REPRESENTATIONS OF INFINITE DIMENSIONAL LIE ALGEBRAS ANDDIRAC’S POSITRON THEORY

Jeremy S. Gresham

A Thesis Submitted to theUniversity of North Carolina Wilmington in Partial Fulfillment

Of the Requirements for the Degree ofMaster of Science

Department of Mathematics and Statistics

University of North Carolina Wilmington

2011

Approved by

Advisory Committee

Mark Lammers Gabriel Lugo

Dijana Jakelic

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 of a∞ . . . . . . . . . . . . . . . . . . . . . . 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

•⟨· | ·⟩

- 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

• ~, ε, 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, τ - algebra homomorphisms

•⊕

, ⊕ - direct sum

•∑

- 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, 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

Schrodinger equation 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

x =

0 1

0 0

, y =

0 0

1 0

, h =

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

g′ = g/I equipped with bracket [x+ I, y + I] = [x, y] + I for all x, y ∈ g. Then g′ is

a Lie algebra called the quotient algebra of g by I. Moreover, there is a surjective

Lie algebra homomorphism φ : g→ g′ 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), ∀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

Poincare-Birkhoff-Witt theorem:

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

sl2(C). This is an associative algebra with a basis of elements xjhkyl 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

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 representa-

tion) 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

a real vector field, please see Appendix C. Now consider X(S1) := {f(θ) ddθ|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:

[f(θ)d

dθ, g(θ)

d

dθ] = (fg′ − f ′g)(θ)

d

dθ(2)

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

X(S1) as a vector space over R : { ddθ, cos(nθ) d

dθ, sin(nθ) d

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

can be written as {dn|n ∈ Z} where dn = −zn+1 ddz

and z = eiθ. This can be seen

through a simple calculation:

dz

dθ= iz

id

dθ= −z d

dz.

Thus dn = ieinθ ddθ

, while ddθ

= −id0, cos(nθ) ddθ

= − i2(dn+d−n), sin(nθ) d

dθ= −1

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:

[dm, dn] = [−iei(m+1)θ d

dθ,−iei(n+1)θ d

dθ]

= i3(n+ 1)ei(m+n+1)θ d

dθ− i3(m+ 1)ei(m+n+1)θ d

dθ

= (m− n)

(− iei(m+n+1)θ d

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)

(c) ω([x, y]) = [ω(y), ω(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)

[dm, dn] = (m− n)dm+n + δm,−nm3 −m

12c for m,n ∈ Z. (4)

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:

d′n =

d0 if n = 0

dn − 1na(0, n)c if n 6= 0

c′ = c.

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

[c′, d′n] = 0

[d′m, d′n] = [dm, dn]

= (m− n)dm+n + a(m,n)c

= (m− n)d′m+n + a′(m,n)c′ (5)

for m,n ∈ Z where a′ : Z× Z→ C is defined by

a′(m,n) =

a(m,n) if m+ n = 0

a(m,n) + m−nm+n

a(0,m+ n) if m+ n 6= 0.

(6)

Since a is antisymmetric, a′ is as well, and therefore a′(0, 0) = 0. From (6) it follows

that a′(0, n) = 0 for any nonzero n. These facts together with (5) show that

[d′0, d′n] = −nd′n.

11

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

[[d′0, d′n], d′m] + [[d′n, d

′m], d′0] + [[d′m, d

′0], d

′n] = 0

[−nd′n, d′m] + [(n−m)d′n+m + a′(n,m)c′, d′0]− [d′n,md′m] = 0

−(n+m)(n−m)d′n+m − (n+m)a′(n,m)c′ + (n−m)(n+m)d′n+m = 0

(n+m)a′(n,m)c′ = 0

which shows that a′(n,m) = 0 unless n + m = 0 and n 6= 0, m 6= 0. Thus, setting

b(m) = a′(m,−m), (5) can be rewritten as

[c′, d′n] = 0

[d′m, d′n] = (m− n)d′m+n + δm+n,0b(m)c′

with b(0) = 0. Again using the Jacobi identity

[[d′n, d′1], d

′−n−1] + [[d′1, d

′−n−1], d

′n] + [[d′−n−1, d

′n], d′1] = 0

[(n− 1)d′n+1, d′−n−1] + [(n+ 2)d′−n, d

′n] + [(−2n− 1)d′−1, d

′1] = 0

(n− 1)(2(n+ 1)d′0 + b(n+ 1)c′) + (n+ 2)(−2nd′0 + b(−n)c′)

+ (−2n− 1)(−2d′0 + b(−1)c′) = 0

(2n2 − 2− 2n2 − 4n+ 4n+ 2)d′0 + {(n− 1)b(n+ 1)− (n+ 2)b(n)

+ (2n+ 1)b(1)}c′ = 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

dn = d′n + δn,0α + β

2c′,

and

c = 12αc′.

13

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

[dm, dn] = [d′m, d′n]

= (m− n)d′m+n + δm+n,0(αm3 + βm)c′

= (m− n)dm+n − (m− n)δm+n,0α + β

2c′ + δm+n,0(αm

3 + βm)c′

= (m− n)dm+n − δm+n,02mα + β

2c′ + δm+n,0(αm

3 + βm)c′

= (m− n)dm+n + δm+n,0(αm3 − αm)c′

= (m− n)dm+n + δm+n,0m3 −m

12c.

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:

[ω(dn), ω(dm)] = [d−n, d−m] = (−n+m)d−n−m − δ−n,mn3 − n

12c

= (m− n)ω(dm+n) + δm,−nm3 −m

12ω(c)

= ω([dm, dn]).

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

14

decomposition of Vir:

V ir = n−⊕h⊕ n+ where

n− =∞⊕i=1

Cd−i, h = Cc⊕ Cd0, n+ =∞⊕i=1

Cdi.

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

n− =∞⊕i=1

Ca−i, h = C~⊕ Ca0, n+ =∞⊕i=1

Cai.

15

3.4 Algebras of Infinite Matrices

Let

V =⊕j∈Z

Cvj

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

vj with the column vector whose ith 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}.

Clearly a∞ is a set of matrices with a finite number of nonzero diagonals and

gl∞ ⊆ a∞. Note that the usual matrix multiplication is well-defined in a∞: 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,

(xy)ij =∑k∈Z

xikykj =∑k∈Z

|i−k|≤M|k−j|≤N

xikykj

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 of a∞, 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 subalge-

bra of a∞.

Proof. This is shown by checking that a∞ 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

Λk =∑i∈Z

Ei,i+k.

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

Λk ∈ a∞. Now,

ΛkΛj =∑i∈Z

Ei,i+k+j = ΛjΛk

18

and so [Λj,Λk] = 0 for j, k ∈ Z, i.e., the Λk form an abelian subalgebra of a∞.

For every pair of nonzero constants α and β there is an inclusion of the Witt

algebra d in a∞ 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 in a∞. We show that these elements generate a subal-

gebra in a∞ 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

dn =∑k∈Z

(k − α− β(n+ 1))Ek−n,k.

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

coefficients. We define the loop algebra gln to be gln(C[t, t−1]), i.e., the complex

Lie algebra of n× n matrices with Laurent polynomials as entries.

An element of gln has the form

a(t) =∑k

tkak with ak ∈ gln

where k runs over a finite subset of Z. The standard basis {eij|0 ≤ i, j ≤ n} of gln

yields a basis for gln via

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

The elements of gln form an associative algebra with multiplication defined on the

basis by

eij;kemn;l = tk+leijemn = δjmein;k+l.

Then the Lie bracket on gln can be defined as the commutator:

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

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

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

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

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

20

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

nvj;k = t−kej with k ∈ Z, j ∈ {1, . . . , n}

form a basis of C[t, t−1]n (over C). Thus C[t, t−1]n is identified with C∞ via nvj;k 7→

vnk+j. Now gln acts on this space by

eij;k(nvl;k) =

nvi;s−k j = l

0 otherwise

.

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

eij;kvkn+l = δjlvn(s−k)+i. (8)

For a(t) ∈ gln denote the corresponding matrix in a∞ by τ(a(t)). Then we have the

following matrix representation of gln in a∞:

τ(eij;k

)=∑s∈Z

En(s−k)+i,ns+j.

For arbitrary a(t) =∑tkak ∈ gln this can be rewritten in block form as

τ(a(t)) =

. . . . . . . . . . . . . . . . . .

. . . a−1 a0 a1 . . . . . .

. . . . . . a−1 a0 a1 . . .

. . . . . . . . . a−1 a0. . .

. . . . . . . . . . . .. . . . . .

.

Clearly this is a matrix in a∞ whose entries on each diagonal parallel to the main

diagonal form an n-periodic sequence.

21

Proposition 2 a) τ : gln → a∞ is an injective homomorphism of associative

algebras, and hence Lie algebras.

b) The image of a(t) =∑

j ajtj under τ is a strictly upper triangular matrix if

and only if

a(t) = a0 + a1t+ a2t2 + · · · with a0 strictly upper triangular.

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

a =n−1∑i=1

ei,i+1, b = en1.

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 gln

to a linearly independent set in a∞. b) This is clear from the block matrix given for

arbitrary a(t) ∈ gln.

For c), recall

Λj =∑i∈Z

Ei,i+j.

Using (8)

τ((a+ tb)j

)=

( n−1∑i=1

τ(ei,i+1

)+ τ(ten1

))j=

( n−1∑i=1

∑s

Ens+i,ns+i+1 +∑s

Ens+n,n(s+1)+1

)j=

(∑s

n∑i=1

Ens+i,ns+i+1

)j=

(∑l

El,l+1

)j= Λj

1 = Λj

22

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

Λjvk = vk−j

and so

Λj1vk = Λj−1

1 vk−1 = · · · = vk−j = Λjvk. �

Define an antilinear involution ω on gln by

ω(Xk

)= t−kX∗,

where Xk = tkX ∈ gln, 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∞.

Proof. It must be shown that τ(t−kX∗) =

(τ(Xk

))∗. Let X = (xij) xij ∈ C,

then

τ(Xk

)∗= τ(tkX)∗ =

(∑ij

xijτ(ekij))∗

=

(∑ij

xij∑s

En(s−k)+i,ns+j

)∗

23

while

τ(t−kX∗) = τ

(∑ij

xjie−kij

)=∑ij

xjiτ(e−kij)

=∑ij

xji∑s

En(s+k)+i,ns+j

=

(∑ij

xji∑s

En(s−k)+j,ns+i

)∗.

Thus the inclusion of gln into a∞ respects ω. �

We also have a triangular decomposition of gl∞ given by

n− =⊕i>j

CEij h =⊕i=j

CEij n+ =⊕i<j

CEij

so that

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|π(ω(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))∗ = π(ω(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∞ and a∞ 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 Poincare-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∑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−ikd0d−ik−1

· · · d−i1(v)

=k∑l=1

ild−ik · · · d−i1(v) + d−ik · · · d−i1d0(v)

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

= (h+ j)d−ik · · · d−i1(v)

and so,

V =⊕j∈Z+

Vh+j. (9)

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

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−i1φ(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 homo-

morphism 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 Poincare-

Birkhoff-Witt theorem. �

4.3 Irreducible Positive Energy Representations

Here we continue discussing the highest weight representations of the Virasoro alge-

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

d0djt · · · dj1w = (λ− (jt + · · ·+ j1))djt · · · dj1w

by using the same technique as in the proof of (9). V is positive energy, and so the

set

M = {j ∈ Z|d0djt · · · dj1w 6= 0 for some t ≥ 0, (jt, . . . , j1) ∈ Zt 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 · · · dj1w. Then

djv = djdjt · · · djtw = 0 for j > 0

31

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

d0v = d0djt · · · djtw = (λ− (jt + · · ·+ j1))djt · · · dj1w = hv

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

V ′ = U(V ir)v.

Now, V ′ =⊕

k∈Z≥0V ′h+k, where V ′h+k is the h+ k eigenspace of d0 acting on V ′. Let

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 ′. Clearly V ′ as defined is a

highest weight representation.

V ′ is nontrivial since 0 6= v ∈ V ′. Therefore, since V is irreducible, V = V ′, 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⊕

U⊥ using unitarity. We have either v ∈ U or v ∈ U⊥ as follows. Let

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

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

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(C, h) =⊕k∈Z≥0

M(C, h)h+k

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⊕

U ′ for some submodules U and U ′.

Now, we use the same argument as in the proof of the previous proposition to show

that either U = 0 or U ′ = 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 ′(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 ′(C, h). Let J ′(C, h) = kerφ. By the

33

First Isomorphism Theorem, we have

V ′(C, h) ∼= M(C, h)/J ′(C, h).

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

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-

lution ω : 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 Poincare-Birkhoff-Witt theorem, 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 = x′v and yv = y′v for some x, x′, y, y′ ∈ U(V ir) then x−x′, y− y′ ∈ I

so that

⟨xv|yv

⟩−⟨x′v|y′v

⟩=⟨(x− x′)v|yv

⟩+⟨x′|(y − y′)v

⟩=⟨ω(y)(x− x′)v|v

⟩+⟨v|ω(x′)(y − y′)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⟩

= (k − l)⟨x|y⟩,

and therefore⟨x|y⟩

= 0.

(c) Let x ∈ rad(⟨· | ·

⟩) and y, z ∈ U(V ir). Then

⟨y|zx

⟩=⟨ω(z)y|x

⟩= 0,

and so rad(⟨· | ·

⟩) is a subrepresentation. Since

⟨v|v⟩

= 1 it is proper, hence

rad(⟨· | ·⟩) ⊆ J(C, h).

Conversely, suppose x ∈ U(V ir) such that

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 con-

travariant 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 irre-

ducible 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 repre-

sentation 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

cn =⟨v|dnd−nv

⟩=⟨v|(d−ndn + 2nd0 +

n3 − n12

c)v⟩

= 2nh+n3 − n

12C.

Then c1 = 2h, and so h ≥ 0. For sufficiently large values of n we have n3 − 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 = εn ∂/∂xn (10)

a−n = ε−n~nxn

a0 = µI

~ = ~I

where the εn are arbitrary pairs of real numbers with ε−n · εn = 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 = εn∂

∂xnε−n~nxn(f)− ε−n~nxnεn

∂

∂xn(f)

= ~n∂

∂xn(xnf)− ~nxn

∂

∂xn(f)

= ~nf + ~nxn∂

∂xn(f)− ~nxn

∂

∂xn(f)

= ~nf

and so the relations are satisfied on A. We’ll call this representation a Fock rep-

resentation.

The following lemma will be used for numerous calculations in this section.

39

Lemma 1 The A-action on B satisfies:

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

Proof. Induction on k. Clearly, [an, a−n] = ~n = ~na0−n. Now, suppose (11) is true

for some k ∈ N. Then:

[an, ak+1−n ] = ana

k+1−n − ak+1

−n an

= (a−nan + ~n)ak−n − ak+1−n an

= a−n(anak−n) + ~nak−n − ak+1

−n an

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

= k~nak−n + ~nak−n + ak+1−n an − ak+1

−n an = (k + 1)~nak−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 xj1i1 · · · xjnin

. Applying the operator

aj1i1 · · · ajnin

to this monomial (using the last lemma repeatedly) yields ~kj1! · · · jn!

with k =∑n

l=1 jk, while it yields 0 when applied to any other monomial in P . Thus

aj1i1 · · · ajninP is a nonzero multiple of 1. Since B is generated by 1, we are done. �

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

satisfying (12) with ~ 6= 0. Then monomials of the form ak1−1 · · · akn−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.

Proof Define a map φ from B to V by φ(P (. . . , xn, . . . )) = P (. . . ,(εn~n

)a−n, . . . )v.

Now:

an(φ(P )) = an(P (. . . ,( εn~n)a−n, . . . )v)

=∑

an(· · · (( εn~n)a−n)kn · · · (v))

=∑

(· · · [( εn~n)kn

anakn−n] · · · )(v)

=∑

(· · · [( εn~n)kn

(kn~nakn−1−n + ak−nan)] · · · )(v)

=∑

[· · · (knεn( εn~n)kn−1

akn−1−n ) · · · ](v) + [· · · (akn−nan) · · · ](v)

where the third step uses the previous lemma. The last term can be rearranged as

[· · · (akn−nan) · · · ](v) = [· · · (akn−n) · · · an](v) which is clearly zero by (12), leaving only∑[· · · (knεn

(εn~n

)kn−1akn−1−n ) · · · ](v). Next apply an to P :

φ(anP (. . . , xn, . . . )) = φ(∑

an(· · · xknn · · · ))

= φ(∑

(· · · εn∂

∂xnxknn · · · ))

= φ(∑

(· · · εnknxkn−1n · · · ))

=∑

(· · · knεn( εn~n)kn−1

akn−1−n · · · )(v)

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

the vacuum vector v. The distinct monomials ak1−1 · · · akn−n(v) (ki ∈ Z+) form an

orthogonal basis with respect to⟨· | ·⟩. These monomials have norms given by

⟨ak1−1 · · · akn−n(v)|ak1−1 · · · akn−n(v)

⟩=

n∏j=1

kj!(~j)kj . (13)

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⟩.

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

42

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

⟨ak′1−1 · · · a

k′n−nv|a

k′1−1 · · · a

k′n−nv⟩

=⟨ak1−1 · · · a−ja

kj−j · · · akn−nv|a

k1−1 · · · a

kj+1−j · · · akn−nv

⟩=⟨a−ja

k1−1 · · · akn−nv|a

k1−1 · · · a

kj+1−j · · · akn−nv

⟩=⟨ak1−1 · · · akn−nv|ω(a−j)a

k1−1 · · · a

kj+1−j · · · akn−nv

⟩=⟨ak1−1 · · · akn−nv|aja

k1−1 · · · a

kj+1−j · · · akn−nv

⟩=⟨ak1−1 · · · akn−nv|a

k1−1 · · · aja

kj+1−j · · · akn−nv

⟩=⟨ak1−1 · · · akn−nv|a

k1−1 · · · ([aj, a

kj+1−j ] + a

kj+1−j aj) · · · akn−nv

⟩=⟨ak1−1 · · · akn−nv|a

k1−1 · · · [aja

kj+1−j ] · · · akn−nv

⟩+⟨ak1−1 · · · akn−nv|a

k1−1 · · · a

kj+1−j · · · akn−najv

⟩=⟨ak1−1 · · · akn−nv|a

k1−1 · · · (kj~j)a

kj−j · · · akn−nv

⟩+ 0 (by Lemma 1)

= kj~jΠnm=1km!(~m)km

= Πnm=1k

′m!(~m)k

′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,

⟨aL1−1 · · · aLn

−nv|aK1−1 · · · aKm

−1 v⟩

=⟨aLj

−jaL1−1 · · · a

Lj−1

−j+1aLj+1

−j−1 · · · aLn−nv|a

K1−1 · · · aKm

−mv⟩

=⟨aL1−1 · · · a

Lj−1

−j+1aLj+1

−j−1 · · · aLn−nv|ω(a

Lj

−j)aK1−1 · · · aKm

−mv⟩

=⟨aL1−1 · · · a

Lj−1

−j+1aLj+1

−j−1 · · · aLn−nv|a

Lj

j aK1−1 · · · aKm

−mv⟩

=⟨aL1−1 · · · a

Lj−1

−j+1aLj+1

−j−1 · · · aLn−nv|a

K1−1 · · · a

Lj

j aKj

−j · · · aKm−mv

⟩.

43

Now we note, using Lemma 2, that

aLj

j aKj

−j = aLj−1j [aja

Kj

−j ] + aKj

−jaj

= (Kj~j)aLj−1j a

Kj−1−j + a

Kj

−jaj.

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. �

Definition 24 The degree of the monomial xj11 · · ·xjkk 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

B =⊕j≥0

Bj,

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:

Lk =1

2

∑j∈Z

: a−jaj+k : (k ∈ Z), (14)

where the colons indicate ‘normal ordering’, defined by

: aiaj :=

aiaj if i ≤ j

ajai if i > j.

(15)

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 :=

a−jaj+k − j ≤ j + 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 εn∂∂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

ψ(x) =

1 |x| ≤ 1

0 |x| > 1.

Then let Ln(ε) = 12

∑j∈Z : a−jaj+n : ψ(jε), for ε ∈ R. This way Ln(ε) is a finite

sum for ε 6= 0, and Ln(ε) → Ln as ε → 0. Also, given v ∈ B, Ln(ε)(v) = Ln(v)

for ε sufficiently small. Now, since Ln(ε) can be written without normal ordering by

adding a finite sum of constant terms,

[ak, Ln(ε)] =1

2

∑j∈Z

[ak, a−jaj+n]ψ(jε)

=1

2

∑j∈Z

[ak, a−j]aj+nψ(jε) +1

2

∑j∈Z

a−j[ak, aj+n]ψ(jε)

=1

2kak+nψ(kε) +

1

2kak+nψ(kε).

Take ε→ 0, giving the required formula. �

Proposition 13 The Lk satisfy the following relations:

[Lm, Ln] = (m− n)Lm+n + δm,−nm3 −m

12. (16)

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

[Lm(ε), Ln] =1

2

∑j

[a−jaj+m, Ln]ψ(jε)

=1

2

∑j

(−j)an−jaj+mψ(jε) +1

2

∑j

(j +m)a−jaj+m+nψ(jε).

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

2

∑j

(−j) : an−jaj+m : ψ(jε) +1

2

∑j

(j +m) : a−jaj+m+n : ψ(jε)

−1

2δm,−n

−m∑j=−1

j(m+ j)ψ(jε)

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

[Lm, Ln] = (m− n)Lm+n + δm,−n(m3 −m)

12,

where −12

∑−mj=−1 j(m+ j) = (m3−m)

12by the following induction argument:

−1

2

−1∑j=−1

j(m+ j) = −1

2(−1)(1− 1) = 0.

48

Now suppose −12

∑−nj=−1 j(n+ j) = n3−n

12for some n ∈ Z.

−1

2

−(n+1)∑j=−1

j((n+ 1) + j) = −1

2

[− 1(n+ 1− 1)

− 2(n+ 1− 2) + · · · − n(n+ 1− n) + (−n+ 1)(n+ 1− n− 1)]

= −1

2

[− 1(n)− 2(n− 1)− 3(n− 2) + · · ·+ (−n− 1)(2)− n(1) + 0

]= −1

2

[− 1(n)− 1(n− 1)− 1(n− 2) + · · · − 1(2)− 1(1) + 0

]− 1

2

[− 1(n− 1)− 2(n− 2) + · · ·+ (−n+ 2)(2) + (−n+ 1)(1) + 0

]=

1

2

[12n(n+ 1)

]− 1

2

[ −n∑j=−1

j(n+ j)]

=n2 + n

4+n3 − n

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 π′ : A → gl(B), is unitary. Then, for any element x ∈ A,

π′(ω(x)) = π′(x)∗.

49

Then

L∗k =∑j∈Z

: π′(a−j)π′(ak+j) :∗

=∑j∈Z

: π′(aj+k)†π′(a−j)

∗ :

=∑j∈Z

: π′(ω(ak+j))π′(ω(a−j)) :

=∑j∈Z

: π′(a−k−j)π′(aj) :

=∑l∈Z

: π′(al)π′(a−k+l) :

= L−k

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

Proposition 15 The unitary representation of the Virasoro algebra from the previ-

ous 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 L0 = µ2/2 +∑

j∈Z+a−jaj. Let L0 act on some monomial of degree j:

L0(xj11 · · ·x

jkk ) = (µ2/2)xj11 · · ·x

jkk +

k∑i=1

jixj11 · · ·x

jkk = (µ2/2 + j)xj11 · · ·x

jkk

and thus the Bj are eigenspaces for L0 with eigenvalue µ2/2 + j. As a result, any

subrepresentation U of B will have a decomposition into eigenspaces:

U =⊕j∈Z+

(U ∩Bj

)

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

U⊥ by

U⊥ =⊕j∈Z+

U⊥j

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)

L0(1) = h · 1 (h = µ2/2)

Let B′ be the span of the vectors

L−ik · · ·L−i2L−i1(1)

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 B′ is an irreducible representation of Vir.

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

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 B′, and the other must be 0, and hence B′ is

irreducible. �

An alternate way to show B′ 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 =⊕

i∈ZCvi. We start with a

space

F (0) = Λ∞0 (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

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.

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

a) F (0) =⊕

F(0)k , F

(0)0 = Cψ0

b) dimF(0)k = p(k)

c) dimqF(0) =

∑k(dimF

(0)k )qk = 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

deg(ψ) =n−1∑i=0

[(ki − i) + i] +∞∑i=n

(−i+ i)

=n−1∑i=0

ki = k

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:

deg(ψ) =∞∑s=0

(im−s + s−m).

This is similar to F (0), F (m) 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 =⊕

m∈Z F(m).

55

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

dimF(m)k = p(k)

dimqF(m)k = 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

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

r is a direct sum of subrepresentations rm on F (m).

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

⟨r(Eij)ψ|ψ′

⟩=⟨ψ|r(E∗ij)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.

Now examine the action of the r(Eij) on the decomposition F (m) =⊕

k≥0 F(m)k ,

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

r(Eij)F(m)k ⊂ F

(m)k+i−j.

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

degree j:

gl∞ =⊕j∈Z

gj

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

r(gj)F(m)k ⊂ F

(m)k+j

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

r(gj)ψm = 0 for j < 0.

Now, since every ψ ∈ F (m)k 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

F(m)k =

∑j1+···+jn=kj1,...jn∈Z+

rm(gj1) · · · rm(gjn)ψm.

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,

n+ =⊕j<0

gj.

Then

rm(n+)ψm = 0,

rm(Eii)ψm = λiψm,

where

λi =

1 if i ≤ m

0 if i > m.

Then for each m ∈ Z we have constructed an irreducible highest weight repre-

sentation 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 fun-

damental 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

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 represen-

tations 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

on∑

i ki using tensor products completes the proof. �

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

60

6.3 Representations of a∞

The matrices in a∞ have a finite number of nonzero diagonals and so are finite linear

combinations of matrices of the form

ak =∑i∈Z

λiEi,i+k

where the λi are arbitrary complex numbers. If the same representation is applied

to a∞ as gl∞, there is a problem:

with ak =∑i∈Z

λiEi,i+k, λi ∈ C

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-

sentation rm by

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

rm(Eii) = rm(Eii)− I if i ≤ 0.

61

Then

rm(a0)ψm =

(∑m

i=0 λi)ψm for m ≤ 1

−(∑m+1

i=0 λi)ψm for m ≤ −1 (and is 0 for m = 0)

For A ∈ a∞ the rm clearly map F (m) into themselves, but they do not satisfy the

original commutation relations for a∞. 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 for rm, since the presence of I in the brackets makes no difference.

For the last

[rm(Eij), rm(Eji)] = rm(Eii)− rm(Ejj) + α(Eij, Eji)I

where

α(Eij, Eji) = −α(Eji, Eij) =

1 for i ≤ 0, j ≥ 1,

0 otherwise

then

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

This can be made into a representation of the central extension of a∞, 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

rm(c) = 1, a representation of a∞ in F (m) 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) makes r

unitary as well. Now we consider the subalgebra of a∞ of shift operators under rm.

[rm(Λn), rm(Λk)] = α(Λn,Λk)I.

In this case

α(Λn,Λk) = nδn,−k

so that

[rm(Λn), rm(Λk)] = nδn,−k.

Also,

rm(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 Schrodinger equation, a wave equation:

i~∂

∂tΨ(t, x) = HΨ(t, x)

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~∂

∂t, p→ i~∇

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 Schrodinger equation is the equation for a one

dimensional simple harmonic oscillator:

i~∂

∂tΨ = − ~2

2m

d2

dx2Ψ +

1

2mω2x2Ψ.

Note that our Hamiltonian can be written as

H =p2

2m+mω2

2x2.

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

a =

√mω

2~(x+

i

mωp)

a† =

√mω

2~(x− i

mωp)

where p = i~ ddx

.

65

Then we have

~ωaa† =

√mω

2~(x+

i

mωp)

√mω

2~(x− i

mωp)

=mω2

2(x2 +

i

mω(px− xp) +

1

m2ω2p2)

=mω2

2(x2 +

i

mω(−i~ d

dxx+ i~x

d

dx) +

1

m2ω2p2)

=mω2

2(x2 +

~mω

+1

m2ω2p2)

=mω2

2x2 +

p2

2m+

~ω2

and similarly

~ωa†a =

√mω

2~(x− i

mωp)

√mω

2~(x+

i

mωp)

=mω2

2(x2 − i

mω(px− xp) +

1

m2ω2p2)

=mω2

2(x2 − i

mω(−i~ d

dxx+ i~x

d

dx) +

1

m2ω2p2)

=mω2

2(x2 − ~

mω+

1

m2ω2p2)

=mω2

2x2 +

p2

2m− ~ω

2.

Then we can write

H = (a†a+1

2)~ω.

Now consider the commutator of a and a†:

[a, a†] = aa† − a†a

=1

~ω(mω2

2x2 +

p2

2m+

~ω2

)− 1

~ω(mω2

2x2 +

p2

2m− ~ω

2) = 1.

66

Using this we also have

[H, a] = [~ωa†a+1

2, a]

= ~ω[a†a, a] +1

2[1, a]

= ~ω(a†a2 − aa†a)

= ~ω(a†a2 − (a†a− [a†, a])a

= ~ω(−a+ a†a2 − a†a2)

= −~ωa

and similarly

[H, a†] = ~ω[a†a+1

2, a†]

= ~ω(a†aa† − (a†)2a)

= ~ω(a†([a, a†] + a†a)− (a†)2a)

= ~ω(a† + (a†)2a− (a†)2a)

= ~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

i~∂

∂tψ(x)φ(t) = − ~2

2m

d2

dx2ψ(x)φ(t) +

1

2mω2x2ψ(x)φ(t)

i~ ∂∂tφ(t)

φ(t)=− ~2

2md2

dx2ψ(x) + 1

2mω2x2ψ(x)

ψ(x)

and so

i~∂

∂tφ(t) = kφ(t)

− ~2

2m

d2

dx2ψ(x) +

1

2mω2x2ψ(x) = kψ(x)

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

toHψ = 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 Schrodinger equa-

tion, 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

∂t2Ψ(t, x) = (−c2~2∇2 +m2c4)Ψ(t, x),

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 equa-

tion.

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

69

square,

(−c~(α1∂

∂x1+ α2

∂

∂x2+ α3

∂

∂x3) + βmc2)2,

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:

(−c~(α1∂

∂x1+ α2

∂

∂x2+ α3

∂

∂x3) + βmc2)2 = −c2~2(α2

1

∂2

∂x21+ α2

2

∂2

∂x22+ α2

3

∂2

∂x23)

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

+ i~mc3[(α1β + βα1) + (α2β + βα2) + (α3β + βα3)]

Comparison yields the equations:

αiαj + αjαi = 2δijI

αiβ + βαi = 0

α2i = 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),

since β = β−1, and, since αiβ = −βαi,

tr(βαiβ) = −tr(β2α) = −tr(αi).

70

Thus,

tr(αi) = −tr(αi) = 0.

Now, since α2i = 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,

β =

I2 0

0 −I2

αi =

0 σi

σi 0

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

σ1 =

0 1

1 0

, σ2 =

0 −i

i 0

, σ3 =

1 0

0 −1

.Let

α =

α1

α2

α3

, σ =

σ1

σ2

σ3

.

71

Then we can write

H0 = −i~cα · ∇+ βmc2 =

mc2I −i~cσ · ∇

−i~cσ · ∇ −mc2I

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

i~∂

∂tΨ(t, x) = H0Ψ(t, x) = (−i~cα · ∇+ βmc2)Ψ(t, x).

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

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

u ≡ u(p) =(mc2 + λ(p))I + βcα · p√

2λ(p)(mc2 + λ(p)),

where λ(p) =√c2p2 +m2c4. The new form of H0 under this transformation is

UFWH0U−1FW =

√−c2∇2 +m2c4 0

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

H(e) = cα · (p− e

cA(t, x)) + βmc2 + eφ(t, x).

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 con-

jugation 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 transfor-

mation 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 quan-

tization. We have some Hilbert space of states, H = H+ ⊕ H−, with orthogonal

positive-energy and negative-energy subspaces. The positive-energy states are par-

ticles, the negative-energy states are antiparticles. We can build a Fock space from

these beginning with

F(1)+ = H+, F

(1)− = CH−

where C represents the antiunitary charge conjugation operator from before. We are

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

on CH−.

Here F(1)− and F

(1)+ are considered as 1-particle states.

Let

F(n)± = Λn

i=1F(1)± ,

and then

F =∞⊕

n,m=0

F (n,m), F (n,m) = F(n)+ ⊗ F (m)

− ,

with F 0± = C. A (normalized) state in F represents a physical system with a vari-

able 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

∑j∈J

mjαj, mj ∈ N.

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 ) =⊕

n∈N TnV is the tensor algebra of V . Multiplication in T (V ) is

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

Ui where i ranges over some (possibly infinite) index set I (i.e., M =⋃i∈I Ui). Let

Rn = {(x1, . . . , xn)|xi ∈ R}, and φi : Ui → Rn 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

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

that for all i, j ∈ I, we have φj ◦ φ−1i : φi(Ui ∩ Uj) → φj(Ui ∩ Uj) is C∞ (i.e., has

continuous partial derivatives of all orders).We add the technical assumption that

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

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

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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

(φ ◦ γ1)′(0) = (φ ◦ γ2)′(0) for some (and hence any) chart φ : U → Rn with x ∈ U .

An equivalence class of curves through x is called a tangent vector at x; the set

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

d

dtγ(t)

∣∣∣∣t=0

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

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 ].

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

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[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.

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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.

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BIOGRAPHICAL SKETCH

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

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