Intermediate Statistics, Parastatistics, Fractionary Statistics and Gentilionic Statistics
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Identical Particles
8.06 Spring 2016 Lecture Notes 4. Identical particles Aram Harrow Last updated: May 19, 2016 Contents 1 Fermions and Bosons 1 1.1 Introduction and two-particle systems .......................... 1 1.2 N particles ......................................... 3 1.3 Non-interacting particles .................................. 5 1.4 Non-zero temperature ................................... 7 1.5 Composite particles .................................... 7 1.6 Emergence of distinguishability .............................. 9 2 Degenerate Fermi gas 10 2.1 Electrons in a box ..................................... 10 2.2 White dwarves ....................................... 12 2.3 Electrons in a periodic potential ............................. 16 3 Charged particles in a magnetic field 21 3.1 The Pauli Hamiltonian ................................... 21 3.2 Landau levels ........................................ 23 3.3 The de Haas-van Alphen effect .............................. 24 3.4 Integer Quantum Hall Effect ............................... 27 3.5 Aharonov-Bohm Effect ................................... 33 1 Fermions and Bosons 1.1 Introduction and two-particle systems Previously we have discussed multiple-particle systems using the tensor-product formalism (cf. Section 1.2 of Chapter 3 of these notes). But this applies only to distinguishable particles. In reality, all known particles are indistinguishable. In the coming lectures, we will explore the mathematical and physical consequences of this. First, consider classical many-particle systems. If a single particle has state described by position and momentum (~r; p~), then the state of N distinguishable particles can be written as (~r1; p~1; ~r2; p~2;:::; ~rN ; p~N ). The notation (·; ·;:::; ·) denotes an ordered list, in which different posi tions have different meanings; e.g. in general (~r1; p~1; ~r2; p~2)6 = (~r2; p~2; ~r1; p~1). 1 To describe indistinguishable particles, we can use set notation. -
Magnetic Materials I
5 Magnetic materials I Magnetic materials I ● Diamagnetism ● Paramagnetism Diamagnetism - susceptibility All materials can be classified in terms of their magnetic behavior falling into one of several categories depending on their bulk magnetic susceptibility χ. without spin M⃗ 1 χ= in general the susceptibility is a position dependent tensor M⃗ (⃗r )= ⃗r ×J⃗ (⃗r ) H⃗ 2 ] In some materials the magnetization is m / not a linear function of field strength. In A [ such cases the differential susceptibility M is introduced: d M⃗ χ = d d H⃗ We usually talk about isothermal χ susceptibility: ∂ M⃗ χ = T ( ∂ ⃗ ) H T Theoreticians define magnetization as: ∂ F⃗ H[A/m] M=− F=U−TS - Helmholtz free energy [4] ( ∂ H⃗ ) T N dU =T dS− p dV +∑ μi dni i=1 N N dF =T dS − p dV +∑ μi dni−T dS−S dT =−S dT − p dV +∑ μi dni i=1 i=1 Diamagnetism - susceptibility It is customary to define susceptibility in relation to volume, mass or mole: M⃗ (M⃗ /ρ) m3 (M⃗ /mol) m3 χ= [dimensionless] , χρ= , χ = H⃗ H⃗ [ kg ] mol H⃗ [ mol ] 1emu=1×10−3 A⋅m2 The general classification of materials according to their magnetic properties μ<1 <0 diamagnetic* χ ⃗ ⃗ ⃗ ⃗ ⃗ ⃗ B=μ H =μr μ0 H B=μo (H +M ) μ>1 >0 paramagnetic** ⃗ ⃗ ⃗ χ μr μ0 H =μo( H + M ) → μ r=1+χ μ≫1 χ≫1 ferromagnetic*** *dia /daɪə mæ ɡˈ n ɛ t ɪ k/ -Greek: “from, through, across” - repelled by magnets. We have from L2: 1 2 the force is directed antiparallel to the gradient of B2 F = V ∇ B i.e. -
Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff Dimension, Percolation, Clusters
Applied Mathematics 2018, 8(1): 5-8 DOI: 10.5923/j.am.20180801.02 Why Fermions and Bosons are Observable as Single Particles while Quarks are not? Sencer Taneri Private Researcher, Turkey Abstract Bosons and Fermions are observable in nature while Quarks appear only in triplets for matter particles. We find a theoretical proof for this statement in this paper by investigating 2-dim model. The occupation numbers q are calculated by a power law dependence of occupation probability and utilizing Hausdorff dimension for the infinitely small mesh in the phase space. The occupation number for Quarks are manipulated and found to be equal to approximately three as they are Parafermions. Keywords Probability and Statistics, Parastatistics, Boson, Fermion, Parafermion, Hausdorff dimension, Percolation, Clusters There may be particles that obey some kind of statistics, 1. Introduction generally called parastatistics. The Parastatistics proposed in 1952 by H. Green was deduced using a quantum field The essential difference in classical and quantum theory (QFT) [2, 3]. Whenever we discover a new particle, descriptions of N identical particles is in their individuality, it is almost certain that we attribute its behavior to the rather than in their indistinguishability [1]. Spin of the property that it obeys some form of parastatistics and the particle is one of its intrinsic physical quantity that is unique maximum occupation number q of a given quantum state to its individuality. The basis for how the quantum states for would be a finite number that could assume any integer N identical particles will be occupied may be taken as value as 1<q <∞ (see Table 1). -
5.1 Two-Particle Systems
5.1 Two-Particle Systems We encountered a two-particle system in dealing with the addition of angular momentum. Let's treat such systems in a more formal way. The w.f. for a two-particle system must depend on the spatial coordinates of both particles as @Ψ well as t: Ψ(r1; r2; t), satisfying i~ @t = HΨ, ~2 2 ~2 2 where H = + V (r1; r2; t), −2m1r1 − 2m2r2 and d3r d3r Ψ(r ; r ; t) 2 = 1. 1 2 j 1 2 j R Iff V is independent of time, then we can separate the time and spatial variables, obtaining Ψ(r1; r2; t) = (r1; r2) exp( iEt=~), − where E is the total energy of the system. Let us now make a very fundamental assumption: that each particle occupies a one-particle e.s. [Note that this is often a poor approximation for the true many-body w.f.] The joint e.f. can then be written as the product of two one-particle e.f.'s: (r1; r2) = a(r1) b(r2). Suppose furthermore that the two particles are indistinguishable. Then, the above w.f. is not really adequate since you can't actually tell whether it's particle 1 in state a or particle 2. This indeterminacy is correctly reflected if we replace the above w.f. by (r ; r ) = a(r ) (r ) (r ) a(r ). 1 2 1 b 2 b 1 2 The `plus-or-minus' sign reflects that there are two distinct ways to accomplish this. Thus we are naturally led to consider two kinds of identical particles, which we have come to call `bosons' (+) and `fermions' ( ). -
Identical Particles in Classical Physics One Can Distinguish Between
Identical particles In classical physics one can distinguish between identical particles in such a way as to leave the dynamics unaltered. Therefore, the exchange of particles of identical particles leads to di®erent con¯gurations. In quantum mechanics, i.e., in nature at the microscopic levels identical particles are indistinguishable. A proton that arrives from a supernova explosion is the same as the proton in your glass of water.1 Messiah and Greenberg2 state the principle of indistinguishability as \states that di®er only by a permutation of identical particles cannot be distinguished by any obser- vation whatsoever." If we denote the operator which interchanges particles i and j by the permutation operator Pij we have ^ y ^ hªjOjªi = hªjPij OPijjÃi ^ which implies that Pij commutes with an arbitrary observable O. In particular, it commutes with the Hamiltonian. All operators are permutation invariant. Law of nature: The wave function of a collection of identical half-odd-integer spin particles is completely antisymmetric while that of integer spin particles or bosons is completely symmetric3. De¯ning the permutation operator4 by Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = ª(1; 2; ¢ ¢ ¢ ; j; ¢ ¢ ¢ ; i; ¢ ¢ ¢ N) we have Pij ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) = § ª(1; 2; ¢ ¢ ¢ ; i; ¢ ¢ ¢ ; j; ¢ ¢ ¢ N) where the upper sign refers to bosons and the lower to fermions. Emphasize that the symmetry requirements only apply to identical particles. For the hydrogen molecule if I and II represent the coordinates and spins of the two protons and 1 and 2 of the two electrons we have ª(I;II; 1; 2) = ¡ ª(II;I; 1; 2) = ¡ ª(I;II; 2; 1) = ª(II;I; 2; 1) 1On the other hand, the elements on earth came from some star/supernova in the distant past. -
Physics Abstracts CLASSIFICATION and CONTENTS (PACC)
Physics Abstracts CLASSIFICATION AND CONTENTS (PACC) 0000 general 0100 communication, education, history, and philosophy 0110 announcements, news, and organizational activities 0110C announcements, news, and awards 0110F conferences, lectures, and institutes 0110H physics organizational activities 0130 physics literature and publications 0130B publications of lectures(advanced institutes, summer schools, etc.) 0130C conference proceedings 0130E monographs, and collections 0130K handbooks and dictionaries 0130L collections of physical data, tables 0130N textbooks 0130Q reports, dissertations, theses 0130R reviews and tutorial papers;resource letters 0130T bibliographies 0140 education 0140D course design and evaluation 0140E science in elementary and secondary school 0140G curricula, teaching methods,strategies, and evaluation 0140J teacher training 0150 educational aids(inc. equipment,experiments and teaching approaches to subjects) 0150F audio and visual aids, films 0150H instructional computer use 0150K testing theory and techniques 0150M demonstration experiments and apparatus 0150P laboratory experiments and apparatus 0150Q laboratory course design,organization, and evaluation 0150T buildings and facilities 0155 general physics 0160 biographical, historical, and personal notes 0165 history of science 0170 philosophy of science 0175 science and society 0190 other topics of general interest 0200 mathematical methods in physics 0210 algebra, set theory, and graph theory 0220 group theory(for algebraic methods in quantum mechanics, see -
The Conventionality of Parastatistics
The Conventionality of Parastatistics David John Baker Hans Halvorson Noel Swanson∗ March 6, 2014 Abstract Nature seems to be such that we can describe it accurately with quantum theories of bosons and fermions alone, without resort to parastatistics. This has been seen as a deep mystery: paraparticles make perfect physical sense, so why don't we see them in nature? We consider one potential answer: every paraparticle theory is physically equivalent to some theory of bosons or fermions, making the absence of paraparticles in our theories a matter of convention rather than a mysterious empirical discovery. We argue that this equivalence thesis holds in all physically admissible quantum field theories falling under the domain of the rigorous Doplicher-Haag-Roberts approach to superselection rules. Inadmissible parastatistical theories are ruled out by a locality- inspired principle we call Charge Recombination. Contents 1 Introduction 2 2 Paraparticles in Quantum Theory 6 ∗This work is fully collaborative. Authors are listed in alphabetical order. 1 3 Theoretical Equivalence 11 3.1 Field systems in AQFT . 13 3.2 Equivalence of field systems . 17 4 A Brief History of the Equivalence Thesis 20 4.1 The Green Decomposition . 20 4.2 Klein Transformations . 21 4.3 The Argument of Dr¨uhl,Haag, and Roberts . 24 4.4 The Doplicher-Roberts Reconstruction Theorem . 26 5 Sharpening the Thesis 29 6 Discussion 36 6.1 Interpretations of QM . 44 6.2 Structuralism and Haecceities . 46 6.3 Paraquark Theories . 48 1 Introduction Our most fundamental theories of matter provide a highly accurate description of subatomic particles and their behavior. -
Talk M.Alouani
From atoms to solids to nanostructures Mebarek Alouani IPCMS, 23 rue du Loess, UMR 7504, Strasbourg, France European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 1/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 2/107 Course content I. Magnetic moment and magnetic field II. No magnetism in classical mechanics III. Where does magnetism come from? IV. Crystal field, superexchange, double exchange V. Free electron model: Spontaneous magnetization VI. The local spin density approximation of the DFT VI. Beyond the DFT: LDA+U VII. Spin-orbit effects: Magnetic anisotropy, XMCD VIII. Bibliography European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 3/107 Magnetic moments In classical electrodynamics, the vector potential, in the magnetic dipole approximation, is given by: μ μ × rˆ Ar()= 0 4π r2 where the magnetic moment μ is defined as: 1 μ =×Id('rl ) 2 ∫ It is shown that the magnetic moment is proportional to the angular momentum μ = γ L (γ the gyromagnetic factor) The projection along the quantification axis z gives (/2Bohrmaμ = eh m gneton) μzB= −gmμ Be European Summer Campus 2012: Physics at the nanoscale, Strasbourg, France, July 01–07, 2012 4/107 Magnetization and Field The magnetization M is the magnetic moment per unit volume In free space B = μ0 H because there is no net magnetization. -
Spontaneous Symmetry Breaking and Mass Generation As Built-In Phenomena in Logarithmic Nonlinear Quantum Theory
Vol. 42 (2011) ACTA PHYSICA POLONICA B No 2 SPONTANEOUS SYMMETRY BREAKING AND MASS GENERATION AS BUILT-IN PHENOMENA IN LOGARITHMIC NONLINEAR QUANTUM THEORY Konstantin G. Zloshchastiev Department of Physics and Center for Theoretical Physics University of the Witwatersrand Johannesburg, 2050, South Africa (Received September 29, 2010; revised version received November 3, 2010; final version received December 7, 2010) Our primary task is to demonstrate that the logarithmic nonlinearity in the quantum wave equation can cause the spontaneous symmetry break- ing and mass generation phenomena on its own, at least in principle. To achieve this goal, we view the physical vacuum as a kind of the funda- mental Bose–Einstein condensate embedded into the fictitious Euclidean space. The relation of such description to that of the physical (relativis- tic) observer is established via the fluid/gravity correspondence map, the related issues, such as the induced gravity and scalar field, relativistic pos- tulates, Mach’s principle and cosmology, are discussed. For estimate the values of the generated masses of the otherwise massless particles such as the photon, we propose few simple models which take into account small vacuum fluctuations. It turns out that the photon’s mass can be naturally expressed in terms of the elementary electrical charge and the extensive length parameter of the nonlinearity. Finally, we outline the topological properties of the logarithmic theory and corresponding solitonic solutions. DOI:10.5506/APhysPolB.42.261 PACS numbers: 11.15.Ex, 11.30.Qc, 04.60.Bc, 03.65.Pm 1. Introduction Current observational data in astrophysics are probing a regime of de- partures from classical relativity with sensitivities that are relevant for the study of the quantum-gravity problem [1,2]. -
International Centre for Theoretical Physics
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS OUTLINE OF A NONLINEAR, RELATIVISTIC QUANTUM MECHANICS OF EXTENDED PARTICLES Eckehard W. Mielke INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1981 MIRAMARE-TRIESTE IC/81/9 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization 3TERKATIOHAL CENTRE FOR THEORETICAL PHTSICS OUTLINE OF A NONLINEAR, KELATIVXSTTC QUANTUM HECHAHICS OF EXTENDED PARTICLES* Eckehard W. Mielke International Centre for Theoretical Physics, Trieste, Italy. MIRAHABE - TRIESTE January 198l • Submitted for publication. ABSTRACT I. INTRODUCTION AMD MOTIVATION A quantum theory of intrinsically extended particles similar to de Broglie's In recent years the conviction has apparently increased among most theory of the Double Solution is proposed, A rational notion of the particle's physicists that all fundamental interactions between particles - including extension is enthroned by realizing its internal structure via soliton-type gravity - can be comprehended by appropriate extensions of local gauge theories. solutions of nonlinear, relativistic wave equations. These droplet-type waves The principle of gauge invarisince was first formulated in have a quasi-objective character except for certain boundary conditions vhich 1923 by Hermann Weyl[l57] with the aim to give a proper connection between may be subject to stochastic fluctuations. More precisely, this assumption Dirac's relativistic quantum theory [37] of the electron and the Maxwell- amounts to a probabilistic description of the center of a soliton such that it Lorentz's theory of electromagnetism. Later Yang and Mills [167] as well as would follow the conventional quantum-mechanical formalism in the limit of zero Sharp were able to extend the latter theory to a nonabelian gauge particle radius. -
A Scientific Metaphysical Naturalisation of Information
1 A Scientific Metaphysical Naturalisation of Information With a indication-based semantic theory of information and an informationist statement of physicalism. Bruce Long A thesis submitted to fulfil requirements for the degree of Doctor of Philosophy Faculty of Arts and Social Sciences The University of Sydney February 2018 2 Abstract The objective of this thesis is to present a naturalised metaphysics of information, or to naturalise information, by way of deploying a scientific metaphysics according to which contingency is privileged and a-priori conceptual analysis is excluded (or at least greatly diminished) in favour of contingent and defeasible metaphysics. The ontology of information is established according to the premises and mandate of the scientific metaphysics by inference to the best explanation, and in accordance with the idea that the primacy of physics constraint accommodates defeasibility of theorising in physics. This metaphysical approach is used to establish a field ontology as a basis for an informational structural realism. This is in turn, in combination with information theory and specifically mathematical and algorithmic theories of information, becomes the foundation of what will be called a source ontology, according to which the world is the totality of information sources. Information sources are to be understood as causally induced configurations of structure that are, or else reduce to and/or supervene upon, bounded (including distributed and non-contiguous) regions of the heterogeneous quantum field (all quantum fields combined) and fluctuating vacuum, all in accordance with the above-mentioned quantum field-ontic informational structural realism (FOSIR.) Arguments are presented for realism, physicalism, and reductionism about information on the basis of the stated contingent scientific metaphysics. -
The Connection Between Spin and Statistics
Sudarshan: Seven Science Quests IOP Publishing Journal of Physics: Conference Series 196 (2009) 012011 doi:10.1088/1742-6596/196/1/012011 The connection between spin and statistics Luis J. Boya Departamento de Física Teórica, Universidad de Zaragoza E - 50 009, Zaragoza, Spain E-Mail: [email protected] Abstract. After some brief historical remarks we recall the first demonstration of the Spin-Statistics theorem by Pauli, and some further developments, like Streater- Wightman’s, in the axiomatic considerations. Sudarshan’s short proof then follows by considering only the time part of the kinetic term in the lagrangian: the identity SU(2)=Sp(1,C) is crucial for the argument. Possible generalization to arbitrary dimensions are explored, and seen to depend crucially on the type of spinors, obeying Bott´s periodicity 8. In particular, it turns out that for space dimentions (7, 8, or 9) modulo 8, the conventional result does not automatically hold: there can be no Fermions in these dimensions. 1. Introduction The idea of this work came from discussions with Sudarshan in Zaragoza some years ago. George was surprised to learn that the properties of spinors crucial for the ordinary connection between spin and statistics would not hold in arbitrary dimensions, because of periodicity 8 of real Clifford algebras. After many discussions and turnarounds, I am happy today to present this collaboration, and I also apologize for the delay, for which I am mainly responsible. In any case, the priviledge to work and discuss physics with Sudarshan is a unique experience, for which I have been most fortunate.