A Categorical Presentation of Quantum Computation with Anyons
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8.1. Spin & Statistics
8.1. Spin & Statistics Ref: A.Khare,”Fractional Statistics & Quantum Theory”, 1997, Chap.2. Anyons = Particles obeying fractional statistics. Particle statistics is determined by the phase factor eiα picked up by the wave function under the interchange of the positions of any pair of (identical) particles in the system. Before the discovery of the anyons, this particle interchange (or exchange) was treated as the permutation of particle labels. Let P be the operator for this interchange. P2 =I → e2iα =1 ∴ eiα = ±1 i.e., α=0,π Thus, there’re only 2 kinds of statistics, α=0(π) for Bosons (Fermions) obeying Bose-Einstein (Fermi-Dirac) statistics. Pauli’s spin-statistics theorem then relates particle spin with statistics, namely, bosons (fermions) are particles with integer (half-integer) spin. To account for the anyons, particle exchange is re-defined as an observable adiabatic (constant energy) process of physically interchanging particles. ( This is in line with the quantum philosophy that only observables are physically relevant. ) As will be shown later, the new definition does not affect statistics in 3-D space. However, for particles in 2-D space, α can be any (real) value; hence anyons. The converse of the spin-statistic theorem then implies arbitrary spin for 2-D particles. Quantization of S in 3-D See M.Alonso, H.Valk, “Quantum Mechanics: Principles & Applications”, 1973, §6.2. In 3-D, the (spin) angular momentum S has 3 non-commuting components satisfying S i,S j =iℏε i j k Sk 2 & S ,S i=0 2 This means a state can be the simultaneous eigenstate of S & at most one Si. -
Many Particle Orbits – Statistics and Second Quantization
H. Kleinert, PATH INTEGRALS March 24, 2013 (/home/kleinert/kleinert/books/pathis/pthic7.tex) Mirum, quod divina natura dedit agros It’s wonderful that divine nature has given us fields Varro (116 BC–27BC) 7 Many Particle Orbits – Statistics and Second Quantization Realistic physical systems usually contain groups of identical particles such as spe- cific atoms or electrons. Focusing on a single group, we shall label their orbits by x(ν)(t) with ν =1, 2, 3, . , N. Their Hamiltonian is invariant under the group of all N! permutations of the orbital indices ν. Their Schr¨odinger wave functions can then be classified according to the irreducible representations of the permutation group. Not all possible representations occur in nature. In more than two space dimen- sions, there exists a superselection rule, whose origin is yet to be explained, which eliminates all complicated representations and allows only for the two simplest ones to be realized: those with complete symmetry and those with complete antisymme- try. Particles which appear always with symmetric wave functions are called bosons. They all carry an integer-valued spin. Particles with antisymmetric wave functions are called fermions1 and carry a spin whose value is half-integer. The symmetric and antisymmetric wave functions give rise to the characteristic statistical behavior of fermions and bosons. Electrons, for example, being spin-1/2 particles, appear only in antisymmetric wave functions. The antisymmetry is the origin of the famous Pauli exclusion principle, allowing only a single particle of a definite spin orientation in a quantum state, which is the principal reason for the existence of the periodic system of elements, and thus of matter in general. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Arxiv:1709.02742V3
PIN GROUPS IN GENERAL RELATIVITY BAS JANSSENS Abstract. There are eight possible Pin groups that can be used to describe the transformation behaviour of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group. 1. Introduction For bosons, the space-time transformation behaviour is governed by the Lorentz group O(3, 1), which comprises four connected components. Rotations and boosts are contained in the connected component of unity, the proper orthochronous Lorentz group SO↑(3, 1). Parity (P ) and time reversal (T ) are encoded in the other three connected components of the Lorentz group, the translates of SO↑(3, 1) by P , T and PT . For fermions, the space-time transformation behaviour is governed by a double cover of O(3, 1). Rotations and boosts are described by the unique simply connected double cover of SO↑(3, 1), the spin group Spin↑(3, 1). However, in order to account for parity and time reversal, one needs to extend this cover from SO↑(3, 1) to the full Lorentz group O(3, 1). This extension is by no means unique. There are no less than eight distinct double covers of O(3, 1) that agree with Spin↑(3, 1) over SO↑(3, 1). They are the Pin groups abc Pin , characterised by the property that the elements ΛP and ΛT covering P and 2 2 2 T satisfy ΛP = −a, ΛT = b and (ΛP ΛT ) = −c, where a, b and c are either 1 or −1 (cf. -
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. -
Algebraic Topology
Algebraic Topology Len Evens Rob Thompson Northwestern University City University of New York Contents Chapter 1. Introduction 5 1. Introduction 5 2. Point Set Topology, Brief Review 7 Chapter 2. Homotopy and the Fundamental Group 11 1. Homotopy 11 2. The Fundamental Group 12 3. Homotopy Equivalence 18 4. Categories and Functors 20 5. The fundamental group of S1 22 6. Some Applications 25 Chapter 3. Quotient Spaces and Covering Spaces 33 1. The Quotient Topology 33 2. Covering Spaces 40 3. Action of the Fundamental Group on Covering Spaces 44 4. Existence of Coverings and the Covering Group 48 5. Covering Groups 56 Chapter 4. Group Theory and the Seifert{Van Kampen Theorem 59 1. Some Group Theory 59 2. The Seifert{Van Kampen Theorem 66 Chapter 5. Manifolds and Surfaces 73 1. Manifolds and Surfaces 73 2. Outline of the Proof of the Classification Theorem 80 3. Some Remarks about Higher Dimensional Manifolds 83 4. An Introduction to Knot Theory 84 Chapter 6. Singular Homology 91 1. Homology, Introduction 91 2. Singular Homology 94 3. Properties of Singular Homology 100 4. The Exact Homology Sequence{ the Jill Clayburgh Lemma 109 5. Excision and Applications 116 6. Proof of the Excision Axiom 120 3 4 CONTENTS 7. Relation between π1 and H1 126 8. The Mayer-Vietoris Sequence 128 9. Some Important Applications 131 Chapter 7. Simplicial Complexes 137 1. Simplicial Complexes 137 2. Abstract Simplicial Complexes 141 3. Homology of Simplicial Complexes 143 4. The Relation of Simplicial to Singular Homology 147 5. Some Algebra. The Tensor Product 152 6. -
Special Unitary Group - Wikipedia
Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Special unitary group In mathematics, the special unitary group of degree n, denoted SU( n), is the Lie group of n×n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U( n), consisting of all n×n unitary matrices. As a compact classical group, U( n) is the group that preserves the standard inner product on Cn.[nb 1] It is itself a subgroup of the general linear group, SU( n) ⊂ U( n) ⊂ GL( n, C). The SU( n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.[1] The simplest case, SU(1) , is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+ I, − I}. [nb 2] SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. Contents Properties Lie algebra Fundamental representation Adjoint representation The group SU(2) Diffeomorphism with S 3 Isomorphism with unit quaternions Lie Algebra The group SU(3) Topology Representation theory Lie algebra Lie algebra structure Generalized special unitary group Example Important subgroups See also 1 of 10 2/22/2018, 8:54 PM Special unitary group - Wikipedia https://en.wikipedia.org/wiki/Special_unitary_group Remarks Notes References Properties The special unitary group SU( n) is a real Lie group (though not a complex Lie group). -
Group Theory
Group theory March 7, 2016 Nearly all of the central symmetries of modern physics are group symmetries, for simple a reason. If we imagine a transformation of our fields or coordinates, we can look at linear versions of those transformations. Such linear transformations may be represented by matrices, and therefore (as we shall see) even finite transformations may be given a matrix representation. But matrix multiplication has an important property: associativity. We get a group if we couple this property with three further simple observations: (1) we expect two transformations to combine in such a way as to give another allowed transformation, (2) the identity may always be regarded as a null transformation, and (3) any transformation that we can do we can also undo. These four properties (associativity, closure, identity, and inverses) are the defining properties of a group. 1 Finite groups Define: A group is a pair G = fS; ◦} where S is a set and ◦ is an operation mapping pairs of elements in S to elements in S (i.e., ◦ : S × S ! S: This implies closure) and satisfying the following conditions: 1. Existence of an identity: 9 e 2 S such that e ◦ a = a ◦ e = a; 8a 2 S: 2. Existence of inverses: 8 a 2 S; 9 a−1 2 S such that a ◦ a−1 = a−1 ◦ a = e: 3. Associativity: 8 a; b; c 2 S; a ◦ (b ◦ c) = (a ◦ b) ◦ c = a ◦ b ◦ c We consider several examples of groups. 1. The simplest group is the familiar boolean one with two elements S = f0; 1g where the operation ◦ is addition modulo two. -
UCLA Electronic Theses and Dissertations
UCLA UCLA Electronic Theses and Dissertations Title Shapes of Finite Groups through Covering Properties and Cayley Graphs Permalink https://escholarship.org/uc/item/09b4347b Author Yang, Yilong Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA Los Angeles Shapes of Finite Groups through Covering Properties and Cayley Graphs A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Yilong Yang 2017 c Copyright by Yilong Yang 2017 ABSTRACT OF THE DISSERTATION Shapes of Finite Groups through Covering Properties and Cayley Graphs by Yilong Yang Doctor of Philosophy in Mathematics University of California, Los Angeles, 2017 Professor Terence Chi-Shen Tao, Chair This thesis is concerned with some asymptotic and geometric properties of finite groups. We shall present two major works with some applications. We present the first major work in Chapter 3 and its application in Chapter 4. We shall explore the how the expansions of many conjugacy classes is related to the representations of a group, and then focus on using this to characterize quasirandom groups. Then in Chapter 4 we shall apply these results in ultraproducts of certain quasirandom groups and in the Bohr compactification of topological groups. This work is published in the Journal of Group Theory [Yan16]. We present the second major work in Chapter 5 and 6. We shall use tools from number theory, combinatorics and geometry over finite fields to obtain an improved diameter bounds of finite simple groups. We also record the implications on spectral gap and mixing time on the Cayley graphs of these groups. -
Arxiv:0805.0285V1
Generalizations of Quantum Statistics O.W. Greenberg1 Center for Fundamental Physics Department of Physics University of Maryland College Park, MD 20742-4111, USA, Abstract We review generalizations of quantum statistics, including parabose, parafermi, and quon statistics, but not including anyon statistics, which is special to two dimensions. The general principles of quantum theory allow statistics more general than bosons or fermions. (Bose statistics and Fermi statistics are discussed in separate articles.) The restriction to Bosons or Fermions requires the symmetrization postu- late, “the states of a system containing N identical particles are necessarily either all symmetric or all antisymmetric under permutations of the N particles,” or, equiv- alently, “all states of identical particles are in one-dimensional representations of the symmetric group [1].” A.M.L. Messiah and O.W. Greenberg discussed quantum mechanics without the symmetrization postulate [2]. The spin-statistics connec- arXiv:0805.0285v1 [quant-ph] 2 May 2008 tion, that integer spin particles are bosons and odd-half-integer spin particles are fermions [3], is an independent statement. Identical particles in 2 space dimensions are a special case, “anyons.” (Anyons are discussed in a separate article.) Braid group statistics, a nonabelian analog of anyons, are also special to 2 space dimensions All observables must be symmetric in the dynamical variables associated with identical particles. Observables can not change the permutation symmetry type of the wave function; i.e. there is a superselection rule separating states in inequivalent representations of the symmetric group and when identical particles can occur in states that violate the spin-statistics connection their transitions must occur in the same representation of the symmetric group. -
Lecture Notes in Physics
Lecture Notes in Physics Volume 823 Founding Editors W. Beiglböck J. Ehlers K. Hepp H. Weidenmüller Editorial Board B.-G. Englert, Singapore U. Frisch, Nice, France F. Guinea, Madrid, Spain P. Hänggi, Augsburg, Germany W. Hillebrandt, Garching, Germany M. Hjorth-Jensen, Oslo, Norway R. A. L. Jones, Sheffield, UK H. v. Löhneysen, Karlsruhe, Germany M. S. Longair, Cambridge, UK M. L. Mangano, Geneva, Switzerland J.-F. Pinton, Lyon, France J.-M. Raimond, Paris, France A. Rubio, Donostia, San Sebastian, Spain M. Salmhofer, Heidelberg, Germany D. Sornette, Zurich, Switzerland S. Theisen, Potsdam, Germany D. Vollhardt, Augsburg, Germany W. Weise, Garching, Germany For further volumes: http://www.springer.com/series/5304 The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opments in physics research and teaching—quickly and informally, but with a high quality and the explicit aim to summarize and communicate current knowledge in an accessible way. Books published in this series are conceived as bridging material between advanced graduate textbooks and the forefront of research and to serve three purposes: • to be a compact and modern up-to-date source of reference on a well-defined topic • to serve as an accessible introduction to the field to postgraduate students and nonspecialist researchers from related areas • to be a source of advanced teaching material for specialized seminars, courses and schools Both monographs and multi-author volumes will be considered for publication. Edited volumes should, however, consist of a very limited number of contributions only. Proceedings will not be considered for LNP. Volumes published in LNP are disseminated both in print and in electronic formats, the electronic archive being available at springerlink.com. -
Chapter 3 Central Extensions of Groups
Chapter 3 Central Extensions of Groups The notion of a central extension of a group or of a Lie algebra is of particular importance in the quantization of symmetries. We give a detailed introduction to the subject with many examples, first for groups in this chapter and then for Lie algebras in the next chapter. 3.1 Central Extensions In this section let A be an abelian group and let G be an arbitrary group. The trivial group consisting only of the neutral element is denoted by 1. Definition 3.1. An extension of G by the group A is given by an exact sequence of group homomorphisms ι π 1 −→ A −→ E −→ G −→ 1. Exactness of the sequence means that the kernel of every map in the sequence equals the image of the previous map. Hence the sequence is exact if and only if ι is injec- tive, π is surjective, the image im ι is a normal subgroup, and ∼ kerπ = im ι(= A). The extension is called central if A is abelian and its image im ι is in the center of E, that is a ∈ A,b ∈ E ⇒ ι(a)b = bι(a). Note that A is written multiplicatively and 1 is the neutral element although A is supposed to be abelian. Examples: • A trivial extension has the form i pr 1 −→ A −→ A × G −→2 G −→ 1, Schottenloher, M.: Central Extensions of Groups. Lect. Notes Phys. 759, 39–62 (2008) DOI 10.1007/978-3-540-68628-6 4 c Springer-Verlag Berlin Heidelberg 2008 40 3 Central Extensions of Groups where A × G denotes the product group and where i : A → G is given by a → (a,1).