A Functorial Construction of Quantum Subtheories
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Infinitely Many Star Products to Play With
DSF{40{01 BiBoS 01-12-068 ESI 1109 hep{th/0112092 December 2001 Infinitely many star products to play with J.M. Gracia-Bond´ıa a;b, F. Lizzi b, G. Marmo b and P. Vitale c a BiBoS, Fakult¨atder Physik, Universit¨atBielefeld, 33615 Bielefeld, Germany b Dipartimento di Scienze Fisiche, Universit`adi Napoli Federico II and INFN, Sezione di Napoli, Monte S. Angelo Via Cintia, 80126 Napoli, Italy [email protected], [email protected] c Dipartimento di Fisica, Universit`adi Salerno and INFN Gruppo Collegato di Salerno, Via S. Allende 84081 Baronissi (SA), Italy [email protected] Abstract While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [xi; xj] = iθij. Here we present new classes of (non-formal) deformed products k associated to linear Lie algebras of the kind [xi; xj] = icijxk. For all possible three- dimensional cases, we define a new star product and discuss its properties. To complete the analysis of these novel noncommutative spaces, we introduce noncom- pact spectral triples, and the concept of star triple, a specialization of the spectral triple to deformations of the algebra of functions on a noncompact manifold. We examine the generalization to the noncompact case of Connes' conditions for non- commutative spin geometries, and, in the framework of the new star products, we exhibit some candidates for a Dirac operator. On the technical level, properties of 2n the Moyal multiplier algebra M(Rθ ) are elucidated. 1 Introduction Over five years ago, Connes gave the first axiomatics for first-quantized fermion fields on (compact) noncommutative varieties, the so-called spectral triples [1]. -
Symplectic Linear Algebra Let V Be a Real Vector Space. Definition 1.1. A
INTRODUCTION TO SYMPLECTIC TOPOLOGY C. VITERBO 1. Lecture 1: Symplectic linear algebra Let V be a real vector space. Definition 1.1. A symplectic form on V is a skew-symmetric bilinear nondegen- erate form: (1) ω(x, y)= ω(y, x) (= ω(x, x) = 0); (2) x, y such− that ω(x,⇒ y) = 0. ∀ ∃ % For a general 2-form ω on a vector space, V , we denote ker(ω) the subspace given by ker(ω)= v V w Vω(v, w)=0 { ∈ |∀ ∈ } The second condition implies that ker(ω) reduces to zero, so when ω is symplectic, there are no “preferred directions” in V . There are special types of subspaces in symplectic manifolds. For a vector sub- space F , we denote by F ω = v V w F,ω(v, w) = 0 { ∈ |∀ ∈ } From Grassmann’s formula it follows that dim(F ω)=codim(F ) = dim(V ) dim(F ) Also we have − Proposition 1.2. (F ω)ω = F (F + F )ω = F ω F ω 1 2 1 ∩ 2 Definition 1.3. A subspace F of V,ω is ω isotropic if F F ( ω F = 0); • coisotropic if F⊂ω F⇐⇒ | • Lagrangian if it is⊂ maximal isotropic. • Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1,L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism ϕ : V V such that ϕ∗ω = ω and ϕ(L )=L . -
On the Complexification of the Classical Geometries And
On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical -
Phase Space Quantum Mechanics Is Such a Natural Formulation of Quantum Theory
Phase space quantum mechanics Maciej B laszak, Ziemowit Doma´nski Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Pozna´n, Poland Abstract The paper develope the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise prod- uct of observables to an appropriate noncommutative ⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The formalism of the phase space quantum mechanics is presented in a very systematic way for the case of any smooth Hamiltonian function and for a very wide class of deformations. The considered class of deformations and the corresponding ⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. -
Symmetries of the Space of Linear Symplectic Connections
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 002, 30 pages Symmetries of the Space of Linear Symplectic Connections Daniel J.F. FOX Departamento de Matem´aticas del Area´ Industrial, Escuela T´ecnica Superior de Ingenier´ıa y Dise~noIndustrial, Universidad Polit´ecnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain E-mail: [email protected] Received June 30, 2016, in final form January 07, 2017; Published online January 10, 2017 https://doi.org/10.3842/SIGMA.2017.002 Abstract. There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen{Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle poly- nomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term. -
Weyl Quantization and Wigner Distributions on Phase Space
faculteit Wiskunde en Natuurwetenschappen Weyl quantization and Wigner distributions on phase space Bachelor thesis in Physics and Mathematics June 2014 Student: R.S. Wezeman Supervisor in Physics: Prof. dr. D. Boer Supervisor in Mathematics: Prof. dr H. Waalkens Abstract This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate. -
Matrix Bases for Star Products: a Review?
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 10 (2014), 086, 36 pages Matrix Bases for Star Products: a Review? Fedele LIZZI yzx and Patrizia VITALE yz y Dipartimento di Fisica, Universit`adi Napoli Federico II, Napoli, Italy E-mail: [email protected], [email protected] z INFN, Sezione di Napoli, Italy x Institut de Ci´enciesdel Cosmos, Universitat de Barcelona, Catalonia, Spain Received March 04, 2014, in final form August 11, 2014; Published online August 15, 2014 http://dx.doi.org/10.3842/SIGMA.2014.086 Abstract. We review the matrix bases for a family of noncommutative ? products based on a Weyl map. These products include the Moyal product, as well as the Wick{Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity. Key words: noncommutative geometry; star products; matrix models 2010 Mathematics Subject Classification: 58Bxx; 40C05; 46L65 1 Introduction Star products were originally introduced [34, 62] in the context of point particle quantization, they were generalized in [9, 10] to comprise general cases, the physical motivations behind this was the quantization of phase space. Later star products became a tool to describe possible noncommutative geometries of spacetime itself (for a review see [4]). In this sense they have become a tool for the study of field theories on noncommutative spaces [77]. In this short review we want to concentrate on one particular aspect of the star product, the fact that any action involving fields which are multiplied with a star product can be seen as a matrix model. -
1 Symplectic Vector Spaces
Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
1. Overview Over the Basic Notions in Symplectic Geometry Definition 1.1
AN INTRODUCTION TO SYMPLECTIC GEOMETRY MAIK PICKL 1. Overview over the basic notions in symplectic geometry Definition 1.1. Let V be a m-dimensional R vector space and let Ω : V × V ! R be a bilinear map. We say Ω is skew-symmetric if Ω(u; v) = −Ω(v; u), for all u; v 2 V . Theorem 1.1. Let Ω be a skew-symmetric bilinear map on V . Then there is a basis u1; : : : ; uk; e1 : : : ; en; f1; : : : ; fn of V s.t. Ω(ui; v) = 0; 8i and 8v 2 V Ω(ei; ej) = 0 = Ω(fi; fj); 8i; j Ω(ei; fj) = δij; 8i; j: In matrix notation with respect to this basis we have 00 0 0 1 T Ω(u; v) = u @0 0 idA v: 0 −id 0 In particular we have dim(V ) = m = k + 2n. Proof. The proof is a skew-symmetric version of the Gram-Schmidt process. First let U := fu 2 V jΩ(u; v) = 0 for all v 2 V g; and choose a basis u1; : : : ; uk for U. Now choose a complementary space W s.t. V = U ⊕ W . Let e1 2 W , then there is a f1 2 W s.t. Ω(e1; f1) 6= 0 and we can assume that Ω(e1; f1) = 1. Let W1 = span(e1; f1) Ω W1 = fw 2 W jΩ(w; v) = 0 for all v 2 W1g: Ω Ω We claim that W = W1 ⊕ W1 : Suppose that v = ae1 + bf1 2 W1 \ W1 . We get that 0 = Ω(v; e1) = −b 0 = Ω(v; f1) = a and therefore v = 0. -
Symplectic and Kaehler Geometry
Chapter 3 Symplectic and Kaehler Geometry Lecture 12 Today: Symplectic geometry and Kaehler geometry, the linear aspects anyway. Symplectic Geometry Let V be an n dimensional vector space over R, B : V V R a bilineare form on V . × → Definition. B is alternating if B(v, w) = B(w, v). Denote by Alt2(V ) the space of all alternating bilinear forms on V . − Definition. Take any B Alt(V ), U a subspace of V . Then we can define the orthogonal complement by ∈ U ⊥ = v V, B(u, v) = 0, u U { ∈ ∀ ∈ } Definition. B is non-degenerate if V ⊥ = 0 . { } Theorem. If B is non-degenerate then dim V is even. Moeover, there exists a basis e1, . , en, f1, . , fn of V such that B(ei, en) = B(fi, fj ) = 0 and B(ei, fj ) = δij Definition. B is non-degenerate if and only if the pair (V, B) is a symplectic vector space. Then ei’s and fj ’s are called a Darboux basis of V . Let B be non-degenerate and U a vector subspace of V Remark: dim U ⊥ = 2n dim V and we have the following 3 scenarios. − 1. U isotropic U ⊥ U. This implies that dim U n ⇔ ⊃ ≤ 2. U Lagrangian U ⊥ = U. This implies dim U = n. ⇔ 3. U symplectic U ⊥ U = . This implies that U ⊥ is symplectic and B and B ⊥ are non-degenerate. ⇔ ∩ ∅ |U |U Let V = V m be a vector space over R we have 2 2 Alt (V ) ∼= Λ (V ∗) is a canonical identification. Let v1, . , vm be a basis of v, then 2 1 Alt (V ) B B(v , v )v∗ v∗ ∋ 7→ 2 i j i ∧ j 2 2 � and the inverse Λ (V ∗) ω B Alt (V ) is given by ∋ 7→ ω ∈ B(v, w) = iW (iV ω) Suppose m = 2n. -
A Student's Guide to Symplectic Spaces, Grassmannians
A Student’s Guide to Symplectic Spaces, Grassmannians and Maslov Index Paolo Piccione Daniel Victor Tausk DEPARTAMENTO DE MATEMATICA´ INSTITUTO DE MATEMATICA´ E ESTAT´ISTICA UNIVERSIDADE DE SAO˜ PAULO Contents Preface v Introduction vii Chapter 1. Symplectic Spaces 1 1.1. A short review of Linear Algebra 1 1.2. Complex structures 5 1.3. Complexification and real forms 8 1.3.1. Complex structures and complexifications 13 1.4. Symplectic forms 16 1.4.1. Isotropic and Lagrangian subspaces. 20 1.4.2. Lagrangian decompositions of a symplectic space 24 1.5. Index of a symmetric bilinear form 28 Exercises for Chapter 1 35 Chapter 2. The Geometry of Grassmannians 39 2.1. Differentiable manifolds and Lie groups 39 2.1.1. Classical Lie groups and Lie algebras 41 2.1.2. Actions of Lie groups and homogeneous manifolds 44 2.1.3. Linearization of the action of a Lie group on a manifold 48 2.2. Grassmannians and their differentiable structure 50 2.3. The tangent space to a Grassmannian 53 2.4. The Grassmannian as a homogeneous space 55 2.5. The Lagrangian Grassmannian 59 k 2.5.1. The submanifolds Λ (L0) 64 Exercises for Chapter 2 68 Chapter 3. Topics of Algebraic Topology 71 3.1. The fundamental groupoid and the fundamental group 71 3.1.1. The Seifert–van Kampen theorem for the fundamental groupoid 75 3.1.2. Stability of the homotopy class of a curve 77 3.2. The homotopy exact sequence of a fibration 80 3.2.1. Applications to the theory of classical Lie groups 92 3.3.