Gravitation and Gauge Theory in the Non Commutative Geometry Formalism

Gravitation and Gauge Theory in the

Non Commutative Geometry Formalism

A thesis submitted to the University of Batna

for the degree of Doctor of Sciences

in the Faculty of Sciences

2009

Slimane Zaim

Contents

1 Introduction 5

1.1 Why Gauge theories ? . . . 5 1.2 Why Non-commutative Geometry? . . . 9 1.3 Plan of thesis . . . 11 2 Non-commutative Gauge field theory and Siberg-Witten maps 13 2.1 Non-commutative Gauge field theory . . . 13 2.1.1 Generalized infinitesimal general coordinate transformations 16 2.1.2 Generalized infinitesimal local Lorentz transformations . . 17 2.1.3 Non-commutative gauge transformations . . . 18 2.2 Seiberg-Witten maps . . . 20

3 Non-commutative gravity and applications 26

3.1 Introduction . . . 26 3.2 QED action in a curved non commutative space-time . . . 28 3.2.1 Generalized field equations and Noether theorem . . . 30 3.2.2 Generalized Dirac equation and particle creation process . 32 3.2.3 The weak field approximation . . . 40 3.2.4 The strong field approximation: . . . 42 3.3 Conclusion . . . 43

4 Gauge Gravity in Non-commutative De Sitter Space-time and

Pair creation 48

4.1 Introduction . . . 48 4.2 Generalized Dirac equation and particle creation process . . . 49 4.3 Conclusion . . . 55

5 Non-commutative Minimal Super-symmetric Standard Model 57

5.1 Introduction . . . 57 5.2 Minimal Super-symmetric Standard Model . . . 63 5.3 Conclusion . . . 78

THE UNIVERSITY OF BATNA

ABSTRACT OF THESIS submitted by Slimane Zaim for the Degree of Doc-tor of Sciences and entitled Gravitation and Gauge Theory in the Non Commu-tative Geometry Formalism.

Month and year of submission: June 2008.

In this thesis we studied quantum field gravity and gauge theory in the frame of non-commutative space-time geometry. We have also tested this study on two problems: quantization of gauge gravity related to 2D De-Sitter universe, and anisotropic Bianchi I Universe of Minkowski geometry with with scalar curva-ture R = 1/t, which gave good results for the understanding of the phenomenon of creation and annihilation of particles as well as the Casimir effect as a di-rect consequence of non-commutative space-time geometry. We also noticed that non-commutative geometry plays the same role as gravity and the Electric field. Furthermore we formulated the MSSM in the frame of non-commutative space-time geometry by writing the non-commutative Lagrangian density and finding the non-commutative transformations which conserve it.

Chapter 1

Introduction

We are interested in the formulation of gauge field theories in the part of the non-commutative geometry and Seiberg-Witten maps , where the gauge theory is a mathematical language in which the fundamental interaction in physics are formulated. We intend to make a point in this introduction on how gauge theories are important, and we shall highlight their principal characteristics. Then we explain how the non-commutative geometry appears to be an appropriate tool to formulate gauge theories and we will also present a short background on its appearance in mathematics and physics. Finally, we will explain many interests in the non-commutative gauge field theories and its generalization and give a detailed plan of this thesis.

1.1

Why Gauge theories ?

The term “gauge” was first introduced by Hermann Weyl in 1919 in an attempt to unify electromagnetism and gravitation, thus, in Weyl theory , the gauge is a measurement reference which permits to calibrate the scale that will be

used to measure a physical quantity. The physical quantities or observables are assumed to be invariant under local transformations of scale (or gauge). The gauge invariance, as it was introduced by Weyl, was directly inspired by the theory of linear connections used by Albert Einstein in his theory of general relativity and it had therefore a geometric status in its first formulation.

At the beginning of wave quantum mechanics developed by Schrodinger in 1926, Weyl, with Vladimir Fock and Fritz London, worked to develop a new type of gauge changing from a gauge-type scale factor to a complex phase gauge.

In retrospect it could be seen that James Clerk Maxwell was the first to introduce gauge theories by formulating the laws of electromagnetism as we know them today. In the theory of Maxwell, the gauge symmetry is associated with a group structure U(1). The local invariance of the Maxwell equations was therefore not really exploited only from the work of Weyl.

In his second proposal, he identified the U(1) group to the invariance under phase transformations of the wave-function of a particle charge in Quantum Me-chanics. Merely asking that the equations of motion are invariant under gauge transformation permits the understanding of the origin of the coupling between the electromagnetic field and the wave-function of a charged particle. This sym-metry, U(1), then takes a fundamental status which is associated to the local conservation of the electric charge.

This model is expressed within a formalism that aims to make compatible the quantum mechanics and special relativity, called quantum field theory. In this theory the charged particles are associated with the irreducible representations of the Poincare group which representations are indexed by two quantum numbers that are mass and spin (or helicity in the states of zero mass). The dynamics of these states is governed by what is called a quantum field.

In the theory of quantum electromagnetism, the first model of quantum the-ory of fields, the photons are the representations of helicity 1 and zero mass of Poincare group, and are described by Maxwell quantum field. Charged particles correspond to representations of spin 1/2 and positive or zero mass and are de-scribed by Dirac quantum field. The coupling of these two fields is obtained only if the Dirac field was also a representation of the group U(1) associated with the gauge invariance of Maxwell equation.

Similarly we learned, through the Emmy Noether theorem, that one can asso-ciate the invariance under local transformation symmetry of conservative quanti-ties called Noether currents which corresponded to currents algebra in quantum field theory. Applied to the symmetry U(1), the identities of Noether express the law of local charge conservation. Thus, in this model, the group structure of U(1) appears as a group of internal symmetry.

Thus, in 1954 C.N. Yang and R. Mills filed introduced a non-Abelian gauge generalizing in a direct way the theory of Maxwell. He replaced the group U(1) by a non-Abelian SU (2) group and the connection of maxwell, prescribing the phases of the wave function of the relationship between them at each point by a connection to values in a non-Abelian algebra. This model was able to highlight a generalization of the theory of connections used in general relativity for linear equations, to a theory of connections for a group of arbitrary compact binds equated in physics to a group of internal symmetries.

Despite the support enabling the description of the fundamental interactions with a principle of gauge invariance, physicists are strongly appealing for a theory forming a real guide to establish new theories. Indeed it is in the theory of weak interactions that resurfaces the model Yang and Mills because of the difficulties in the theory of S.L. Glashow and J. Schwinger who found the analogy between

the symmetry of isospin forming the basis of Yang-Mills theory and the symmetry observed between leptons.

It is finally the mechanism of spontaneous symmetry breaking proposed by Pe-ter Higgs in 1964, which canceled out the situation and allowed Steven Weinberg (1967) and Abdus Salem (1968) to develop a unified theory of weak and electro-magnetic interactions, this model has been constructed with the band structure of SU(2) ∗ U(1) and predicted the existence of intermediate bosons W and Z. In this model, the electron neutrino is a partner and they should both be zero mass in a gauge invariant formalism. The presence of a mixture called Wein-berg angle, while allowing the Higgs mechanism, assigns a weight to the electron and intermediate bosons. This mixture angle can also characterize the sub-group

U(1) corresponding to the portion of the electromagnetic gauge symmetry which

is not broken at the same time. In the years 1960-1970 the theory of Quantum Chromodynamics was developed. It is a gauge theory built with the internal symmetry SU(3) group which describes the strong interactions between quarks and gluons.

Until the end of the 20th century the Standard Model was drafted describing all the elementary particles known to date and the three fundamental interactions (electroweak and strong interactions). This model is a gauge theory with group structure SU(2) ∗ U(1) ∗ SU(2) all the particles of this model were found or brought to light with the exception of the Higgs boson which remains the missing coin.

The mechanism of symmetry breaking of the Higgs appears to be fundamental and became responsible for the mass of all elementary particles that the laws of the dynamic nature seem independent of the gauge chosen even if it is local.

1.2

Why Non-commutative Geometry?

In a way, one can think that the non-commutative geometry was born with Quan-tum Mechanics. Indeed, in its first formulation, which was that of Heisenberg quantum mechanics, appears in a form comparable to the mechanical Hamil-tonian when the coordinates of the space are replaced by operators which do not commute with each other.

In physics, the principle that the coordinates and momentum of space-time may not commute was issued by Heisenberg in 1930 with the hope that this could solve the problem of ultraviolet divergences in quantum field theory. This idea was then adopted in an article by Snyder in 1947.

Finally, the term ”non-commutative geometry” was introduced by Alain Connes in the 1980’s as a program to generalize different concepts borrowed from the reg-ular geometry concepts equivalent to ordinary non-commutative algebras and in particular the concepts from differential geometry.

The non-commutative geometry today represents a set of techniques on alge-bra operators to handle issues such as very different mathematical representations of the group or the study of space, seen as pathological in regular geometry.

From the perspective of physics, non-commutative geometry mathematics pro-vides a framework in which a number of physical concepts can be expressed and unified. The remark made by Dirac on the analogy between quantum mechanics and Poisson-bracket–Hamiltonian mechanics for example, can be implemented by the introduction of a non-commutativity. It was shown by Michel Dubois [1] that the quantum mechanics can be understood in this context as a Poisson bracket.

In 1985, M.Dubois-Violtte, R. Kerner and J. Madore [2 − 5] Showed that the gauge theories formulated on algebra functions naturally gives rise to a mechanism

for breaking of gauge symmetry similar to the one proposed by Higgs. This model was then generalised by R. Coquereaux [6 − 7] and A.connes, J.lott [8] to using other types of differential calculations.

In particular, it was demonstrated [9, 10] that it is possible to express the Standard Model Lagrangian in a totally algebraic method calculated using a differential form built from a triplet composed of a spectral algebra.

Finally, it was highlighted by Dirkkreimer and Alain Connes that the group structure for perturbative normalisation can be understood in terms of Hof al-gebra (generalization of the concept of non-commutative group geometry).

The non-commutative geometry is a central notion in both Mathematics and physics. There are many important mathematical structures which do not com-mute, most abstractly, nonabelian groups, which have numerous applications. In quantum mechanics, non-commutative algebras are one of the most important features if two Hermitian operators do not commute then there is an uncertainty relation between their corresponding observables. In quantum mechanics linear position and momentum along the same direction do not commute giving the famous Heisenberg uncertainty relation. In certain situations, it is possible to have linear momentum along different directions not commuting.

In the 1980’s, non-commutative geometry was considered as a way of extend-ing the Standard Model in a number of different ways [11] .

In condensed matter, non-commutative geometry describes, for the example, electrons in a magnetic field at lowest energy level which is related to the quantum hall effect [12 − 13]. Recent interest in non-commutative geometry is strongly motivated by the discovery that gives rise to string theory [14] .

Despite all of these recent motivations, non-commutative geometry remains a nontrivial subject approached from numerous angles and allowing for numerous

generalizations from the simplest. On a fundamental level, the interest in non-commutative geometry results from the fact that it is a non-locality theory and the theory of quantum gravity seems to require non-locality.

1.3

Plan of thesis

In the first part of this thesis we describe the coordinate frame in the non-commutative geometry, which we choose the same as a covariant derivative. On the basis of this frame we introduce generalised general coordinates and local lorentz transformations at the level of non-commutative space time, which are exact symmetries of the canonical non-commutative space time commutation relations. Finally we present the Seibarg-Witten maps for the different fields (matter field, gauge field, virbien fiekd, etc) and we solve the non-commutative transformation equation for these fields in the non-commutative space-time to the order θ.

In the second part we show how the canonical non-commutative space time relations invariance ensures that the scalar Lagrangian density preserves the same form under the above transformations, and we derive the generalized Euler-Lagrange field equation, Noether theorem and modified Dirac equation in a non-commutative curved space-time. In this part we present a work not yet published, which represents the calculation of particle creation density in a non-commutative cosmological anisotropic Bianchi I Universe, by solving the modified Dirac equa-tion in the presence of a constant electric field in a non-commutative space time. The third part, which is published, discusses particle creation process in the non-commutative two dimensional De Sitter Universe.

super-symmetric standard model(MSSM). We propose an extension of the MSSM action for the non-Abelian gauge fields in the context of the non-commutative geometry. Finally we propose a model which is invariant under the non-commutative super-symmetric transformations of first order in θ.

Chapter 2

Non-commutative Gauge field

theory and Siberg-Witten maps

2.1

Non-commutative Gauge field theory

The non-commutative gauge theories are equivalent to commutative ones and in particular there exists a map from a commutative gauge field to a non-commutative one, which is compatible with the gauge structure of each.

Here we describe a slightly modified procedure based on the idea that the structure equations of the gauge group of the non-commutative theory are a deformation of those of the gauge group of the commutative theory.

we will consider gauge theories on the non-commutative space defined by [15]:

[ˆxµ_{, ˆx}ν_] ∗ = iθµν, (1-1) where h ˆ A, bB i

∗ = bA ∗ bB − bB ∗ bA and θ is a constant anti-symmetric Matrix. The

as[16]: f ∗ g = f exp µ i 2θ µν←−∂µ−→∂ν ¶ g (1-2)

We still require some basic properties: Associativity is preserved:

(f ∗ g) ∗ h = f ∗ (g ∗ h) (1-3)

Linearity and distributativity:

af ∗ bg = abf ∗ g (1-4)

where ab ∈ c and f g are function of ˆxµ

The product rule is preserved:

∂µ(f ∗ g) = (∂µf ) ∗ g + f ∗ (∂µg) (1-5)

The star product is combatable with integration in sense that for the functions

f g that vanish rapidly enough at infinity so that one can integrate by parts in

evaluating the following integrals; one has : Z

trf ∗ g =

Z

trg ∗ f (1-6)

Here T r is the ordinary trace.

Now we consider Axto be the algebra of formal power series in the coordinates

modulo the relations

Ax = c

££ ˆ

x1_{· · · bx}n¤¤±_{R (1-7)}

let Ax be an associative algebra ”in non-commutative space” with stare product.

we consider fields ψ (ˆx) = ψ (ˆx1_{· · · ˆx}n_{) as elements of the algebra A} x

The infinitesimal gauge transformation of a field ψ is

which we call a covariant transformation law of a field. It follows then of course that δψ ∈ Ax. Since λ is an element of Ax it is the equivalent to an Abelian gauge

transformation . If it belonged to an algebra Mn(Ax) of matrices with elements

in Ax then it would be equivalent of a non-Abelian gauge transformation.

An essential concept is that the coordinates are invariant under the infinites-imal transformation δˆxi _{= 0, so that}

δ¡xˆi_ψ¢_{= ibx}i_{λ ∗ ψ (1-9)}

since the product of a function and fields is not covariant on a non-commutative space:

δ (f ∗ ψ) = f ∗ δψ = if ∗ λ ∗ ψ 6= iλ ∗ f ∗ ψ (1-10)

the right-hand sid in (1 − 10) is not equal to iλ ∗ ˆxi_{ψ. Following the ideas of}

ordinary gauge theory we introduce covariant coordinates ˆXi _{such that}

δ ³ ˆ Xi_ψ´_{= iλ ∗ ˆX}i_{ψ (1-11)} i.e; δ ³ ˆ Xi´ _{= i}h_{λ, ˆX}ii

∗, to find the relation between ˆX

i _{and ˆx}i _{we make an}

Ansatz of the form

ˆ

Xi _{= ˆx}i_{+ ˆA}i_{; ˆA ∈ A}

x. (1-12)

This is quite analogous to the expression of a covariant derivative as the sum of an ordinary derivative plus a gauge potential. In order to preserve this relation we have to generalize the general coordinates and local Lorentz transformations at the level of the non-commutative space-time manifold.

2.1.1

Generalized infinitesimal general coordinate

trans-formations

Under generalized general coordinate transformations with an infinitesimal non-commutative parameter ˆξµ_{, the coordinates ˆx}µ _{which has as a representation:}

ˆ xµ_{= x}µ₊ i 2θ µρ_∂ ρ (1-13) becomes ˆ xµ _{→ ˆx}0µ _{= ˆx}µ_{+ ˆξ}µ _(1-14) with ˆ ξµ= ξµ+ eξµ+ O(θ2) (1-15)

where eξµ _{is an operator which depends on θ and ξ}µ _{(the ordinary general}

coor-dinate transformations parameter). Using eqs. (1 − 13) − (1 − 15) and requiring that: [ˆx0µ_{, bx}0ν_{] = iθ}µν_{. (1-16)} we arrive at : [xµ_{, eξ}ν_{] +} i 2θ µρ_∂ ρξν = i 2θ νρ_∂ ρξµ+ [xν, eξµ] + O(θ2, ξ2). (1-17)

It is to be noted that this last equation, has as a solution: e ξµ = −i 2θ ρα_∂ αξµ∂ρ+ i 2θ µα_∂ αξρ∂ρ (1-18)

Thus, the generalized coordinate transformations which preserve the canonical non-commutative space-time commutation relations take the form:

ˆ x0µ_{= bx}µ₋ i 2θ ρα_∂ αξµ∂ρ+ i 2θ µα_∂ αξρ∂ρ. (1-19)

Notice that, and contrary to ref.[4], we do not have constraint of unimodularity and consequently ˆξµ _{is not is not of O(θ).}

2.1.2

Generalized infinitesimal local Lorentz

transforma-tions

Under generalized infinitesimal local Lorentz transformations, the non-commutative space-time coordinates ˆxµ _{transform as}

ˆ xµ→ ˆx0µ = ˆΛµLσxˆσ+ ˆΩµ (1-20) where ˆ Λµ_Lσ = δµ σ + λµσ + eλµσ (1-21) and ˆΩµ _{and eλ}µ

σ are operators depending on θ and λ. Here λµσ denotes the

ordi-nary Lorentz transformation parameter. Now, imposing the preservation of the canonical non-commutative commutation relation of eq.(1 − 1), leads to:

[ ˆΩµ, xν] + [xµ, bΩν] + iθµσλν_σ+ iθσνλµ_σ + [xµ, eλν_σ]xσ+ [eλµ_σ, xν]xσ+ + · −i 2θ µρ_∂ ρ, λνσ ¸ xσ₊ · λµ σ, − i 2θ νρ_∂ ρ ¸ xσ _{+ O}¡_θ2_{, λ}2¢ _{= 0 (1-22)}

when the solution is given by: ˆ Ωµ_{= −}i 2θ µσ_λρ σ∂ρ+ i 2θ ρσ_λµ σ∂ρ (1-24) e λµ_σ = −i 4θ ρα_∂ αλµσ∂ρ+ i 4θ µα_∂ αλρσ∂ρ (1-25)

Note that eq. (1 − 20) is similar to Poincar´e or an inhomogeneous Lorentz trans-formation where the translation parameter ˆΩµ_{is related to the rotation parameter}

λρ

σ (see eq.(1 − 23)). This means that the generalized local Lorentz

transforma-tions in the non-commutative space-time induce at the same time a curvature and a torsion [7].

2.1.3

Non-commutative gauge transformations

The non-commutative gauge fields and gauge parameters are given by local dif-ferential expressions in the ordinary fields and parameters. At first sight it seems that we want a local field redefinition ˆA = ˆA, (A, ∂A, ∂2_{A, · · · θ) of the gauge}

fields, and a simultaneous re-parametrization ˆΛ = ˆΛ (Λ, ∂Λ, ∂2_{Λ, · · · , θ) of the}

gauge parameters that maps one gauge invariance to the other. The non-commutative gauge invariance which acts by

ˆ δˆ_ΛVˆi = ∂iΛ +ˆ h ˆ Λ, ˆVi i ∗ (1-26)

is non-Abelian ,wile the ordinary gauge invariance which acts by

δλVi = ∂iλ + [λ, V ] (1-27)

is Abelian. An Abelian group cannot be isomorphic to a non-Abelian group so no redefinition of the gauge parameter can map the ordinary gauge parameter to the non-commutative one while intertwining with the gauge symmetries.

We will find a mapping from ordinary gauge fields V to non-commutative gauge fields ˆV which is local to any finite order in θ and has the following further

property. Suppose that two ordinary gauge fields V and V0 are equivalent by an ordinary gauge transformation by exp (iΛ). Then, the corresponding non-commutative gauge fields ˆV and ˆV0 will also be gauge equivalent, by a non-commutative gauge transformation exp

³

iˆΛ ´

. However ˆΛ will depend on both Λ and V , where V denotes the U (1) gauge field Aµ or the spin connection ωµ or

virbbaien ea µ.

Following the general structure of gauge theories, we require in the non-commutative space-time that under:

Infinitesimal local U (1) gauge transformations : ˆ δˆ_λGAˆi = ∂iλˆi+ h ˆ λi, ˆAi i ∗ (1-28) ˆ δˆ_λGeˆaµ= 0 (1-29) ˆ δˆ_λGωˆµab= 0 (1-30) where: ˆλG = ˆλaTa

Infinitesimal local Lorentz transformations:

ˆ δˆ_λLAˆi = ˆAj ∗ ³ ˆ λ−1 L ´_j i (1-31) ˆ δˆ_λLˆea µ= h ˆ λL, ˆeaµ i ∗ (1-32) ˆ δˆ_λLωˆab µ = ˜∂µλˆabL + h ˆ λai L, ˆωibµ i ∗ (1-33)

where ˆλL= λabLΣab and ˜∂µ= ˆeaµ∂a, with Σab = 1₂[γa, γb] and [Σab, Σcd] = f_abcdij Σij

Infinitesimal local translation transformations:

ˆ δξψ = ˆˆ ξa∗ ∂aψˆ ˆ δξeˆb = ˆξa∗ ∂aˆeb (1-34) ˆ δξωˆµbc= ˆξa∗ ∂aωˆbcµ

Composition of two gauge transformations: h ˆ λ1, ˆλ2 i ∗ ˆϕA _{= ˆλ} ˆ λ1×λ2 ∗ ˆϕ A _(1-35) or h ˆ δλ1, ˆδλ2 i ∗ ˆϕA _{= ˆδ} ˆ λ1×λ2 ∗ ˆϕ A _(1-36)

2.2

Seiberg-Witten maps

The Seiberg-Witten map is one of the most important transformations of non-commutative Gauge theories. It is map from a non-non-commutative theory with parameter θ to an equivalent field theory with a different non-commutative pa-rameter θ. Since commutative gauge theories are fairly well understood, the Seiberg-witten map from non-commutative theories to commutative theories with

θ = 0 is an important tool to understanding non-commutative Gauge theory.

Now we look for a mapping V ¡Aµ, Ψ, Φ, eaµ

¢

for a generic field (Aµ, Ψ, Φ, eaµ

standing for the Gauge field, spinor , scalar filed and veirbein respectively) and

λ (λG, λL). Now we look for a mapping V → ˆV (V ) and λ → ˆλ (λ, V ) such that

[14]:

ˆ

V (V ) + ˆδˆ_λV (V ) = ˆˆ V (V + δλV ) (1-37)

where δλ and ˆδλ denote the ordinary Gauge transformation and non-commutative

Gauge transformation. We first work to first order in θ. We write ˆV (V ) =

V + V0(V ) and ˆλ (λ, V ) = λ + λ0(λ, V ) with V0 and λ0 as a local function of λ

and V of order θ. Expanding (1 − 37) in powers of θ, we find that :

δλV 0 µ− i h λ, V_µ0 i − i h λ0, Vµ i = θµν_∂ νλ 0 − 1 2θ σρ_(∂ σλ∂ρVµ+ ∂σVµ∂ρλ) (1-38)

In arriving at this formula, we used the expansion of Poisson: f ∗ g = f g + i 2θ µν_∂ µf ∂νg + O ¡ θ2¢_{. (1-39)}

All products in(1 − 38) are therefore ordinary matrix products, for example £

λ0, Vµ

¤

= λ0Vµ − Vµλ

0

, where (as λ0 is of order θ), the multiplication on the

right hand side should be interpreted as ordinary matrix multiplication at θ = 0. If V is a Gauge field Aµ, the solution to eq (37) is given by

A0_µ= −1 4θ σρ_{A σ, ∂ρAµ+ Fρµ} + O ¡ θ2¢ _(1-40) λ0 = 1 4θ σρ_{∂ σλ, Aρ} + O ¡ θ2¢ _(1-41)

where again the products on the right hand side, such as {Aσ, ∂ρAµ} = Aσ∂ρAµ+

∂ρAµAσ, are ordinary matrix products; and Fρµ is the classical strength Fρµ =

∂ρAµ− ∂µAρ+ i [Aρ, Aµ].

From the formula for ˆAµ it follows that:

ˆ Fµν = Fµν+ 1 4θ σρ_{(2 {F} µσ, Fνρ} − {Aσ, DρFµν+ ∂ρFµν}) + O ¡ θ2¢. (1-42) If Aµ is the spinor connection which is given in this form: ˆAµ= 1₄ωˆabµγab, then we

have γab = 1₂[γa, γb] as the six generators of Lorentz group. The Gauge fields ˆωabµ

subject to the conditions [17 − 18] ¡ ˆ ωab µ ¢+ = −ˆωba µ (1-43) ¡ ˆ ω_µab¢r = −ˆω_µba (1-44)

Expanding the Gauge fields in powers of θ, we have ˆ ωab µ = ωµab+ ω 0_ab µ (1-45) with ω0ab

Expanding (1 − 38) where the infinitesimal transformation of ωab µ is given by: δλωabµ = ∂µλab+ £ ωac µ, λcb ¤ (1-46.a) and for the non-commutative field it is

ˆ δˆ_λωˆab µ = ∂µλˆab+ h ˆ ωac µ , ˆλcb i ∗ (1-46.b) where ˆλab _{= λ}ab_{+ λ}0_ab

with λ0ab _{is the local function of λ}ab _{and ω}ab

µ of order θ, we find that: δλω 0_ab µ − i h λac, ω_µ0cb i − i h λ0ac, ω_µcb i = θµν∂νλ 0ab − 1 2θ σρ_(∂ σλac∂ρωµcb +∂σωacµ∂ρλcb) (1-47)

The solution to this equation is given by

ω0ab µ = − i 4θ νρ_{ω ν, ∂ρωµ+ Rρµ}ab+ O ¡ θ2¢ _(1-48) λ0ab ₌ i 4θ νρ_{∂ νλ, ωρ}ab+ O ¡ θ2¢ _(1-49)

where we have divided the anti-commutator {A, B}ab = Aac_Bcb _{+ B}ac_Acb _and

Rab

µν = ∂µωνab − ∂νωabµ + [ωµ, ων]ab , where we have divided the commutator

[A, B]ab= Aac_Bcb_{− B}ac_Acb_.

If ˆAµ is the vierbein beaµ as the gauge field of the translational generator of the

inhomogeneous Lorentz group, the infinitesimal transformation of ea

µ is given by δλeaµ= £ λ, ea µ ¤ (1-50) and for the non-commutative field it is

δˆ_λeˆa µ= h ˆ λ, ˆea µ i ∗ (1-51) where ˆea µ = eaµ+ e0aµ ¡ ea µ, ωµab ¢

and ˆλ = λ + λ0. The primed fields are first order in

Expanding (1 − 38) in powers of θ, we find that δλe 0_a µ − i h λ, e0_µa i − i h λ0, ea_µ i = θµν∂νλ 0 −1 2θ σρ¡_∂ σλ∂ρeaµ+ ∂σeaµ∂ρλ ¢ (1-52) the solution to this equation is given by

e0_µa= 1 4θ kl_²a bcd µ ω_kcd∂lebµ− 1 4ω b kieiµωcdl ¶ + O¡θ2¢ = −i 4θ kl©_ωab k ∂lebµ+ ¡ ∂lωµab+ Rablµ ¢ eb_kª+ O¡θ2¢ (1-53) If we have the fundamental matter field, which under an ordinary Gauge transformations transforms as follows

δλΨ = [Ψ, λ] (1-54)

then it is reasonable to postulate the following non-commutative gauge transfor-mation for the corresponding non-commutative field

δˆ_λΨ =ˆ h ˆ Ψ, ˆλ i ∗, (1-55)

where ˆΨ = Ψ + Ψ0_{(Ψ, A) is a non-commutative spinor field ; with Ψ}0_{(Ψ, A) being}

a local function Ψ and A of order θ.

Expanding (36) in powers of θ , we find that

δλΨ0 = [λ, Ψ0] + h λ0, Ψ i − i 2θ σρ_(∂ σλ∂ρΨ + ∂σΨ∂ρλ) + O ¡ θ2¢ _(1-56)

The solution to this equation is given by: Ψ0 = −i 2θ σρ µ {Aσ, ∂ρΨ} + 1 2{[Ψ, Aσ] , Aρ} ¶ + O¡θ2¢ (1-57) If we have an ordinary real scalar field ϕ ,which under an non-commutative Gauge transformations transforms as follows:

δ_λˆϕ =ˆ h ˆ ϕ, ˆλ i ∗ (1-58)

where ˆϕ = ϕ + ϕ0_{(ϕ, A) is a non-commutative real scalar field ; with ϕ}0_{(ϕ, A)}

local function ϕ and A order θ .

Expanding (1 − 38) and we have used the Moyal product of first order in θ , we find that we need :

ϕ0_{(ϕ + δϕ) − ϕ}0_{(ϕ) = −θ}ρσ_∂

ρλ∂σϕ (1-59)

The solution to this equation is given by:

ϕ0 _{= −θ}ρσ_A

ρ∂σϕ + O

¡

θ2¢ _(1-60)

Let ˆϕ an complex scalar field , which for under au non-commutative gauge

transformations transforms as followers

δˆ_λϕ = iˆˆ λ ∗ ˆϕ (1-61)

where ˆϕ = ϕ + ϕ0_{(ϕ, A), with ϕ}0 _{being a local function of ϕ and A of order θ.}

Expanding (1 − 38) and we use the Moyal product of first order in θ, we find that we need :

ϕ0(ϕ + δϕ) − ϕ0(ϕ) = −1 2θ

ρσ_∂

ρλ∂σϕ (1-62)

The solution to this equation is given by:

ϕ0 = −1 2θ ρσ_A ρ∂σϕ + O ¡ θ2¢ (1-63)

When the non-commutativity is induced by the Moyal product we can use the Seiberg-Witten map in order to deal with ordinary fields. The Seiberg-Witten map is usually written in terms of ordinary commutation functions in the star-product formulation To use the operator form of the map, Gauge transformations are easily written in the operator language and are thus amenable to explicit analysis. An important property of non-commutative theories induced by the

Moyal product, which distinguishes them from the conventional ones, is that transformations in the non-commutative directions are equivalent to the gauge transformations [19]. The only other field theory which has a similar property is general relativity where local translations are gauge transformations associated to general coordinate transformations. This remarkable property shows that, as in general relativity, there are no local gauge invariant observables in non-commutative theories. In the description obtained through the Seiberg-Witten map the theory is presented as a series expansion in θ. In this way a local field theory is obtained at the expense of introducing a large number of non-renormalizable interactions [20].

At the classical level, on the other hand, it is possible to understand very clearly the breakdown of Lorentz invariance induced by the non-commutativity. In this case non-commutative field theories can be interpreted as ordinary theo-ries immersed in a gravitational background generated by the Gauge field. We have seen that it is possible to regard non-commutative theories as conventional theories embedded in a gravitational background product by the gauge field. This brings a new connection between non-commutativity and gravitation.

Chapter 3

Non-commutative gravity and

applications

3.1

Introduction

The standard concept of space-time as a geometric manifold is based on the no-tion of a manifold whose points are locally labeled by a finite number of real coordinates. However, it is generally believed that this picture of space-time as a manifold should break down at very short distances of the order of the Planck length. This implies that the mathematical concepts of High energy physics has to be changed or more precisely our classical geometric concepts may not be well suited for the description of physical phenomenon at short distances. Non-commutativity is a mathematical concept expressing uncertainty in quantum me-chanics, where it applies to any pair of conjugate variables such as position and momentum. The most concrete motivation for space-time non-commutativity, where the commutation relations for the canonical variables (coordinate- mo-mentum operators) were generalized to non-trivial ones of coordinate operators,

which came from the work of ref. [21], where it was shown that combining Heisenberg uncertainty principle with Einstein’s theory of classical gravity leads to the conclusion that ordinary space-time loses any operational meaning at short distances; that is a space-time coordinate with a great accuracy ε causes an un-certainty in momentum or energy of the order of 1

², which is transmitted to the

system and concentrated at some time in the localization region. The associated energy-momentum tensor Tµν generates a gravitational field according to Einstein

equations. The smaller the uncertainty ∆xµ, the stronger will the gravitational

field generated by the measurement of coordinates be. When this field becomes so strong as to prevent light from leaving the region in question, an operational meaning can no longer be attached to the localization. Exploring the limitations of localization measurements which are due to the possible black hole creation by concentration of energy, one arrives at uncertainty relations among space-time coordinates which can be traced back to the commutation relations eq. (1 − 1) . From a theoretical point of view and model building, it is a challenge to formu-late a theory of gravitation like general relativity on non-commutative manifold, and there are different approaches in the literature. The main problem is that it is difficult to implement symmetries such as general coordinate covariance and local Lorentz invariance and to define derivatives which are torsion-free and sat-isfy the metricity condition. In a flat space-time, to get non commutative local gauge theories, but with Lorentz violation symmetry, a formulation within the en-veloping algebra approach has been proposed [22 − 24]. Following a similar idea a gauge formulation of gravity is proposed. It is a theory of general relativity on curved space-time with canonical preservation of non-commutative space-time commutation relations and based partially in implementing symmetries on flat non-commutative space-time [25 − 27]. The resulting theory appears to be a

non-commutative extension of unimodular theory of gravitation.

The goal of this chapter is to introduce generalized general coordinates and local Lorentz transformations at the level of non-commutative space-time, which are exact symmetries of the canonical non-commutative space-time commutation relations and to keep the proposed non-commutative theory of gravitation as general as possible (not unimodular, etc...). It is worth to mentioning that the canonical non-commutative space-time commutation relations invariance assures that the scalar Lagrangian density preserves the same form under the above transformations.

This chapter is organized as follows. In section 2 we propose an invariant action of the pure non-commutative Gauge gravity and spinor matter in inter-action with electromagnetic and spin connection fields. In section 3, we derive generalized Euler-Lagrange field equations and Noether theorem. In section 4, we deduce a modified Dirac equation in a non-commutative curved space-time and in the presence of a constant electric field and compute the number den-sity of particle creation in a non-commutative cosmological anisotropic Bianchi I universe. Finally, in section 5, we draw our conclusions.

3.2

QED action in a curved non commutative

space-time

For the non commutative action of QED in a curved non-commutative space-time (where gravity is treated as a gauge theory), we propose the following action:

S = 1

2κ2

Z

d4_{x (£}

where £G and £M are the pure gravity and matter scalar densities respectively

in the non-commutative curved space-time and are given in the holonomic coor-dinates by: £G = ˆe ∗ ˆR (3-2) and £M = ˆe ∗ψ ∗ ˜¯ˆ γµ∗ ˆDµ∗ ˆψ (3-3) with: ˆ e = det∗((ˆeaµ)) ≡ 1 4!² µνρσ_ε abcdˆeaµ∗ ˆebν ∗ ˆecρ∗ ˆedσ (3-4) ˆ R = ˆeµ ∗a∗ ˆeν∗b∗ ˆRabµν (3-5) Dµ ≡ ∆µ− iQAµ, ˆDµ ≡ ˆ∆µ− iQAµ (3-6)

Q is the charge of the electron. In what follows we consider a metric ˆgµνsuch

that: ˆgµν = 1 2(ˆe b µ∗ ˆeνb+ ˆebν∗ ˆeµb) (3-7)

As a consequence, the first order in the non-commutative parameter θαβ _{of the}

scalar curvature ˆR vanishes and ˆR can be rewritten as:

ˆ

R = R + R(2)+ O¡θ3¢ (3-8)

where

R(2) _{= R}rs

rs (3-9)

and the expression of Rmn

rs is given explicitly in ref.[6].

Now, using the generic fields infinitesimal transformations of eqs.(1 − 28) − (1 − 34) and the ∗−tensor relations (see Appendix), it is easy to show that the action of eq.(3 − 1) is invariant [28].

3.2.1

Generalized field equations and Noether theorem

In a general framework of a non commutative space-time geometry and under an infinitesimal Gauge variation of any Siebeg-Witten map of a dynamical field ˆϕA_,

one can write:

ˆ

δ ˆϕA= ˆλ ∗ ˆGA+ ∂µλ ∗ ˆˆ TAµ (3-10)

Moreover, the scalar density £ is a function of the fields and their first and second derivatives i.e.

£ = £¡ϕˆA_{, ∂}

µϕˆA, ∂µ∂νϕˆA

¢

+ O(θ2_{) (3-11)}

and thus, the variation of the scalar density under the infinitesimal gauge trans-formation of eq.(3 − 11) reads:

ˆ δ£ =ˆδ ˆϕA ∂£ ∂ ˆϕA + ˆδ ¡ ∂µϕˆA ¢ _∂£ ∂ (∂µϕˆA) + ˆδ¡∂µ∂νϕˆA ¢ _∂£ ∂ (∂µ∂νϕˆA) + O(θ2_{) (3-12)}

Now, it is easy to show that the vanishing of the variation of the action lead to the modified field equations [7]:

ˆ δ£ ˆ δ ˆϕA ≡ ∂£ ∂ ˆϕA − ∂µ ∂£ ∂ (∂µϕˆA) + ∂µ∂ν ∂£ ∂ (∂µ∂νϕˆA) + O(θ2_{) = 0 (3-13)}

and the following off-shell equations (up to O(θ2_)):

ˆ GA ∂£ ∂ ˆϕA − ∂µ µ ˆ TAµ ∂£ ∂ ˆϕA ¶ + ∂µLµ = 0 (3-14) Lν_α+ ∂µKανµ− θνγΣγ = 0 (3-15) Ξµν _{+ Ξ}νµ _{= 0 (3-16)} and θβγ_∂ γTˆAµ( ∂£ ∂ (∂µ∂ρϕˆA) ) = 0 (3-17) where

Lµ_{= ˆG}A µ ∂£ ∂ (∂µϕˆA) − ∂ν ∂£ ∂ (∂µ∂νϕˆA) ¶ + +∂νGˆA ∂£ ∂ (∂µ∂νϕˆA) + ˆTAµ δ£ δ ˆϕA (3-18) Σγ = ∂γGˆA∂µ ∂£ ∂ (∂µϕˆA) + +³∂ρ∂γGˆA ´ ∂µ ∂£ ∂ (∂µ∂ρϕˆA) (3-19) Kνµ = ˆTAν µ ∂£ ∂ (∂µϕˆA) ¶ + ˆGA µ ∂£ ∂ (∂µ∂νϕˆA) ¶ − ˆ TAµ∂ρ µ ∂£ ∂ (∂µ∂ρϕˆA) ¶ + ∂ρTˆAµ µ ∂£ ∂ (∂µ∂ρϕˆA) ¶ + θνγ_∂ γGˆA∂µ ∂£ ∂ (∂µϕˆA) +³∂ρ∂γGˆA ´ ∂µ ∂£ ∂ (∂µ∂ρϕˆA) (3-20) and Ξνµ ₌ 1 2 · Kνµ_{+ ∂} ρ( ˆTAµ ∂£ ∂ (∂µ∂ρϕˆA) ) + θβγ ½ ∂γTˆAν ∂£ ∂ ˆϕA+ (η_µν∂γGˆA+ ∂µ∂γTˆAν) ∂£ ∂ (∂µϕˆA) + (2ην ρ∂µ∂γGˆA+ ∂ρ∂µ∂γTˆAν) ∂£ ∂ (∂µ∂ρϕˆA) ¾¸ (3-21) Now, the Modified Noether current ˆJν _{is defined as:}

ˆ

Jν = Lν_α+ ∂µKανµ− ∂µΞνµ− θνγΣγ (3-22)

Notice that using the field equations and eqs. (3 − 14) − (3 − 21), one can show easily that:

ˆ

Jν _{= −∂}

µΞνµ (3-23)

and from the anti-symmetry of Ξνµ _{we obtain:}

∂νJˆν = 0 (3-24)

Thus, the non commutative space-time Noether current defined in eq.(3 − 24) is by construction a conserved quantity.

3.2.2

Generalized Dirac equation and particle creation

process

During the last few years a great effort has been made in understanding quantum processes in strong fields, where the associated vacuum instability leads to an additional source of quantum processes and could enhance the particle creation mechanism and produces deviations from the thermal spectrum. An interesting scenario for discussing particle creation process is the non-commutative version of the Bianchi I anisotropic universe associated with the metric:

ds2 = −dt2+ t2¡dx2+ dy2¢+ dz2. (3-25) with dimensionless space-time coordinates. Since the metric presents a space-like singularity at t = 0, it is difficult to define the particle states within the adiabatic approach [29]. To overcome this problem, we follow the quasi-classical approach of ref. [30]. First we identify the positive and negative frequency modes by solving the classical Hamilton-Jacobi equation and look for the asymptotic behaviour of the solutions when t → 0 and t → ∞. Second, we solve the Dirac equation by comparing it with the above quasi-classical limits, we identify the positive and negative frequency states. Third, using Bogouliubov transformations we compute the particle number of the created particles. It turns out that in the quasi-classical limit, the positive and negative frequency modes have positive (resp. negative) eigenvalues.

Now, regarding the Dirac equation in a curved non-commutative space-time and in the presence of an electromagnetic field and using the modified field equa-tions of eq.(3 − 13), with the generic fieldψ such that:¯ˆ

one can find in the co-moving coordinates: h ˆe D − m + i 2θαβ n [(∂αγ˜µ)(∂βDeµ) + ˜γµ(∂αDeµ)∂β]+ (∂αln ˆe)[∂β(˜γµA˚µ) + ˜γµA˚µ∂β+ m ∂β] o + O(θ2₎i_{Ψ = 0}ˆ , (3-27) with e D = ˜γµ∗ eDµ (3-28) and ˚ Aµ = ˜ωµ+ iQ eAµ (3-29)

and for a convenience in the notations, we denote by: ˜γµ_{, e∆}

µ, ˜ωµ, and eAµ etc...,

the γµ_{, ∆}

µ, ωµ, and Aµ etc... in the curved space-time. The matrices ˜γµ satisfy

the clifford algebra:

{eγµ∗_,eγν_{} = 2g}µν _(3-30)

where {.∗_{,.} stands for the anti-commutator with the star product.}

Notice that, in ref. [31], the authors derived the Dirac equation in a commutative curved space-time by just taking the Dirac equation in a non-commutative flat space-time used in ref. [32] and replacing the ordinary deriv-ative ∂µ by the covariant one Dµ. However, this procedure does not work in a

non-commutative space-time when one deals with ∗-product (see appendix). It is worth to mention that since the metric of eq. (3 − 30) is diagonal, and for further simplifications, we choose to work with a diagonal tetrad too. Moreover, the vector potential eAµ is chosen such that:

e

Aµ = (0, 0, 0, −Et) (3-31)

which corresponds to a constant electric field along the z-direction. This means that the system possesses a rotational symmetry along the z-axis, preserving the

invariance of the metric of eq. (3 − 30). To visualize the influence of the non-commutativity, we use the parametrization that determines the elements of the

θ-matrix from the direction of the background electromagnetic field [33]:

Cµν ₌         

0 sin α cos β cos α sin β cos α

− sin α cos β 0 sin γ − cos γ sin β

− cos α sin β − sin γ 0 − sin γ cos β

− cos α cos γ sin β sin γ cos β 0

         . (3-32)

and again keep the background electric field parallel to the z-axis to take benefit from the existed rotational symmetry in order to be sure that the particles number density remains unchanged under this symmetry. Geometrically, we have α = β =

γ = 0 to obtain θ03 _{= −θ}30 _{= θ, and set all the remaining components equal to}

zero. Now, using the fact that: ˚ A0 = 0, ˚A3 = −iQEt, ˚A1 = 1 2γ 0_γ1_{, ˚A} 2 = 1 2γ 0_γ2_{, (3-33)} θαβ_(∂ αγ˜µ)(∂βDeµ) = − θ t2 ¡ γ2_∂ 2∂3+ γ1∂1∂3 ¢ (3-34) θαβ_γ˜µ_(∂ αDeµ)∂β = iθQEγ3∂3 (3-35) and θρσ_∂ ρln ˆe∂σ = 2θ t ∂3 (3-36)

the Dirac equation and up to the O(θ2_{) takes the form:}

[γ0 µ ∂0− i θ t2∂3 ¶ + γ3³_eQ₂θE_∂ 3+ iQEte− 1 2θE ´ +1 t ¡ γ1_∂ 1 + γ2∂2 ¢ −i θ 2t2 ¡ γ2_∂ 2∂3+ γ1∂1∂3 ¢ − m(1 − 1 tiθ∂3)]t ˆΨ0 = 0 (3-37) where ˆ Ψ = t ˆΨ0 (3-38)

The factor t was introduced in order to cancel the contribution due to the spin connection in the term proportional to γ0_{. It is important to mention that eq.}

(3 − 37) can be rewritten as a sum of two first order differential equations as follows [34]: ³ ˆ K1+ ˆK2 ´ ˆ Φ = 0 (3-39) with ˆ K2Φ = k ˆˆ Φ (3-40) ˆ K1Φ = −k ˆˆ Φ (3-41) where ˆ Φ = γ3γ0Ψˆ0 (3-42)

and k is a separation constant. The operators ˆK1 and ˆK2 have as expressions:

ˆ K1 = t · [γ3 µ ∂0− iθ t2∂3 ¶ + γ0 ³

eQ2θE∂3+ iQEte−12θE

´ − i θ 2t2 ¡ γ2_∂ 2∂3+ γ1∂1∂3 ¢ γ3_γ0_{− γ}3_γ0_{m(1 −} 1 tiθ∂3)] ¸ , (3-43) and ˆ K2 = (γ1∂1+ γ2∂2)γ3γ0, (3-44)

Notice that in order to keep our results compact and transparent, the approxi-mation

1 + θg ≈ eθg (3-45)

was made (g is an arbitrary regular function). The spinor ˆΦ can be written as: ˆ

where ˆΦ0 is a bi-spinor. Working in the representation where the Dirac matrices

in the tangent space are given by:

γ0 ₌   −iσ1 0 0 iσ1   , γ1 ₌   0 i −i 0   , γ2 ₌   σ2 0 0 −σ2   , γ3 ₌   σ3 0 0 −σ3   (3-47) and the σi’s denote the 2×2 Pauli matrices. We deduce that eqs.(3−40)−(3−41)

reduce to algebraic equations that permit us to determine the relation between the components of the bi-spinor ˆΦ0 such that:

ˆ Φ0 =   Φˆ1 ˆ Φ2   =   Φˆ1 σ2iky_kx−kΦˆ1   =   Φˆ1 σ2_ikk_yx_+kΦˆ1   _(3-48)

and the eigenvalue k is given by:

k = ik⊥ ≡ i

q

k2

x+ ky2 (3-49)

By using the representation in eq.(3 − 47) and taking into account the spinor structure of eq.(3 − 48), we deduce that for kz = 0, eq.(3.37) together with

eq.(3 − 39) reduces to the following system of coupled differential equations:

d ˆΦ1 dt + k tΦˆ1− ³ m + iQEte−12Eθ ´ ˆ Φ2 = 0, (3-50) −d ˆΦ2 dt + k tΦˆ2− ³ m − iQEte−1 2Eθ ´ ˆ Φ1 = 0, (3-51)

In this way, we have reduced the problem of solving eq.(3 − 39) to that of finding the solutions of eqs. (3 − 50) − (3 − 51). From eqs. (3 − 50) and (3 − 51), we obtain the second order differential equation:

  d 2 dt2 − k2 t2 + 2k t2 − 3/4 ³ t − im QEe 1 2Eθ ´₂ + m2_{+ Q}2_E2_t2_e−1₂Eθ    ˆΦ2 = 0, (3-52)

where the variable ˆΨ2 such that: ˆ Ψ2 = µ t + im QEe 1 2Eθ ¶−1 2 ˆ Φ2 (3-53)

was introduced. Now, if we neglect the mass in the first order variation of ˆΦ2,

eq.(3 − 52) becomes: · d2 dt2 − k2_{− 2k + 3/4} t2 + m 2_{+ Q}2_E2_t2_e−1 2Eθ ¸ ˆ Ψ2 = 0, (3-54)

Following ref.[10], the solution ˆΨ2of eq.(2−54) can be expressed as a combination

of Whittacker functions Mλ,µ and Wλ,µ:

ˆ

Ψ2 = C1Mλ,µ(iQEt2e−

1

2Eθ) + C₂W_λ,µ(iQEt2e−12Eθ) (3-55)

where λ = + imt 4QEe 1 2Eθ (3-56) and µ = k − 1 2 (3-57)

(C1 and C2 are normalization constants). Similarly, after introducing the new

variable ˆΨ1 such that:

ˆ Ψ1 = µ t + im QEe 1 2Eθ ¶−1 2 ˆ Φ1 (3-58)

and neglecting the mass in the first order variation of ˆΦ1, we obtain the following

second order differential equation: · d2 dt2 − k2_{+ 2k + 3/4} t2 + m 2_{+ Q}2_E2_t2_e−1 2Eθ ¸ ˆ Ψ1 = 0, (3-59)

where the solution is: ˆ Ψ1 = C3Mλ,µ+1(iQEt2e− 1 2Eθ) + C₄W_λ,µ+1(iQEt2e− 1 2Eθ) (3-60)

(C3 and C4 are normalization constants). In order to construct the positive and

negative frequency modes, we use the asymptotic behavior of the solutions ˆΨ1 and

ˆ

Ψ2 and compare the result with that obtained by solving the Hamilton-Jacobi

relativistic equation. In fact, for t → 0, one can show that the positive and negative frequency solutions ˆΨ+_{(t → 0) and ˆΨ}−_{(t → 0) respectively are given by}

the following asymptotic forms: ˆ Ψ+_{(t → 0) ≈ C}+ 0 Mλ,µ(iQEt2e− 1 2Eθ) (3-61) and ˆ Ψ−(t → 0) ≈ C₀+(−1)−µ+1/2Mλ,−µ(iQEt2e− 1 2Eθ) (3-62)

where the Whittacker function Mλ,µ(z) has the asymptotic behaviour:

Mλ,µ(z) ≈ e−z/2zµ+1/2 ; z ¿ 1 (3-63)

and C+

0 is a normalization function. Similarly, for t → ∞, the corresponding

positive and negative frequency modes are: ˆ Ψ+_{(t → ∞) ≈ C}+ ∞Wλ,µ(iQEt2e− 1 2Eθ) (3-64) and ˆ Ψ−(t → ∞) ≈ C_∞−W−λ,µ(−iQEt2e− 1 2Eθ) (3-65) (C±

0 are normalization functions). Now, using the fact that [35]: Mλ,µ(z) = Γ(2µ + 1) Γ(µ − λ + 1/2)e −iπλ_W −λ,µ(−z) + Γ(2µ + 1) Γ(µ + λ + 1/2)e −iπ(λ−µ−1/2)_W λ,µ(z) (3-66) and W−λ,µ(−z) = (Wλ,µ(z))∗ (3-67)

(Γ (x) is the Euler Gamma function) we deduce that: ˆ Ψ−_{(t → 0) =} Γ(2µ + 1) Γ(µ − λ + 1/2)e −iπλ_Ψˆ−_{(t → ∞)} + Γ(2µ + 1) Γ(µ + λ + 1/2)(−1) 1/4_e−iπ(λ−µ−1/2)_{( ˆΨ}−_{(t → ∞))}∗ _(3-68)

Now, since we are able to obtain the single particle states in the vicinity of t → 0 and t → ∞, we can then compute the density of the created particles ˆn by the

non-commutative curved space-time and electromagnetic field. In fact, with the help of the Bogouliugov transformations [36 − 38]:

ˆ

Ψ−_{(t → 0) = ˆα ˆΨ}−_{(t → ∞) + ˆβ ˆΨ}+_{(t → ∞) (3-69)}

and the use of eq. (3 − 69), as well as the normalization condition:

|ˆα|2+ ¯ ¯ ¯ ˆβ ¯ ¯ ¯2 = 1 (3-70) we obtain: ˆ n =       ¯ ¯ ¯ ˆβ ¯ ¯ ¯2 |ˆα|2    −1 + 1    −1 (3-71) where _¯ ¯ ¯ ˆβ ¯ ¯ ¯2 |ˆα|2 = e 2iπµ ¯ ¯Γ¡1 2 + µ − λ ¢¯_¯2 ¯ ¯Γ¡1 2 + µ + λ ¢¯_¯2 (3-72)

Using the relation:

|Γ (ix)|2 = π

x sinh πx (3-73)

Direct simplifications lead to: ¯ ¯ ¯ ˆβ ¯ ¯ ¯2 |ˆα|2 = e −πk⊥ µ k⊥ 2 − m 2 4QEe− 12 Eθ ¶ µ k⊥ 2 + m 2 4QEe− 12 Eθ ¶ . sinh π µ k⊥ 2 − m 2 4QEe− 12 Eθ ¶ sinh π µ k⊥ 2 + m 2 4QEe− 12 Eθ ¶ , (3-74)

and the number density of the created particles ˆn reads

ˆ

where n denotes the ordinary number density of created particles in the presence of an electric field and has as expression:

n = e−πk⊥ K − ⊥sinh πK⊥− K+ ⊥sinh πK⊥++ K⊥−e−πk⊥sinh πK⊥− , (3-76)

and nθ is the generated non-commutative correction of order θ given by:

nθ = −θ (k⊥)   k⊥ 2 eπk⊥+ m2 2QEsinh π ³ m2 2QE ´ −k⊥ 2 cosh π ³ m2 2QE ´ ¡ K+ ⊥sinh πK⊥++ K⊥−e−πk⊥sinh πK⊥− ¢₂   , (3-77) where θ (k⊥) = θπm 2 8Q e−πk⊥ and K⊥± = k2⊥ ± m 2 4QE

Notice that our results differ from the one obtained in ref. [31] at the order θ. In fact, if one takes kz = 0 in eq. (2−37) of this reference (as in our case), the

non-commutative correction vanishes . As it is mentioned before, the reason lies in the fact that ref. [31] did not use the right expression of the non-commutative space-time Dirac equation which should be derived from the modified field equations (based on Moyal product) and the use of the Seiberg-Witten maps.

It is also worth to consider the weak and strong electric field limits and see the behavior of the number density and derive some of the related thermodynamic quantities.

3.2.3

The weak field approximation

In this limit, if we set:

x = 2QE πm2(1 − E θ 2 ) (3-78) such that: |x| < 1 and E θ 2 < 1 (3-79)

It is easy to show that the number density ˆn takes the form:

ˆ

n = 1

This density is thermal and looks like a two-dimensional Fermi-Dirac distribution with a temperature correction by a factor of 1/(1 + x).

To get the total number ˆN of the created particles per a unit volume, we

have to integrate the particle density bn over the momentum space . Taking into

account the fact that bn does not depend on kz [34], the total number bN reads:

ˆ

N ≈ QE

4π4_T.Γ(2) ζ(2)/(1 + x)

2 _(3-81)

where T is the time the external field interaction, Γ(b) the Gamma function and

ζ(b) has the following expression:

ζ(b) = ∞ X l=1 1 lb (3-82) with ζ(2l) = 2 2l_{− 1} 2l π 2l_B l (3-83)

for l is an integer and Bl reads for the Bernouli numbers (B1 = 1₆, B2 = ₃₀1, etc...).

Regarding the non-commutative internal energy ˆU and the grand potential ˆΩ per a unit volume, it is straightforward to show that:

ˆ U = QE 8π5_T.Γ(3)ζ(3)/(1 + x) 3 _(3-84) and ˆ Ω ∝ QE 8π5_T.Γ(3)ζ(3)/(1 + x) 2 _(3-85)

Notice that, since the gravitational density is proportional to 1/T 2 _it

de-creases faster than ˆN and consequently the particle creation mechanism

effec-tively isotropises also in the presence of a constant electric field of the anisotropic Bianchi I universe of the non commutative space-time.

3.2.4

The strong field approximation:

In this limit, if we set:

y = m2 4QE(1 + E θ 2 ) (3-86) such that: |y| < 1 and E θ 2 < 1 (3-87)

Direct simplifications show that the number density ˆn takes the form:

ˆ n = 1 1 + f (k, y)eπk⊥ (3-88) where f (k, y) = v w ev _{− 1} 1 − e−w (3-89) with v = 2π(k⊥ 2 + y) (3-90) and w = 2π(k⊥ 2 − y) (3-91)

Notice that at very high energies (ultra relativistic limit), the number density takes the form:

ˆ

n ≈ e−2π(k⊥₂ +y)_≈ 1

1 + e2πy_eπk⊥ (3-92)

and it look like a two-dimensional Fermi-Dirac distribution with a fugacity z and a chemical potential µ given by:

z = e−2πy _(3-93)

and

3.3

Conclusion

Thought out this work, we have first constructed generalized infinitesimal local coordinates and Lorentz transformations which preserve the non-commutative coordinates canonical commutation relations. It turns out that the form of the latter, looks like classical inhomogeneous Lorentz transformations. This suggests that non-commutativity induces gravity [31]. Second, and contrary to ref. [21], we do not have unimodular theory of gravity. Third, using Seiberg-Witten fields and Moyal product, we have generalized the equations of motion and Noether theorem. Fifth, we have constructed an invariant action of a pure non-commutative gravity with a symmetric non-commutative metric and a Seiberg-Witten spinor matter field in interaction with a Seiberg-Witten electromagnetic field potential in a curved non-commutative space-time with respect to modified infinitesimal local Poincar´e and U(1) transformations. As an application to the generalized field equations, we have derived the modified Dirac equation and studied the particle creation process in a non-commutative space-time anisotropic Bianchi I universe. After straightforward calculations, using the Bogouliubov transformations and the quasi-classical limit for identifying the positive and negative frequency modes, we have deduced the corresponding number density as a function of θ. We have studied both the weak and strong field limits and obtained under certain circum-stances a Fermi-Dirac like distribution. In the weak electric field approximation certain thermodynamic quantities like the total number of the created particles, internal and grand potential per a unit volume were calculated. Moreover, and as in the ordinary case [30], we have found also that in this limit the gravitational density decreases faster than the total number ˆN. Consequently, the particle

an anisotropic Bianchi I universe in the presence of an electric field. It is worth to mention that in the limit θ → 0 and high energies our results coincide with those or ref. [30]. As a conclusion, the non-commutativity plays the same role as the electric field and gravity and contribute to the pair creation process.

Appendix

Tensor calculus

Under a space-time local transformation infinitely closed to the identity with a parameter ˆΛ = ˆΛ(x) such that:

ˆ

Λ = I + λ + O(ˆλ2_{) (A1)}

we define a *-tensor Ti1...ip

j1...jq (x) of the type (q, p) transformations as:

Ti1....ip j1....jq(x) −→ T 0i1....ip j1....jq (x 0_{) = ˆΛ}i1 l1 ∗ ˆΛ i2 l2 ∗ ...ˆΛ ip lp ∗ T l1....lp k1....kq(x) ∗ ³ ˆ Λ−1 ´_kq jq ∗ .. ∗ ³ ˆ Λ−1 ´_k1 j1 (A2) where ˆΛ−1 _{is the inverse of ˆΛ.}

Up to the the first order in the non-commutative parameter θ the following relations hold:

A ∗ B = B¯∗A (A3)

ˆ

Λ_ρµ∗ ˆΛν_σ = ˆΛµ_ρ∗0Λˆν_σ = ˆΛν_σ∗ ˆΛµ_ρ (A4) and

ˆ Λµ ρ ∗ ³ ˆ Λ−1´ρ σ = ³ ˆ Λ−1´ρ σ∗ ˆΛ µ ρ = ˆΛµρ∗0 ³ ˆ Λ−1´ρ σ = ³ ˆ Λ−1´ρ σ ∗ 0_Λˆµ ρ = δσµ (A5) where A ∗ B = AB +1 2θ µν_∂ µA∂νB + O(θ2) A¯∗B = AB − 1 2θ µν_∂

µA∂νB + O(θ2) (A6)

A ∗0_{B = AB +}1 2θ µν_∂0 µA∂ν0B + O(θ2) We notice that A ∗ B = AB +1 2θ µν_∂ µA∂νB + O(θ2) = BA −1 2θ νµ_∂

νB∂µA + O(θ2) = B¯∗A (A7)

Using eq.(A1), and

δν

σ ∗ ˆλµρ = δνσ∗0λˆµρ = ˆλµρ∗ δνσ (A8)

Up to the O(ˆλ2_{), one can show easily that:}

ˆ

Λµ_ρ ∗ ˆΛν_σ = ˆΛµ_ρ∗0Λˆν_σ = ˆΛν_σ∗ ˆΛµ_ρ = I + ˆλµ_ρ + ˆλν_σ+ O(ˆλ2) (A9) Similarly, using the fact that:

³ ˆ Λ−1 ´_ρ σ = I − ˆλ ρ σ + O(ˆλ2) (A10)

and the relation (A8), we deduce relation (A5).

If ˆΛ is a local Lorentz transformation parameter infinitely closed to the iden-tity , then up to the the first order in the non-commutative parameter θ we have:

ˆ U ∗ ˆU−1 ∗ = ˆU∗−1∗ ˆU = ˆU ∗0 Uˆ∗−1 = I (A11) ˆ U−1 ∗ γk∗0Λˆkj ∗0U = γˆ j if and only if Σab = 1 4[γa, γb] (A12) where ˆ U = exp µ 1 2λˆ ab_Σ ab ¶ (A13) Since ˆλab _{is an infinitesimal parameter eq.(A13) can be rewritten as:}

ˆ

U = I + 1

2λˆ

ab_Σ

ab+ O(ˆλ2) (A14)

It is easy to define the ∗−inverse ˆU−1

∗ of ˆU as ˆ U = I − 1 2ˆλ ab_Σ ab+ O(ˆλ2) (A15)

Using relations (A8), we obtain ˆ U−1 ∗ γk∗0Λˆjk∗0U = −Σˆ abγkλˆab∗0δkj ∗0I + γkI ∗0δkj ∗0λˆabΣab +γk_{I ∗}0 _λj k∗0 I + γkI ∗0δ j k∗0I + O(ˆλ2) = −Σabγj + γjΣab+ γj + O(ˆλ2) (A16)

Now, the relation (A12) holds if £ γj_{, Σ} ab ¤ = 0 (A17) as a solution Σab = 1 4[γa, γb] (A18)