Univerzita Komenského v Bratislave

Fakulta matematiky, fyziky a informatiky

Relativistic methods in non-commutativequantum mechanics

Diploma thesis

2018Adam Hložný

Comenius University in Bratislava, Slovakia

Faculty of mathematics, physics and computer science

Relativistic methods in non-commutativequantum mechanics

Diploma thesis

Workplace: Department of theoretical physicsThesis advisor Prof. RNDr. Peter Prešnajder, DrSc.

Bratislava, 2018Adam Hložný

iii

Thanks to: My thesis advisor, Peter Prešnajder, for his always friendly attitude,patience and exceptional understanding of mathematics and physics.

iv

Abstrakt

Venujeme sa konštrukcii Diracovho operátora častice so spinom 12. Diracov operá-

tor je dôležitým objektom nekomutatívnej geometrie a tiež relativistickej kvantovejteórie poľa. Staviame na fakte, že Poincarého algebra je pod-algebrou su (2, 2). Potommôžeme použiť su (2, 2) na explicitnú konštrukciu reprezentácií Poincarého algebry.Ukazuje sa, že fundamentálna reprezentácia sama o sebe nestačí, pretože z nej nemožnovybudovať hmotnú teóriu. Toto má pôvod v tom, že Casimirov element m̂2 = p̂µp̂µ

reprezentácie Poincarého algebry je rovný nule. Obdobná situácia nastáva pre duálnureprezentáciu: m̃2 = p̃µp̃µ = 0. Vezmeme priamy súčet týchto dvoch reprezentácií a op-erátor hybnosti definujeme ako Pµ = p̂µ+ p̃µ. Casimirov elementM je teraz nenulový ainterpretovaný ako hmotnosť. Toto nám umožnňuje zadefinovať relativistický Diracovoperátor na fuzzy priestore. Hľadáme jeho vlstné funkcie a ukazuje sa, že netriviálnačasť v porovnaní sa komutatívnym Diracovym operátorom spočíva v hľadaní vlstnýchfunkcií operátora hybnosti Pµ.

Kľúčové slová: Fuzzy priestor, su(2,2), Diracov operátor, oscilátorová reprezentácia,relativistický

v

Abstract

We present approach of constructing relativistic Dirac operator of spin 12particle in

fuzzy space, which is very important object for both non-commutative geometry andrelativistic quantum field theory. We build on the fact that Poincaré algebra is sub-algebra of su (2, 2). We can thus use oscillator representation of su (2, 2) to explicitlyconstruct representation of Poincaré algebra. During the process, we find out that justfundamental representation does not suffice, since it is impossible to define massivetheory. This is due to the fact that Casimir element m̂2 = p̂µp̂µ of representation ofPoincaré algebra is equal to zero. The same situation arises for dual representation:m̃2 = p̃µp̃

µ = 0. We take direct sum of them, the momentum operator is then defined asPµ = p̂µ + p̃µ. Casimir element M is now non-vanishing and interpreted as mass. Thisallows us to define relativistic Dirac operator in fuzzy space. Eigen-problem for theoperator is then investigated. The non-trivial part in comparison with commutativeDirac operator is to find eigenfunctions of momentum operator Pµ.

Keywords: Fuzzy space, relativistic, Dirac operator, su (2, 2), oscillator representa-tion

Contents

Introduction 1

1 Fuzzy spacess 21.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Fuzzy spaces in our case . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 spin (4, 2), su (2, 2), oscillator representation 52.1 Basic definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . 52.2 su (2, 2) algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Review of Poincaré algebra . . . . . . . . . . . . . . . . . . . . . 62.2.2 spin(4,2) and su(2,2) . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Gauss decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 Gauss decomposition for sl (2,C) . . . . . . . . . . . . . . . . . 102.3.2 Calculating boost . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Momentum operator 143.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Eigenvalue problem for the momentum operator . . . . . . . . . . . . . 16

4 Noncommutative Dirac operator 194.1 Dirac operator and Dirac equation . . . . . . . . . . . . . . . . . . . . 194.2 Non-commutative version . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Conclusions 21

6 Appendix: calculations and formulas 22

vi

Introduction

It is well known fact that current well established theory of fundamental interac-tions, quantum field theory, is not UV finite and it fails to describe gravity since it isnon-renormalizable. Lack of advances in qft approach to quantum gravity led to devel-opment of different approaches such as string theory and non-commutative geommetry- both of them often combined together.

Aspiring non-commutative theories often start by defining Dirac operator since onecan learn a lot about underlying geometry by studying solutions of Dirac equation.This is due to the fact, that Alain Connes proved spin manifold theorem (see [5]) thatgiven spectral triple and Charge conjugation operator, we can reconstruct whole man-ifold from spectrum of Dirac operator. We will pursue the path of constructing Diracoperator too, our aim is to define relativistic-covariant Dirac operator and investigatesolutions to the free Dirac equation.

Our approach stands on the fact, that su (2, 2) algebra contains Poincaré algebraas sub-algebra. To obtain massive theory, we have to take direct sum of two su (2, 2)representations - fundamental and dual representation. This approach relies on oscilla-tor representations of su (2, 2) (see [12] or [9]). We give detailed description of masslessoscillator representation of su (2, 2), massive(constructed from two massless represen-tations) as well as Poincaré algebra which it contains as sub-algebra. In our case, wedefined 4-momentum operator Pµ which allows us to define free Dirac operator in fuzzyspace.

Chapter 1 contains brief introduction into the problematics. Chapter 2 containsintroduction to the su (2, 2) as well as explicit construction of infinite-dimensional rep-resentation of su (2, 2). In "physical" part of this work - chapters 4 and 3 we omittedmost of the lengthy calculations and moved them to chapter 6 to highlight conceptualpart and also to make reading little easier.

1

Chapter 1

Fuzzy spacess

1.1 Motivation

Reason to study non-commutative spaces is to make description of space fuzzy - thatis to introduce some fundamental scale which determines the shortest observable length.This is usually done by making coordinates non-commutative, possible "positions" arethen determined by their spectra. Non-commutativity necessarily leads to uncertaintyrelations between coordinates. This fact can be utilized e.g. to get regular descriptionof quantum field theory. Another reason to introduce non-commutativity to the spaceis to naturally incorporate fundamental scale relevant for physics at Planck length.Very important theorem in non-commutative geometry is theorem of Alain Connes for(commutative) manifolds.

Theorem 1. (Alain Connes) Given spectral triple

1. C∞ (M) smooth functions on manifold

2. L2 (M,S) spinor fileds on manifold

3. D Dirac operator

together with charge conjugation and chirality operator, then metric g can be recon-structed from spectrum of the Dirac operator.

It is usually good practice to define non-commutative analogue of Dirac operator inthe spirit of this theorem. We will, however, do this just very loosely.

To develop some intuition about non-commutative spaces, let us consider followingtwo-dimensional non-commutative toy model. Define two non-commutative coordi-nates x1,x2 with commutation relation given by

[x1, x2] = λ (1.1)

2

CHAPTER 1. FUZZY SPACESS 3

This coordinates can be constructed from annihilation and creation operators x1 =√λa, x2 =

√λa†. Recall that for n-tuple of annihilation and creation operators[

ai, a†j

]= δij holds. These operators act on auxiliary Hilbert space. Now we define

non-commutative (analytic) function as:

φ (x1, x2) =∞∑

m,n=0

cmnxn1x

m2 (1.2)

So far it does not seem much extra-ordinary, since it is obvious that the set of com-mutative analytic functions is of the same size as the set of non-commutative analyticfunctions. Let’s look at product of the functions now and where commutativity takesplace. We will use coherent states to get the job done. Recall that coherent state iseigenstate of annihilation operator i.e. a |α〉 = α |α〉. |α〉 can be expressed as (see [15]):

|α〉 = e−|α|2∞∑n=0

(α)n√n!|n〉 (1.3)

but for our purposes here it is useful to redefine this as:

|√λα〉 = e−|α

√λ|2

∞∑n=0

(α√λ)n

√n!

|n〉 (1.4)

for whichx1 |√λα〉 = λα |

√λα〉 (1.5)

We now take symbol of non-commutative function φ (χ̄, χ) = 〈χ|φ |χ〉 where χ = αλand we denote |

√λα〉 as |χ〉. Assignment of symbol is bijective. !!!!tuto nieco doplnit!!!!

Let’s take two non-commutative functions ζ and ψ. Their product is:

ζψ =∞∑

m,n,i,j=0

cmnkijxm1 x

n2x

i1x

j2 (1.6)

Using Wick’s theorem for x1 and x2 and assigning symbol to the product of functionswe get the following structure:

〈χ| ζψ |χ〉 = ζ (χ̄, χ) + λ (single contractions) + λ2 (double contractions) + ... (1.7)

We can see now, that as λ→ 0 we get commutative point-wise product.

1.2 Fuzzy spaces in our case

In this work we will not constrain ourselves to the specific non-commutative model,since our considerations are more general. Typically, non-commutativity (or fuzziness)

CHAPTER 1. FUZZY SPACESS 4

is introduced as non-commutativity forced to coordinate functions (for example see[6]):

[xk, xl] = Θkl (1.8)

Where Θkl is not specified.This in turn leads to non-commutativity of algebra of functions(since they are functionsof coordinates) and uncertainty relations for them as well. Different commutationrelations define, of course, different geometries. Common example of fuzzy space (with"fuzzy" Dirac operator) is fuzzy sphere or it’s variations, see for example [8].Often there is quantum theory constructed on fuzzy space to investigate some im-plications of non-commutativity. To get some picture of non-commutative space weshall construct space of functions. This is done by using two pairs of annihilation andcreation operators ai and a′i (reasons for doing so will be discussed later). Recall that:

[ai, a†j] = [a

′i , a′†j ] = δij

[ai, a′j] = 0 (1.9)

Space of functions then consists of normal-ordered states of the form:

Ψ = Ψ(a, a†, a′, a′†

)And more precisely:

Ψ =∑

CmnijC′opkl

(a†1

)m (a†2

)n (a′†1

)o (a′†2

)p(a1)

i (a2)j (a′1)

k(a′2)

l (1.10)

Where aα and a′α are two pairs of annihilation (together with corresponding creation)operators. We additionally put one constraint for Ψ: it has to have the same numberof annihilation and creation operators (of the same kind) - this effectively reduces twodegrees of freedom from both C and C ′.

Chapter 2

spin (4, 2), su (2, 2), oscillatorrepresentation

We shall begin this section by mentioning some important definitions and theorems.We give introduction to fundamental representations of spin (4, 2) and su (2, 2). In theend, we give oscillator representation of those mentioned algebras.

2.1 Basic definitions and theorems

Definition 1. An enveloping algebra A of lie algebra g is an associative algebra inwhich can g be embedded (denoting embedding as Φ) such that abstract brackets [f, g](f , g ∈ g) are realized as XY − Y X for X = Φ (f) and Y = Φ (g).

Proposition 1. The Universal enveloping algebra U (g) is the most general (max-imal) enveloping algebra.

This is rather vague statement, but we will give more precise definition soon. The keyto this is to give precise meaning of "the most general". We will not, however, givecomplete theory behind universal enveloping algebra. For thorough description see [10]chapter 9.

Definition 2. The Universal enveloping algebra U (g) is Tensor algebra over gwith condition A ⊗ B − B ⊗ A = [A,B] imposed. That is quotient of Tensor algebraover g with respect to ideal generated by elements of the form A⊗B−B⊗A− [A,B].

It is very intuitive that "the most general enveloping algebra" should be unique to bewell defined. From Theorem 9.7. of [10] we can take this as a fact and in addition wecan deduce that for any enveloping algebra A there exist (unique) homomorphism h :A→ U (g) such that for φ : g→ A it holds that h ◦ φ is the desired embedding. Thisgives rise to the fact, that we can compute Casimir operators (central operators for

5

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 6

U (g)) in given representation entirely in terms of the basis matrices (of that algebra ingiven matrix representation), since embedding into algebra of matrices is embeddinginto associative algebra. Existence (and uniqueness) of mentioned homomorphism thengives clear relationship to U (g). This allows us to express Casimir elements in termsof some finite-dimensional associative algebra.

Theorem 2. Poincaré-Birkhoff-Witt (PBW) Let g be finite-dimensional Lie al-gebra with basis e1...en and i : g → U (g) be injective homomorphism, and Xj = i (ej)then elements of the form:

Xn11 ⊗ ...⊗Xnkk

are linearly independent and span U (g).

Definition 3. The real Clifford algebra Cl (p, q) is associative algebra with unit forwhich it additionally holds:

eiej = ηij (2.1)

where ηij is bilinear form with signature (p, q). We will assume that ηij is diagonal andnon-degenerate.

Definition 4. spin (m,n) is lie algebra of Spin (m,n) group that is positive-determinant invertible part of Clifford algebra.

For us, spin (4, 2) is relevant, since it is isomorphic to su (2, 2).

2.2 su (2, 2) algebra

2.2.1 Review of Poincaré algebra

Recall (see [18], we use, however, different signature convention) that for Poincarégroup generators following commutation relations hold:

[Ji, Jj] = i�ijkJk (2.2)

[Ji, Kj] = i�ijkKk (2.3)

[Ki, Kj] = −i�ijkJk (2.4)

[Ji, Pj] = i�ijkPk (2.5)

[Ki, Pj] = −iδijP0 (2.6)

[Ji, P0] = i [Pi, P0] = 0 (2.7)

[Ki, P0] = −iPi (2.8)

where Ji, Ki are generators of rotations and boosts respectively. Pµ is generator oftranslations, four-vector and can be considered as momentum operator (given the rightrepresentation).

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 7

2.2.2 spin(4,2) and su(2,2)

Importance of su (2, 2) for this work lies in the fact, that it contains Poincaré algebraas an sub-algebra. SU (2, 2) group is real Lie group which preserves scalar product〈ψ†φ〉 with signature (2, 2) where ψ and φ are four-dimensional complex vectors. Wecan thus write element of su (2, 2) algebra as:

S =

(A B

B† D

)(2.9)

where A, D are hermitian and B, B† are general matrices. su (2, 2) algebra in funda-mental representation is then given by following matrices:

Sij =1

2εijk

(σk 0

0 σk

), Sk4 =

1

2

(σk 0

0 −σk

),

S0k =i

2

(0 σk

σk 0

), S45 =

i

2

(0 1

1 0

),

Sk5 =1

2

(0 σk

−σk 0

), S04 =

1

2

(0 1

−1 0

),

S05 =1

2

(1 0

0 −1

). (2.10)

Where i, j, k = 1..3.Now where can one see Poincaré group? Eagle-eyed reader should have already spot-ted that for example Sij are generators of rotations. Boost matrices and generators oftranslations (momentum) are more intricate. One can straightforwardly (calculatingcommutators) verify that S0k correspond to boosts and Sµ5− Sµ4 correspond to trans-lation generators, let’s call them pµ. We have to stress out, that this is not the singleoption how to choose Poincaré subgroup, rather most obvious. From complexified ver-sion of the algebra we can get even richer structure.

Let’s briefly discuss anti-fundamental (dual) representation of su (2, 2) denoted asS ′ab. For dual (matrix) representation of any Lie algebra holds that ρ? (g) = −ρT (g)where g is an element of Lie algebra. It then follows that Sa5 = −S ′a5. These repre-sentations are inequivalent. From anti-fundamental representation we choose Poincarégenerators in the same manner as in fundamental representation and so p′µ = S ′µ5−S ′µ4.As it has already been mentioned, we will take S ⊕ S ′ for our considerations. This is,however, fundamental representation of spin (4, 2). Let us explicitly write generators

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 8

of the dual representation.

S ′ij = −1

2εijk

(σTk 0

0 σTk

), S ′k4 = −

1

2

(σTk 0

0 −σTk

),

S ′0k = −i

2

(0 σTkσTk 0

), S ′45 = −

i

2

(0 1

1 0

),

S ′k5 =1

2

(0 σTk−σTk 0

), S ′04 =

1

2

(0 1

−1 0

),

S ′05 =1

2

(−1 00 1

). (2.11)

And we can write down an element of fundamental representation of spin (4, 2) as:

Σab =

(Sab 0

0 S ′ab

)(2.12)

where Sab and S ′ab denote fundamental and anti-fundamental representation of su (2, 2).This is very analogous to situation when one constructs spinor representation of Lorentzgroup. One can roughly say that su (2, 2) is to so (4, 2) (or spin (4, 2)) what sl (2,C)is to so (3, 1).Now we construct oscillator representation of su (2, 2) (and also spin (4, 2) as conse-quence). Define operators on NC states âi, â+i , b̂i, â

+i for i = 1, 2 as:

âiΨ = aiΨ

b̂iΨ = Ψai (2.13)

and analogously for + operators. Commutation relations for these operators are:[âi, â

+j

]= −

[b̂i, b̂

+j

]= δij (2.14)

Now we can arrange these operators into quadruple(s):

ÂT = (â1, â2, b̂1, b̂2) , Â+ = (â+1 , â

+2 , b̂

+1 , b̂

+2 ) (2.15)

And for oscillator representation we can write:

Ŝab = Â+ ΓSab  a, b = 0..5 (2.16)

where Γ =

(1 0

0 −1

)is 4× 4 matrix with 1 and −1 being 2× 2 matrices.

This (or similar) approach can be seen for example in [12] or [9], we build mostly onthe first one.To construct oscillator realization Sab of S ′ab we proceed very similarly. We now usesecond pair of a/c operators a′ and a′† introduced in (1.9). Operators b̂i and b̂+i are

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 9

defined for a′i and a′†i in the same manner as operators in (2.13). There is one differencein operator realization for S̃ab- Γ matrix has now swapped position to ensure thatrepresentation is dual:

S̃ab = Ã+S ′abΓÃ a, b = 0..5 (2.17)

where à and Ã+ are:

ÃT = (ã1, ã2, b̃1, b̃2) , Ã+ = (ã+1 , ã

+2 , b̃

+1 , b̃

+2 ) (2.18)

Theorem 3. S̃ab and Ŝab obey the same commutation relations as Sab and S ′ab and thusthey are indeed representations.

Proof. We proceed for Ŝ. Let S, K be from su (2, 2). Define Ā = A+Γ, then[Ai, Āi

]= δij where i, j = 1..4. We write Ŝ = SijĀiAj and K̂ = KmnĀmAn. Cal-

culating commutator:[Ŝ, K̂

]=SijKmn

{ĀiAjĀmAn − ĀmAnĀiAj

}=SijKmn

{Āi(δmj − Ām

)AjAn − Ām

(δin − ĀiAn

)Aj}

=Ā [S,K]A

(2.19)

Theorem 4. This representation is also unitary.

Proof. Let us write the most general element of given lie algebra as:

Ŝ =(u+,−v+

)(A BB† D

)(u1, v1)

T = u+Au+ u+Bv − v+B†u− v+Dv.

Now computing hermitian conjugate of this operator we get

Ŝ+ = −u+Au− u+Bv + v+B†u+ v+Dv = −Ŝ

Ŝ is thus anti-hermitian which is desired result. Since exponential of anti-hermitianoperator gives unitary one. We now used mathematician’s convention (this is the onlytime we use it).

Remark: These considerations also make clear, why we defined NC wave functionswith two pairs of annihilation and creation operators. The first pair ai correspondsto fundamental representation and the second pair a′i corresponds to anti-fundamentalrepresentation. This makes direct sum structure very visible.

2.3 Gauss decomposition

We now show technique how to boost non-commutative functions in operator repre-sentation.

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 10

2.3.1 Gauss decomposition for sl (2,C)

For SL (2,C) we have only one constraint. Let g be general element of the group, thenit is necessary for det (g) = 1 to hold. I the terms of Lie algebra sl (2,C) it means thatTr (X) = 0 where X is from Lie algebra. We now choose (very common) basis forsl (2,C).

H =1

2

(−1 00 1

)K+ =

(0 0

1 0

)K− =

(0 1

0 0

)

[H,K±] =±K±

(2.20)

Every element of sl (2,C) (technically not every, however exception is set of zero mea-

sure) can be expressed as:

X = exp (tK−) exp (τH) exp (sK+)

=

(e−

12τ + ste

12τ te

12τ

se12τ e

12τ

)

=

(1 t

0 1

)(γ−1 0

0 γ

)(1 0

s 1

) (2.21)

From this we can see that for (almost, for reasons given above) general element ofsl (2,C) it is possible to write:(

d c

b a

)=

(1 0

bd−1 1

)(d 0

0 d−1

)(1 cd−1

0 1

)(2.22)

We will use rather special case:(C S

S C

)=

(1 SC−1

0 1

)(C−1 0

0 C

)(1 0

SC−1 1

)(2.23)

We should remember that unit determinant condition is C2 − S2 = 0 so we choosenatural parametrization C = cosh (t) and S = sinh (t). Additionally we introduce newvariable T = SC−1 = tanh (T ) and thus(

C S

S C

)=

(1 T

0 1

)(C−1 0

0 C

)(1 0

T 1

)(2.24)

2.3.2 Calculating boost

Since oscillator representation introduced in this work is complex, we are allowedto work also with complex linear combinations of generators introduced in (2.9) and

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 11

thus to work with complexified algebra su (2, 2)⊗C. This is due to fact that complexrepresentations are the same as representations of complexified algebra. We can thenchoose full sl (2,C) from it. It is then given by matrices S45, S04 and S05.

We shall now examine how boost B̂45 acts on non-commutative functions.

B̂45 = e−2iT Ŝ45 = eTK̂−e2i ln(C)Ŝ05eTK̂+ (2.25)

where K+ =

(0 1

0 0

)and K− =

(0 0

1 0

). In oscillator representation then K̂+ = â+b̂,

K̂− = b̂+â, iŜ05 = 12i

(â+â+ b̂+b̂

). We now introduce new labels r̂ = −iŜ05 and

τ = 12

ln (C). So we can rewrite above equation as:

B̂45 = e−2iT Ŝ45 = eTK̂−eτ r̂eTK̂+ (2.26)

We will now choose more convenient basis for non-commutative functions generated by|n1, n2〉 〈m1,m2|. Consequently we need to calculate matrix elements 〈m1,m2| B̂45 |n1, n2〉.The calculation we are going to do is approached in somewhat simplified form, just forone pair of annihilation/creation operators, however without loss of generality, sincecalculation is done in exactly the same way with two pairs of operators. We calculatematrix elements for K̂− acting on ψ.

〈n′| eTK̂−ψ |n〉 =∞∑k=0

T k

k!〈n′| akψa†k |n〉

=∞∑k=0

T k√

(k + n′)! (k + n)!

k!n′!k!n!〈n′ + k|ψ |n+ k〉

=∞∑k=0

T k

√(k + n′

k

)(k + n

k

)〈n′ + k|ψ |n+ l〉

(2.27)

And in the similar fashion for K̂+:

〈n′| eTK̂+ψ |n〉 =∞∑k=0

T k

k!〈n′| a†kψak |n〉

=∞∑k=0

T k

√n′!n!

k! (n− k)!k! (n′ − k)!〈n′ − k|ψ |n− k〉

=∞∑k=0

T k

√(k + n′

k

)(k + n

k

)〈n′ + k|ψ |n+ l〉

(2.28)

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 12

Let’s investigate action of K̂± on non-commutative functions now.

ΦTK+ = eTK̂+Φ =

∞∑k=0

T k

k!

(â+α b̂α

)kΦ

=∞∑k=0

T k

k!

k∑l=0

(k

l

)(â+1 b̂1

)l (â+2 b̂2

)k−lΦ

=∞∑k=0

T k∑

k1+k2=k

a†k11 a†k22√

k1!k2!Φak11 a

k22√

k1!k2!

(2.29)

and practically the same calculation goes for K̂−:

ΦTK− = eTK̂−Φ =

∞∑k=0

T k

k!

(âαb̂

+α

)kΦ

=∞∑k=0

T k∑

k1+k2=k

ak11 ak22√

k1!k2!Φa†k11 a

†k22√

k1!k2!

(2.30)

Due to the fact that for Φ we have a constraint regarding equal number of annihilationand creation operators of the same kind, it is easy to see, that Φ acts diagonally onauxiliary Fock space i.e. Φ |n1n2〉 = Φn1n2 |n1n2〉. Since (2.26) preserves this property,it is natural to investingate the action in terms of basis |n1n2〉 〈n1n2|. Using similartechnique as in (2.27) and (2.29) we derive how constituents of (2.26) act on Ψ.

ΦTK+ |n1, n2〉 =∞∑k=0

T k∑

k1+k2=k

(n1k1

)(n2k2

)Φn1−k1,n2−k2 |n1, n2〉

ΦTK− |n1, n2〉 =∞∑k=0

T k∑

k1+k2=k

(k1 + n1k1

)(k2 + n2k2

)Φn1+k1,n2+k2 |n1, n2〉

e−τ r̂Φ |n1, n2〉 =e−τ(n1+n2+1)Φn1,n2 |n1, n2〉

(2.31)

Putting everything together, we present grand formula for boosted solution:

B̂45Φ =∑n1,n2

∞∑k=0

∞∑j=0

∑k1+k2=k

∑i1+i2=j

(k1 + n1k1

)(k2 + n2k2

)×

×(k1 + n1i1

)(k2 + n2i2

)T k+jΦn1+k1−i1,n2+k2−i2 |n1, n2〉 〈n1, n2|

(2.32)

To get the function which will eventually correspond to non-zero three-momentum, we

need to rotate this solution. We choose rotation given by R̂i =

(σi 0

0 σi

)Oscillator

representation for this operator then gives:

R̂ = â+σiâ− b̂+σib̂ (2.33)

and rotation, using BCH formula:

eitR̂Ψ = eitRiΨe−itRi (2.34)

CHAPTER 2. SPIN (4, 2), SU (2, 2), OSCILLATOR REPRESENTATION 13

where Ri = a†σa and ai are two annihilation operators on the auxiliary Fock space. Westress out that for dual representation approach is the same, it is merely interchangeof some symbols and labels.

Chapter 3

Momentum operator

3.1 Definition

Since we need oscillator representation of spin (4, 2) on NC wave functions (or fields,if one deals with the field theory) we can define two "small" momentum operators p̂µand p̃µ from translation generators of fundamental and anti-fundamental representa-tions of su (2, 2):

p̂µ = Ŝµ5 − Ŝµ4p̃µ = S̃µ5 − S̃µ4 (3.1)

Let’s look on the operator somewhat closer, first we take generator of translations:

p0 =1

2(S05 − S04) =

1

4

(1 −11 −1

)(3.2)

for time translations and

pk =1

2(Sk5 − Sk4) =

1

4

(−σi σi−σi σi

)(3.3)

for space translations.Now for operator representation we get for p̂0:

p̂0 =(â+1 , â

+2 ,−b̂+1 ,−b̂+2

)(1 −11 −1

)(â1, â2, b̂1, b̂2

)T= 1

4

{â+α âα + b̂

+α b̂α + â

+α b̂α + b̂

+α âα

}(3.4)

and for spatial part:

p̂i =1

4σiαβ

{â+α âβ + b̂

+α b̂β + â

+α b̂β + b̂

+α âβ

}(3.5)

14

CHAPTER 3. MOMENTUM OPERATOR 15

Splitting into components momentum operator acts on non-commutative functions φas:

p̂0φ =1

4{[a+1 , [a1, φ]

]+[a+2 , [a2, φ]

]}

p̂1φ =−1

4{[a+2 , [a1, φ]

]+[a+1 , [a2, φ]

]}

p̂2φ =−1

4i{[a+2 , [a1, φ]

]−[a+1 , [a2, φ]

]}

p̂3φ =−1

4{[a+1 , [a1, φ]

]−[a+2 , [a2, φ]

]}

(3.6)

Similarly for S̃ representation we get:

p̃0 =(ã+1 , ã

+2 , b̃

+1 , b̃

+2

)(−1 −11 1

)(ã1, ã2,−b̃1,−b̃2

)T= −1

4

{ã+α ãα + b̃

+α b̃α + ã

+α b̃α + b̃

+α ãα

}(3.7)

and for spatial part:

p̃1,3 =−1

4σ1,3αβ

{ã+α ãβ + b̃

+α b̃β + ã

+α b̃β + b̃

+α ãβ

}p̃2 =

1

4σ2αβ

{ã+α ãβ + b̃

+α b̃β + ã

+α b̃β + b̃

+α ãβ

} (3.8)And again, splitting into components results in:

p̃0φ′ =− 1

4{[a′+1 , [a

′1, φ′]]

+[a′+2 , [a

′2, φ′]]}

p̃1φ′ =

1

4{[a′+2 , [a

′1, φ′]]

+[a′+1 , [a

′2, φ′]]}

p̃2φ′ =− 1

4i{[a′+2 , [a

′1, φ′]]−[a′+1 , [a

′2, φ′]]}

p̃3φ′ =

1

4{[a′+1 , [a

′1, φ′]]−[a′+2 , [a

′2, φ′]]}

(3.9)

Where φ′ does not mean derivative, it denotes that function corresponds to dual rep-resentation. It can be proven (see appendix) that:

p̂µp̂µ = p̃µp̃µ = 0 (3.10)

Once again analogy with Lorentz group shows up - it is impossible to define massivetheory with just fundamental or anti-fundamental representation alone. In the caseof Lorentz group we have massless left-handed or right-handed spinors, in our case ofspin (4, 2) we have "left-handed" or "right-handed" twistors. Twistor is 4-dimensionalanalogue of spinor.In the end, we use p̂µ and p̃µ to construct momentum operator of the massive theory:

Pµ = p̂µ + p̃µ (3.11)

CHAPTER 3. MOMENTUM OPERATOR 16

Since Pµ has been constructed from generators of translations it is also generator oftranslations. From the fact, that pµ and p′µ are also 4-vectors, Pµ is 4-vector too. Let’sdefine operator of mass M = P µPµ. Simplifying (p̂µ + p̃µ) (p̂µ + p̃µ) we get:

M2 = 2p̂µp̃µ (3.12)

and thus M has, of course, the same eigenfunctions as Pµ.

3.2 Eigenvalue problem for the momentum operator

In addition to solve eigen-problem for spinors as in commutative case, we have to findeigenfunctions of momentum operator (momentum in NC Dirac equation is operatoron NC states) if we want to find eigenfunctions of the Dirac operator. We need to dothat just for p̂0 and p̃0. This is justified by the fact, that we can consider states withzero three-momentum and then boost them in the right direction to obtain solution forany three-momentum. It is, of course, impossible to have zero three-momentum (fornonzero four-momentum) just for "small momentum operators", since they correspondto massless fields. For P µ it is possible and we will indeed capitalize this property.

Since NC states posses direct sum structure and in addition [a1, a2] = [a′1, a′2] = 0holds, together with assumption of normal ordering of NC states and equal number ofannihilation and creation operators, we can introduce possibilities for ansatz:

Φ =

φ1 (N1) φ̃1 (N′1)

φ2 (N2) φ̃2 (N′2)

φ1 (N1) φ̃2 (N′2)

φ2 (N2) φ̃1 (N′1)

(3.13)

where N1 = a†1a1, N2 = a†2a2 and analogously for N ′i . φi and φ̃i are normally ordered.

Now we approach just for φ1. This is without loss of generality, considering that taskis the same for φ2 and φ̃i - it is merely just interchange of symbols. p̂0 acts on φ1 as:

p̂0Ψ = −1

4

[a+1 , [a1, φ1]

](3.14)

And we want to solve:p̂0Ψ = EΨ (3.15)

Considering the fact that commutators act as derivatives (by "erasing" a/c operators)this equation can be rewritten into differential form:

xχ,, (x) + χ, (x) + 4Eχ (x) = 0 (3.16)

CHAPTER 3. MOMENTUM OPERATOR 17

This equation has the solution:

χ (x) = C1J0

(4√Ex)

+ C2Y0

(4√Ex)

(3.17)

where J0 and Y0 are Bessel functions of the first and second kind. The full operatorsolution is then normal ordered operator-evaluated χ i.e. φα (Nα) = : χ (Nα) : . ForN1 then holds(see appendix):

φ1 (N1) : χ (N1) : =∞∑l=0

(−16E)l

22l (l!)2a†l1 a

l1+

+2

π

{∞∑k=0

k∑i=0

∞∑j=0

(k

i

)(−1)2k+j−i−1 42j+iEl+i

k22j (j!)2

(a†1

)i+jai+j1

}+

+2

π

{∞∑k=1

(−1)k−1Hk4kE

(k!)2(a+1)kak1

} (3.18)

What does this solution correspond to? Take a look at how operators p̂i act on φ1 (N1).Since [aα, aβ] = 0 and

[aα, a

†β

]= δαβ it directly follows that (considering 3.14):

p̂1φ1 (N1) =0

p̂2φ1 (N1) =0

p̂3φ1 (N1) =− Eφ1 (N1)

(3.19)

and for φ2 (N2) we have (again, see 3.14):

p̂1φ2 (N2) =0

p̂2φ2 (N2) =0

p̂3φ2 (N2) =Eφ1 (N1)

(3.20)

There is obvious interpretation for this - one gets solution for particle moving alongz axis together with solution for particle "moving" in opposite direction. The samebehavior we get for p̃µ:

p̃1φ̃1 (N1) =0

p̃2φ̃1 (N1) =0

p̃3φ̃1 (N1) =− Eφ̃1 (N1)

(3.21)

and for φ̃2 (N2) the similar relations hold

p̃1φ̃2 (N2) =0

p̃2φ̃2 (N2) =0

p̃3φ̃2 (N2) =Eφ̃1 (N1)

(3.22)

CHAPTER 3. MOMENTUM OPERATOR 18

Finally, for possible solution(s) we write:

Ψ (N1, N2, N′1, N

′2) =

φ1 (N1) φ̃ (N′1)

φ2 (N2) φ̃2 (N′2)

φ1 (N1) φ̃2 (N′2)

φ2 (N2) φ̃1 (N′1)

(3.23)

Look at eigenfunctions of Pµ now. They are, of course, eigenfunctions of p̂µ and p̃µ.We are especially interested for solutions of motionless particle. Those are Ψ1 =φ2 (N2) φ̃1 (N

′1) and Ψ2 = φ1 (N1) φ̃2 (N ′2). Then Pµ acts as:

P0Ψ1 (N1, N2, N′1, N

′2) =2EΨ1 (N1, N2, N

′1, N

′2)

P0Ψ2 (N1, N2, N′1, N

′2) =2EΨ2 (N1, N2, N

′1, N

′2)

P3Ψ1 (N1, N2, N′1, N

′2) =0

P3Ψ2 (N1, N2, N′1, N

′2) =0

(3.24)

and it, of course, also vanishes on for P1 and P2. This is solution with zero threemomentum. To obtain full solution, formula derived in the last section chapter 2should be used to boost and rotate solution.

Chapter 4

Noncommutative Dirac operator

The work on non-commutative Dirac operator has already been done. The nontriv-ial part, comparing to commutative Dirac operator, is eigen-problem for momentumoperator Pµ. We will briefly discuss Dirac equation for the sake of completeness.

4.1 Dirac operator and Dirac equation

Recall that Dirac operator in Minkowski space is γµ∂µ and then the Dirac equation:

(iγµ∂µ −m)ψ (x) = 0 (4.1)

and fourier-transforming equation we get:

(γµpµ −m)ψ (p) = 0 (4.2)

this is the form that non-commutative version shall take too (with pµ, however, beingan operator). Dirac equation is typically solved in a following way:

1. take fourier transform of Dirac equation (4.2)

2. take ansatz u (p)T = (u1, u2) and solve for u1, u2

3. write solution in symmetric form u (p) =

( √p · σζ√p · σ̄ζ

)where ζ is two-componnent spinor.

4.2 Non-commutative version

We introduce Dirac equation right in p-representation in the similar way as in commu-tative theory - we have no choice but ordinary gamma matrices since we want spin 1

2

Lorentz invariant equation.(γµPµ −M) Ψp = 0 (4.3)

19

CHAPTER 4. NONCOMMUTATIVE DIRAC OPERATOR 20

Taking MΨp = mΨp and PµΨp = pµΨp we see it takes form of ordinnary Dirac equa-tion. The major difference here is that components of bi-spinor Ψp are noncommuta-tive functions, solutions from the (3.23) or their boosted/rotated versions (see 2.34 and2.32).

Chapter 5

Conclusions

Logic of our work relies on the fact, that su (2, 2) contains Poincaré algebra as sub-algebra. We proceeded straightforwardly in the beginning with fundamental repre-sentation of su (2, 2). We took translation generators Sµ4 − Sµ5 to define momentumoperator. This is done by oscillator representation on non-commutative functions. Therepresentation acts on space of NC functions, constructed from two pairs of annihilationand creation operators. First pair for fundamental and second pair for anti-fundamental(dual) representation. Lesson gathered from representation theory of so (3, 1) (massmixes left-handed with right-handed components of spinors) can also be seen here.Momentum operator p̂µ has vanishing square p̂µp̂µ and we need to take direct sumwith dual representation. The momentum operator Pµ is then sum of operators fromfundamental and dual representation p̂µ and p̃µ. In this representation we possessCasimir element of Poincaré algebra constructed in terms of operators p̂µ and p̃µ thisis naturally interpreted as mass. In chapters 2, 3 and 4 we investigated (and mostlysolved) eigen-problem for Dirac operator. The spinor part is the same as in com-mutative case, however non-commutativity of functions introduces some difficulties.We overcame them and developed method how to deal with the problem, at least inP-representation.

21

Chapter 6

Appendix: calculations and formulas

We now calculate the result (3.10). For the sake of clarity we define two-dimensional

complex vectors â = (â1, â2)T and b̂ =

(b̂1, b̂2

)T. Now momentum operator can be

written as follows:

p̂0 =1

2

(â+,−b̂+

)(1 −11 −1

)(â, b̂)T

=1

2

(â+ − b̂+

)(a− b)

p̂i =1

2

(â+,−b̂+

)(−σi σi−σi σi

)(â, b̂)T

=− 12

(â+ − b̂+

)σi

(â− b̂

)(6.1)

Lorentz square of p̂µ is:

p̂µp̂µ = p̂0p̂0 − p̂1p̂1 − p̂2p̂2 − p̂3p̂3 (6.2)

Since σ3 is diagonal matrix it is natural to group p̂0 (contains identity matrix) withp̂3 (which contains σ3) and also to group p̂0 and p̂0 since they contain anti-diagonalmatrices σ1 and σ2.

p̂µp̂µ = (p̂0p̂0 − p̂3p̂3)− (p̂2p̂2 + p̂1p̂1) (6.3)

Now it is natural to rewrite the equation using basic product formulas:

p̂µp̂µ = (p̂0 − p̂3) (p̂0 + p̂3)− (p̂1 − ip̂2) (p̂1 + ip̂2) (6.4)

We shall now calculate first and second term separately. For the first one:

p̂0 ± p̂3 =1

2

(â+ − b̂+

)(1∓ σ3)

(â− b̂

)=

1

2

(â+ − b̂+

)(1∓ 1 00 1± 1

)σ3

(â− b̂

)

=

(â+2 − b̂+2

)(â2 − b̂2

)case +(

â+1 − b̂+1)(

â1 − b̂1)

case −

(6.5)

22

CHAPTER 6. APPENDIX: CALCULATIONS AND FORMULAS 23

and for the second one:

p̂1 ± ip̂2 =1

2

(â+ − b̂+

)(1∓ σ3) (a− b)

=1

2

(â+ − b̂+

)( 0 1± 11∓ 1 0

)σ3 (a− b)

=

−(â+1 − b̂+1

)(â2 − b̂2

)case +

−(â+2 − b̂+2

)(â1 − b̂1

)case −

(6.6)

And thus we finally arrive at desired conclusion

p̂µp̂µ =

(â+2 − b̂+2

)(â2 − b̂2

)(â+1 − b̂+1

)(â1 − b̂1

)−(â+2 − b̂+2

)(â1 − b̂1

)(â+1 − b̂+1

)(â2 − b̂2

)= 0

(6.7)

Where we used that(â1 − b̂1

)commutes with

(â+1 − b̂+1

). To prove this, we use

commutation relations for a and b[âi, â

+j

]= δij and

[b̂i, b̂

+j

]= −δij. Simple calculation

then shows that:(â+1 − b̂+1

)(â1 − b̂1

)=â+1 â1 − b̂+1 â1 − â+1 b̂1 + b̂+1 b̂1

=â1â+1 − 1− b̂+1 â1 − â+1 b̂1 + b̂1b̂+1 + 1

=(â1 − b̂1

)(â+1 − b̂+1

) (6.8)This proof, of course, passes for general product

(â+i − b̂+i

)(âi − b̂i

). We now repeat

previous procedure for p̃µ with some minor modifications. Recall that p′ = S ′µ5 − S ′µ4and in oscillator representation p̃0 = A+p′µΓA. For momentum operator then holds:

p̃0 =1

2

(ã+, b̃+

)(−1 −11 1

)(ã,−b̃

)=− 1

2

(ã+ − b̃+

)(ã− b̃

) (6.9)p̃1 =

1

2

(ã+, b̃+

)( σ1 σ1−σ1 −σ1

)(ã,−b̃

)=

1

2

(ã+ − b̃+

)σ1

(ã− b̃

) (6.10)p̃2 =

1

2

(ã+, b̃+

)(−σ2 −σ2σ2 σ2

)(ã,−b̃

)=− 1

2

(ã+ − b̃+

)σ2

(ã− b̃

) (6.11)p̃3 =

1

2

(ã+, b̃+

)( σ3 σ3−σ3 −σ3

)(ã,−b̃

)=

1

2

(ã+ − b̃+

)σ3

(ã− b̃

) (6.12)

CHAPTER 6. APPENDIX: CALCULATIONS AND FORMULAS 24

And to calculate Lorentz square p̃µp̃µ we proceed similarly as in previous case.

p̃µp̃µ = (p̃0p̃0 − p̃3p̃3)− (p̃2p̃2 + p̃1p̃1) (6.13)

Now it is natural to rewrite the equation using basic product formulas:

p̃µp̃µ = (p̃0 − p̃3) (p̃0 + p̃3)− (p̃1 − ip̃2) (p̃1 + ip̃2) (6.14)

We, again, calculate first and second term separately. For the first one:

p̃0 ± p̃3 =−1

2

(ã+ − b̃+

)(1∓ σ3)

(ã− b̃

)=

1

2

(ã+ − b̃+

)(1∓ 1 00 1± 1

)σ3

(ã− b̃

)

=

−(ã+2 − b̃+2

)(ã2 − b̃2

)case +

−(ã+1 − b̃+1

)(ã1 − b̃1

)case −

(6.15)

and for the second one:

p̃1 ± ip̃2 =1

2

(ã+ − b̃+

)(1∓ σ3)

(ã− b̃

)=

1

2

(ã+ − b̃+

)( 0 1∓ 11± 1 0

)(ã− b̃

)

=

(ã+1 − b̃+1

)(ã2 − b̃2

)case +(

ã+2 − b̃+2)(

ã1 − b̃1)

case −

(6.16)

Let’s derive differential equation introduced in (3.16). We first calculate the com-mutator [a, φ] where φ is normal ordered function of N1.[

a†1,(a†1

)m(a1)

n]

=a†1

(a†1

)m(a1)

n −(a†1

)m(a1)

n a†1

=(a†1

)m+1(a1)

n −(a†1

)m(a1)

n−1(

1 + a†1a1

)= ...

=− n(a†1

)m(a1)

n−1

(6.17)

where ellipsis denotes repetitive use of commutation relation[ai, a

†j

]= δij until one

arrives to the final formula. Using the same approach we calculate:[a1,(a†1

)m(a1)

n]

=m(a†1

)m−1(a1)

n (6.18)

This means that commutator acts as derivative (which is not at all surprising sincecommutators obey Leibniz rule). Let us now ask question how this commutator acts

CHAPTER 6. APPENDIX: CALCULATIONS AND FORMULAS 25

on normal ordered functions of number operator N1. This is done by setting m = n inprevious equations (6.17 and 6.18). One can now easily see that:

[a1, : Nn1 :] = n : N

n−11 : a1 =⇒ : [a1, χ (N1)] : =: χ′ (N1) a1 :[

: Nn1 :, a†1

]= na†1 : N

n−11 : =⇒ :

[χ (N1) , a

†1

]: =: a†1χ

′ (N1) :(6.19)

Now consider expression φ (N1) a1 where normal ordered function φ is χ′. By usingLeibniz rule we have:[

a†1, φ (N1) a1

]=[a†1, φ (N1)

]a1 + φ (N1)

[a†, a1

](6.20)

and after normal ordering:

:[a†1, φ (N1) a1

]: = − : a†1φ′a1 : −φ

= − : N1χ′′ (N1)− χ′ (N1) : (6.21)

Recall that:

p̂0 =14

{â+α âα + b̂

+α b̂α + â

+α b̂α + b̂

+α âα

}p̂i =

14σiαβ

{â+α âβ + b̂

+α b̂β + â

+α b̂β + b̂

+α âβ

}From this, one can finally write equivalent differential equation for p̂0φ1 = Eφ1 as

p̂0φ1 = −14{: N1φ′′1 (N1) + φ

′1 (N1) :} = Eφ1 (N1)

=⇒ ζχ′′ (ζ) + χ (ζ) + 4Eχ (ζ) = 0 (6.22)

And operator solution φ1 (N1) is then obtained from normal ordered operator evaluatedsolution χ (ζ). Finally φ1 (N1) =: χ (N1) : .To prove result in (3.18) we first write identities for Bessel functions(exhausting

information can be found in [1]):

J0 (z) =∞∑l=0

(−1)l

22l (l!)2z2l

Y0 (z) =2

π

{ln

(1

2z

)0

(z) +∞∑1

(−1)k+1Hk4−kz2k

(k!)2

} (6.23)where Hk =

∑ki=1

1iare harmonic numbers. The first term in expression for Y0 does

not look good since it contains logarithm. This can be circumvented by exponentiatingargument of logarithm by 2 and multiplicating by factor 1

2. As consequence, we do not

have to deal with square root of operator. Writing logarithm as power series

ln (X) =∞∑k=0

(−1)k−1 (x− 1)k

k(6.24)

and using binomial theorem

(X − 1)k =k∑i=0

(k

i

)(−1)k−iX i. (6.25)

Inserting 4√EN1 combining everything together we get the desired result.

Bibliography

[1] Milton Abramowitz. Handbook of Mathematical Functions, With Formulas,Graphs, and Mathematical Tables,. Dover Publications, Inc., New York, NY,USA, 1974.

[2] A. P. Balachandran, Giorgio Immirzi, Joohan Lee, and Peter Presnajder. Diracoperators on coset spaces. J. Math. Phys., 44:4713–4735, 2003.

[3] Ursula Carow-Watamura and Satoshi Watamura. Chirality and dirac operator onnoncommutative sphere. Comm. Math. Phys., 183(2):365–382, 01 1997.

[4] M. Chaichian, P. P. Kulish, K. Nishijima, and A. Tureanu. On a Lorentz-invariantinterpretation of noncommutative space-time and its implications on noncommu-tative QFT. Phys. Lett., B604:98–102, 2004.

[5] Alain Connes. Noncommutative geometry. 1994.

[6] Veronika Galikova and Peter Presnajder. Coulomb problem in NC quantum me-chanics: Exact solution and non-perturbative aspects. 2013.

[7] H. Grosse, C. Klimcik, and P. Presnajder. Field theory on a supersymmetriclattice. Commun. Math. Phys., 185:155–175, 1997.

[8] H. Grosse and P. Presnajder. The Dirac operator on the fuzzy sphere. Lett. Math.Phys., 33:171–182, 1995.

[9] M. Gunaydin, D. Minic, and Marco Zagermann. Novel supermultiplets ofSU(2,2|4) and the AdS(5) / CFT(4) duality. Nucl. Phys., B544:737–758, 1999.

[10] Brian Hall. Lie Groups, Lie Algebras, and Representations. Springer, 2003.

[11] Samuel Kováčik and Peter Prešnajder. Magnetic monopoles and symmetries innoncommutative space. J. Phys. Conf. Ser., 965(1):012024, 2018.

[12] Samuel Kováčik and Peter Prešnajder. Magnetic monopoles in noncommutativequantum mechanics 2. 2018.

26

BIBLIOGRAPHY 27

[13] G. Mack. All unitary ray representations of the conformal group SU(2,2) withpositive energy. Commun. Math. Phys., 55:1, 1977.

[14] G. Mack and I. Todorov. Irreducibility of the ladder representations of u(2,2)when restricted to the poincare subgroup. J. Math. Phys., 10:2078–2085, 1969.

[15] A. M. Perelomov. Generalized coherent states and their applications. 1986.

[16] Michael E. Peskin and Daniel V. Schroeder. An Introduction to quantum fieldtheory. Addison-Wesley, Reading, USA, 1995.

[17] I. T. Todorov. CONFORMAL DESCRIPTION OF SPINNING PARTICLES.1986.

[18] Steven Weinberg. The Quantum theory of fields. Vol. 1: Foundations. CambridgeUniversity Press, 2005.