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 gl∞ ...... 56

6.3 Representations ofa ¯∞ ...... 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 representations 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

• · | · - a symmetric bilinear form

• [·, ·] - a Lie bracket, usually the matrix commutator

• V , W - a vector space

• x, y, z, v, ψ - vectors

• gl∞,a ¯∞, g, X, d, etc - Lie algebras

• Vir - the Virasoro Algebra

•A - the oscillator (Heisenberg) algebra

• G, GLn, GL∞ - 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

• ω - an antilinear anti-involution

v • π, φ, r,r ˆ, 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

• C∞(M) - smooth functions on a manifold M

• TxM - tangent space at a point x ∈ M

• X(M) - algebra of vector ﬁelds on M

• F - space of semi-inﬁnite monomials

• Res0[·] - the residue at zero of some Laurent polynomial

• A† 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, infinite 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 theory 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 mathematics – 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 representations 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 Schrödingerequation 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 ∈ g

c) Jacobi identity: [[y, z], x] + [[z, x], y] + [[x, y], z] = 0, for all x, y ∈ 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 ∈ 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 {x, h, y} 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

      0 1 0 0 1 0       x =   , y =   , h =   . 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] ∈ e, ∀x, y ∈ e. A subalgebra e is abelian provided [x, y] = 0, ∀x, y ∈ e.

Deﬁnition 3 An ideal of a Lie algebra g is a subspace I such that for any x ∈ I and any y ∈ g we have [x, y] ∈ 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 ∈ 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 subalgebra 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), ∀x, y ∈ 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 {x1, x2, ···} 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 ∈ 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)] ∀x, y ∈ 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. This operation must satisfy the following conditions:

(ax + by)v = a(xv) + b(yv)

x(av + bw) = a(xv) + b(xw)

[x, y]v = xyv − yxv

5 for all x, y ∈ g, v, w ∈ V and a, b ∈ K.

Note that if φ : g → gl(V ) is a representation of g, then V is a module for g by the action xv = φ(x)(v) and vice versa. We will use both terminologies interchangeably.

Deﬁnition 8 Two representations φ : g → gl(V ) and ψ : g → gl(W ) of a Lie algebra g are homomorphic if there exists a linear map T : V → W such that for all x ∈ g and v ∈ V we have T (φ(x)(v)) = ψ(x)(T (v)). If T is also invertible, the two representations are said to be isomorphic.

Deﬁnition 9 Let h be an abelian subalgebra of g, h∗ be its dual space, and let V be

∗ a representation of g. For λ ∈ h , the λ-weight space Vλ of V (relative to h) is deﬁned as

Vλ = {v ∈ V |h · v = λ(h)v, ∀h ∈ h}.

If Vλ 6= 0, we say λ is a weight of V . Moreover, V is called a weight module if L V = λ∈h∗ Vλ.

Deﬁnition 10 A highest weight vector with highest weight λ (relative to h) of a g-module V is a nonzero vector v ∈ V such that

xv = 0, ∀x ∈ n+

hv = λ(h)v, ∀h ∈ h

where λ ∈ h∗. Furthermore, V is called a highest weight module (or representation) provided V = U(g)v for some highest weight vector v ∈ V .

Deﬁnition 11 A Verma module L is a highest weight module for g with highest weight λ ∈ h∗ and highest weight vector v ∈ L with a universal property: for any

6 highest weight representation V of g with highest weight λ and highest weight vector w there exists a unique surjective homomorphism φ : L → V of g-modules which maps v to w.

7 3 Algebras

3.1 Witt Algebra

The Witt algebra is the complexification of the Lie algebra of (real) vector fields on the unit circle. More precisely, consider the unit circle in the complex plane S1 := {eiθ|θ ∈ [0, 2π]}. Obviously, S1 is a group under multiplication and it is a 1-manifold, so clearly S1 is a Lie group. For the definitions of a Lie group and

1 d a real vector field, please see Appendix C. Now consider X(S ) := {f(θ) dθ |f ∈ C∞(S1,S1), f(θ + 2π) = f(θ)}, the vector space of real vector fields on S1, where C∞(S1,S1) denotes the smooth (infinitely differentiable) functions from S1 to S1. The Lie bracket on X(S1) is given by:

d d d [f(θ) , g(θ) ] = (fg0 − f 0g)(θ) (2) dθ dθ dθ making X(S1) into a Lie algebra. Using Fourier series, a basis of the 2π - periodic

C∞(S1,S1)-functions is : {1, cos(nθ), sin(nθ)|n ∈ Z}. This in turn gives a basis for

1 d d d X(S ) as a vector space over R : { dθ , cos(nθ) dθ , sin(nθ) dθ |n ∈ Z}. The Witt algebra d is the complexiﬁcation of X(S1), that is, the span of the above basis over C. Using Euler’s formula ( eiθ = cos(θ) + i sin(θ) ), a new basis

n+1 d iθ can be written as {dn|n ∈ Z} where dn = −z dz and z = e . This can be seen through a simple calculation:

dz = iz dθ d d i = −z . dθ dz

inθ d d d i d 1 Thus dn = ie dθ , while dθ = −id0, cos(nθ) dθ = − 2 (dn +d−n), sin(nθ) dθ = − 2 (dn − d−n). Taking the span of these vectors over C returns the algebra d. These basis

8 vectors satisfy the following commutation relations:

d d [d , d ] = [−iei(m+1)θ , −iei(n+1)θ ] m n dθ dθ d d = i3(n + 1)ei(m+n+1)θ − i3(m + 1)ei(m+n+1)θ dθ dθ d = (m − n) − iei(m+n+1)θ dθ

= (m − n)dm+n.

Deﬁnition 12 An antilinear involution on a Lie algebra g is a map ω : g → g such that

(a) ω(ω(x)) = x

(b) ω(λx) = λω¯ (x)

∀x, y ∈ g and ∀λ ∈ C where λ¯ is the complex conjugate of λ.

An antilinear involution ω can be deﬁned on d by setting

ω(dn) = d−n, n ∈ Z.

An easy calculation shows that the set of real elements of d, i.e., d ∩ X(S1), consists of the elements ﬁxed under −ω.

3.2 Virasoro Algebra

Deﬁnition 13 Two elements x and y of a Lie algebra g are said to commute provided [x, y] = 0. A subalgebra e of g is called central if every element of e commutes with all elements of g.

Note that any central subalgebra is actually an ideal.

9 Deﬁnition 14 A central extension of a Lie algebra g is a Lie algebra A with a subalgebra e such that e is central and the quotient of A by e is g. Using short exact sequences, we have

0 → e → A → g → 0.

Theorem 2 The Witt algebra d has a unique nontrivial one-dimensional central extension d˜ = d ⊕ Cc¯ up to Lie algebra isomorphism. This extension has a basis

{c} ∪ {dn|n ∈ Z} where c ∈ Cc¯, such that the following relations are satisﬁed:

[c, dn] = 0 for n ∈ Z (3) m3 − m [d , d ] = (m − n)d + δ c for m, n ∈ . (4) m n m+n m,−n 12 Z

The extension d˜ is called the Virasoro algebra, and is denoted by Vir. Proof. To prove existence, it is enough to check that the relations (3)-(4) deﬁne a Lie algebra, which is easy. We give a proof of uniqueness. Suppose d˜ = d ⊕ Cc¯ is a ¯ nontrivial one-dimensional central extension of d. Let {dn|n ∈ Z} denote the basis elements of d from the last section, then we have

¯ ¯ ¯ [dm, dn] = (m − n)dm+n + a(m, n)¯c ¯ [¯c, dn] = 0 for m, n ∈ Z, where a : Z × Z → C is some function. Note that a(m, n) = −a(n, m) because d˜ is a Lie algebra and thus has anti-symmetric product:

¯ ¯ ¯ ¯ ¯ 0 = [dm, dn] + [dn, dm] = (m − n + n − m)dm+n + (a(m, n) + a(n, m))¯c.

10 Deﬁne new elements:

  ¯ d0 if n = 0 0  dn =  ¯ 1 dn − n a(0, n)¯c if n 6= 0

c0 =c. ¯

0 0 ˜ Then {c } ∪ {dn|n ∈ Z} is a new basis for d. The new commutation relations are:

0 0 [c , dn] = 0

0 0 ¯ ¯ [dm, dn] = [dm, dn] ¯ = (m − n)dm+n + a(m, n)¯c

0 0 0 = (m − n)dm+n + a (m, n)c (5) for m, n ∈ Z where a0 : Z × Z → C is deﬁned by

  a(m, n) if m + n = 0 a0(m, n) = (6)  m−n a(m, n) + m+n a(0, m + n) if m + n 6= 0.

Since a is antisymmetric, a0 is as well, and therefore a0(0, 0) = 0. From (6) it follows that a0(0, n) = 0 for any nonzero n. These facts together with (5) show that

0 0 0 [d0, dn] = −ndn.

11 Now, using the Jacobi identity in d˜, we have:

0 0 0 0 0 0 0 0 0 [[d0, dn], dm] + [[dn, dm], d0] + [[dm, d0], dn] = 0

0 0 0 0 0 0 0 0 [−ndn, dm] + [(n − m)dn+m + a (n, m)c , d0] − [dn, mdm] = 0

0 0 0 0 −(n + m)(n − m)dn+m − (n + m)a (n, m)c + (n − m)(n + m)dn+m = 0

(n + m)a0(n, m)c0 = 0 which shows that a0(n, m) = 0 unless n + m = 0 and n 6= 0, m 6= 0. Thus, setting b(m) = a0(m, −m), (5) can be rewritten as

0 0 [c , dn] = 0

0 0 0 0 [dm, dn] = (m − n)dm+n + δm+n,0b(m)c with b(0) = 0. Again using the Jacobi identity

0 0 0 0 0 0 0 0 0 [[dn, d1], d−n−1] + [[d1, d−n−1], dn] + [[d−n−1, dn], d1] = 0

0 0 0 0 0 0 [(n − 1)dn+1, d−n−1] + [(n + 2)d−n, dn] + [(−2n − 1)d−1, d1] = 0

0 0 0 0 (n − 1)(2(n + 1)d0 + b(n + 1)c ) + (n + 2)(−2nd0 + b(−n)c )

0 0 + (−2n − 1)(−2d0 + b(−1)c ) = 0

2 2 0 (2n − 2 − 2n − 4n + 4n + 2)d0 + {(n − 1)b(n + 1) − (n + 2)b(n)

+ (2n + 1)b(1)}c0 = 0, which is equivalent to

(n − 1)b(n + 1) = (n + 2)b(n) − (2n + 1)b(1). (7)

Next, b(m) = m and b(m) = m3 are shown to be two solutions of (7). First we show

12 b(m) = m is consistent with the (7):

(n − 1)(n + 1) = (n + 2)(n) − (2n + 1)(1)

n2 − 1 = n2 + 2n − 2n − 1

0 = 0 and now for b(m) = m3:

(n − 1)(n + 1)3 = (n + 2)(n)3 − (2n + 1)(1)

(n − 1)(n3 + 3n2 + 3n + 1) = n4 + 2n3 − 2n − 1

n4 + 2n3 − 2n − 1 = n4 + 2n3 − 2n − 1

0 = 0.

Since (7) is a second order linear recurrence relation in b and the above two solutions are linearly independent, then there are α, β ∈ C such that

b(m) = αm3 + βm.

Finally, set

α + β d = d0 + δ c0, n n n,0 2 and

c = 12αc0.

13 If α 6= 0, this is again a change of basis. Then,

0 0 [dm, dn] = [dm, dn]

0 3 0 = (m − n)dm+n + δm+n,0(αm + βm)c α + β = (m − n)d − (m − n)δ c0 + δ (αm3 + βm)c0 m+n m+n,0 2 m+n,0 α + β = (m − n)d − δ 2m c0 + δ (αm3 + βm)c0 m+n m+n,0 2 m+n,0 3 0 = (m − n)dm+n + δm+n,0(αm − αm)c m3 − m = (m − n)d + δ c. m+n m+n,0 12

The proof of uniqueness is finished. Notice also that α = 0 corresponds to the trivial extension. An antilinear involution ω can be defined similarly to the one defined for d, by requiring

ω(dn) = d−n ∀n ∈ Z

ω(c) = c.

We only need to check the following:

n3 − n [ω(d ), ω(d )] = [d , d ] = (−n + m)d − δ c n m −n −m −n−m −n,m 12 m3 − m = (m − n)ω(d ) + δ ω(c) m+n m,−n 12

= ω([dm, dn]).

Vir has the following decomposition into Lie subalgebras, called a triangular

14 decomposition of Vir:

V ir = n−⊕h ⊕ n+ where ∞ ∞ − M + M n = Cd−i, h = Cc ⊕ Cd0, n = Cdi. i=1 i=1

3.3 Oscillator Algebra

The oscillator (Heisenberg) algebra A is deﬁned as the complex Lie algebra with generators ~ and an (n ∈ Z), and relations:

[~, an] = 0, ∀n ∈ Z

[am, an] = δm,−nm~, ∀m, n ∈ Z.

It is easy to see that A is well-deﬁned. Note that [a0, an] = 0 for all n ∈ Z, so that a0 is central. An antilinear involution ω can be deﬁned for A as follows:

ω(an) = a−n

ω(~) = ~.

The oscillator algebra has a triangular decomposition similar to the Virasoro algebra:

A = n−⊕h ⊕ n+ where ∞ ∞ − M + M n = Ca−i, h = C~ ⊕ Ca0, n = Cai. i=1 i=1

15 3.4 Algebras of Inﬁnite Matrices

Let

M V = Cvj j∈Z be an inﬁnite dimensional complex vector space with ﬁxed basis {vj|j ∈ Z}. Identify

th vj with the column vector whose i entry is δij. Any vector in V is a finite sum of multiples of vj, j ∈ Z, and thus it has only a finite number of nonzero coordinates. This identifies V with C∞. Now we define two Lie algebras:

Deﬁnition 15 Let gl∞ be the vector space of matrices deﬁned by:

gl∞ = {(aij)i,j∈Z|aij = 0 for all but ﬁnitely many i, j ∈ Z}

and let a¯∞ be the vector space deﬁned by:

a¯∞ = {(aij)i,j∈Z|aij = 0 for |i − j| 0}.

Clearlya ¯∞ is a set of matrices with a ﬁnite number of nonzero diagonals and gl∞ ⊆ a¯∞. Note that the usual matrix multiplication is well-deﬁned ina ¯∞: Let x, y ∈ a¯∞. Then xkl = 0 for |k − l| > M for some M ≥ 0, and ymn = 0 for

|m − n| ≥ N for some N > 0. Then for i, j ∈ Z,

X X (xy)ij = xikykj = xikykj k∈Z k∈Z |i−k|≤M |k−j|≤N

16 which is clearly a ﬁnite sum. Furthermore,

|i − j| = |i − k + k − j| ≤ |i − k| + |k − j| ≤ M + N

by the triangle inequality. This means that (xy)ij = 0 for |i − j| > M + N, thus the bracket [x, y] = xy − yx is an element ofa ¯∞, since each product is well-deﬁned and each summand has ﬁnitely many nonzero diagonals.

Proposition 1 The vector spaces gl∞ and a¯∞ are Lie algebras, with gl∞ a subalgebra of a¯∞.

Proof. This is shown by checking thata ¯∞ is an associative algebra, and thus a

Lie algebra under the commutator bracket, and by checking that gl∞ is closed under the commutator bracket. However, for illustration purposes, we demonstrate that gl∞ is a Lie algebra.

Let Eij be the matrix with 1 as the (i, j) entry and zeros elsewhere. The Eij obviously form a basis for gl∞. Clearly

Eijvk = δjkvi and

EijEmn = δjmEin.

Then the commutation relations (using the usual matrix commutator) are:

[Eij,Emn] = δjmEin − δniEmj.

17 This is clearly antisymmetric, and we can check the Jacobi identity. First, we have

[[Eij,Emn],Ekl] = δjm[Ein,Ekl] − δni[Emj,Ekl]

= δjm{δnkEil − δliEkn} − δni{δjkEml − δlmEkj}

= δjmδnkEil − δjmδliEkn − δniδjkEml + δniδlmEkj and then

[[Eij,Emn]Ekl] + [[Emn,Ekl],Eij] + [[Ekl,Eij],Emn] =

δjmδnkEil − δjmδliEkn − δniδjkEml + δniδlmEkj

+δnkδliEmj − δnkδjmEil − δlmδniEkj + δlmδjkEin

+δliδjmEkn − δliδnkEmj − δjkδlmEin + δjkδniEml

= 0.

Thus gl∞ is a Lie algebra.

Deﬁne the shift operators Λk by

Λkvj = vj−k.

Then we have

X Λk = Ei,i+k. i∈Z

Thus Λk is the matrix with 1 at each entry of the k-th diagonal and 0 elsewhere, so

Λk ∈ a¯∞. Now,

X ΛkΛj = Ei,i+k+j = ΛjΛk i∈Z

18 and so [Λj, Λk] = 0 for j, k ∈ Z, i.e., the Λk form an abelian subalgebra ofa ¯∞. For every pair of nonzero constants α and β there is an inclusion of the Witt algebra d ina ¯∞ as a subalgebra via

dn = (k − α − β(n + 1))Λn or, acting on Cn,

dn(vk) = (k − α − β(n + 1))vk−n.

The elements dn are clearly ina ¯∞. We show that these elements generate a subalgebra ina ¯∞ isomorphic to d with the following calculation:

[dm, dn](vk) = dmdn(vk) − dndm(vk)

= dm(k − α − β(n + 1))vk−n − dn(k − α − β(m + 1))vk−m

= (k − α − β(n + 1))((k − n) − α − β((k − n) + 1))vk−n−m

− (k − α − β(m + 1))((k − m) − α − β((k − m) + 1))vk−m−n

= [−kn − βk(m + 1) + αn − β(n + 1)(k − n)

+ km + βk(n + 1) − αm + β(m + 1)(k − m)]vk−m−n

= (m − n)(k − α − β(m + n + 1))vk−n−m

= (m − n)dm+n(vk).

An alternate way to write these elements is

X dn = (k − α − β(n + 1))Ek−n,k. k∈Z

19 3.5 Loop Algebras

Deﬁnition 16 Let gln denote the Lie algebra of all n × n matrices with complex entries and let C[t, t−1] denote the ring of Laurent polynomials in t with complex

−1 coeﬃcients. We deﬁne the loop algebra gle n to be gln(C[t, t ]), i.e., the complex Lie algebra of n × n matrices with Laurent polynomials as entries.

An element of gle n has the form

X k a(t) = t ak with ak ∈ gln k where k runs over a ﬁnite subset of Z. The standard basis {eij|0 ≤ i, j ≤ n} of gln yields a basis for gle n via

k eij;k = t eij (1 ≤ i, j ≤ n and k ∈ Z).

The elements of gle n form an associative algebra with multiplication deﬁned on the basis by

k+l eij;kemn;l = t eijemn = δjmein;k+l.

Then the Lie bracket on gle n can be deﬁned as the commutator:

[eij;k, emn;l] = δjmein;k+l − δniemj;k+l.

k+l Note that [eij;k, emn;l] = t [eij, emn].

n The Lie algebra gln has a natural representation on the space C given by matrix

n multiplication. Let {e1, . . . , en} denote the standard basis of C where eij is the n×1

−1 n column vector with entries (ej)i = δij, i, j ∈ {1, . . . , n}. The vector space C[t, t ]

20 consists of n×1 column vectors with Laurent polynomials in t as entries. The vectors

n −k vj;k = t ej with k ∈ Z, j ∈ {1, . . . , n}

−1 n −1 n ∞ n form a basis of C[t, t ] (over C). Thus C[t, t ] is identiﬁed with C via vj;k 7→ vnk+j. Now gle n acts on this space by

 n  vi;s−k j = l n  eij;k( vl;k) = .  0 otherwise

n Using the identiﬁcation of vi;j with vjn+k, this means

eij;kvkn+l = δjlvn(s−k)+i. (8)

For a(t) ∈ gle n denote the corresponding matrix ina ¯∞ by τ(a(t)). Then we have the following matrix representation of gle n ina ¯∞:

X τ eij;k = En(s−k)+i,ns+j. s∈Z

P k For arbitrary a(t) = t ak ∈ gle n this can be rewritten in block form as

 . . .  ......     . . . a−1 a0 a1 ......      τ(a(t)) = ...... a a a ... .  −1 0 1     ..  ...... a−1 a0 .  . .  ......

Clearly this is a matrix ina ¯∞ whose entries on each diagonal parallel to the main diagonal form an n-periodic sequence.

21 Proposition 2 a) τ : gle n → a¯∞ is an injective homomorphism of associative algebras, and hence Lie algebras.

P j b) The image of a(t) = j ajt under τ is a strictly upper triangular matrix if and only if

2 a(t) = a0 + a1t + a2t + ··· with a0 strictly upper triangular.

j c) The shift operator Λj is the image under τ of (a + tb) , where

n−1 X a = ei,i+1, b = en1. i=1

Proof. a) τ is an algebra homomorphism and respects the Lie bracket, making it a Lie algebra homomorphism, and is injective since it maps the basis {eij;k} of gle n to a linearly independent set ina ¯∞. b) This is clear from the block matrix given for arbitrary a(t) ∈ gle n. For c), recall

X Λj = Ei,i+j. i∈Z

Using (8)

n−1 j j X τ (a + tb) = τ ei,i+1 + τ ten1 i=1 n−1 j X X X = Ens+i,ns+i+1 + Ens+n,n(s+1)+1 i=1 s s n j j X X X = Ens+i,ns+i+1 = El,l+1 s i=1 l

j = Λ1 = Λj

22 where the last step can be seen from the diﬁntion of Λj:

Λjvk = vk−j and so

j j−1 Λ1vk = Λ1 vk−1 = ··· = vk−j = Λjvk.

Deﬁne an antilinear involution ω on gle n by

−k ∗ ω Xk = t X ,

k where Xk = t X ∈ gle n, X ∈ gln, where k ∈ Z, with ∗ denoting the usual Hermitian conjugate (conjugate transpose).

Proposition 3

∗ τ ω Xk = τ Xk

where the ∗ on the right-hand side denotes the Hermitian conjugate in a¯∞.

∗ −k ∗ Proof. It must be shown that τ(t X ) = τ Xk . Let X = (xij) xij ∈ C, then

∗ ∗ k ∗ X k τ Xk = τ(t X) = xijτ eij ij ∗ X X = xij En(s−k)+i,ns+j ij s

23 while

−k ∗ X −k X −k τ(t X ) = τ x¯jieij = x¯jiτ eij ij ij X X = x¯ji En(s+k)+i,ns+j ij s ∗ X X = xji En(s−k)+j,ns+i . ij s

Thus the inclusion of gle n intoa ¯∞ respects ω.

We also have a triangular decomposition of gl∞ given by

− M M + M n = CEij h = CEij n = CEij i>j i=j i

− + gl∞ = n ⊕ h ⊕ n .

24 4 Representations of the Virasoro Algebra

So far we have a set of algebras and certain inclusions between them. Before contin- uing to the speciﬁcs of their representations, we need some important general ideas from representation theory.

4.1 Hermitian Forms

One property of certain representations that helps link them to physical concepts is the existence of an Hermitian form on the representation. This gives a notion of relative magnitude and orthogonality for the vectors in the representation.

Deﬁnition 17 A Hermitian form on a vector space V is a map ·|· : V ×V → C such that

λ1v + λ2w|x = λ1 v|x + λ2 w|x

v|λ1w + λ2x = λ1 v|w + λ2 v|x

v|x = x|v

for all v, x, w ∈ V , λ1, λ2 ∈ C.

In particular, we want to look at representations that are unitary. This essentially means that the action of the algebra preserves the Hermitian form of the vector space on which it acts. Each of the algebras deﬁned are equipped with a function that plays a role similar to the Hermitian conjugation and gives a way to deﬁne unitary representations.

Deﬁnition 18 Let g be a Lie algebra and ω an antilinear involution on g. Let π : g → gl(V ) be a representation of g. A Hermitian form · | · on V is called contravariant with respect to ω if π(x)(u)|v = u|πω(x)(v) , ∀x ∈ g, and

25 ∀u, v ∈ V and it is called non-degenerate if

v|w = 0 for all w ∈ V then v = 0.

For a non-degenerate Hermitian form, let L∗ denote the Hermitian conjugate of the operator L. A representation is called unitary if it is equipped with a contravariant non-degenerate Hermitian form and

(π(x))∗ = πω(x) for all x ∈ g.

We recall that we have already deﬁned an antilinear involution for d, Vir, the oscillator algebra, and the loop algebras.

For the algebras gl∞ anda ¯∞ we take ω to be the Hermitian conjugate, which will be denoted as a superscript ∗. Each antilinear involution will be denoted by ω, but that should create no con- fusion. Context will always clarify which ω is being referenced. Unitarity of a representation will always be shown in terms of the aformentioned ω.

4.2 Highest Weight Representations of Vir

Recall that for a given Lie algebra g there is an unique associative algebra with unit called the universal enveloping algebra containing g such that there is a unique homomorphism from the universal enveloping algebra to any representation of g. This is clearly of interest since all representations of a given Lie algebra could then be studied simply by studying quotients of the universal enveloping algebra associated to it. The construction will be covered for the Virasoro algebra in detail. The universal enveloping algebra of a Lie algebra g is denoted U(g) and when g

26 has triangular decomposition

g = n− ⊕ h ⊕ n+ the Poincar´e-Birkhoﬀ-Witt theorem (see section 2.1) gives immediately the following result:

U(g) = U(n−)U(h)U(n+).

It has been demonstrated in section 3.2 that the Virasoro algebra has such a decomposition, so we can write

U(V ir) = U(n−)U(h)U(n−).

Deﬁnition 19 Let C, h ∈ C.A highest weight representation of Vir with highest weight (C, h) is a representation V generated by a nonzero vector v, called a highest weight vector, such that:

c(v) = Cv

d0(v) = hv

n+v = 0.

Let V be a highest weight representation of Vir with highest weight (C, h) and highest weight vector v. Using the triangular decomposition and rewriting U(n+) =

C · 1 + U(n+)n+, we see that

− + + − − U(V ir)v = U(n )U(h)(C · 1 + U(n )n )v = U(n )U(h)v = U(n )v.

27 Moreover, all of the vectors of the form d−ik ··· d−i1 (v) (0 < i1 ≤ · · · ≤ ik) with P im = j span the eigenspace Vh+j of d0 with eigenvalue h + j:

d0(d−ik ··· d−i1 (v)) = [d0, d−ik ]d−ik−1 ··· d−i1 (v) + d−ik d0d−ik−1 ··· d−i1 (v) k X = ild−ik ··· d−i1 (v) + d−ik ··· d−i1 d0(v) l=1

= jd−ik ··· d−i1 (v) + hd−ik ··· d−i1 (v)

= (h + j)d−ik ··· d−i1 (v) and so,

M V = Vh+j. (9) j∈Z+

Clearly dim Vh+j ≤ p(j) where p(j) is the partition function, assigning the number of partitions of an integer to that integer, and equality holds when all the vectors P d−ik ··· d−i1 (v) with im = j are linearly independent.

Deﬁnition 20 A highest weight representation M(C, h) of Vir with highest weight vector v and highest weight (C, h) is called a Verma module if it satisﬁes the following universal property: For any highest weight representation V of Vir with highest weight vector u and highest weight (C, h) there exists a unique surjective homomorphism φ : M(C, h) → V of Vir-modules which maps v to u.

Proposition 4 For each C, h ∈ C there exists a unique Verma module M(C, h) of Vir with highest weight (C, h) and the map U(n−) → M(C, h) sending x to x · v is injective.

Proof. Existence: Let 1 = 1U(V ir) be the identity element of U(V ir) and let

I = I(C, h) denote the left ideal in U(V ir) generated by the elements {dn|n >

28 0} ∪ {d0 − h · 1, c − C · 1}. Set M(C, h) = U(V ir)/I and deﬁne a map π : V ir → gl(M(C, h)) by

π(x)(u + I) = xu + I.

Then π is a highest weight representation of Vir with highest weight vector v = 1+I and highest weight (C, h). This is seen by the following calculations:

π(dn)(1 + I) = dn + I = I for n > 0

π(d0)(1 + I) = d0 + I = h · 1 + I = h · (1 + I)

π(c)(1 + I) = c + I = C · 1 + I = C · (1 + I).

Now it can be shown that M(C, h) is a Verma module. Let ρ : V ir → gl(V ) be any highest weight representation with highest weight (C, h) and highest weight vector u. Since U(V ir) can be viewed as a left Vir-module, the action of U(V ir) on V given by α : U(V ir) → V , where x 7→ xu, is seen to be a Vir-module homomorphism.

It is claimed that α(I) = 0. It is enough to check the generators {dn|n > 0} ∪ {d0 − h · 1, c − C · 1} of the left ideal are mapped to zero. This follows since V is a highest weight representation with highest weight vector u and highest weight (C, h) (simply compare this requirement with the deﬁnition for a highest weight representation). Thus α induces a Vir-module epimorphism φ : U(V ir)/I = M(C, h) → V which clearly maps v to u. This shows the existence of the map φ. Next it is proved that there can exist at most one Vir-module epimorphism φ : M(C, h) → V which maps v to u. Since M(C, h) is a highest weight module,

any element is a linear combination of elements of the form d−is ··· d−i1 · 1 + I = d−is ··· d−i1 · v where ij ≥ 0 and s ≥ 0. Since φ is a Vir-module homomorphism,

φ(d−is ··· d−i1 · v) = d−is ··· d−i1 φ(v) = d−is ··· d−i1 · u. Thus φ is uniquely deﬁned

29 on M(C, h) and so M(C, h) is a Verma module. For uniqueness suppose there exists another such Verma module V with highest weight (C, h) and highest weight vector u. Then both M(C, h) and V have the universal property of Verma modules, and so there exists a unique surjective homomorphism φ : M(C, h) → V sending v to u and a unique surjective homomorphism ψ : V → M(C, h) sending u to v. Then there are surjective maps φ ◦ ψ and ψ ◦ φ from M(C, h) → M(C, h) and V → V respectively, sending v 7→ v and u 7→ u. Since these are homomorphisms and they send the highest weight vector to itself and each representation is generated by the action of U(V ir) on that vector, they must be isomorphisms. Furthermore they must be the identity map in each case, and thus φ = ψ−1, and each is an isomorphism. The map x 7→ π(x)(1 + I) = x + I for x ∈ U(n−) is injective by the Poincar´e-

Birkhoﬀ-Witt theorem.

4.3 Irreducible Positive Energy Representations

Here we continue discussing the highest weight representations of the Virasoro algebra. We introduce some desirable properties for the representations to have and give conditions on the representations for these properties to exist. The main material in this section is heavily based on [2].

Deﬁnition 21 A representation of an algebra g is called irreducible if it contains no nonzero proper subrepresentations.

Deﬁnition 22 A representation of an algebra g is called indecomposable if it cannot be written as the direct sum of two or more proper subrepresentations.

Clearly if a representation is irreducible then it is indecomposable. Irreducible and indecomposable representations are building blocks for larger representations,

30 and so are one of the main subjects to study when classifying representations of a given algebra.

Deﬁnition 23 Let π : V ir → gl(V ) be a representation of Vir on a vector space V such that

a) V has a basis of eigenvectors of π(d0)

b) all π(d0) eigenvalues of the basis vectors are non-negative real numbers

c) the eigenspaces of π(d0) are ﬁnite-dimensional.

Then (π, V ) is said to be a positive-energy representation of Vir.

Proposition 5 An irreducible positive energy representation of Vir is a highest weight representation.

Proof. Let V be an irreducible positive energy representation of Vir, let w ∈ V be a nontrivial eigenvector for d0. Then d0w = λw for some λ ∈ R≥0. Now for any

t t ∈ Z≥0 and (jt, . . . , j1) ∈ Z we have

d0djt ··· dj1 w = (λ − (jt + ··· + j1))djt ··· dj1 w by using the same technique as in the proof of (9). V is positive energy, and so the set

t M = {j ∈ Z|d0djt ··· dj1 w 6= 0 for some t ≥ 0, (jt, . . . , j1) ∈ Z with jt + ··· + j1 = j} is bounded from above by λ. It is also nonempty, because 0 ∈ M. let t ≥ 0 and

jt ··· + j1 = max(M) be such that v = djt ··· dj1 w. Then

djv = djdjt ··· djt w = 0 for j > 0

31 since otherwise j + max(M) = j + jt + ··· + j1 ∈ M. We also have

d0v = d0djt ··· djt w = (λ − (jt + ··· + j1))djt ··· dj1 w = hv

where h = λ − (jt + ··· + j1). Consider the submodule deﬁned by

V 0 = U(V ir)v.

0 L 0 0 0 Now, V = V , where V is the h + k eigenspace of d0 acting on V . Let k∈Z≥0 h+k h+k w ∈ Vh+k for some k ∈ Z≥0. Then dncw = cdnw for all n ∈ Z because [c, dn] = 0 for all n ∈ Z. Thus by Schur’s lemma, c is constant on V 0. Clearly V 0 as deﬁned is a highest weight representation. V 0 is nontrivial since 0 6= v ∈ V 0. Therefore, since V is irreducible, V = V 0, and so V is a highest weight representation.

Proposition 6 A unitary highest weight representation V of Vir is irreducible.

Proof. Let V be a unitary highest weight representation of Vir with highest weight (C, h) and highest weight vector v. If U is a subrepresentation of V, then V = U L U ⊥ using unitarity. We have either v ∈ U or v ∈ U ⊥ as follows. Let

0 0 ⊥ 0 0 v = u + u for some u ∈ U and u ∈ U . Then d0u + d0u = d0v = hv = hu + hu .

0 ⊥ 0 0 Thus, (d0 − h)u = −(d0 − h)u . Since U ∩ U = {0}, then d0u = hu and d0u = hu .

0 0 It follows that u, u ∈ Vh and therefore we have either u = 0 or u = 0 which proves the claim.

Since V = U(V ir)v, either U = V or U = 0.

Proposition 7 a) The Verma module M(C,h) has the decomposition

M M(C, h) = M(C, h)h+k

k∈Z≥0

32 where M(C, h)h+k is the (h + k)-eigenspace of d0 of dimension p(k) spanned by vectors of the form

d−is ··· d−i1 (v) with 0 < i1 ≤ · · · ≤ is = k.

b) M(C,h) is indecomposable. c) M(C,h) has a unique maximal proper submodule J(C, h), and

V (C, h) := M(C, h)/J(C, h) is the unique irreducible highest weight representation with highest weight (C, h), up to isomorphism.

Proof. Part (a) is a restatement of an earlier result. To prove part (b), suppose M(C, h) = U L U 0 for some submodules U and U 0. Now, we use the same argument as in the proof of the previous proposition to show that either U = 0 or U 0 = 0. To prove (c), observe from the proof of part (b) that a subrepresentation of M(C, h) is proper if and only if it does not contain the highest weight vector v. Deﬁne J(C, h) as the sum of all proper subrepresentations of M(C, h). Since v∈ / J(C, h), it is a proper subrepresentation of M(C, h). Clearly J(C, h) is a maximal proper subrepresentation. It is also unique, because it contains and is contained in any other maximal proper subresentation of M(C, h). Since J(C, h) is a maximal proper submodule of M(C, h), then V (C, h) is clearly irreducible. For the uniqueness part, let V 0(C, h) be any irreducible highest weight module with the same highest weight. Then by deﬁnition of the Verma module there is a surjective homomorphism φ : M(C, h) → V 0(C, h). Let J 0(C, h) = kerφ. By the

33 First Isomorphism Theorem, we have

V 0(C, h) ∼= M(C, h)/J 0(C, h).

Since V 0(C, h) is irreducible, J 0(C, h) must be a maximal proper submodule, and so

0 ∼ equal to J(C, h). Thus V (C, h) = V (C, h). The antilinear involution ω : V ir → V ir extends uniquely to an antilinear invo- lutionω ˜ : U(V ir) → U(V ir) as follows:

ω˜(x1 ··· xm) = ω(xm) . . . ω(x1) where xi ∈ Vir.

Proposition 8 Let C, h ∈ R. Then

a) there is a unique contravariant Hermitian form · | · on M(C, h) such that v|v = 1,

b) the eigenspaces of d0 are pairwise orthogonal with respect to this form,

c) J(C, h) = rad( · | · ) ≡ {u ∈ M(C, h)| u|w = 0 for all w ∈ M(C, h)}.

This form is called Shapovalov’s form.

Proof. (a): If x, y ∈ U(V ir), then

xv|yv = v|ω˜(x)yv since the form is contravariant. From the Poincar´e-Birkhoﬀ-Witttheorem, the universal enveloping algebra U(V ir) of Vir has the following decomposition:

U(V ir) = (n−U(V ir) + U(V ir)n+) ⊕ U(h).

34 Since U(h) is commutative it is the same as S(h), the symmetric algebra on the vector space h = Cc ⊕ Cd0. Let P : U(V ir) → U(h) be the projection onto the second component of the direct sum, and let e : U(h) → C be the algebra homomorphism determined by

e(c) = C e(d0) = h.

Then for x ∈ U(V ir),

P (x)v = e(P (x))v.

Since M(C, h) is a highest weight representation,

v|n−U(V ir)v + U(V ir)n+v = n+v|U(V ir)v + v|U(V ir)n+v = 0

Therefore

xv|yv = v|ω˜(x)yv = v|P (˜ω(x)y)v = e(P (˜ω(x)y)).

This shows the form is unique, if it exists. To show existence, recall the construction of M(C, h) as a quotient of U(V ir) by a left ideal I generated by the set {dn|n > 0} ∪ {c − C · 1, d0 − h · 1}. Clearly, P (n+) = P (n−) = 0, but also

e(P (c − C · 1)) = e(c − C · 1) = C − C = 0

e(P (d0 − h · 1)) = e(d0 − h · 1) = h − h = 0

35 where 1 = 1U(V ir). Note further that

P (xy) = P (x)y P (yx) = yP (x) for x ∈ U(V ir), y ∈ U(h). Combining these observations,

e(P (x)) = 0 for x ∈ I or x ∈ ω˜(I).

It is now clear that xv|yv = e(P (˜ω(x)y)) can be taken as the deﬁnition of the form, because if xv = x0v and yv = y0v for some x, x0, y, y0 ∈ U(V ir) then x − x0, y − y0 ∈ I so that

xv|yv − x0v|y0v = (x − x0)v|yv + x0|(y − y0)v

= ω˜(y)(x − x0)v|v + v|ω˜(x0)(y − y0)v

= 0.

It is clear that the form is Hermitian since P and e are linear functions, and ω is antilinear. Contravariance is clear as well:

xyv|zv = e(P (˜ω(xy)z)) = e(P (˜ω(y)˜ω(x)z)) = yv|ω˜(x)v with x, y, z ∈ U(V ir). Finally, we have

v|v = e(P (1 · 1)) = 1, ending the proof of (a).

36 (b) If x ∈ M(C, h)h+k and y ∈ M(C, h)h+l with k 6= l we have

(h + k)x|y − x|(h + l)y = d0x|y − x|d0y

= x|ω(d0)y − x|d0y

= x|d0y − x|d0y

= 0. also

(h + k)x|y − x|(h + l)y = (h + k) x|y − (h + l) x|y

0 6= yv|xv = e(C,h)(P (˜ω(y)x)).

Since J(C, h) is a representation of Vir, z =ω ˜(y)xv ∈ J(C, h) with a nonzero component in M(C, h)h = Cv. Therefore v ∈ J(C, h). This contradicts J(C, h) 6=

M(C, h) and the proof is ﬁnished.

Corollary 1 If C, h ∈ R, then V (C, h) = M(C, h)/J(C, h) carries a unique contravariant Hermitian form · | · such that v + J(C, h)|v + J(C, h) = 1.

Remark 1 Note that the representations V (C, h) with h ≥ 0 are precisely all irreducible positive energy representations of Vir.

37 Proposition 9 For a given highest weight (C, h) there exists at most one unitary highest weight representation of V ir, V (C, h).

Proof. We have already shown that a unitary highest weight representation of V ir is irreducible, and that V (C, h) is the unique irreducible highest weight representation with highest weight (C, h).

Proposition 10 If V (C, h) is unitary, then C ≥ 0 and h ≥ 0.

Proof. Let cn = d−nv|d−nv for n > 0. Unitarity requires that cn ≥ 0. Con- travariance requires that

n3 − n n3 − n c = v|d d v = v|(d d + 2nd + c)v = 2nh + C. n n −n −n n 0 12 12

3 Then c1 = 2h, and so h ≥ 0. For suﬃciently large values of n we have n − n >

2nh > 0 and so C ≥ 0.

38 5 Oscillator Representations

5.1 Representations of the Oscillator Algebra

Let B = C[x1, x2,... ] the space of polynomials in inﬁnitely many variables. Often B is called the Fock space.

Given µ, ~ ∈ R, we claim that a representation of A on B is given by:

an = n ∂/∂xn (10)

a−n = −n~nxn

a0 = µI

~ = ~I

where the n are arbitrary pairs of real numbers with −n · n = 1, and the symbol

~ is used for both the operator and the eigenvalue of the operator. It is clear that a0 and ~ are central (commute with all the other operators), and [an, am](f) = 0 for n 6= −m, f ∈ B. Now, for every f ∈ B:

∂ ∂ [an, a−n]f = n −n~nxn(f) − −n~nxnn (f) ∂xn ∂xn ∂ ∂ = ~n (xnf) − ~nxn (f) ∂xn ∂xn ∂ ∂ = ~nf + ~nxn (f) − ~nxn (f) ∂xn ∂xn

= ~nf and so the relations are satisﬁed on A. We’ll call this representation a Fock representation. The following lemma will be used for numerous calculations in this section.

39 Lemma 1 The A-action on B satisﬁes:

k k−1 [an, a−n] = k~na−n , k ≥ 1, n ≥ 0. (11)

0 Proof. Induction on k. Clearly, [an, a−n] = ~n = ~na−n. Now, suppose (11) is true for some k ∈ N. Then:

k+1 k+1 k+1 [an, a−n ] = ana−n − a−n an

k k+1 = (a−nan + ~n)a−n − a−n an

k k k+1 = a−n(ana−n) + ~na−n − a−n an

k−1 k k k+1 = a−n(k~na−n + a−nan) + ~na−n − a−n an

k k k+1 k+1 k = k~na−n + ~na−n + a−n an − a−n an = (k + 1)~na−n.

Lemma 2 If ~ 6= 0, then the representation (10) is irreducible.

Proof. Let P be an arbitrary polynomial in B. Take any monomial in P with highest degree. Suppose, without loss of generality, it is xj1 ··· xjn . Applying the operator i1 in aj1 ··· ajn to this monomial (using the last lemma repeatedly) yields kj ! ··· j ! i1 in ~ 1 n Pn with k = l=1 jk, while it yields 0 when applied to any other monomial in P . Thus aj1 ··· ajn P is a nonzero multiple of 1. Since B is generated by 1, we are done. i1 in The constant polynomial v = 1 is called the vacuum vector of B, and has the following properties:

an(v) = 0 for n > 0 (12)

a0(v) = µv

~(v) = ~v.

Note v is actually a highest weight vector.

40 Proposition 11 Let V be a representation of A which admits a nonzero vector v

k1 kn satisfying (12) with ~ 6= 0. Then monomials of the form a−1 ··· a−n(v) (ki ∈ Z+) are linearly independent. If these monomials span V, then V is isomorphic to the representation (10). In particular, this is the case if V is irreducible.

n Proof Deﬁne a map φ from B to V by φ(P (. . . , xn,... )) = P (..., a−n,... )v. ~n Now:

n an(φ(P )) = an(P (..., a−n,... )v) ~n X n kn = an(··· ( a−n ··· (v)) ~n X n kn kn = (··· [ ana−n] ··· )(v) ~n X n kn kn−1 k = (··· [ (kn~na−n + a−nan)] ··· )(v) ~n X n kn−1 kn−1 kn = [··· (knn a−n ) ··· ](v) + [··· (a−nan) ··· ](v) ~n where the third step uses the previous lemma. The last term can be rearranged as

kn kn [··· (a−nan) ··· ](v) = [··· (a−n) ··· an](v) which is clearly zero by (12), leaving only k −1 P n n kn−1 [··· (knn a ) ··· ](v). Next apply an to P : ~n −n

X kn φ(anP (. . . , xn,... )) = φ( an(··· xn ··· ))

X ∂ kn = φ( (··· n xn ··· )) ∂xn

X kn−1 = φ( (··· nknxn ··· )) X n kn−1 kn−1 = (··· knn a−n ··· )(v) ~n

and so an(φ(P )) = φ(an(P )) for arbitrary P ∈ B. Since B is irreducible, ker(φ) = 0, and thus φ is an isomorphism if φ is onto.

Proposition 12 Let V be as in Proposition 11. Then V carries a unique Hermitian form · | · which is contravariant with respect to ω, and such that v|v = 1 for

41 k1 kn the vacuum vector v. The distinct monomials a−1 ··· a−n(v)(ki ∈ Z+) form an orthogonal basis with respect to · | · . These monomials have norms given by

n k1 kn k1 kn Y kj a−1 ··· a−n(v)|a−1 ··· a−n(v) = kj!(~j) . (13) j=1

Proof. If · | · is a contravariant Hermitian form, then both the orthogonality and (13) are proved by induction on k1 + ··· + kn, giving uniqueness as follows: Let k1 + ··· + kn = 1. Without loss of generality, say km = 1. Then

amv|amv = 0|0 and so

a−mv|a−mv = a−mv|a−mv − amv|amv

= v|ω(a−m)a−mv − v|ω(am)amv

= v|ama−mv − v|a−mamv

= v|[am, a−m]v

= v|~mv

= ~m v|v .

0 0 Suppose the given formula is true for k1 + ··· + kn = r, and let k1 + ··· + kn = r + 1.

42 0 0 Without loss of generality let kj = kj + 1 and km = km for m 6= j. Now,

0 0 0 0 k1 kn k1 kn k1 kj kn k1 kj +1 kn a−1 ··· a−nv|a−1 ··· a−nv = a−1 ··· a−ja−j ··· a−nv|a−1 ··· a−j ··· a−nv

k1 kn k1 kj +1 kn = a−ja−1 ··· a−nv|a−1 ··· a−j ··· a−nv

k1 kn k1 kj +1 kn = a−1 ··· a−nv|ω(a−j)a−1 ··· a−j ··· a−nv

k1 kn k1 kj +1 kn = a−1 ··· a−nv|aja−1 ··· a−j ··· a−nv

k1 kn k1 kj +1 kn = a−1 ··· a−nv|a−1 ··· aja−j ··· a−nv

k1 kn k1 kj +1 kj +1 kn = a−1 ··· a−nv|a−1 ··· ([aj, a−j ] + a−j aj) ··· a−nv

k1 kn k1 kj +1 kn = a−1 ··· a−nv|a−1 ··· [aja−j ] ··· a−nv

k1 kn k1 kj +1 kn + a−1 ··· a−nv|a−1 ··· a−j ··· a−najv

k1 kn k1 kj kn = a−1 ··· a−nv|a−1 ··· (kj~j)a−j ··· a−nv + 0 (by Lemma 1)

n km = kj~jΠm=1km!(~m)

0 n 0 km = Πm=1km!(~m)

and thus the formula holds by induction. Now for orthogonality: Let L1, ··· ,Ln and K1, ··· ,Km be arbitrary distinct ﬁnite sequences of positive integers. Then for some j, Lj 6= Kj. Then, without loss of generality, let Lj > Kj. Then,

L1 Ln K1 Km Lj L1 Lj−1 Lj+1 Ln K1 Km a−1 ··· a−nv|a−1 ··· a−1 v = a−ja−1 ··· a−j+1a−j−1 ··· a−nv|a−1 ··· a−mv

L1 Lj−1 Lj+1 Ln Lj K1 Km = a−1 ··· a−j+1a−j−1 ··· a−nv|ω(a−j)a−1 ··· a−mv

L1 Lj−1 Lj+1 Ln Lj K1 Km = a−1 ··· a−j+1a−j−1 ··· a−nv|aj a−1 ··· a−mv

L1 Lj−1 Lj+1 Ln K1 Lj Kj Km = a−1 ··· a−j+1a−j−1 ··· a−nv|a−1 ··· aj a−j ··· a−mv .

43 Now we note, using Lemma 2, that

Lj Kj Lj −1 Kj Kj aj a−j = aj [aja−j ] + a−j aj

Lj −1 Kj −1 Kj = (Kj~j)aj a−j + a−j aj.

Each time the multiplication is reversed, we end up with a term with aj to the right, which will yield zero since we can shift it to the right all the way to v. The remaining part will have a−j with power one less than before, until we have exhausted all copies of a−j. Since Lj > Kj, this leaves us with terms only containing aj to a power, and these similarly yield zero. Thus, all of the vectors generated this way are orthogonal. One checks directly that the Hermitian form, for which monomials are orthogonal and have norms given by (13), is contravariant, proving existence. Note that assuming contravariance also forces the representation to be unitary.

Corollary 2 The contravariant Hermitian form on V such that v|v = 1 is positive- deﬁnite if and only if ~ > 0.

j1 jk Deﬁnition 24 The degree of the monomial x1 ··· xk is deﬁned to be j1 + 2j2 +

··· + kjk. Let Bj be the subspace of B spanned by the monomials of degree j. Bj is clearly ﬁnite dimensional, and dim(Bj) = p(j) where p(j) is the number of partitions of j ∈ Z+ into a sum of positive integers with p(0) = 1.

We have

M B = Bj, j≥0 the principle gradation of B.

44 5.2 Oscillator Representations of Vir

Using the results of section 4 we can introduce the Virasoro operators Lk. These are deﬁned in the Fock representation B with ~ = 1 by:

1 X L = : a a :(k ∈ ), (14) k 2 −j j+k Z j∈Z where the colons indicate ‘normal ordering’, deﬁned by

  aiaj if i ≤ j : aiaj := (15)  ajai if i > j.

When Lk is applied to any vector of B, only a ﬁnite number of terms in the sum contribute as can be seen in the following: For j ∈ Z we have

  a−jaj+k − j ≤ j + k : a−jaj+k :=  aj+ka−j − j > j + k.

Splitting the terms this way means if an element with a non-positive subscript comes second in a product (the first applied) then either −j ≤ j + k and j + k ≤ 0 or −j > j + k and −j ≤ 0. Combining the inequalities for each case, we have −j ≤ j + k ≤ 0 or j + k < −j ≤ 0. Rearranging terms we have 0 ≤ 2j + k ≤ j or 2j + k < 0 ≤ j. In each case, there are a finite number of choices for j given any k. If an element with a positive subscript comes second in a product, when that product is applied to a polynomial in B it decreases the degree of the polynomial. For a given polynomial there are only finitely many indices such that applying ∂ n ∂xn on the polynomial produces a nonzero result. These considerations tell us that each

Lk yields only a ﬁnite number of terms when applied to any polynomial in B, and thus these operators are well-deﬁned on B.

45 Lemma 3 [ak,Ln] = kak+n

Proof. A so-called “Cutoﬀ Procedure” can be introduced to simplify the calculations.

Deﬁne ψ : R → {0, 1} by

  1 |x| ≤ 1 ψ(x) =  0 |x| > 1.

Then let L () = 1 P : a a : ψ(j), for ∈ . This way L () is a finite n 2 j∈Z −j j+n R n sum for 6= 0, and Ln() → Ln as → 0. Also, given v ∈ B, Ln()(v) = Ln(v) for sufficiently small. Now, since Ln() can be written without normal ordering by adding a finite sum of constant terms,

1 X [a ,L ()] = [a , a a ]ψ(j) k n 2 k −j j+n j∈Z 1 X 1 X = [a , a ]a ψ(j) + a [a , a ]ψ(j) 2 k −j j+n 2 −j k j+n j∈Z j∈Z 1 1 = ka ψ(k) + ka ψ(k). 2 k+n 2 k+n

Take → 0, giving the required formula.

Proposition 13 The Lk satisfy the following relations:

m3 − m [L ,L ] = (m − n)L + δ . (16) m n m+n m,−n 12

46 Proof. Using the same cutoﬀ procedure as the Lemma and the fact that for f ∈ B

[a−jaj+m,Ln](f) = a−jaj+mLn(f) − Lna−jaj+m(f)

= (a−j[aj+m,Ln] + a−jLnaj+m)(f) − Lna−jaj+m(f)

= ((j + m)a−jaj+m+n + [a−jLn]aj+m)(f) + Lna−jaj+m(f) − Lna−jaj+m(f)

= ((j + m)a−jaj+m+n + (−j)an−jaj+m)(f), it follows that

1 X [L (),L ] = [a a ,L ]ψ(j) m n 2 −j j+m n j 1 X 1 X = (−j)a a ψ(j) + (j + m)a a ψ(j). 2 n−j j+m 2 −j j+m+n j j

Here split the sums, the ﬁrst into parts j ≥ (n−m)/2 and j < (n−m)/2, the second into parts j ≥ −(n+m)/2 and j < −(n+m)/2, reversing the order of multiplication on the appropriate part to rewrite in the normal order:

1 X 1 X (−j): a a : ψ(j) + (j + m): a a : ψ(j) 2 n−j j+m 2 −j j+m+n j j −m 1 X − δ j(m + j)ψ(j) 2 m,−n j=−1 where the extra terms come from the cases where −n + j = j + m for the ﬁrst sum and j = j + n + m for the second:

an−jaj+m = [an−j, aj+m] + aj+man−j

= −(j + m)δ−n+j,j+m + aj+man−j

47 and similarly

a−jaj+m+n = [a−jaj+m+n] + aj+m+na−j

= (−j)δj,j+m+n + aj+m+na−j,

where both kronecker deltas reduce to δm,−n. This yields two overlapping sums with opposite signs and the same summand. The nonoverlapping portion is the interval 0 < j ≤ −m, giving the required sum. Using the transformation j → j + n in the ﬁrst sum of the now normal ordered sums, this leaves

(m3 − m) [L ,L ] = (m − n)L + δ , m n m+n m,−n 12

1 P−m (m3−m) where − 2 j=−1 j(m + j) = 12 by the following induction argument:

−1 1 X 1 − j(m + j) = − (−1)(1 − 1) = 0. 2 2 j=−1

48 1 P−n n3−n Now suppose − 2 j=−1 j(n + j) = 12 for some n ∈ Z.

−(n+1) 1 X 1 − j((n + 1) + j) = − − 1(n + 1 − 1) 2 2 j=−1

− 2(n + 1 − 2) + · · · − n(n + 1 − n) + (−n + 1)(n + 1 − n − 1) 1 = − − 1(n) − 2(n − 1) − 3(n − 2) + ··· + (−n − 1)(2) − n(1) + 0 2 1 = − − 1(n) − 1(n − 1) − 1(n − 2) + · · · − 1(2) − 1(1) + 0 2 1 − − 1(n − 1) − 2(n − 2) + ··· + (−n + 2)(2) + (−n + 1)(1) + 0 2 −n 1 1 1 X = n(n + 1) − j(n + j) 2 2 2 j=−1 n2 + n n3 − n = + 4 12 3n2 + 3n + n3 − n = 12 (n + 1)3 − (n + 1) = . 12

Proposition 14 The representation of the Virasoro algebra π : V ir → gl(B),

π(dn) = Ln, π(c) = 1, is unitary.

Proof The representation of the Oscillator algebra given before, here for clarity called π0 : A → gl(B), is unitary. Then, for any element x ∈ A,

π0(ω(x)) = π0(x)∗.

49 Then

∗ X 0 0 ∗ Lk = : π (a−j)π (ak+j): j∈Z

X 0 † 0 ∗ = : π (aj+k) π (a−j) : j∈Z

X 0 0 = : π (ω(ak+j))π (ω(a−j)) : j∈Z

X 0 0 = : π (a−k−j)π (aj): j∈Z

X 0 0 = : π (al)π (a−k+l): l∈Z

= L−k

∗ and since π(dk) = Lk, π(ω(dk)) = π(dk) .

Proposition 15 The unitary representation of the Virasoro algebra from the previous proposition is a direct sum of irreducible representations.

Proof. From the equation for the operators Lk it is clear that L0 can be written in the form L = µ2/2 + P a a . Let L act on some monomial of degree j: 0 j∈Z+ −j j 0

k j1 jk 2 j1 jk X j1 jk 2 j1 jk L0(x1 ··· xk ) = (µ /2)x1 ··· xk + jix1 ··· xk = (µ /2 + j)x1 ··· xk i=1

2 and thus the Bj are eigenspaces for L0 with eigenvalue µ /2 + j. As a result, any subrepresentation U of B will have a decomposition into eigenspaces:

M U = U ∩ Bj j∈Z+

Denote U ∩ Bj by Uj. The representation of Vir on B is unitary, and so the eigenspaces Bj are orthogonal with respect to · | · , and so the Uj are as well.

50 ⊥ Taking Uj to be the orthogonal complement of Uj in B, we can deﬁne a subspace U ⊥ by

⊥ M ⊥ U = Uj j∈Z+

Then, clearly,

B = U ⊕ U ⊥ since

U ⊥ = {v ∈ B| U|v = 0}.

⊥ ⊥ U is then an invariant subspace for Vir, since U|U = 0 and LjU ⊂ U imply

⊥ ⊥ that 0 = LjU|U = U|LjU . A subrepresentation of B can be constructed as follows: Clearly for the vacuum vector 1

Lk(1) = 0 (k > 0)

2 L0(1) = h · 1 (h = µ /2)

Let B0 be the span of the vectors

L−ik ··· L−i2 L−i1 (1)

0 where the 0 < i1 ≤ i2 ≤ · · · ≤ ik are arbitrary ﬁnite sequences. It is clear that B is invariant under the Lj, and so we have a subrepresentation of B, called the highest component of B. This is an example of a highest weight representation of Vir.

51 Proposition 16 B0 is an irreducible representation of Vir.

Proof. If B0 were reducible, it could be written as a direct sum of subrepresentations

0 ⊥ ⊥ B = U ⊕U as above. Each of U and U then has an L0-eigenspace decomposition.

The vacuum vector spans the h-eigenspace of L0, and so can only belong to one of the two. Then that summand is B0, and the other must be 0, and hence B0 is irreducible. An alternate way to show B0 is irreducible is to note it is a unitary highest weight representation of V ir, and is irreducible by proposition 6 of the section 4.2. It is also a positive energy representation.

52 6 Dirac Positron Theory

The Dirac positron theory can be expressed in terms of representations of infinite matrices and the infinite wedge space, or “semi-infinite polynomials”. Here we de- velop representations of various algebras of infinite matrices, and relate them to the Oscillator (Heisenberg) and Virasoro algebras.

6.1 Inﬁnite Wedge Space

To build the Dirac positron theory, a few constructions from algebra are needed: the tensor product, and the wedge product. These are reviewed in Appendix B. Fermions are particles that obey the Pauli exclusion principle, which states that no two fermions can occupy the same state. For example, electrons are fermions. This requirement can be seen in the construction of Λ(V ), where x ∧ x = 0 may be understood as saying that no state appears twice in a product. This way a vector can be written as a wedge of occupied states will have no repeated states, meaning no two electrons share a state. Dirac’s theory requires a larger space than Λ(V ); the inﬁnite wedge space Λ∞(V ). This allows an inﬁnite number of particles. Let V = L v . We start with a i∈Z C i space

(0) ∞ F = Λ0 (V ) where F (0) is the vector space consisting of elements

ψ = vi0 ∧ vi−1 ∧ vi−2 ∧ · · · where

a) i0 > i−1 > . . .

53 b) ik = k for k 0.

Deﬁne the vector ψ0 = v0 ∧v−1 ∧v−2 ∧· · · to be the vacuum vector. The vectors vi P∞ are states, and the degree (energy) of a ψ can be deﬁned as deg(ψ) = s=0(i−s + s).

This gives a degree (energy) of 0 for ψ0. The condition ik = k for k 0 placed on the ψ means every ψ ∈ F (0) has ﬁnite degree. Moreover, the positive subscript states in ψ always correspond to missing negative subscript states in ψ. In terms of physics, this means F (0) consists of charge conserving excitations of the vacuum state ψ0 - for every electron of positive energy, we have a hole of negative energy.

(0) (0) Proposition 17 Let Fk denote the span of all vectors of degree k in F . Then

(0) L (0) (0) a) F = Fk , F0 = Cψ0

(0) b) dimFk = p(k)

(0) P (0) k c) dimqF = k(dimFk )q = 1/φ(q).

Proof. a) Let k be any non-negative integer, and {k0, k1, . . . , kn−1} be a partition of k in non-increasing order. Then deﬁne a unique ψ as follows:

ψ = vj0 ∧ vj−1 ∧ · · · ∧ vj−n+1 ∧ v−n ∧ v−n−1 ∧ · · · where

j−i = ki − i for i = 0, . . . , n − 1.

The set of all ψ deﬁned as above are linearly independent. The degree is calculated

54 to be

n−1 ∞ X X deg(ψ) = [(ki − i) + i] + (−i + i) i=0 i=n n−1 X = ki = k i=0 as required. b) The number of partitions of k in non-increasing order is p(k) by deﬁnition. Now deﬁne F (m) to be the span of vectors of the form

ψ = vim ∧ vim−1 ∧ · · · where

a) im > im−1 > . . .

b) ik = k + m for k 0.

The degree can be calculated in the same way as before:

∞ X deg(ψ) = (im−s + s − m). s=0

(0) (m) This is similar to F , F having a reference vector ψm = vm ∧ vm−1 ∧ · · · with all other vectors being higher degree excitations of this vector. In this case m is called the charge number, and is the number of occupied positive energy states without corresponding holes. Vectors of this form, with a finite number of unoccu- pied negative energy states (holes), are called semi-infinite monomials. The space of all semi-infinite monomials is F = L F (m). m∈Z

55 Clearly the same dimension formulas apply to F (m) as F (0):

(m) dimFk = p(k)

(m) dimqFk = 1/φ(k).

6.2 Highest Weight Representations of gl∞

We can deﬁne a representation of gl∞ on F by the following:

r(a)(vi1 ∧ vi2 ∧ · · · ) = a(vi1 ) ∧ vi2 ∧ · · · + vi1 ∧ a(vi2 ) ∧ · · · + ··· where the action of the matrix on an individual vector is the usual one. The action of the basis elements Eij of gl∞ on F is particularly simple:

r(Eij)(vi1 ∧ vi2 ∧ · · · ) = 0 if j∈ / {i1, i2,... },

= vi1 ∧ · · · ∧ vik−1 ∧ vi ∧ vik+1 ∧ · · · if j = ik,

where the right hand side is zero if there is a repeated index. The r(Eij) map a vector ψ from F (m) into a vector with a single index changed and with a reordering and appropriate negative sign we have another vector clearly satisfying im > ··· > i1 and

(m) ik = k + m for k >> 0, and so r(Eij) maps F into itself. Thus the representation

(m) r is a direct sum of subrepresentations rm on F . Now, since each vector in F (m) is a linear combination of ψ of the form

ψ = vim ∧ · · · ∧ vim−k ∧ vim−k−1 ∧ · · · ,

56 there is the following formula:

ψ = r(Eim,m) . . . r(Eim−k,m−k)ψm,

and so the representations rm are irreducible.

Theorem 3 The representation r of gl∞ in F is a direct sum of irreducible unitary subrepresentations.

Proof. Define a positive-definite Hermitian form · | · on F by declaring the semi-infinite monomials to be an orthonormal basis. Let ω be the standard antilinear anti-involution of gl∞:

ω(a) = a∗, where a∗ denotes the Hermitian adjoint of the matrix a. Then, for the basis vectors

Eij of gl∞ and hence for all of gl∞, then

0 ∗ 0 r(Eij)ψ|ψ = ψ|r(Eij)ψ and so the form · | · is contravariant with respect to ω and the representation r is unitary. It has been shown r is the direct sum of irreducible subrepresentations rm, and so these must be unitary as well.

For the algebra gl∞ we can deﬁne a highest weight representation directly.

Deﬁnition 25 Given a collection of numbers λ = {λi|i ∈ Z}, called a highest weight deﬁne the highest weight representation πλ of the Lie algebra gl∞ as an irreducible representation on a vector space L(λ) generated by a non-zero vector vλ,

57 called a highest weight vector, such that

πλ(n+)vλ = 0,

πλ(Eii)vλ = λivλ

It is not hard to show that L(λ) is determined by λ.

(m) L (m) Now examine the action of the r(Eij) on the decomposition F = k≥0 Fk , subspaces of ﬁxed degree. From the calculations above r(Eij) either replaces vi with vj or yields zero as a result. The replacement of vi by vj changes the degree of the vector by i − j. Thus

(m) (m) r(Eij)Fk ⊂ Fk+i−j.

gl∞ can be decomposed into a direct sum of homogeneous components gj of degree j:

M gl∞ = gj j∈Z where a matrix in gj has nonzero entries only on the j-th diagonal above or below the principle diagonal. This is called the principal gradation of gl∞. Then

(m) (m) r(gj)Fk ⊂ Fk+j

Also, since r(Eij)ψm = 0 for i < j, we have

r(gj)ψm = 0 for j < 0.

(m) Now, since every ψ ∈ Fk is in the span of vectors r(Eim,m) . . . r(Eim−k,m−k)ψm, and

58 every Eik,k has degree ik − k, then

(m) X Fk = rm(gj1 ) ··· rm(gjn )ψm. j1+···+jn=k j1,...jn∈Z+

Now a representation-theoretic interpretation of Dirac’s deﬁnition of energy can be given.

Len n+ be the subalgebra of gl∞ consisting of strictly upper triangular matrices. Clearly,

M n+ = gj. j<0

Then

rm(n+)ψm = 0,

rm(Eii)ψm = λiψm, where

  1 if i ≤ m λi =  0 if i > m.

Then for each m ∈ Z we have constructed an irreducible highest weight representation rm of gl∞ with highest weight

$m = {λi = 1 for i ≤ m, λi = 0 for i > m}.

The rm are called the fundamental representations of gl∞ and the $m the fundamental weights. Thus F is a direct sum of all fundamental representations of

59 gl∞. In particular, the fundamental representations are unitary by Theorem 3.

Proposition 18 The irreducible highest weight representations of gl∞ with highest P weight of the form i ki$i where the ki are nonnegative integers are unitary.

Proof. We begin the proof by showing that tensor products of unitary representations are unitary.

Let π1 : g → gl(V1), π2 : g → gl(V2) be unitary representations of a Lie algebra g with antilinear anti-involution ω. We have a natural representation on the tensor product π : g → gl(V1 ⊗ V2) by π(x)(v ⊗ w) = π1(x)v ⊗ w + v ⊗ π2(x)w. This representation is unitary:

π(ω(x))(v ⊗ w) = π1(ω(x))v ⊗ w + v ⊗ π2(ω(x))w

∗ ∗ = π1(x) v ⊗ w + v ⊗ π2(x) w

= π(x)∗(v ⊗ w).

It is not diﬃcult to see that if V1 and V2 are irreducible unitary highest weight representations with highest weight vectors v1 and v2 then the vector v1 ⊗ v2 is a highest weight vector of an irreducible subrepresentations of V1 ⊗ V2, its highest component. It has highest weight equal to the sum of the two highest weights. Since the fundamental representations are unitary, an obvious induction argument P on i ki using tensor products completes the proof.

It is easy to see that the highest weight representation of gl∞ of highest weight P ki$i, with ki ∈ Z and ki ≥ 0, is unitary.

60 6.3 Representations ofa ¯∞

The matrices ina ¯∞ have a ﬁnite number of nonzero diagonals and so are ﬁnite linear combinations of matrices of the form

X ak = λiEi,i+k i∈Z where the λi are arbitrary complex numbers. If the same representation is applied toa ¯∞ as gl∞, there is a problem:

X with ak = λiEi,i+k, λi ∈ C i∈Z then

r(ak)ψm = ak(vm) ∧ vm−1 ∧ · · · + vm ∧ ak(vm−1) ∧ · · · + ··· ,

a ﬁnite linear combination of vectors for k 6= 0, since the terms in ak vanish for i + k > m or i ≤ m. For k = 0

r(a0)ψm = (λm + λm−1 + ··· )ψm.

The sum on the right-hand side can diverge. This can be ﬁxed by deﬁning a repre- sentationr ˆm by

rˆm(Eij) = rm(Eij) if i 6= j or i = j > 0,

rˆm(Eii) = rm(Eii) − I if i ≤ 0.

61 Then

  Pm ( i=0 λi)ψm for m ≤ 1 rˆm(a0)ψm =  Pm+1 −( i=0 λi)ψm for m ≤ −1 (and is 0 for m = 0)

(m) For A ∈ a¯∞ ther ˆm clearly map F into themselves, but they do not satisfy the original commutation relations fora ¯∞. The original relations can be written as

i) [Eij,Ekl] = 0 for j 6= k, l 6= i

ii) [Eij,Ejl] = Eil for l 6= i

iii) [Eij,Eki] = −Ekj for j 6= k

iv) [Eij,Eji] = Eii − Ejj.

All but (iv) hold forr ˆm, since the presence of I in the brackets makes no diﬀerence. For the last

[ˆrm(Eij), rˆm(Eji)] =r ˆm(Eii) − rˆm(Ejj) + α(Eij,Eji)I where

  1 for i ≤ 0, j ≥ 1, α(Eij,Eji) = −α(Eji,Eij) =  0 otherwise then

rˆm([Eij,Ekl]) = [ˆrm(Eij), rˆm(Ekl)] − α(Eij,Ekl)I.

This can be made into a representation of the central extension ofa ¯∞, a∞ =a ¯∞ ⊕Cc

62 where c is central and

[a, b] = ab − ba + α(a, b)c,

where α(a, b) is linear in each variable and deﬁned on the Eij as before. Letting

(m) rˆm(c) = 1, a representation of a∞ in F can be obtained. ω can also be extended to a∞ by letting ω(c) = c. Since the rm are unitary, the choice of ω(c) makesr ˆ unitary as well. Now we consider the subalgebra ofa ¯∞ of shift operators underr ˆm.

[ˆrm(Λn), rˆm(Λk)] = α(Λn, Λk)I.

In this case

α(Λn, Λk) = nδn,−k so that

[ˆrm(Λn), rˆm(Λk)] = nδn,−k.

Also,

rˆm(Λ0) = mI.

These are the commutation relations for the Oscillator algebra A. The antilinear involution ω here is consistent with that of A as well, so it is a unitary representation of A.

63 7 Some Physics

To understand what is going on in our construction for Dirac’s positron theory it may be useful to give an overview of the analogous construction used in physics. This section contains statements without proof, as it is a summary of a large amount of work.

7.1 Dirac Equation and First Quantization

First, we have the time-dependent Schr¨odingerequation, a wave equation:

∂ i Ψ(t, x) = HΨ(t, x) ~∂t where Ψ is a complex valued function of t ∈ R, and x ∈ Rn, ~ the reduced Planck’s constant, and H is the Hamiltonian operator for some physical system, which usually corresponds to the total energy. The function Ψ needs to be square-integrable over the spatial variable x. This equation comes from a process now known by the name ﬁrst quantization, in which observables (measurable quantities) of a physical system are replaced by self-adjoint operators. For example, in many situations the replacements

∂ E → i , p → i ∇ ~∂t ~ are used where E and p are the classical energy and momentum, and ∇ is the partial

64 diﬀerential operator,

  ∂  ∂x1     ∂   ∂x2  ∇ =  .  .  .      ∂ ∂xn

One very important form of the Schr¨odingerequation is the equation for a one dimensional simple harmonic oscillator:

∂ 2 d2 1 i Ψ = − ~ Ψ + mω2x2Ψ. ~∂t 2m dx2 2

Note that our Hamiltonian can be written as

p2 mω2 H = + x2. 2m 2

Deﬁne operators a and a†, known as ladder operators, by

rmω i a = (x + p) 2~ mω rmω i a† = (x − p) 2~ mω

d where p = i~ dx .

65 Then we have

r r † mω i mω i ~ωaa = (x + p) (x − p) 2~ mω 2~ mω mω2 i 1 = (x2 + (px − xp) + p2) 2 mω m2ω2 mω2 i d d 1 = (x2 + (−i x + i x ) + p2) 2 mω ~dx ~ dx m2ω2 mω2 1 = (x2 + ~ + p2) 2 mω m2ω2 mω2 p2 ω = x2 + + ~ 2 2m 2 and similarly

r r † mω i mω i ~ωa a = (x − p) (x + p) 2~ mω 2~ mω mω2 i 1 = (x2 − (px − xp) + p2) 2 mω m2ω2 mω2 i d d 1 = (x2 − (−i x + i x ) + p2) 2 mω ~dx ~ dx m2ω2 mω2 1 = (x2 − ~ + p2) 2 mω m2ω2 mω2 p2 ω = x2 + − ~ . 2 2m 2

Then we can write

1 H = (a†a + ) ω. 2 ~

Now consider the commutator of a and a†:

[a, a†] = aa† − a†a 1 mω2 p2 ω 1 mω2 p2 ω = ( x2 + + ~ ) − ( x2 + − ~ ) = 1. ~ω 2 2m 2 ~ω 2 2m 2

66 Using this we also have

1 [H, a] = [ ωa†a + , a] ~ 2 1 = ω[a†a, a] + [1, a] ~ 2 † 2 † = ~ω(a a − aa a)

† 2 † † = ~ω(a a − (a a − [a , a])a

† 2 † 2 = ~ω(−a + a a − a a )

= −~ωa

and similarly

1 [H, a†] = ω[a†a + , a†] ~ 2 † † † 2 = ~ω(a aa − (a ) a)

† † † † 2 = ~ω(a ([a, a ] + a a) − (a ) a)

† † 2 † 2 = ~ω(a + (a ) a − (a ) a)

† = ~ωa .

The operators H, a, a†, and identity then form a Lie algebra with commutation relations

[H, a] = −~ωa

† † [H, a ] = ~ωa

[a, a†] = 1.

67 Compare this with the deﬁnition of the Heisenberg algebra in section 3.3. So far we have only explored the operator on the right-hand side of the original equation. The reason for this will soon be evident. Going back to the original equation we can suppose a seperation of variables: Ψ(x, t) = ψ(x)φ(t), and so

∂ 2 d2 1 i ψ(x)φ(t) = − ~ ψ(x)φ(t) + mω2x2ψ(x)φ(t) ~∂t 2m dx2 2 2 2 ∂ ~ d 1 2 2 i~ φ(t) − 2 ψ(x) + mω x ψ(x) ∂t = 2m dx 2 φ(t) ψ(x) and so

∂ i φ(t) = kφ(t) ~∂t 2 d2 1 − ~ ψ(x) + mω2x2ψ(x) = kψ(x) 2m dx2 2 for some seperation constant k ∈ R. It is clear that φ(t) = e−i~/k. This leaves us to solve

Hψ = kψ.

The details of ﬁnding a solution to this equation are not relevant, so we will skip it. Given any solution to this equation Ψa, we can generate a family of solutions using the algebraic relations found earlier. This is done by noting that HΨa = kaΨa for some ka ∈ R and as a result

HaΨa = aHΨa − ~ωaΨa

= akaΨa − ~ωaΨa

= (ka − ~ω)aΨa

68 † using the commutation relations for H and a. This shows that a Ψa is also a solution to Hψ = kψ. Further constraints on the types of functions allowed as solutions (these constraints, which make the domain of the problem a Hilbert space, can be found in any basic quantum physics text) lead to the conclusion that applying a repeatedly leads to a nonzero solution, Ψ0, such that aΨ0 = 0. We know that HΨ0 = k0Ψ0 and that HaΨ0 = (k0 − ~ω)Ψ0, and thus k0 = ~ω. Looking at the earlier solutions, this means each ka is some multiple of ~ω. The Dirac equation comes from the same starting point, the Schr¨odingerequation, but with diﬀerent assumptions about the total energy of the system described. From relativity we have well known relationship between energy and momentum

E2 = p2c2 + m2c4 with c the speed of light in vacuum, and m the mass of the object in question. Dirac was one of many to attempt combining relativity and quantum mechanics. One notable result in this area was the Klein-Gordon equation

∂2 − 2 Ψ(t, x) = (−c2 2∇2 + m2c4)Ψ(t, x), ~ ∂t2 ~ which was obtained by replacing the energy and momentum with the appropriate operators in the relativistic energy equation. Dirac found a way to rectify the dif- ﬁculties of this particular equation including, but not limited to, having a second derivative in the time variable. He had studied Heisenberg’s matrix mechanics, and realized that a square root for the right-hand side of the equation could be found under the assumption of a matrix equation instead of a one-dimensional wave equation.

This is done by assuming the quantity −c2~2∇2 +m2c4 can be written as a perfect

69 square,

∂ ∂ ∂ 2 2 (−c~(α1 + α2 + α3 ) + βmc ) , ∂x1 ∂x2 ∂x3

where the new quantities αi and β are matrices. By computing this square and comparing it to the original equation, we can ﬁnd relations for the new matrix quantities:

2 2 2 ∂ ∂ ∂ 2 2 2 2 2 ∂ 2 ∂ 2 ∂ (−c~(α1 + α2 + α3 ) + βmc ) = −c ~ (α1 2 + α2 2 + α3 2 ) ∂x1 ∂x2 ∂x3 ∂x1 ∂x2 ∂x3

+ ic~[(α1α2 + α2α1) + (α1α3 + α3α1) + (α2α3 + α3α2)]

3 + i~mc [(α1β + βα1) + (α2β + βα2) + (α3β + βα3)]

Comparison yields the equations:

αiαj + αjαi = 2δijI

αiβ + βαi = 0

2 αi = I

β2 = I where I is the identity matrix. The dimension of such a system has to be an even number, which can be seen as a result of the following calculation:

tr(αi) = tr(βαiβ),

−1 since β = β , and, since αiβ = −βαi,

2 tr(βαiβ) = −tr(β α) = −tr(αi).

70 Thus,

tr(αi) = −tr(αi) = 0.

2 Now, since αi = I, all eigenvalues of αi are ±1. As a matrix is similar to its Jordan canonical form, the trace is the sum of its eigenvalues, and so there must be an even number of eigenvalues to have a sum of zero. There are many choices of dimension to use now, but the standard representation is a 4x4 system,

  I 0  2  β =   0 −I2   0 σ  i αi =   σi 0

where I2 is the 2x2 identity matrix, and σi are the Pauli matrices, deﬁned as follows:

      0 1 0 −i 1 0       σ1 =   , σ2 =   , σ3 =   . 1 0 i 0 0 −1

Let

    α1 σ1         α = α  , σ = σ  .  2  2     α3 σ3

71 Then we can write

  mc2I −i cσ · ∇ 2  ~  H0 = −i~cα · ∇ + βmc =   . −i~cσ · ∇ −mc2I

H0 is called the Dirac operator and it will be the Hamiltonian in Dirac’s equation:

∂ i Ψ(t, x) = H Ψ(t, x) = (−i cα · ∇ + βmc2)Ψ(t, x). ~∂t 0 ~

The Ψ functions here have four components, and each is a function of the variables t ∈ R and x ∈ R3. According to the basic principles of quantum mechanics one deﬁnes a Hilbert space H for each quantum mechanical system. Every observable must be a self-adjoint operator on this space so that the eigenvalues, or possible measurements, are real numbers. The vectors in the Hilbert space are states of the system, and the state of a system at some initial time t0, Ψ(t0, x) is assumed to be normalized. Here we are taking H = L2(R3, C4). The Dirac operator can be written in a 2x2 block form under the Foldy-Wouthuysen

−1 transformation UFW = F uF, where F is the Fourier transform and

(mc2 + λ(p))I + βcα · p u ≡ u(p) = , p2λ(p)(mc2 + λ(p))

p 2 2 2 4 where λ(p) = c p + m c . The new form of H0 under this transformation is

√  −c2∇2 + m2c4 0 U H U −1 =   , FW 0 FW  √  0 − −c2∇2 + m2c4

which means the Dirac equation can be interpreted as two two-component square- root Klein-Gordon equations. One equation will yield a subspace of positive-energy states, and the other a subspace of negative-energy states. We can, in fact, write H

72 as a direct sum of these orthogonal subspaces,

H = H+ ⊕ H− .

We need one more thing to begin constructing the space used for the many- particle theory. So far we have only seen the Dirac operator for a free particle. The Dirac operator for a charge e in an external electromagnetic ﬁeld (φ, A) is given by

e H(e) = cα · (p − A(t, x)) + βmc2 + eφ(t, x). c

Now consider the antiunitary transformation

CΨ = UcΨ where

βUc = Ucβ

αkUc = Ucαk.

In the standard representation we can take Uc = iβα2. C is called the charge conjugation operator. If Ψ is a solution for the Dirac equation with Hamiltonian H(e) then CΨ is a solution for the Dirac equation with Hamiltonian H(−e). Moreover,

CH(e)C−1 = −H(−e), and so the negative energy subspace for H(e) is connected by a symmetry transformation to the positive energy subspace for H(−e), the Dirac operator for a particle with opposite charge. This is the antiparticle, or positron.

73 7.2 Second Quantization and Fock space

From here we skip to quantization of the Dirac ﬁeld, also known as second quantization. We have some Hilbert space of states, H = H+ ⊕ H−, with orthogonal positive-energy and negative-energy subspaces. The positive-energy states are particles, the negative-energy states are antiparticles. We can build a Fock space from these beginning with

(1) (1) F+ = H+,F− = CH− where C represents the antiunitary charge conjugation operator from before. We are

−1 restricting the use of the Hamiltonian H to H+ and using the hamiltonian −CHC on CH−.

(1) (1) Here F− and F+ are considered as 1-particle states. Let

(n) n (1) F± = Λi=1F± , and then

∞ M (n,m) (n,m) (n) (m) F = F ,F = F+ ⊗ F− , n,m=0

0 with F± = C. A (normalized) state in F represents a physical system with a variable number of particles and antiparticles. A (normalized) state that is a constant function is the vacuum vector, representing a state with no particles or antiparticles.

74 Two useful operators on this new Hilbert space are

NΨ = (n + m)Ψ

QΨ = (n − m)Ψ for Ψ ∈ F (n,m). N is called the number operator, giving the total number of particles in a given state. Q is the charge operator, giving the total imbalance between the number of particles and antiparticles. F can be broken down into eigenspaces of Q,

Fq, where QΨ = qΨ for Ψ ∈ Fq. These are called q-charge sectors of F . The inﬁnite wedge space we constructed for Dirac’s theory in section 7 can be compared to this, and the component F (k) of the inﬁnite wedge space from section

7 corresponds to Fk, the k-charge sector here. The number operator, counting the total number of particles, corresponds indirectly to the degree function from section 7. The degree accounts for both the number of particles and their energy.

75 APPENDIX

Triangular Decomposition

Let g be a Lie algebra over a ﬁeld K.A triangular decomposition of g consists of an abelian subalgebra h 6= 0 and two subalgebras n+ and n− such that

a) g = n+ ⊕ h ⊕ n−

b) n+ 6= 0, [h, n+] ⊆ n+, and n+ admits a weight space decomposition relative to h (under the adjoint representation) with weights α 6= 0 lying in the free

∗ additive semigroup Q+ ⊂ h ;

c) there exists an anti-involution σ (i.e., antiautomorphism of period 2) on g such that

σ(n+) = n−

σ|h = idh;

d) there exists a basis {αj}j∈J of Q+ consisting of linearly independent elements

∗ of h . In particular, Q+ consists of all nonzero ﬁnite sums of the form

X mjαj, mj ∈ N. j∈J

Tensor Products and Related Algebras

Tensor Algebra

Definition 26 Let V be a vector space over a field K. Define V ⊗ V to be

V ⊗ V = (V × V )/I

76 Where I is the left ideal generated by all elements of the form:

(x + z, y) − (x, y) − (z, y)

(x, y + z) − (x, y) − (x, z)

c(x, y) − (cx, y)

(cx, y) − (x, cy) where x, y, z ∈ V and c ∈ K. Denote the image of (x, y) in V ⊗ V by x ⊗ y.

Let T 0(V ) = K, T 1(V ) = V , and T n(V ) = V ⊗ V ⊗ · · · ⊗ V with n-copies of V . Then T (V ) = L T nV is the tensor algebra of V . Multiplication in T (V ) is n∈N deﬁned by concatenation. Alternating Algebra

Deﬁnition 27 Let V be a vector space over K. Deﬁne Λ(V ), called the alternating algebra over V , to be

Λ(V ) = T (V )/J where T (V ) is the tensor algebra of V , and J is the two-sided ideal generated by all elements of the form x ⊗ y + y ⊗ x. Denote the image of x ⊗ y in Λ(V ) as x ∧ y.

Notice that x ∧ x = 0 for all x ∈ V . If {xi}i∈I is a basis of V where I is a totally

ordered set, then a basis of Λ(V ) is given by {xi1 ∧ xi2 ∧ · · · ∧ xik |i1 < ··· < ik}. Universal Enveloping Algebra The deﬁnition of a universal enveloping algebra was given in Introduction. With tensor products deﬁned, we can say a little bit more about this construction. Let T (g) be the tensor algebra over a Lie algebra g. Let J be the ideal generated by elements of the form x ⊗ y − y ⊗ x − [x, y] for x, y ∈ g. Then U(g) = T (g)/J.

77 Geometry

Although the Witt algebra can be deﬁned directly using generators and relations, its use in physics is tied to its relation to the Lie group S1, and so here we give a brief overview of the geometric construction needed for the deﬁnition of the Witt algebra using material from Bleecker ([1]).

Deﬁnition 28 Let M be a set, and suppose M is the union of a number of subsets S Ui where i ranges over some (possibly inﬁnite) index set I (i.e., M = i∈I Ui). Let

n 1 n i n R = {(x , . . . , x )|x ∈ R}, and φi : Ui → R be an injective function such that

φi(Ui) is open. A subset V ⊂ M is open if φi(Ui ∩ V ) is open for all i ∈ I. The collection of open sets is called the topology of M relative to {φi|i ∈ I}.

Deﬁnition 29 Let M be a set with a topology. Then the topology of M is called

Hausdorﬀ if for x, y ∈ M with x 6= y, there are disjoint open sets Vx and Vy with x ∈ Vx and y ∈ Vy.

Deﬁnition 30 Let M be a set with a topology as deﬁned above, and assume the

n topology of M is Hausdorﬀ, and given by functions {φi|i ∈ I}, φi : Ui → R . Assume

−1 ∞ that for all i, j ∈ I, we have φj ◦ φi : φi(Ui ∩ Uj) → φj(Ui ∩ Uj) is C (i.e., has continuous partial derivatives of all orders).We add the technical assumption that S M = k Uik where ik ∈ I, k = 1, 2, 3,... . Under these assumptions, {φi|i ∈ I} is called an atlas of M. Two atlases are equivalent if their union is an atlas. An equivalence class of atlases is called a diﬀerentiable structure on M. M together with a diﬀerential structure is called a C∞ n-manifold, where n is the dimension

n of M, and any φ : Ui → R in some atlas is called a chart or coordinate system.

78 Figure 1: Diﬀerentiable Structure

Deﬁnition 31 A curve through a point x ∈ M is a map γ :(a, b) → M (a <

0 < b) such that γ(0) = x. Curves γ1 and γ2 through x are called equivalent if

0 0 n (φ ◦ γ1) (0) = (φ ◦ γ2) (0) for some (and hence any) chart φ : U → R with x ∈ U. An equivalence class of curves through x is called a tangent vector at x; the set

0 of all tangent vectors at x is denoted by TxM. We write γ (0) or

d γ(t) dt t=0

for the vector in TxM determined by γ. Note that TxM has a natural vector space

0 ∞ 0 structure. If Yx ∈ TxM (say Yx = γ (0)) and f ∈ C (M), then (f ◦ γ) (0) ∈ R is called the derivative of f along Yx, and it is denoted Yx[f].

S Deﬁnition 32 Let TM = x∈M TxM.A vector ﬁeld on M is a function Y :

∞ M → TM such that Y (x) = Yx ∈ TxM and (for all f ∈ C (M)) the function

∞ x 7→ Yx[f] is in C (M); we denote this function by Y [f]. The set of all vector ﬁelds on M is denoted X(M). If Y,Z ∈ X(M), then [Y,Z] is that vector ﬁeld such that

79 [Y,Z]x[f] = Yx[Z[f]] − Zx[Y [f]]. The proof of existence and uniqueness of [Y,Z] is omitted. Observe that [Y,Z] = −[Z,Y ], and [[Y,Z],W ]+[[Z,W ],Y ]+[[W, Y ],Z] = 0. The latter identity is the Jacobi identity.

80 REFERENCES

[1] David Bleecker, Gauge Theory and Variational Principles, Global Analysis, Pure and Applied; no.1; Series A, Addison-Wesley, 1981.

[2] Jonas T. Hartwig, ”Generalized Derivations on Algebras and Highest Weight Representations of the Virasoro Algebra”, Master thesis, Lund University, http://sites.google.com/site/jonashartwig/papers, 2002.

[3] James E. Humphries, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, 1972.

[4] Victor G. Kac, A. K. Raina, Bombay Lectures on Highest Weight Representa- tions of Inﬁnite Dimensional Lie Algebras, Advanced Series in Mathematical Physics Vol.2, World Scientiﬁc, 1987.

[5] R. V. Moody, Arturo Pianzola, Lie Algebras with Triangular Decompositions, Canadian Mathematical Society series of monographs and advanced texts, John Wiley & Sons, 1995

[6] Bernd Thaller, The Dirac Equation, Texts and Monographs in Physics, Springer-Verlag, 1952.

81 BIOGRAPHICAL SKETCH

Born, Oct. 22 1984 Graduated from Mt. Tabor High School, 2003 Completed B.S. Mathematics at North Carolina State University, 2007