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REPRESENTATIONS of INFINITE DIMENSIONAL LIE ALGEBRAS and DIRAC's POSITRON THEORY Jeremy S. Gresham a Thesis Submitted to the U

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REPRESENTATIONS of INFINITE DIMENSIONAL LIE ALGEBRAS and DIRAC's POSITRON THEORY Jeremy S. Gresham a Thesis Submitted to the U REPRESENTATIONS OF INFINITE DIMENSIONAL LIE ALGEBRAS AND DIRAC'S POSITRON THEORY Jeremy S. Gresham A Thesis Submitted to the University of North Carolina Wilmington in Partial Fulfillment Of the Requirements for the Degree of Master of Science Department of Mathematics and Statistics University of North Carolina Wilmington 2011 Approved by Advisory Committee Mark Lammers Gabriel Lugo Dijana Jakeli´c Chair Accepted by Dean, Graduate School TABLE OF CONTENTS ABSTRACT . iv LIST OF SYMBOLS . v 1 INTRODUCTION . 1 2 Preliminaries . 3 3 Algebras . 8 3.1 Witt Algebra . 8 3.2 Virasoro Algebra . 9 3.3 Oscillator Algebra . 15 3.4 Algebras of Infinite Matrices . 16 3.5 Loop Algebras . 20 4 Representations of the Virasoro Algebra . 25 4.1 Hermitian Forms . 25 4.2 Highest Weight Representations of Vir . 26 4.3 Irreducible Positive Energy Representations . 30 5 Oscillator Representations . 39 5.1 Representations of the Oscillator Algebra . 39 5.2 Oscillator Representations of Vir . 45 6 Dirac Positron Theory . 53 6.1 Infinite Wedge Space . 53 6.2 Highest Weight Representations of gl1 . 56 6.3 Representations ofa ¯1 ...................... 61 7 Some Physics . 64 7.1 Dirac Equation and First Quantization . 64 7.2 Second Quantization and Fock space . 74 APPENDIX . 76 ii A Triangular Decomposition . 76 B Tensor Products and Related Algebras . 76 C Geometry . 78 REFERENCES . 81 iii ABSTRACT Representation theory and physics interact in complex and often unexpected ways, with one discipline building upon the work of the other. We present a number of rep- resentations for the Witt, Virasoro, and Heisenberg algebras building up to the Dirac theory for relativistic electrons, once from the representation theory perspective and then, less rigorously, from a physics perspective. iv LIST OF SYMBOLS • · j · - a symmetric bilinear form • [·; ·] - a Lie bracket, usually the matrix commutator • V , W - a vector space • x, y, z, v, - vectors • gl1,a ¯1, g, X, d, etc - Lie algebras • Vir - the Virasoro Algebra •A - the oscillator (Heisenberg) algebra • G, GLn, GL1 - Lie groups • Z - the integers • i,j,k,l,m,n - integers • R - the reals • ~, , q - real numbers • C - the complex numbers • λ, µ, α, β - complex numbers • λ¯ - the complex conjugate of λ • B = C[x1; x2; :::] - the space of polynomials over C in infinitely many variables • C[t; t−1] - the space of Laurent polynomials, polynomials in t and t−1 • ! - an antilinear anti-involution v • π, φ, r,r ^, τ - algebra homomorphisms • L, ⊕ - direct sum • P - sum • ⊗ - the tensor product • ^ - the antisymmetric (or wedge) product • : aiaj : - the normal ordering of ai, aj (Section 4.2) • I - identity matrix • tr(·) - the trace • exp(·) - the exponential map • δi;j = δij - the Kronecker delta • M - a manifold • C1(M) - smooth functions on a manifold M • TxM - tangent space at a point x 2 M • X(M) - algebra of vector fields on M • F - space of semi-infinite monomials • Res0[·] - the residue at zero of some Laurent polynomial • Ay is the Hermitian conjugate of A This list is not all-inclusive, but covers most of the symbols in their most common usage. vi 1 INTRODUCTION Representation theory has very strong connections to physics. In particular, infi- nite dimensional Lie algebras are important for conformal field theory and exactly solvable models. The Witt, Virasoro, and Heisenberg algebras all have interesting representations which can be used to describe Dirac's positron theory. Dirac's the- ory is an attempt to combine quantum mechanics and relativity. It contained the first prediction of antimatter, in the form of positrons. In this thesis we will describe some of these representations along with Dirac's original theory with the aim of giving a taste of the interplay of representation theory and physics. We will also give some representations of loop algebras in the same context. The physics included will be presented in a more informal way than the mathe- matics { rather like a summary of relevant ideas. We point out that the Heisenberg or oscillator algebra defined here is an infinite dimensional algebra, and not directly related to the usual finite dimensional Heisenberg group. The Virasoro algebra, the unique central extension of the Witt algebra, is used for vertex operator algebras and has applications in conformal fields and string theory. Our main focus, however, is Dirac's positron theory. This is an interesting model, but still simple enough to see some of the real interplay between physics and representation theory. Sections 3, 4, 5, 6 and 8 are devoted to representation theory and Dirac's positron theory. These sections were based on material in Kac and Raina's book \Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras" ([4]). Subsection 4.3 which covers positive energy and highest weight representa- tions of the Virasoro algebra follows also \Generalized Derivations on Algebras and Highest Weight Representations of the Virasoro Algebra" ([2]) by Jonas Hartwig. Section 7 addresses Dirac's theory from a physics perspective, beginning with the Schr¨odingerequation and the relativistic energy equation. It draws material from Bernd Thaller's \The Dirac Equation" ([6]), a physics text. 2 2 Preliminaries Definition 1 A Lie algebra is a vector space g over a field K equipped with a multiplication [·; ·]: g × g ! g having the following properties: a) [·; ·] is K-bilinear, b) [x; y] = −[y; x] for all x; y 2 g c) Jacobi identity: [[y; z]; x] + [[z; x]; y] + [[x; y]; z] = 0, for all x; y 2 g. In this thesis we will mostly have K = C. Example 1 An example of a Lie algebra is the set of all linear transformations of a vector space V , denoted gl(V ), where [x; y] = xy − yx for x; y 2 gl(V ). This bracket is called the commutator bracket. A common way of defining an algebra is by specifying a set of generators and relations among those generators. For Lie algebras this is done by giving a basis and specifying the Lie bracket on those elements. Example 2 The Lie algebra sl2(C) is a vector space over C with basis fx; h; yg and Lie bracket defined by the relations [x; y] = h [h; x] = 2x [h; y] = −2y: This can be realized as a matrix algebra, taking 2 3 2 3 2 3 0 1 0 0 1 0 6 7 6 7 6 7 x = 4 5 ; y = 4 5 ; h = 4 5 : 0 0 1 0 0 −1 3 Definition 2 A subalgebra of a Lie algebra g is a subspace e of g that is closed under the bracket, i.e., [x; y] 2 e, 8x; y 2 e. A subalgebra e is abelian provided [x; y] = 0, 8x; y 2 e. Definition 3 An ideal of a Lie algebra g is a subspace I such that for any x 2 I and any y 2 g we have [x; y] 2 I. The algebra sl2(C) defined above has no nontrivial (i.e. nonzero) proper ideals. sl2(C) can be seen as a proper ideal of gl2(C), the four dimensional Lie algebra of 2x2 matrices with the commutator bracket. Given a Lie algebra g and an ideal I of g, consider the quotient vector space g0 = g=I equipped with bracket [x + I; y + I] = [x; y] + I for all x; y 2 g. Then g0 is a Lie algebra called the quotient algebra of g by I. Moreover, there is a surjective Lie algebra homomorphism φ : g ! g0 such that kerφ = I. A triangular decomposition of a Lie algebra g consists of an abelian subal- gebra h 6= 0 and two subalgebras n+ and n− such that g = n− ⊕ h ⊕ n+ with a few conditions imposed on these subalgebras that we will not elaborate on. For a detailed definition, refer to Appendix A. Definition 4 Given a Lie algebra g, the universal enveloping algebra of g is a pair (U(g); i), where U(g) is an associative algebra with unit (denoted by 1) over K, i : g !U(g) is a linear map satisfying i([x; y]) = i(x)i(y) − i(y)i(x); 8x; y 2 g (1) and the following holds: for any associative algebra L over K with unit 1 and any linear map j : g ! L satisfying (1) there exists a unique algebra homomorphism φ : U(g) ! L with φ(1) = 1 such that φ ◦ i = j. 4 It can be proved that the map i is injective. We also state but do not prove the Poincar´e-Birkhoff-Witttheorem: Theorem 1 Let fx1; x2; · · · g be any ordered basis of g. Then the elements xi(1) ··· xi(m), m > 0, i(1) ≤ i(2) ≤ · · · ≤ i(m) along with 1 form a basis of U(g). For an explicit construction of the universal enveloping algebra, please refer to Appendix B. A simple example of a universal enveloping algebra is the one associated with j k l sl2(C). This is an associative algebra with a basis of elements x h y where j; k; l 2 N. Definition 5 A Lie algebra homomorphism is a map of Lie algebras φ : g1 ! g2 such that φ is linear and φ([x; y]) = [φ(x); φ(y)] 8x; y 2 g1: Furthermore, a Lie algebra isomorphism is a homomorphism that is both injective and surjective. Definition 6 A representation of a Lie algebra g is a Lie algebra homomorphism π : g ! gl(V ) where V is a vector space over K and gl(V ) the algebra of endomor- phisms of V . Definition 7 A module for a given Lie algebra g is a vector space V over K along with an operation g × V ! V sending (x; v) to x · v, usually denoted as xv.
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