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Dipl.-Pliys. Elena Gnbaiikova

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FLOW EQUATIONS FOR THE QUANTUM ELECTRODYNAMICS ON THE LIGHT-FRONT

Gutachter: Prof. Dr. Franz Wegner

Prof. Dr.Hans-Christian Pauli

Abstract

The method of flow equations is applied to QED on the light front. Requiring that the particle number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal an effective Hamiltonian is obtained which reduces the positronium problem to a two-particle problem, since the particle number violating contributions are eliminated.

Using an effective electron-positron Hamiltonian, obtained in the second order in coupling, we analyze the positronium bound state problem analytically and numerically. The results obtained for Bohr spectrum and hyperfine splitting coincide to a high accuracy with experimental values. The rotational invariance, that is not manifest symmetry on the light-front, is recovered for positronium mass spectrum.

Except for the longitudinal infrared divergences, that are special for the light-front gauge calculations, no infrared divergences appear. The ultraviolet renormalization in the second order in coupling constant is performed simultaneously. To preserve boost invariance we take into account the diagrams arising from the normal ordering of instantaneous interactions. Using flow equations and coupling coherence we obtain the counterterms for electron and photon masses, which are free from longitudinal infrared divergences.

Abstrakt

In dieser Arbeit wird die Methode von Flußgleichungen im Lichtkegel-Formalismus auf die QED angewandt. Wir konstruieren einen effektiven, block-diagonalen Hamiltonian, wobei wir fordern, daß diejenigen Terme, welche die Teilchenzahl erhalten, diagonal sind, während alle anderen Terme nichtdiagonal sind. Dieser effektive Hamiltonian vereinfacht das Positronium-Problem auf ein Zweikörper-Problem, weil die Anteile, die eine Teilchenzahländerung verursachen, eliminiert sind.

Mittels des effektiven Elektron-Positron-Hamiltonians, der in der elektromagnetischen Kopplung von zweiter Ordnung ist, werden die Bindungszustände des Positroniums analytisch und numerisch untersucht. Unsere Resultate für das Bohr-Spektrum und die Hyperfeinaufspaltung stimmen sehr gut mit den experimentellen Werten überein. Die Rotationsinvarianz, welche auf dem Lichtkegel nicht mehr gewährleistet ist, kann für das Massenspektrum wieder hergestellt werden.

Außer longitudinalen Infrarot-Divergenzen, die für die Lichtkegel-Eichung spezifisch sind, treten bei den Rechnungen keine Infrarot-Divergenzen auf. Die Renorrnierung der Ultraviolet-Divergenzen bis zur zweiten Ordnung in der Kopplungskonstanten erhält man mit der Konstruktion des Hamiltonians automatisch. Um die Invarianz des renormierten Hamiltonian unter Boosts sicherzustellen, berücksichtigen wir auch die Diagramme, welche aus der Normalordnung von instantanen Wechselwirkungen entstehen. Die aus den Flußgleichungen und der Kopplungskohärenz resultierenden Counterterme der Photon- und Elektronmasse erhalten dann keine longitudinalen Infrarot-Divergenzen.

Contents

1 Introduction 5

2 Flow equations 7

2.1 Flow equations for Hamiltonian matrices............................................7

2.2 Similarity transformation..............................................................8

2.3 Flow equations for solid state physics................................................10

2.4 Flow equations in field theory........................................................10

3 Light-front field theory 13

3.1 Introduction............................................................................13

3.2 Preceding work........................................................................14

3.2.1 Models and methods in the light-front field theory..........................14

3.2.2 Results for light front QED3+1 ..............................................15

3.3 Canonical QED Hamiltonian..........................................................17

3.3.1 Canonical light-front QED3+1 Hamiltonian..................................17

3.3.2 QED Hamiltonian in second quantization....................................20

4 Hamiltonian bound state problem on the light-front 22

4.1 Introduction............................................................................22

4.2 Flow equations in the perturbative frame............................................23

4.3 Effective low-energy Hamiltonian....................................................25

4.3.1 Effective electron-positron interaction........................................25

4.3.2 Positronium model (general computational strategy) ......................30

4.3.3 Effective, renormalized QED Hamiltonian ..................................31

5 Positronium spectrum (analytically) 34

5.1 Bound state perturbative theory (BSPT)............................................34

5.2 Effective electron-positron interaction in light-front and instant frames ..........37

5.3 Positronium's ground state spin splitting............................................39

6 Positronium spectrum (numerically) 44

6.1 Light-front bound state equation ....................................................44

6.2 Brodsky-Lepage prescription of the light-front dynamics: effective electron-positron interaction..............................................................................46

6.2.1 First and second order solutions of the flow equations......................46

6.2.2 Effective electron-positron interaction........................................48

6.3 Mass spectrum and wave functions of positronium..................................54

7 Renormalization in the light-front QED 57

7.1 Introduction............................................................................57

7.2 Flow equations for renormalization issues............................................60

7.3 Mass renormalization..................................................................67

8 Conclusions and outlook 69 A Calculation of the commutator Heei] in the electron-positron sector 71

B Matrix elements of the effective inter action. Exchange channel 75

B.l The general helicity table..............................................................76

B.2 The helicity table for arbitrary Jz......................................................78

C Fermion and photon self energy terms 80

Bibliography 84

List of Figures 87

Figure 1: Block-diagonalization of Hamiltonians..........................................88

Table 1: Effective light-front QED Hamiltonian..........................................89

Figure 2: Matrix elements of the effective Hamitonian....................................90

Figure 3: Effective electron-positron interaction..........................................92

Figure 4: Effective electron-positron interaction (another choice of coordinates) .... 93

Figure 5: Positronium spectrum............................................................94

Figure 6: Stability of positronium spectrum..............................................95

Figure 7: Deviation of corresponding eigenvalues for Jz = 0 and Jz = 1................96

Figure 8: Electron self energy..............................................................97

Figure 9: Photon self energy................................................................98

Acknowledgments 99

Chapter 1 Introduction

Quantum chromodynamics (QCD) is widely accepted as the microscopic theory of strong interaction, where quarks and gluons are considered as the elementary degrees of freedom. But we are still unable to solve this theory on the macroscopic level (in the low energy domain) and to obtain an accurate description of the structure of hadrons, which are the strongly interacting particles observed in nature.

A central goal of modern theoretical particle physics is to build a bridge between microscopic (high energy) and macroscopic (low energy) domains of QCD.

At the microscopic level the theory is defined completely, and no further development seems to be necessary. On the other side, when it comes to the macroscopic hadron world, there is no rigorous way to calculate hadron properties immediately from QCD. One still has to resort to different phenomenological methods and QCD-inspired models of covariant field theory to calculate the hadron mass spectrum, form factors, wave functions, etc. Also lattice calculations serving as a model-independent method in covariant field theory improved our understanding of the hadronic structure. However, besides different problems, this approach is still strongly limited by the available computer power.

In this work we use the light-front formulation of field theory, which is best suited for solving relativistic bound state problems in QCD [1], because of the simplified vacuum structure. We believe that light-front field theory together with the renormalization group approach for Hamiltonians [2],[3] provides a good strategy to reach the above mentioned goal. The physical idea behind this approach is to use the renormalization concept for Hamiltonians on the lightfront to get an effective, low-energy Hamiltonian.

Recently, Glazek and Wilson suggested the renormalization scheme for Hamiltonians, called similarity renormalization by the authors, where they developed a renormalization group and the basic elements of renormalization group calculations for Hamiltonians on the light-front [3]. An alternative approach for Hamiltonian renormalization, the method of flow equations, was proposed by Wegner [2]. It is common for both methods that they renormalize the canonical Hamiltonian of light-front field theory to a given order of perturbation theory. A basic advantage of the method of flow equations in comparison to the similarity renormalization scheme is that one obtains an effective, renormalized Hamiltonian for a limited (truncated) Fock space.

The method of flow equations is based on the following idea: performing a set of infinitesimal unitary transformations, with the condition that the particle number conserving terms in the Hamiltonian are considered to be diagonal and the other terms off-diagonal, it is possible to get the block-diagonal effective Hamiltonian, where the particle number in each block is conserved. This reduces the bound state problem to a few-body problem, since the particle number violating contributions are eliminated. This procedure is similar to the Tamm-Dancoff space

truncation [33] in the sense that also in this truncation particle number changing interactions are eliminated. The effective Hamiltonian constructed by flow equations is automatically renor-malized to the given order in the coupling constant, since an elimination of particle number changing sectors can not be achieved in one step but rather sequently for transition amplitudes from large to small energy differences.

Low-energy QCD is challenging and explicit calculations become complicated due to the nonperturbative nature of this theory. To test and to illustrate our approach we consider QEDz+i in the light-front dynamics and investigate the corresponding positronium bound state problem.

This work is organized as follows: In chapter 2 we outline the theoretical framework by giving the key ingredients of the flow equation method and by enumerating their applications to problems of solid state physics and statistical mechanics done so far. The approach of similarity renormalization is also considered. In chapter 3 we review the methods and results known for light-front QEDz+\. To second order in the coupling constant we obtain the effective, renormalized QED Hamiltonian (chapter 4), which reduces the positronium problem to a two-particle problem to be analyzed further analytically (chapter 5) and numerically (chapter 6) for positronium bound states. The renormalization issues of light-front QED are considered in chapter 7.

Chapter 2 Flow equations

In this section we give the key ingredients of flow equations method and set up the framework to use flow equations for the problems of high-energy physics.

Flow equations are introduced in order to bring Hamiltonians closer to diagonalization. The method is based on the numerical recipe by Jacobi, consisting of unitary transformations between two states which makes the connecting off-diagonal matrix elements vanishing. If this is repeated for all off-diagonal matrix elements again and again then the off-diagonal matrix elements will become arbitrarily small. It is characteristic for these equations that matrix elements between degenerate or almost degenerate states do not decay or decay very slowly. In the next section we follow the original work of Wegner [2] to introduce flow equations for Hamiltonian matrices.

2.1 Flow equations for Hamiltonian matrices

The aim is to find continuous unitary transformation, that brings Hamiltonian to diagonalization. Continuous unitary transformation depend on the flow parameter I. We call it Uil). In what follows we assume that U(0) = 1 is valid and that U(oo) brings the given Hamiltonian operator H or the given matrix H to the diagonal form. The operator H acquires the /-dependence through the unitary transformation U(/)

If the transformation U(l) were known, one would find H(oo) and the problem would be solved. Generally, one does not know the transformation, which diagonalizes the given matrix. Therefore the transformation is formulated in infinitesimal form

Here r) is the generator of transformation; rj is antihermitian t]+ = —17. The connection between the unitary transformation and its generator is given

H(l) = U+{l)HU(l)

(2.1)

(2.2)

where the ordering along '1-axis' is imposed. One has for the matrix elements

(2.3)

^f1 = E(%p(0 W) - W)»M0)

(2.4)

The generator rj must be chosen in a way, that the matrix H is more and more diagonal as I increases. We demand, that falls monotonously. We separate H = Hd + Hr and use

that use that TrH2 is invariant under the unitary transformation

TrHj + TrH2r = TrH2 = const (2.5)

Y^k^q ^Iq = TrH2 monoton falls when TrHj increases. One has

cITrH2 d

» d = iyI>L = 2Y,Ki ZX»79>p,<I - Kv^i) = 2]C W^OP,? - Ki) (2-6)

at at ? g p p,q

The right hand side must be negative. The possible choice for the generator is r\VA = hptq(hp^p — hq,q) or

ri = [Hd,Hr] (2.7)

We have

= YXhk,k{i) + MO - 2hp}p(i))hk,p(i)hp,q(i)

= = -2 = (2-8)

k^q k k,q k,q

Since Ylk^q hi q faHs monotonously and is restricted from below, the derivative must vanish in the limit I —y oo. Therefore one has

v{l) = [Hd,H]^oo0 (2.9)

Practically we have reached the aim, the matrix commute with its diagonal part as I —y oo. The eqs. (2.2) and (2.9) are called flow equations for Hamiltonians and are the basis for the work presented further.

2.2 Similarity transformation

Another method to diagonalize Hamiltonians continuously was suggested independently by Glazek and Wilson [3], which has been called similarity transformation by the authors. In this subsection we follow the original work of Glazek and Wilson [3] to give the key ingredients of this scheme.

We define the unitary transformation, that brings the Hamiltonian operator in a form, where no transitions between the states with energy difference larger than A are present. The role of 'flow parameter' plays A, which corresponds to the ultraviolet (UV)-cutoff (see chapter 7) and is changed continuously.

The Hamiltonian operator H\ (and other quantities) depend then on the continuous parameter A. The Hamiltonian is given as a sum H\ = H0\ + Hi\, where H0\ is the free Hamiltonian and Hi\ contains (not renormalized) interactions and counterterms.

We separate each matrix M into two parts, M = D(M) + R(M). Let EiX are the eigenvalues of Hq\, and

_ | Ej a — Ejx | " EiX + Ej\ + A

uij\ — u{xija), rtj\ = 1 - Uijx = r(xij\) (2.10)

and u(x) is the function, which is 1 for small arguments and vanishes for large arguments. Explicit one can choose for u(x)

u(x) = 1 for 0 < x < -

1 2

u(x) falls monotonously from 1 to 0 for — < x < — 2

u(x) = 0 for - < x < 1 (2.11)

o

We define D(M)ij — Uij\Mij and corresponding R(M)ij = rij\Mij. As we introduce Uij\ and fija, we have the continuous transition between D(M) and R(M). It is important in order to avoid the divergences in renormalization equations. When one chooses u(x) = 9(xo — x), then one has the original definition for D(M) and R(M). The continuous transformation for Hamiltonian operator can be written as before in the form

J ZT

^ = (2.12)

The generator rjx is chosen in a way that D(H\) = H\. For practical purpose to distinguish in the calculations between the Hamiltonian operator H\ and its 'D'-part, it is useful to introduce the operator Q, such that

Hx = D(Q\) (2.13)

Then one has for the matrix elements

^f Qax + «iiA^f = flMx ~ EtX) + [nx, Hjx]i3 (2.14)

One is not able to find Qx and rjx from this equation. The reason is, that for the given A the Hamiltonian operator H\ can be additionally transformed unitary without breaking the condition D(Hx) = Hx- Thus one has one more condition. We rewrite the above equation in the form

UijX<~~d}T ~ ~ ~ ^ijX (2.15)

where we define

G^x = [vx,HIxh-d^Qi3x (2.16)

= D(Gx)i3 (2.17)

Now let

and

rHj\(EjX - Eix) = -R(Gx)ij (2-18)

Now all functions are defined. One has

rija | it dujjx Hjjx \

and

^ = <%A[»A, H, x}it + (2.20)

dX aA Uijx

There are no small energy denominators in these equations. These equations should be solved iteratively.

2.3 Flow equations for solid state physics

Flow equations were successfully applied to different systems in solid state physics. Unitary transformation in the form of exact diaginalization of Hamiltonian operator has been tested in

(1) model of impurity in the electron band [5];

(2) dissipative quantum systems, in particular for spin-boson problem [6];

(3) Lipkin model [7];

(4) problem of interacting electrons and phonons in a solid (referred to as BCS-theory) [10]. In all these models a system of interest couples to its environment Hamiltonian (for example, in (1) single impurity couples to the band of electrons; in (2) a small quantum system couples to the thermodynamical bath). The aim is then to decouple 'small' system from its 'large' environment to find a behavior of the system. It is different from what is usually done: most theoretical work starts off by tracing out the bath degrees of freedom and then using suitable approximation schemes for the time evolution of the reduced density matrix of the small quantum system. In general the approach of Hamiltonian diagonalization is particularly suited for studying low-energy properties of the system, thereby being complementary to most other approximation schemes.

In some cases the aim to get the diagonal Hamiltonian operator for I —> oo can not be reached. It was shown in the original work of Wegner [2] for the model of interacting fermions in one dimension, that a literal use of the concept of Hamiltonian diagonalization can lead to convergency problems. In the case discussed there the divergences appeared as / —> oo. The way out, as proposed by Wegner [2], is to bring Hamiltonian operator instead of diagonal to the block-diagonal form, where the number of quasiparticles is conserved in each block.

In many cases it is enough to transform the given Hamiltonian to the block-diagonal form. In particular it is so, when the block, which describes the states of interest, can be treated further with other methods. There are many known transformations, that construct in this way from the initial Hamiltonian operator an effective Hamiltonian operator, acting in a smaller Hilbert space and which is simpler to consider. Flow equations, where block-diagonalization is performed, have been compared with the following transformations

(1) the Schrieffer-Wolff transformation, which reduces the Anderson model with single magnetic impurity to the Kondo problem [8];

(2) the Foldy-Wouthuysen transformation, which decouples the Dirac equation into two two-component equations, one of which gives in the nonrelativistic limit the known Pauli equation

(3) the Fröhlich transformation, which constructs from the electron-phonon interaction an effective electron-electron interaction [10]. In all these cases flow equations re-examine the transformations used before.

2.4 Flow equations in field theory

In this section we set up the framework to use flow equations for the problems of high-energy physics. We remind, that flow equations perform the unitary transformation, which brings the Hamiltonian to a block-diagonal form with the number of particles (or Fock state) conserving in each block. In what follows we distinguish between the 'diagonal' (here Fock state conserving) and 'rest' (Fock state changing) sectors of the Hamiltonian. We break the Hamiltonian as

H = Hod + Hd + Hr (2.21)

where Hod is the free Hamiltonian; and the indices 'd','r' correspond to 'diagonal','rest' parts of the Hamiltonian, respectively. The flow equation for the Hamiltonian eq. (2.2) and the generator of unitary transformation eq. (2.7) are written [2]

^ = [V, Hd + Hr] + [[Hd, Hr], Hod] + [[#0* Hrl Hod]

77 = [Hod, Hr} + [Hd, Hr] (2.22)

In the basis of the eigenfunctions of the free Hamiltonian

Hod\i >= Ei\i > (2.23)

one obtains for the matrix-elements between the many-particle states

dHi

Î3

dl

= [77, Hd + Hr]ij - (Ei - E3)[Hd, Hr]^ - {Ei - Ej)2H,

rtj

t]ij = (Ei - Ej)Hrij + [Hd, Hr]%J (2.24)

The energy differences are given by

n 1

Ei - ^ = E Ei* ~ E (2.25)

k=1 k=1

where and Äj^ are the energies of the created and annihilated particles, respectively. The generator belongs to the 'rest' sector, i.e. rjij = r]rij,r]dij = 0. In what follows we use

[Ör,Hd]d = 0

[Ör,Hd]r^0 (2.26)

where Or is the operator from the 'rest' sector (for example Hr or rjr) and Hd is the diagonal part of Hamiltonian.

For the 'diagonal' (rai = n2) and 'rest' (ni ^ n2) sectors of the Hamiltonian one has correspondingly

dHdij

dl

= bh Hr

—TT~ — [Vi Hd + Hr]rij — (Ei — Ej)[Hd, Hr]rij -\--(2.27)

al ' at Uij

where we have introduced the cutoff function Uij(l)

ui}(l) = (2.28)

The energies Ei(l) start to depend on the flow parameter I in the second order of perturbation theory, that is taken into account by the renormalization of single-particle electron and photon energies (see chapter 7).

The main difference between these two sectors is the presence of the third term in the 'rest' sector ^ilEiIl which insures the band-diagonal structure for the 'rest' part

Ctl Uij

Hrij — utj H„3 (2.29)

i.e. in the 'rest' sector the matrix elements with the energy differences larger than 1 j\fl are suppressed. In the similarity renormalization scheme [3] the width of the band corresponds to the UV cutoff A. The connection between the two quantities is given

I = ^ (2-30)

The matrix elements of the interactions, which change the Fock state, are strongly suppressed,if the energy difference exceeds A, while for the Fock state conserving part of the Hamiltonian the matrix elements with all energy differences are present. As the flow parameter I —> oo (or A —» 0) the 'rest' part is completely eliminated, except maybe for the matrix elements with i = j. One is left with the block-diagonal effective Hamiltonian. Generally, the flow equations are written

^ = [77, Hd + Hr]t3 - (Ei - E3)[Hd, Hr]ii + al al Uij

where the following condition on the cutoff function in 'diagonal' and 'rest' sectors, respectively, is imposed

Udij = 1

Urij — II ij

(2.32)

One recovers with this condition the flow equations eq. (2.27) for both sectors. Other unitary transformations, which bring the Hamiltonian to the block-diagonal form, with the Fock state conserving in each block are used [3]

dHij r TT, duij Hij

= nd + ha,, + r„——

and [4]

dH{j 1 // 1 duij Hij

——- = ui:j[T], Hd + Hr\ij + —--

UA ClA Uij

where Uij + rij = 1; and the constrain eq. (2.32) on the cutoff function in both sectors is implied. One can choose the sharp cutoff function u^ = 9(X — |A,j|).

Chapter 3

Light-front field theory

3.1 Introduction

The development of light-front field theory dates back to the work of Dirac [11], where he introduced the light-front coordinates (the coordinate vector is x — (x+,x~,x±) with x± — x° ± x3 and = (xi,x2)) and the concept of front form dynamics for Hamiltonians. Dirac suggested, that a Hamiltonian operator can 'propagate' a physical system either in the usual time x° (instant form dynamics) or in the light-front time x+ (front form dynamics). The latter form of relativistic dynamics combines "the restricted principle of relativity with the Hamiltonian formulation of dynamics" [11].

Later the rules for front form perturbation theory were formulated [12], and the equivalence of this theory with the Feynman rules of covariant perturbation theory was established [13],[14].

Recent interest in light-front coordinates is driven mainly by two topics: low-energy bound state problem in QCD, where light-front coordinates offer a scenario in which a constituent picture of hadron structure can emerge from QCD, because of the simplified vacuum on the light-front [20], [31], [22]; and high-energy scattering processes, where light-front coordinates are the natural coordinates of the system [49], [22]. For an extensive list of light-front references through the early 1990's see [1], for the list of recent reviews see [21], [22] and references within.

Below we give briefly an introduction to light-front field theory (for introduction see also

[23]).

Light-Front (LF) quantization is very similar to canonical equal time (ET) quantization (here we closely follow Ref. [15]). Both are Hamiltonian formulations of field theory, where one specifies the fields on a particular initial surface. The evolution of the fields off the initial surface is determined by the Lagrangian equations of motion. The main difference is the choice of the initial surface, x° = 0 for ET and x+ = 0 for the LF respectively. In both frameworks states are expanded in terms of fields (and their derivatives) on this surface. Therefore, the same physical state may have very different wave functions1 in the ET and LF approaches because fields at x° = 0 provide a different basis for expanding a state than fields at x+ = 0. The reason is that the microscopic degrees of freedom — field amplitudes at x° = 0 versus field amplitudes at x+ = 0 — are in general quite different from each other in the two formalisms.

From the purely theoretical point of view, various advantages of LF quantization derive from properties of the ten generators of the Poincare group (translations i3^, rotations L and boosts K) [15]. Those generators which leave the initial surface invariant (P and L for ET and P\_, Ls and K for LF) are "simple" in the sense that they have very simple representations in

JBy "wave function" we mean here the collection of all Fock space amplitudes.

terms of the fields (typically just sums of single particle operators). The other generators, which include the "Hamiltonians" (Pq, which is conjugate to x° in ET and P+, which is conjugate to the LF-time x+ in LF quantization) contain interactions among the fields and are typically very complicated. Generators which leave the initial surface invariant are also called kinematic generators, while the others are called dynamic generators. Obviously it is advantageous to have as many of the ten generators kinematic as possible. There are seven kinematic generators on the LF but only six in ET quantization.

The fact that the generator of x~ translations, is kinematic (obviously it leaves x+ = 0 invariant!) and positive has striking consequences for the LF vacuum[15]. For free fields p2 = m2 implies for the LF energy p+ = (m2 + p±) /p_. Hence positive energy excitations have positive p-. After the usual re-interpretation of the negative energy states this implies that for a single particle is non-negative [which makes sense, considering that = p0 — p3]. P_ being kinematic means that it is given by the sum of single particle momenta . Combined with the non-negativity of this implies that, even in the presence of interactions, the physical vacuum (ground state of the theory) differs from the Fock vacuum (no particle excitations) only by so-called zero-mode excitations, i.e. by excitations of modes which are independent of the longitudinal LF-space coordinate x~. Due to this simplified vacuum structure, the LF-framework seems to be the only framework, where a constituent quark picture in a strongly interacting relativistic field theory has a chance to make sense [16, 17, 18, 19]. This is the most attractive feature of LF-frame to approve constituent quark model desription for QCD.

3.2 Preceding work

As far as progress is concerned, the light-front approach is not so far along; most research effort has occurred since late 1980's [1]. Progress is currently limited by conceptual issues, mainly by problems in renormalization program.

Further the methods available in the light-front field theory to solve the bound state problem are discussed. Then the results for positronium problem on the light-front follow.

3.2.1 Models and methods in the light-front field theory

The ultimate goal of the light-front field theory is to start with QCD Lagrangian and, with a minimum of approximation, calculate the hadron spectrum. The basic idea behind this approach is to use Hamiltonian techniques in the coordinate system best suited for relativistic dynamics. For light-front field theory, physically