then if the particle is initially in the state the state of the combined system (observer + particle) can be written as In the Copenhagen interpretation [2], the state of the system after the measurement will be either with a 50% probability each, where and are the states of the observer in which he has obtained a measurement of 0 or 1 respectively. -
Inflation Without Quantum Gravity
Inflation without quantum gravity Tommi Markkanen, Syksy R¨as¨anen and Pyry Wahlman University of Helsinki, Department of Physics and Helsinki Institute of Physics P.O. Box 64, FIN-00014 University of Helsinki, Finland E-mail: tommi dot markkanen at helsinki dot fi, syksy dot rasanen at iki dot fi, pyry dot wahlman at helsinki dot fi Abstract. It is sometimes argued that observation of tensor modes from inflation would provide the first evidence for quantum gravity. However, in the usual inflationary formalism, also the scalar modes involve quantised metric perturbations. We consider the issue in a semiclassical setup in which only matter is quantised, and spacetime is classical. We assume that the state collapses on a spacelike hypersurface, and find that the spectrum of scalar perturbations depends on the hypersurface. For reasonable choices, we can recover the usual inflationary predictions for scalar perturbations in minimally coupled single-field models. In models where non-minimal coupling to gravity is important and the field value is sub-Planckian, we do not get a nearly scale-invariant spectrum of scalar perturbations. As gravitational waves are only produced at second order, the tensor-to-scalar ratio is negligible. We conclude that detection of inflationary gravitational waves would indeed be needed to have observational evidence of quantisation of gravity. arXiv:1407.4691v2 [astro-ph.CO] 4 May 2015 Contents 1 Introduction 1 2 Semiclassical inflation 3 2.1 Action and equations of motion 3 2.2 From homogeneity and isotropy to perturbations 6 3 Matching across the collapse 7 3.1 Hypersurface of collapse 7 3.2 Inflation models 10 4 Conclusions 13 1 Introduction Inflation and the quantisation of gravity. -
Effects of Primordial Chemistry on the Cosmic Microwave Background
A&A 490, 521–535 (2008) Astronomy DOI: 10.1051/0004-6361:200809861 & c ESO 2008 Astrophysics Effects of primordial chemistry on the cosmic microwave background D. R. G. Schleicher1, D. Galli2,F.Palla2, M. Camenzind3, R. S. Klessen1, M. Bartelmann1, and S. C. O. Glover4 1 Institute of Theoretical Astrophysics / ZAH, Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany e-mail: [dschleic;rklessen;mbartelmann]@ita.uni-heidelberg.de 2 INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy e-mail: [galli;palla]@arcetri.astro.it 3 Landessternwarte Heidelberg / ZAH, Koenigstuhl 12, 69117 Heidelberg, Germany e-mail: [email protected] 4 Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany e-mail: [email protected] Received 27 March 2008 / Accepted 3 July 2008 ABSTRACT Context. Previous works have demonstrated that the generation of secondary CMB anisotropies due to the molecular optical depth is likely too small to be observed. In this paper, we examine additional ways in which primordial chemistry and the dark ages might influence the CMB. Aims. We seek a detailed understanding of the formation of molecules in the postrecombination universe and their interactions with the CMB. We present a detailed and updated chemical network and an overview of the interactions of molecules with the CMB. Methods. We calculate the evolution of primordial chemistry in a homogeneous universe and determine the optical depth due to line absorption, photoionization and photodissociation, and estimate the resulting changes in the CMB temperature and its power spectrum. Corrections for stimulated and spontaneous emission are taken into account. -
Derivation of the Born Rule from Many-Worlds Interpretation and Probability Theory K
Derivation of the Born rule from many-worlds interpretation and probability theory K. Sugiyama1 2014/04/16 The first draft 2012/09/26 Abstract We try to derive the Born rule from the many-worlds interpretation in this paper. Although many researchers have tried to derive the Born rule (probability interpretation) from Many-Worlds Interpretation (MWI), it has not resulted in the success. For this reason, derivation of the Born rule had become an important issue of MWI. We try to derive the Born rule by introducing an elementary event of probability theory to the quantum theory as a new method. We interpret the wave function as a manifold like a torus, and interpret the absolute value of the wave function as the surface area of the manifold. We put points on the surface of the manifold at a fixed interval. We interpret each point as a state that we cannot divide any more, an elementary state. We draw an arrow from any point to any point. We interpret each arrow as an event that we cannot divide any more, an elementary event. Probability is proportional to the number of elementary events, and the number of elementary events is the square of the number of elementary state. The number of elementary states is proportional to the surface area of the manifold, and the surface area of the manifold is the absolute value of the wave function. Therefore, the probability is proportional to the absolute square of the wave function. CONTENTS 1 Introduction .............................................................................................................................. 2 1.1 Subject .............................................................................................................................. 2 1.2 The importance of the subject ......................................................................................... -
Quantum Trajectory Distribution for Weak Measurement of A
Quantum Trajectory Distribution for Weak Measurement of a Superconducting Qubit: Experiment meets Theory Parveen Kumar,1, ∗ Suman Kundu,2, † Madhavi Chand,2, ‡ R. Vijayaraghavan,2, § and Apoorva Patel1, ¶ 1Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012, India 2Department of Condensed Matter Physics and Materials Science, Tata Institue of Fundamental Research, Mumbai 400005, India (Dated: April 11, 2018) Quantum measurements are described as instantaneous projections in textbooks. They can be stretched out in time using weak measurements, whereby one can observe the evolution of a quantum state as it heads towards one of the eigenstates of the measured operator. This evolution can be understood as a continuous nonlinear stochastic process, generating an ensemble of quantum trajectories, consisting of noisy fluctuations on top of geodesics that attract the quantum state towards the measured operator eigenstates. The rate of evolution is specific to each system-apparatus pair, and the Born rule constraint requires the magnitudes of the noise and the attraction to be precisely related. We experimentally observe the entire quantum trajectory distribution for weak measurements of a superconducting qubit in circuit QED architecture, quantify it, and demonstrate that it agrees very well with the predictions of a single-parameter white-noise stochastic process. This characterisation of quantum trajectories is a powerful clue to unraveling the dynamics of quantum measurement, beyond the conventional axiomatic quantum -
Quantum Theory: Exact Or Approximate?
Quantum Theory: Exact or Approximate? Stephen L. Adler§ Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA Angelo Bassi† Department of Theoretical Physics, University of Trieste, Strada Costiera 11, 34014 Trieste, Italy and Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy Quantum mechanics has enjoyed a multitude of successes since its formulation in the early twentieth century. It has explained the structure and interactions of atoms, nuclei, and subnuclear particles, and has given rise to revolutionary new technologies. At the same time, it has generated puzzles that persist to this day. These puzzles are largely connected with the role that measurements play in quantum mechanics [1]. According to the standard quantum postulates, given the Hamiltonian, the wave function of quantum system evolves by Schr¨odinger’s equation in a predictable, de- terministic way. However, when a physical quantity, say z-axis spin, is “measured”, the outcome is not predictable in advance. If the wave function contains a superposition of components, such as spin up and spin down, which each have a definite spin value, weighted by coe±cients cup and cdown, then a probabilistic distribution of outcomes is found in re- peated experimental runs. Each repetition gives a definite outcome, either spin up or spin down, with the outcome probabilities given by the absolute value squared of the correspond- ing coe±cient in the initial wave function. This recipe is the famous Born rule. The puzzles posed by quantum theory are how to reconcile this probabilistic distribution of outcomes with the deterministic form of Schr¨odinger’s equation, and to understand precisely what constitutes a “measurement”. -
What Is Quantum Mechanics? a Minimal Formulation Pierre Hohenberg, NYU (In Collaboration with R
What is Quantum Mechanics? A Minimal Formulation Pierre Hohenberg, NYU (in collaboration with R. Friedberg, Columbia U) • Introduction: - Why, after ninety years, is the interpretation of quantum mechanics still a matter of controversy? • Formulating classical mechanics: microscopic theory (MICM) - Phase space: states and properties • Microscopic formulation of quantum mechanics (MIQM): - Hilbert space: states and properties - quantum incompatibility: noncommutativity of operators • The simplest examples: a single spin-½; two spins-½ ● The measurement problem • Macroscopic quantum mechanics (MAQM): - Testing the theory. Not new principles, but consistency checks ● Other treatments: not wrong, but laden with excess baggage • The ten commandments of quantum mechanics Foundations of Quantum Mechanics • I. What is quantum mechanics (QM)? How should one formulate the theory? • II. Testing the theory. Is QM the whole truth with respect to experimental consequences? • III. How should one interpret QM? - Justify the assumptions of the formulation in I - Consider possible assumptions and formulations of QM other than I - Implications of QM for philosophy, cognitive science, information theory,… • IV. What are the physical implications of the formulation in I? • In this talk I shall only be interested in I, which deals with foundations • II and IV are what physicists do. They are not foundations • III is interesting but not central to physics • Why is there no field of Foundations of Classical Mechanics? - We wish to model the formulation of QM on that of CM Formulating nonrelativistic classical mechanics (CM) • Consider a system S of N particles, each of which has three coordinates (xi ,yi ,zi) and three momenta (pxi ,pyi ,pzi) . • Classical mechanics represents this closed system by objects in a Euclidean phase space of 6N dimensions. -
Chapter 4: Linear Perturbation Theory
Chapter 4: Linear Perturbation Theory May 4, 2009 1. Gravitational Instability The generally accepted theoretical framework for the formation of structure is that of gravitational instability. The gravitational instability scenario assumes the early universe to have been almost perfectly smooth, with the exception of tiny density deviations with respect to the global cosmic background density and the accompanying tiny velocity perturbations from the general Hubble expansion. The minor density deviations vary from location to location. At one place the density will be slightly higher than the average global density, while a few Megaparsecs further the density may have a slightly smaller value than on average. The observed fluctuations in the temperature of the cosmic microwave background radiation are a reflection of these density perturbations, so that we know that the primordial density perturbations have been in the order of 10−5. The origin of this density perturbation field has as yet not been fully understood. The most plausible theory is that the density perturbations are the product of processes in the very early Universe and correspond to quantum fluctuations which during the inflationary phase expanded to macroscopic proportions1 Originally minute local deviations from the average density of the Universe (see fig. 1), and the corresponding deviations from the global cosmic expansion velocity (the Hubble expansion), will start to grow under the influence of the involved gravity perturbations. The gravitational force acting on each patch of matter in the universe is the total sum of the gravitational attraction by all matter throughout the universe. Evidently, in a homogeneous Universe the gravitational force is the same everywhere. -
Quantum Trajectories for Measurement of Entangled States
Quantum Trajectories for Measurement of Entangled States Apoorva Patel Centre for High Energy Physics, Indian Institute of Science, Bangalore 1 Feb 2018, ISNFQC18, SNBNCBS, Kolkata 1 Feb 2018, ISNFQC18, SNBNCBS, Kolkata A. Patel (CHEP, IISc) Quantum Trajectories for Entangled States / 22 Density Matrix The density matrix encodes complete information of a quantum system. It describes a ray in the Hilbert space. It is Hermitian and positive, with Tr(ρ)=1. It generalises the concept of probability distribution to quantum theory. 1 Feb 2018, ISNFQC18, SNBNCBS, Kolkata A. Patel (CHEP, IISc) Quantum Trajectories for Entangled States / 22 Density Matrix The density matrix encodes complete information of a quantum system. It describes a ray in the Hilbert space. It is Hermitian and positive, with Tr(ρ)=1. It generalises the concept of probability distribution to quantum theory. The real diagonal elements are the classical probabilities of observing various orthogonal eigenstates. The complex off-diagonal elements (coherences) describe quantum correlations among the orthogonal eigenstates. 1 Feb 2018, ISNFQC18, SNBNCBS, Kolkata A. Patel (CHEP, IISc) Quantum Trajectories for Entangled States / 22 Density Matrix The density matrix encodes complete information of a quantum system. It describes a ray in the Hilbert space. It is Hermitian and positive, with Tr(ρ)=1. It generalises the concept of probability distribution to quantum theory. The real diagonal elements are the classical probabilities of observing various orthogonal eigenstates. The complex off-diagonal elements (coherences) describe quantum correlations among the orthogonal eigenstates. For pure states, ρ2 = ρ and det(ρ)=0. Any power-series expandable function f (ρ) becomes a linear combination of ρ and I . -
Planck 'Toolkit' Introduction
Planck ‘toolkit’ introduction Welcome to the Planck ‘toolkit’ – a series of short questions and answers designed to equip you with background information on key cosmological topics addressed by the Planck science releases. The questions are arranged in thematic blocks; click the header to visit that section. Planck and the Cosmic Microwave Background (CMB) What is Planck and what is it studying? What is the Cosmic Microwave Background? Why is it so important to study the CMB? When was the CMB first detected? How many space missions have studied the CMB? What does the CMB look like? What is ‘the standard model of cosmology’ and how does it relate to the CMB? Behind the CMB: inflation and the meaning of temperature fluctuations How did the temperature fluctuations get there? How exactly do the temperature fluctuations relate to density fluctuations? The CMB and the distribution of matter in the Universe How is matter distributed in the Universe? Has the Universe always had such a rich variety of structure? How did the Universe evolve from very smooth to highly structured? How can we study the evolution of cosmic structure? Tools to study the distribution of matter in the Universe How is the distribution of matter in the Universe described mathematically? What is the power spectrum? What is the power spectrum of the distribution of matter in the Universe? Why are cosmologists interested in the power spectrum of cosmic structure? What was the distribution of the primordial fluctuations? How does this relate to the fluctuations in the CMB? -
Understanding the Born Rule in Weak Measurements
Understanding the Born Rule in Weak Measurements Apoorva Patel Centre for High Energy Physics, Indian Institute of Science, Bangalore 31 July 2017, Open Quantum Systems 2017, ICTS-TIFR N. Gisin, Phys. Rev. Lett. 52 (1984) 1657 A. Patel and P. Kumar, Phys Rev. A (to appear), arXiv:1509.08253 S. Kundu, T. Roy, R. Vijayaraghavan, P. Kumar and A. Patel (in progress) 31 July 2017, Open Quantum Systems 2017, A. Patel (CHEP, IISc) Weak Measurements and Born Rule / 29 Abstract Projective measurement is used as a fundamental axiom in quantum mechanics, even though it is discontinuous and cannot predict which measured operator eigenstate will be observed in which experimental run. The probabilistic Born rule gives it an ensemble interpretation, predicting proportions of various outcomes over many experimental runs. Understanding gradual weak measurements requires replacing this scenario with a dynamical evolution equation for the collapse of the quantum state in individual experimental runs. We revisit the framework to model quantum measurement as a continuous nonlinear stochastic process. It combines attraction towards the measured operator eigenstates with white noise, and for a specific ratio of the two reproduces the Born rule. This fluctuation-dissipation relation implies that the quantum state collapse involves the system-apparatus interaction only, and the Born rule is a consequence of the noise contributed by the apparatus. The ensemble of the quantum trajectories is predicted by the stochastic process in terms of a single evolution parameter, and matches well with the weak measurement results for superconducting transmon qubits. 31 July 2017, Open Quantum Systems 2017, A. Patel (CHEP, IISc) Weak Measurements and Born Rule / 29 Axioms of Quantum Dynamics (1) Unitary evolution (Schr¨odinger): d d i dt |ψi = H|ψi , i dt ρ =[H,ρ] . -
A873: Cosmology Course Notes VIII. Linear Fluctuations Linear
A873: Cosmology Course Notes VIII. Linear Fluctuations Linear perturbation theory in general Start with an \unperturbed" solution for quantities xi, i = 1; 2; ::: Write equations for x~i = xi + δxi: Expand, keeping terms linear in δxi, and subtracting off unperturbed solution to get equations for δxi. Hope you can solve them. In GR, the quantities xi might be metric coefficients, densities, velocities, etc. GR pertur- bation theory can be tricky for two reasons. (1) The equations are very complicated. This is a problem of practice, not of principle. (2) In a perturbed universe, the \natural" coordinate choice is no longer obvious. Different coordinate choices have different equations. Results for physically measurable quantities should be the same, but the descriptions can look quite different. For scales l cH−1 and velocities v c, one can usually get by with Newtonian linear perturbationtheory. Newtonian perturbation theory in an expanding universe (This discussion follows Peebles, pp. 111-119.) Basic unperturbed equations In an inertial frame, the equations governing the density ρ(r; t) and velocity u(r; t) of an ideal fluid are: @ρ Continuity equation : + ~ r (ρu) = 0 (mass conservation) @t r r · @u Euler equation : + (u ~ r)u = ~ rΦ (momentum conservation) @t r · r −r 2 Poisson equation : rΦ = 4πGρ. r We have ignored pressure gradients in the Euler equation. Transformation of variables We would like to have these equations in comoving coordinates x = r=a(t) with \peculiar" velocity v = u (a=a_ )r = (a_x) a_x = ax_ : − − We want to use a quantity that will be small when perturbations are small, so define the dimensionless density contrast δ(x; t) by ρ = ρb(t) [1 + δ(x; t)], ρb = background density 1=a3.