An Intro duction to Light-Front Dynamics for

Pedestrians

Avaroth Harindranath

Saha Institute of Nuclear Physics, Sector I, Blo ck AF, Bidhan Nagar,

Calcutta 700064, India

Abstract. In these lectures we hop e to provide an elementary intro duction to selected

topics in light-front dynamics. Starting from the study of free eld theories of scalar

b oson, fermion, and massless vector b oson, the canonical eld commutators and propa-

gators in the instant and front forms are compared and contrasted. Poincare algebra is

describ ed next where the explicit expressions for the Poincare generators of free scalar

theory in terms of the eld op erators and Fock space op erators are also given. Next,

to illustrate the idea of Fock space description of b ound states and to analyze some of

the simple relativistic features of b ound systems without getting into the wilderness

of light-front renormalizatio n, Quantum Electro dynamics in one space - one time di-

mensions is discussed along with the consideration of anomaly in this mo del. Lastly,

light-frontpower counting is discussed. One of the consequences of light-frontpower

counting in the simple setting of one space - one time dimensions is illustrated using

massive Thirring mo del. Next, motivation for light-frontpower counting is discussed

and p ower assignments for dynamical variables in three plus one dimensions are given.

Simple examples of tree level Hamiltonians constructed bypower counting are pro-

vided and nally the idea of reducing the numb er of free parameters in the theory by

app ealing to symmetries is illustrated using a tree level example in Yukawa theory.

1 Preliminaries

1.1 What Is a Light-Front?

According to Dirac 1949 \ ... the three-dimensional surface in space-time

formed by a plane wave front advancing with the velo city of light. Such a surface

will b e called front for brevity". An example of a light-front is given by the

+ 0 3

equation x = x + x =0.

1.2 Light-Front Dynamics: De nition

A dynamical system is characterized by ten fundamental quantities: energy, mo-

mentum, angular momentum, and b o ost. In the conventional Hamiltonian form

of dynamics one works with dynamical variables referring to physical conditions

0

at some instant of time, the simplest instant b eing given by x = 0. Dirac found

that other forms of relativistic dynamics are p ossible. For example, one may set

up a dynamical theory in which the dynamical variables refer to physical condi-

+

tions on a front x = 0. The resulting dynamics is called light-front dynamics,

which Dirac called front-form for brevity.

2 Avaroth Harindranath

Fig. 1. Light-Front and Light Cone

+ 0 3 0 3

The variables x = x + x and x = x x are called light-front time

? 1 2

and longitudinal space variables resp ectively.Transverse variable x =x ;x .

Beware that many di erent conventions are in use in the literature. For our

conventions, notations, and some useful relations see App endix A.

A note on the nomenclature:

Instead of light-front eld theory one will also nd in the literature eld

theory in the in nite momentum frame, nul l plane eld theory, and light-cone

eld theory.We prefer the word light-front since the quantization surface is a

light-front tangential to the light cone.

1.3 Disp ersion Relation

In analogy with the light-front space-time variables, we de ne the longitudinal

+ 0 3 0 3

momentum k = k + k and light-front energy k = k k .

2 2 +

For a free massive particle k = m leads to k  0 and the disp ersion

? 2 2

k  +m

. relation k =

+

k

The ab ove disp ersion relation is quite remarkable for the following reasons:

1 Even though wehave a relativistic disp ersion relation, there is no square ro ot

?

factor. 2 The dep endence of the energy k on the transverse momentum k is

+

just like in the nonrelativistic disp ersion relation. 3 For k p ositive negative,

is p ositive negative. This fact has several interesting consequences. 4 The k

? +

dep endence of energy on k and k is multiplicative and large energy can result

? +

from large k and/or small k . This simple observation has drastic consequences

for renormalization asp ects Wilson 1990, Wilson et al. 1994.

1.4 Brief History upto 1980

In the following we provide a very brief history of light-front dynamics in particle

physics up to 1980 with randomly selected highlights. We note that light-front

has also b een put to use in other areas such as optics, strings, etc.

An Intro duction to Light-Front Dynamics for Pedestrians 3

As wehave already noted Dirac intro duced light-front dynamics in 1949.

In particle physics, light-front dynamics was rediscovered in the guise of eld

theory at in nite momentum byFubini and Furlan 1964 in an attempt to

2

derive \ xed q " sum rules in the context of current algebra. Adler 1965

and Weisb erger 1965 utilized in nite momentum frame in their formulation of

the sum rule for axial vector coupling constant. In nite momentum limit was

also considered by Dashen and Gell-Mann 1966 for the representation of lo cal

current algebra at in nite momentum.For an intro ductory treatment of current

algebra and light-likecharges, see, Leutwyler 1969. Motivated by the work

on current algebra, Weinb erg 1966 studied the in nite momentum limit of

old-fashioned p erturbation theory diagrams and found some simpli cations and

also investigated the structure of b ound state equations with particle truncation

\Tamm-Danco " approximation Tamm 1945, Danco 1950 in this limit.

In 1969, by combining the high energy q ! i1 limit with the in nite

0

momentum limit P !1 Bjorken 1969 predicted the scaling of deep inelastic

structure functions. Immediately following the exp erimental discovery of scaling

in deep inelastic scattering, the celebrated parton mo del of Feynman came into

b eing, whichwas formulated in the in nite momentum frame. Subsequently,

the study of emergence of scaling in canonical eld theories was carried out

see Drell, Levy, and Yan 1970 exploiting the sp ecial features of the in nite

momentum limit. Meanwhile the connection b etween in nite momentum limit

and light-frontvariables b ecame clear Susskind 1968, Bardacki and Halp ern

1968, Leutwyler 1968, Chang and Ma 1969, Jersak and Stern 1969. This

prompted the investigation of eld theories in light-front quantization.

Sp ecial asp ects of light-front quantization were p ointed out by Leutwyler,

Klauder, and Streit 1970. Kogut and Sop er 1970, Bjorken, Kogut, and Sop er

1971, and Neville and Rohrlich 1971 studied Quantum Electro dynamics in

the light-front formulation. Cornwall and Jackiw 1971 studied the canonical

+

equal x current commutators relevant for deep inelastic scattering the phenom-

ena of whichwas also studied in the context of light cone current algebra program

of Fritzsch and Gell-Mann 1971. Chang, Yan and collab orators Chang, Ro ot,

and Yan 1973, Chang and Yan 1973, Yan 1973a, Yan 1973b systemat-

ically investigated scalar, Yukawa, and massivevector b oson theories and the

connection with deep inelastic scattering.

't Ho oft 1974 exploited light-frontvariables and light-front gauge to exhibit

con nementintwo-dimensional Quantum Chromo dynamics QCD in the large

N limit. Subsequently Marinov, Perelomov, and Terent'ev 1974 initiated the

c

study of the sp ectrum of this mo del in the light-front Hamiltonian framework.

The intuitive picture of scaling violations in parton distributions was devel-

op ed by Kogut and Susskind 1974 in the in nite momentum frame.

Investigations on the relationship b etween the constituent picture and the

current picture in the context of classi cation schemes in the quark mo del Close

1979 lead to Melosh Transformation Melosh 1974. The nontrivial issues

asso ciated with angular momentum on the light-front came into full view with

4 Avaroth Harindranath

studies in light-front constituent quark mo dels Casher and Susskind 1973,

Leutwyler 1974, Terent'ev 1976.

+

The problem of P = 0 in light-front theory the now famous \zero mo de

problem" was rst considered by Maskawa and Yamawaki 1976 and Nakanishi

and Yamawaki 1977.

For the non-p erturbative study of QCD, Bardeen and Pearson 1976 in-

tro duced the Hamiltonian transverse lattice formulation in 1976. Thorn Thorn

1979a, Thorn 1979b, Thorn 1979c studied various asp ects of Light-Front

QCD including asymptotic freedom for the pure Yang-Mills theory.

In the late 70's and b eginning of 80's Bro dsky, Lepage and collab orators

Lepage and Bro dsky 1980 initiated the study of the application of light-front

p erturbation theory to various exclusive pro cesses.

1.5 What Is Covered in these Lectures

In these lectures we hop e to provide an elementary intro duction to selected

topics in light-front dynamics. Starting from the study of free eld theories of

scalar b oson, fermion and massless vector b oson, the canonical eld commutators

and propagators in the instant and front forms are compared and contrasted.

Poincare algebra is describ ed next where the explicit expressions for the Poincare

generators of free scalar theory in terms of the eld op erators and Fock space

op erators are also given. Next, to illustrate the idea of Fock space description

of b ound states and to analyze some of the simple relativistic features of b ound

systems without getting into the wilderness of light-front renormalization, Quan-

tum Electro dynamics in one space - one time dimensions is discussed along with

the consideration of anomaly in this mo del. Lastly, light-frontpower counting is

discussed. One of the consequences of light-frontpower counting in the simple

setting of one space - one time dimensions is illustrated using massive Thirring

mo del. Next, motivation for light-frontpower counting is discussed and p ower as-

signments for dynamical variables in three plus one dimensions are given. Simple

examples of tree level Hamiltonians constructed bypower counting are provided

and nally the idea of reducing the numb er of free parameters in the theory by

app ealing to symmetries is illustrated using a tree level example in Yukawa the-

ory. The notations, conventions and some useful relations are given in App endix

A. A list of review articles on light-front dynamics and a list of b o oks where

light-front has app eared are provided in App endix B.

1.6 Acknowledgements

I thank Stan Glazek, Daniel Mustaki, Rob ert Perry, Steve Pinsky, Junko Shige-

mitsu, James Vary, Ken Wilson, Tim Walhout, and Wei-Min Zhang for fruitful

collab oration and for helping me over several years to understand the wonder-

ful/terrible features of light-front dynamics. I thank James Vary and Jian-Wei

Qiu for making my long-term visit to the International Institute of Theoretical

and Applied Physics at Iowa State University in the rst half of 1996 p ossible

An Intro duction to Light-Front Dynamics for Pedestrians 5

and pro table. I also thank Frank Wo elz and James Vary for providing me the

opp ortunity to deliver the lectures on which these notes are based.

2 Free Fields

In this section we consider free eld theories of scalar b oson, fermion and massless

+

vector b oson in the light-front formulation. In particular we discuss equal-x

commutation relations and propagators.

2.1 Scalar Field

The Lagrangian density expressed in light-frontvariables is

1 1 1

+ ? ? 2 2

L = @ @  @ :@    : 1

2 2 2

The equation of motion is

 

+ ? 2 2

@ @ @  +   =0: 2

The quantized free scalar eld can b e written as Leutwyler, Klauder, and Streit

1970, Rohrlich 1971, Chang, Ro ot, and Yan 1973

Z

1

+ 2 ?

 

dk d k

ik :x y ik :x

x= ak  e + a k  e : 3

+ 3

2k 2 

+

0

The commutators are

 

y 0 3 + 3 0

ak ;a k  = 22  k  k k ;

 

0 y y 0

[ak ;ak ] = a k ;a k  =0: 4

Single particle state

y

j k i = a k  j 0i 5

and has the normalization

0 3 + 3 0

hk j k i = 22  k  k k : 6

+

First let us derive the canonical equal x commutation relation for the scalar

eld. For free eld theory, the commutator of x and y isknown for arbitrary

x and y .Wehave see for example Bjorken and Drell 1965,

[x;y] = i
x y  7

where

Z

4

d k

2 2 0 ik :xy 

2 k  k e : 8
x y =i

4

2 

6 Avaroth Harindranath

+ 3

k k

+ 0 3

Wehavek = k +k .Thus =1+ >0 on the mass shell and hence

0 0

k k

0 +

k  ! k . Thus in terms of light-frontvariables

Z Z Z

+1 +1

2 ?

i d k

+ ? 2 2 +

dk  k k k   
x y = dk

3

2 2 

1 1

+ + + ? ? ?

1 1

+ i k x y + k x y k :x y 

2 2

k e : 9

From 7 and 9 it is easy to show that

i

2 ? ?

x y  x y  10 [x;y] =

+ +

x =y

4

where  is the antisymmetric step function, x= x x.

The ab ove commutation relation is to b e contrasted with the corresp onding

commutation relation in equal-time theory, namely,

[x;y] =0: 11

0 0

x =y

0 0

We note that for x = y , the two elds are separated by a space-likeinterval,

+ +

the commutator has to vanish condition of microscopic causality. For x = y ,

? ?

if x 6= y , the two elds are separated by a space-like distance and hence the

+ + ? ?

commutator has to vanish. On the other hand, for x = y and x = y , the

two elds are separated by a light-like distance and hence the commutator need

not vanish.



Next we consider the scalar eld propagator. Let S denote scalar eld prop-

B

agator in light-front theory.Wehave

+



iS xy=<0jT xyj0>

B

+ +

= x y <0jxyj0>

+ +

+ y x <0jyxj0>: 12

Using 3 and 4 one can show that

Z

4

dk i

ik :xy 



iS x y = e

B

4 2 2

2  k  + i

F

= iS x y  13

B

F

where S is the Feynman propagator for the scalar eld. Thus for a scalar eld,

B

light-front propagator is the same as the Feynman propagator.

2.2 Fermion Field

The equation of motion



i @ m =0 14 

An Intro duction to Light-Front Dynamics for Pedestrians 7

in light-frontvariables is

i i

+ + ? ?

@ + @ i :@ m  =0: 15

2 2

De ne

 

 =  ; 16

1

  

. where  =

4

From 15, it follows that

1

? ? 0 +

 = i :@ + m : 17

+

i@

+ +

Thus  is a constrained eld since at any x it is determined by  . The

+

equation of motion for the dynamical eld  is

? 2 2

@  + m

+ +

i@  =  : 18

+

i@

Note that the fermion mass app ears quadratically in the ab ove equation.

+ +

Consider now the equal x commutation relation for the dynamical eld  .

We start from the solution of the free spin-half eld theory in equal time:

r

Z

3

X

 

m d k

ik :x y ik :x

bk; suk; se + d k; svk; se  x; t= :

3

E

2

2  k

s

19

It follows that see for example, Bjorken and Drell 1965

Z

3

1 d k

y 0

f x; t; y; t g =

3

2  2E

k

h i

0 ik :xy  0 ik :xy 

6 k + m e +6km e

0

=i6@ +m i
x y : 20

x

+

From the ab ove equation it is easy to show that the equal x commutation

y

+ +

relation of  and  is

y

+ + + 2 ? ?

+ +

f x; yg =  x y  x y : 21

x =y

Free fermion eld op erator in light-front theory can b e written as Kogut

and Sop er 1970, Chang, Ro ot, and Yan 1973

Z

i h

+ 2 ?

X

dk d k

y

ik :x ik :x

22 k  v k  e  x= b k  u k  e + d

  



+ 3

2k 2  

8 Avaroth Harindranath



Let S denote fermion eld propagator Chang and Yan 1973 in light-front

F

theory.

+

 

iS x y = <0jT xy j0>

F

+ +



= x y <0jxyj0>

+ +



y x <0jyx j0>: 23

Using 22 for the eld op erator, we can show that the light-front propagator

for the fermion eld is

Z

4

dk 6 k + m

ik :xy  on



iS x y =i e

F

4 2 2

2  k m + i

 

Z

+ 4

1 1 d k

ik :xy 

= i e

4 +

2  6 k m + i 2 k

+

+ + 2 ? ?

= iS x y   x y  x y x y  24

F

4

? 2 2

k  +m

1 1

+ +

+ k where S is the Feynman propagator and 6 k =

+ F

on

2 k 2

? ?

:k .We note that for the fermion eld, light-front propagator di ers from

the Feynman propagator by an instantaneous propagator.

2.3 Massless Vector Field

The equation of motion in light-frontvariables is

 

1 1

2

+ + + ? ? + ? +

@ @ A + @ A @ :A @ @ @ A =0; 25

2 2

 

1 1

2

+ + ? ? + ? i i

@ A + @ A @ :A @ @ @ A =0; 26 @

2 2

 

1 1

2

+ + ? ? + ?

@ A + @ A @ :A @ @ @ @ A =0: 27

2 2

Cho ose the gauge Kogut and Sop er 1970, Neville and Rohrlich 1971

+

A =0: 28

This gauge choice is known as in nite-momentum gauge, null-plane gauge, light-

cone gauge and light-front gauge. From 25, wehave

+ ? ? + ?

@ A =2@ :A + F x ;x  29

Thus A is not a dynamical variable. Cho osing F to b e zero, the dynamical

i

variables A ob ey massless Klein-Gordon equation.

An Intro duction to Light-Front Dynamics for Pedestrians 9

i

Since the dynamical variable A ob ey massless Klein-Gordon equation, we

can follow the same route wehave taken for the free scalar eld and write the

eld op erator in quantum theory as

Z

i h

+ 2 ?

X

dk d k

y

ik :x ik :x j

30  a ke + a k e A x=

j 



+ 3

2k 2 



with

 

y 0 3 + 3 0

a k ;a k  = 22  k   k k ;

 



h i

y

0 y 0

[a k ;a k ] = 0; a k ;a k  =0: 31

 





+

The equal x commutation relation is

 

i

i j 2 ? ?

A x;A y =  x y   x y : 32

ij

+ +

x =y

4

With F =0,wehave,

Z

1

? i i ?

A x ;x = dy x y @ A y ;x : 33

2

Explicitly, using 30, wehave,

Z

h i

j + 2 ?

X

2k dk d k

y

ik :x ik :x

A x=  a ke + a k e : 34

j 



+ 3 +

2k 2  k



Intro ducing the p olarization vectors

1 1

 

1 + 2 +

0; 2k ;k ;0;k= 0; 2k ; 0;k ; 35  k =

2 1

+ +

k k

we can write

Z

h i

+ 2 ?

X

dk d k

 y

ik :x ik :x 

 k  a k e + a k e A x= : 36



 

+ 3

2k 2 



Note that,



@ A =0: 37



?

Intro ducing the four-vector =0;2;0 wehave the relation

2    

X

k k + k



   

: 38  k  k =g +





+ + 2

k k 





Let S denote the massless vector eld propagator Yan 1973b in light-

V

front theory.Wehave

 +  



iS  x y =h0jT AxAyj0i

V

+ +  

= x y h0jAxAyj0i

+ +  

+ y x h0jAyAxj0i: 39

10 Avaroth Harindranath

Using the expansion 36 wehave

Z

4

i d k

ik :xy  



e iS  x y =

V

4 2

2  k + i

 

    2

k + k k

  

g + : 40

+ + 2

k k 

3 Poincare Generators and Algebra

3.1 Lorentz Group

Let us rst consider a pure b o ost along the negative 3-axis. The relationship

~

between space and time of two systems of co ordinates, one S in uniform motion

0

along the negative 3-axis with sp eed v relative to other S is given by~x =

v 1

0 3 3 3 0

p

x + x ,x ~ = x + x , with = and = .Intro duce the

c

2

1

parameter  such that = cosh , = sinh . In terms of the light-front

variables,

+  + 

x~ = e x ; x~ = e x : 41

+

Thus b o ost along the 3-axis b ecomes a scale transformation for the variablesx ~

+

andx ~ and x = 0 is invariant under b o ost along the 3-axis.

i

Let us denote the three generators of b o osts by K and the three generators

i 1 1 2 2 2

of rotations by J in equal-time dynamics. De ne E = K + J , E = K

1 1 1 2 2 2 1

J , F = K J , and F = K + J . The explicit expressions for the 6

3 1 2 3 1 2

generators K , E , E , J , F , and F in the nite dimensional representation

using the conventions of Ryder 1985 are

1 0 0 1

0 1 0 0 0 0 0 1

1 0 0 1 0 0 0 0

C B B C

1 3

; ;E =i K = i

A @ @ A

0 0 0 0 0 0 0 0

0 1 0 0 1 0 0 0

0 1 0 1

0 0 1 0 0 0 0 0

0 0 0 0 0 0 1 0

B C B C

2 3

E =i ;J =i ;

@ A @ A

1 0 0 1 0 1 0 0

0 0 1 0 0 0 0 0

0 1 0 1

0 1 0 0 0 0 1 0

1 0 0 1 0 0 0 0

B C B C

1 2

F =i ;F =i :

@ A @ A

0 0 0 0 1 0 0 1

0 1 0 0 0 0 1 0

3 1 2 3 +

Note that K , E , E , and J leave x =0invariant and are kinematical

1 2

generators while F and F do not and are dynamical generators.

It follows that

1 2 3 i ij j

[F ;F ]= 0;[J ;F ]= i F : 42

An Intro duction to Light-Front Dynamics for Pedestrians 11

3 1 2

Thus J , F and F form a closed algebra. Also

1 2 3 i i

[E ;E ]= 0;[K ;E ]= iE : 43

3 1 2

Thus K , E and E also form a closed algebra.

3.2 Algebra



From the Lagrangian density one may construct the stress tensor T and from



the stress tensor one may construct a four-momentum P and a generalized



angular momentum M .

Z

1

 2 ? +

P = dx d x T ; 44

2

Z

1

2 ?  +  + 

dx d x [x T x T ]: 45 M =

2



Note that M is antisymmetric and hence has six indep endent comp onents.

 

Poincare algebra in terms of P and M is see for example, Ryder 1985

 

[P ;P ]= 0; 46

     

[P ;M ]= i[g P g P ]; 47

         

[M ;M ]= i[g M + g M g M + g M ]: 48

+ i

In light-front dynamics P is the Hamiltonian and P and P i =1;2 are

+ 3 +i i 12 3

the momenta. M =2K and M = E are the b o osts. M = J and

i i

M = F are the rotations. The following table summarizes the commutation

relations b etween the Poincare generators in light-front dynamics.

+ 1 2 3 1 2 3 1 2

P P P K E E J F F P

+ + 1 2

P 0 0 0 iP 0 0 0 2iP 2iP 0

1 + 2

P 0 0 0 0 iP 0 iP iP 0 0

2 + 1

P 0 0 0 0 0 iP iP 0 iP 0

3 + 1 2 1 2

K iP 0 0 0 iE iE 0 iF iF iP

1 + 1 2 3 3 1

E 0 iP 0 iE 0 0 iE 2iK 2iJ 2iP

2 + 2 1 3 3 2

E 0 0 iP iE 0 0 iE 2iJ 2iK 2iP

3 2 1 2 1 2 1

J 0 iP iP 0 iE iE 0 iF iF 0

1 1 1 3 3 2

F 2iP iP 0 iF 2iK 2iJ iF 0 0 0

2 2 2 3 3 1

F 2iP 0 iP iF 2iJ 2iK iF 0 0 0

1 2

P 0 0 0 iP 2iP 2iP 0 0 0 0

12 Avaroth Harindranath

3.3 Free Scalar Field: Generators in Fock Representation

In this section, as an example, we explicitly construct the Poincare generators

of free scalar eld theory in Fock representation Flory 1970.

From the Lagrangian density,we obtain the conserved symmetric stress ten-

sor. The stress tensor

   

T = @ @  g L: 49

with

1 1

 2 2

L = @ @    : 50



2 2

The momentum op erators are given by

Z

1

+ 2 ? + +

P = dx d x @ @ : 51

2

Z

1

i 2 ? + i

P = dx d x @ @ : 52

2

The Hamiltonian op erator

Z

 

1

2 ? i i 2 2

P = dx d x @ @  +   : 53

2

+

The generators of b o osts are at x = 0,

Z

1

2 ? + + 3

dx d x x @ @ ; 54 K =

4

and

Z

1

i 2 ? i + +

E = dx d x x @ @ : 55

2

The generators of rotations are

Z

 

1

2 ? + 1 2 2 1 3

dx d x @  x @  x @  56 J =

2

and

Z

 

1

i 2 ? + i i ? ? 2 2

F = dx d x x @ @  x @ :@  +    : 57

2

In terms of Fock space op erators, wehave,

Z

+ 2 ?

dk d k

+ + y

P = k a k ak : 58

+ 3

2k 2 

An Intro duction to Light-Front Dynamics for Pedestrians 13

Z

+ 2 ?

dk d k

i i y

P = k a k ak : 59

+ 3

2k 2 

Z

+ 2 ? 2 ? 2

dk d k  +k 

y

P = a kak: 60

+ 3 +

2k 2  k

Z

+ 2 ?

dk d k @

y + 3

a k k ak: 61 K = i

+ 3 +

2k 2  @k

Z

+ 2 ?

dk @ d k

i y +

E = i a k k ak: 62

+ 3 i

2k 2  @k

Z

+ 2 ?

dk d k @ @

1 2 y 3

[k k ]a k ak: 63 J = i

+ 3 2 1

2k 2  @k @k

Z

2

+ 2 ? 2 ?

dk d k  + k @

i y

F = i a k ak

+ 3 + i

2k 2  k @k

Z

+ 2 ? y

dk d k @a k

i

2i k ak: 64

+ 3 +

2k 2  @k

For a single particle, wehave,

+ +

P j pi = p j pi; 65

i i

P j pi = p j pi; 66

? 2 2

p  + 

P j pi = j pi; 67

+

p

@

3 +

j pi; 68 K j pi = ip

+

@p

@

i +

E j pi = ip j pi; 69

i

@p

 

@ @

3 2 1

J j pi = i p p j pi; 70

1 2

@p @p

? 2 2

 

p  +  @ @

i i

F j pi = i +2ip j pi: 71

+ i +

p @p @p

14 Avaroth Harindranath

4 Two-Dimensional Quantum Electro dynamics

4.1 Intro duction

In this lecture we discuss two dimensional one space-one time Quantum Elec-

tro dynamics QED in light-front dynamics. Our main purp ose is to exhibit

some of the simple features of relativistic b ound states in the simplest setting.

We also discuss some asp ects of renormalization and anomaly.

We study the b ound state dynamics of QED in the truncated space of one

2

+

fermion-anti fermion pair. In this mo del, with the gauge choice A =0on

the light-frontwehave fermions and antifermions interacting via instantaneous

interactions. It turns out that just with one pair wehave a reasonably go o d

description of the ground state in b oth weak coupling non-relativistic and

strong coupling relativistic domains.

Just for notational convenience we omit the sup erscript + for longitudinal

momenta in this section.

4.2 Hamiltonian

The Lagrangian density for QED is given by

1





F F +  iD6 m 72 L =

 QED

4

       

with F = @ A @ A and D = @ + ieA .We pick the light-front gauge

+

A =0.From the equations of motion

iD6 m =0; 73

 



@ F = e ; : 74



we get the constraint equations

1

0 +

 = m ; 75

+

i@

4e

y

+ +

  ; : 76 A =

+ 2

@ 

+

The equation of motion for the dynamical variable  is

 

1 1

y

+ 2 + 2 + + +

i@  = m  4e    : 77

+ + 2

i@ @ 

The symmetric energy momentum tensor is

1

      



T = F F +   D + D  



2

1

 



g F F +  iD6 m : 78

 4

An Intro duction to Light-Front Dynamics for Pedestrians 15

+

In the gauge A = 0, the momentum

Z

1

y

+ + + +

P = dx 2i @  79

2

and the Hamiltonian is given by

Z

1 1

y y y

2 + + 2 + + + +

m  : 80 P = dx  2e    

+ + 2

i@ @ 

Note that the Hamiltonian has only fermion degrees of freedom which drastically

simpli es Fock space structure. In the following rst we truncate the Fock space

to a fermion-antifermion pair. We give the relevant terms in the Hamiltonian

also in terms of Fock space op erators.

By pro jecting the eigenvalue equation

+ 2

P P j i = M j i 81

on to a pair of free states, we arrive at the b ound state equation in QED. The

b ound state equation is shown to repro duce the well-known results for the ground

state in the massless ultra-relativistic limit. The b ound state equation is also

shown to repro duce the well-known results in the heavy mass non-relativistic

limit.

4.3 Bound State Equation in QED

The eld op erator is

Z

 

dk

+ ik :x y ik :x

p

 x= bk e + d k e 82

4 k

with

 

y 0 y 0 0

bk ;b k  = dk;d k  =4k k k  : 83

The relevant terms in the Hamiltonian are

P = P + P 84

int

free

where

Z

 

dk

y y

b kbk+ d kdk = P

free

4k

 

Z

2

m 1 1 dk

1

2

 +2e ; 85

2 2

k 4 k k  k + k 

1 1

Z Z Z Z

dk dk dk dk

2 3 4 1

2

p p p p

4 k k k + k  P = 4e

1 2 3 4

int

4 k 4 k 4 k 4 k

1 2 3 4

 

1 1

y y

b k bk d k dk  : 86

1 2 4 3

2 2

k k  k + k 

1 2 1 4

16 Avaroth Harindranath

Note that wehave generated self-energy contributions to the mass 85 by normal

ordering the instantaneous four-fermion interaction.

We expand the state vector j >in terms Fock space states and truncate

to a fermion-antifermion pair:

Z

dp dp

1 2

y y

p p

j P  > =  p ;p b p d p  j 0 >

2 1 2 1 2

4p 4p

1 2

p

22 PP p p : 87 

1 2

By pro jecting the eigenvalue equation 81 on to a pair of free states and intro-

p

1

1

p

ducing the momentum fraction variables x = ,  p ;p =  x etc. 

2 1 2 2

P

P

we arrive at the b ound state equation

Z Z

1

2 2 2

e  y   x e m

2 2

2

 x dy + dy  y  M  x=

2 2 2

2

x1 x  x y  

0

88

The factor prop ortional to  x in the third term is the self-energy contribution.

4.4 Relativistic Limit

1

The b ound state equation 88 would have exhibited severe divergences com-

2

x

ing from the instantaneous gauge b oson exchange if self-energy contributions

were ignored. Such divergences are present in the eigenvalue equation for single

fermion. A detailed and excellent discussion of these divergences and corresp ond-

ing regulators in the context of con nement and asymptotic freedom in QCD

2

can b e found in Callan, Co ote and Gross 1976 and Einhorn 1976.

In the extreme relativistic limit m ! 0, 88 shows that  = x 1 x

2

2

e

2

is a solution with eigenvalue M = . This is the well-known Schwinger result



in two-dimensional massless electro dynamics Schwinger mo del.

The result that a single fermion-antifermion pair repro duces the well-known

result in the extreme strong coupling limit in light-front quantization is in fact

3

nontrivial. In equal-time quantization, in A = 0 gauge for example, restriction

to a single pair is a valid approximation only in the extreme nonrelativistic limit.

For a comparison of b ound state equations in equal-time and light-front cases in

the context of QCD see Hanson, Peccei, and Prasad 1976.

2

4.5 Nonrelativistic Limit

In the nonrelativistic limit fermion mass !1 , the last term in 88 which

corresp onds to the \annihilation channel" can b e ignored. Then the b ound state

equations for QED and QCD are identical except for a rescaling of the coupling

constant. Let us start from 88 without the last term.

Z

2 2

e  y   x  m

2 1 2 1

2

 x + dy =0: 89 M  x 

2 1 1 2 1

2

x 1 x   x y 

1 1 1 1

An Intro duction to Light-Front Dynamics for Pedestrians 17

Intro duce the variable q via

q 1

1 90 x =

1

2 q 

p

2 2

with q = q +m . Note that the range of q is 1

fact that   m,

0 2

q q 

2

x y   : 91

1 1

2

4m

2

B



Intro ducing B = B=m where B =2mM,wehave,

2

4m

Z

2 0

e  q   q 

2 0



[B + q ] q = P dq : 92

0 2

 q q 

The second term on r.h.s. is the self-energy correction which also vanishes in the

nonrelativistic limit.

R

0

 q 

1

0

The Fourier transform of jz jz leads to P dq ; and we arrive

0 2

2 q q 

at the co ordinate space equation

 

2

@

+ j u j u= u 93

2

@u

2 4



3 3

where u = e z and  = Be . The solution to 93 are the well known Airy

functions. A discussion of 93 is given by Hamer 1977.

4.6 Anomaly

In this subsection we follow the discussion in Bergkno 1977. Classically,in

the massless limit, chiral symmetry of the QED Lagrangian leads to the con-

2

 





servation of axial vector current j =  , @ j = 0. Let us calculate the

5 

5 5

divergence of the axial vector current in the quantum theory.

Wehave

1 1



+

+

@ j = @ j + @ j : 94



5 5 5

2 2

 



In one space - one time dimensions, the vector current j =   and the axial



vector current j are related by

5





j =  j ; 95



5

 +

where  is the antisymmetric tensor,  = 2. Thus

+

+

j = j and j = j : 96

5 5



From the conservation of the vector current j ,wehave

+ +

@ j =@ j 97

18 Avaroth Harindranath

Thus

 



+ +

@ j : 98 = @ j = i j ;P



5

Thus we need to calculate the commutator of the plus comp onent of the vec-

tor current and the Hamiltonian. This evaluation is most easily carried out in

+

momentum space utilizing the Fourier mo de expansion of the eld  .

In the massless limit, the Hamiltonian can b e written as

Z

+1

2

dp e

+ +

~ ~

j pj p; 99 P =

2

8 p

1

where wehaveintro duced the Fourier transform of the current,

Z

+1

1

1

+ i + px

~

2

j x= dpe j p: 100

4

1

Thus we need to calculate the commutator of the plus comp onent of the currents,

 

+ +

~ ~

j p; j q  .

Using the Fourier mo de expansion of the eld 82, it is easily shown that,

Z Z

1 1 h i

 

dk 1 dk

1 2

+ + i k +k x y 

1 2

2

h0 j j x;j y j 0i =4 e c:c: :

4 4

0 0

101

Thus, wehave,

 

+ +

~ ~

j p; j q  j 0i =4q p + q: 102 h0 j

 

+ +

~ ~

j p; j q  =4q p + q. In the absence of any q-numb er structure, wehave,

An explicit evaluation, then, leads to

2 +

~

 

e j p

+

~

j p;P = : 103

 p

From 98 wehave

2 +

~

@ e j p

+

~

j = i 104

+

@x 2 p



which shows that @ j is not zero. In p osition space the ab ove equation leads to



5

2 + 2

@ j x e

+

= j : 105

+

@x @x 4

Thus we see that 1 in the quantum theory, divergence of the axial vector

+

current is nonzero, even though it is zero in the classical theory, 2 j ob eys

2

e

2

the Klein-Gordon equation for a massive scalar eld with m = . 

An Intro duction to Light-Front Dynamics for Pedestrians 19

5 Light-FrontPower Counting and its Consequences

In this section we discuss the light-frontpower counting intro duced by Wilson

Wilson 1990, Wilson et al. 1994. To illustrate its consequences in a sim-

ple example in one plus one dimensions we rst discuss the massive Thirring

mo del. Then we discuss the motivation for light-frontpower counting and give

the p ower assignments for dynamical variables and the Hamiltonian in three plus

one dimensions. Simple examples of Hamiltonians involving scalars and fermions

are given at the tree level. App ealing to p ower counting alone leads to a large

numb er of free parameters in the theory. The idea of reducing the number of

free parameters by implementing the symmetries is illustrated using a simple

example in Yukawa theory.

5.1 Massive Thirring Mo del

Power counting is di erent in light-front dynamics. For example, in two dimen-

+

sions,  has no mass dimension whereas in equal-time theory  has mass di-

1

mension . In b oth cases the scalar eld  has no mass dimension. Thus in

2

light-front theory in one plus dimensions in nite numb er of terms are p ossible in

the interaction. However, in two-dimensional gauge theories and two-dimensional

Yukawa mo del, the coupling constante and g resp ectively has the dimension

of mass. By dimensional analysis, the Hamiltonian P has dimension twoin

units of mass. Accordingly, in gauge theory case the highest p ower of coupling

2

allowed bypower counting is e and in Yukawa mo del highest p owers of coupling

allowed are g must b e accompanied by a mass m to balance dimensions and

2

g . Explicit construction of the canonical light-front Hamiltonian in these cases

shows that the interaction terms ob ey these p ower counting rules.

2

If the coupling g is dimensionless in nite numb er of terms app ear in P for

theories in two dimensions. In equal-time theory, four-fermion interactions have

1

dimensionless coupling constant. Since  carry mass dimension , six-fermion

2

interactions etc. are not allowed bypower counting. On other hand, in light-

+

front theory  carry no mass dimension, and hence in nite numb er of terms

are allowed for fermionic interactions in P bypower counting just like b osonic

interactions in equal-time theory in one plus one dimensions. By dimensional

2

arguments a constant with dimensions of m has to app ear as a overall mul-

tiplicative factor in front of the interaction Hamiltonian. In the following we

illustrate these features in the context of massive Thirring mo del.

The Lagrangian density for massive Thirring mo del is given by

1

 2  2

 

L =  i @ m g   : 106



2

The equation of motion is

y

+ + 0 2

  : 107 i@  = m  +2g 

20 Avaroth Harindranath

To get the true equation of motion, wehave to eliminate the constraintvariable

 which ob eys the equation of constraint:

y

+ 0 + 2 + +

i@  = m  +2g    : 108

As was mentioned b efore, the equation of constraint is nonlinear, in contrast to

the situation in gauge theories and Yukawa mo del.

The Hamiltonian densityis

i h

y y y y y

2 + + + + 0 0 +

+2g     : H =i @  + m   +  

y

+ 0

= m  : 109

In order to express the Hamiltonian in terms of the physical degree of freedom

+

 ,we need to solve the constraint equation 108.

Following Domokos 1971, intro duce the Green function

2

1

2g Bx By 

[ ]

Gx ;y = x y e ; 110

4i

where

Z

1

y

+ +

dz x z  z  z : 111 B x =

2

One can easily verify that

Z

0 +

 x =m dy Gx ;y  y  112

satis es the constraint equation 108. Thus the constraint equation is explicitly

solved using the ab ove ansatz.

The Hamiltonian

Z Z

y

2 + +

P = m dx dy  x Gx ;y  y : 113

Thus we see explicitly that 1 there are in nite numb er of terms in the Hamil-

tonian which, in this particular case, exp onentiates resulting in a closed form

2 2

and 2 m app ears as an overall multiplicative factor. For g =0 we repro duce the free eld theory result.

An Intro duction to Light-Front Dynamics for Pedestrians 21

5.2 Light-FrontPower Counting: Motivation

In conventional Lagrangian eld theory, one starts with the terms allowed by

power counting in the Lagrangian density.Power counting alone may lead to

a large numb er of arbitrary parameters in the theory. When restrictions from

Lorentz invariance and gauge invariance in the case of gauge theories are im-

p osed, this numb er is drastically reduced. By analyzing arbitrary orders of p er-

turbation theory, one discovers that the counterterms are all of the form as the

canonical ones, provided the cuto s resp ect the imp osed symmetries. Following

the same path, in QCD for example, we need to construct the most general form

including the canonical terms and counterterms of the light-front Hamiltonian

for QCD. In our case, wehave to use the light-frontpower counting to construct

the Hamiltonian. Further, to reduce the numb er of arbitrary parameters we can

imp ose light-front symmetries.

Why light-frontpower counting is di erent? Light-frontpower counting is

?

in terms of the longitudinal co ordinate x and the transverse co ordinate x .It

?

has b een noticed that x and x have to b e treated di erently.Wemay give

three reasons for doing so: 1 The energy k scales di erently with x and

? 2 2

k  +m

?

x scaling. i.e., from the free particle disp ersion relation k = , k

+

k

scales as x b oth are the minus comp onent of four-vectors and k scales as

1

?

. 2 x do es not carry inverse mass dimension, only x do es. 3 Longi-

? 2

x 

tudinal scale transformation is op erationally identical to the longitudinal b o ost

transformation which is a Lorentz symmetry.

5.3 Canonical Power Assignments

Analysis of the canonical light-front Hamiltonian shows that indeed it scales

?

di erently under x and x scaling. To determine the scaling prop erties of the

Hamiltonian, rst we need to determine the scale dimensions of the dynami-

+

cal variables scalar eld , the plus comp onent of the fermion eld  , the

?

transverse comp onent of the gauge eld, A , etc.. From the scaling analysis of

canonical commutation relations Wilson et al. 1994, the p ower assignments

are

1

 :

?

x

1

?

A :

?

x

1

+

p

 : : 114

?

x x

The p ower assignments for the derivatives are

1

?

@ :

?

x

1

+

@ : : 115

x

22 Avaroth Harindranath

1

?

Since @ carry mass dimension is not allowed in the canonical Hamiltonian

?

@

+ +

whereas @ do not carry mass dimension and hence inverse p owers of @ are

allowed in the canonical Hamiltonian. The interaction Hamiltonian density H

1

has the p ower assignment . The Hamiltonian do es not have a unique scal-

? 4

x 

ing b ehavior in the transverse plane when parameters with dimensions of the

mass are present whereas longitudinal scaling b ehavior is una ected by mass

parameters. For dimensional analysis we assign

1

H :

? 4

x 

x

H : : 116

? 2

x 

Let us consider some examples of canonical Hamiltonians constructed using the

power counting rules.

1

Scalar Theory. Since the p ower assignment for the scalar eld is  : ,

?

x

2 2 ? ? 3 4

the allowed terms are   , @ :@ , c , and  where  and c have mass

dimension. Hence the most general form of the canonical Hamiltonian for the

scalar eld is

? ? 2 2 3 4

H = c @ :@  + c   + c  + c  ; 117

1 2 3 4

where c , c , and c are dimensionless and c has mass dimension.

1 2 4 3

Fermions Interacting with Scalar Yukawa Mo del. Let us rst con-

sider the interaction free parts of the Hamiltonian density. Since the dynamical

1

+ +

p

fermion eld  has the p ower assignment  : and the Hamiltonian

?

x x

1

density has the p ower assignment H = , the inverse longitudinal deriva-

? 4

x 

tive o ccurs in the free parts to balance longitudinal scale dimensions. The al-

? 2

y y

@ 

1

+ + 2 + +

lowed free parts are   , m   where m is a mass parameter.

+ +

@ @

? ?

y y y

:@

1 1

+ + + + + +

   ,   ,   , The interaction terms allowed are 

+ + +

@ @ @

? ?

y y

:@

1

+ + + +

    and   . The presence of nonlo cal two fermion - two

+ +

@ @

b oson interaction is a consequence of light-frontpower counting. Note that in

this catalogue wehave ignored terms which app ear as surface terms in the Hamil-

tonian. By adding the terms for the scalar eld Hamiltonian density given in the

previous section, we get the most general form of the canonical Hamiltonian

density allowed bypower counting.

? ? 2 2 3 4

H = c @ :@  + c   + c  + c 

pc 1 2 3 4

? 2

@  1

y y

+ + 2 + +

+c   + c m  

5 6

+ +

@ @

1 1

y y

+ + + +

+c    + c   

7 8

+ +

@ @

An Intro duction to Light-Front Dynamics for Pedestrians 23

? ? ? ?

:@ :@

y y

+ + + +

+c    + c   

9 10

+ +

@ @

1

y

+ +

+c    : 118

11

+

@

It is worthwhile to compare the ab ove catalogue with the Hamiltonian density

of the Yukawa mo del obtained from the Lagrangian density via the standard

canonical pro cedure. It takes the form

1

3 4 ? ? 2 2

+   +   H = @ :@  +  

3 4 can

2

? 2 2

@  + m

1 1

y y

+ + + + +

+  + gm   +  

+ + +

i@ i@ i@

? ? ? ?

:@ 1 :@

y y

+ + 2 + + +

   + g    : +g 

+ + +

@ @ i@

119

Comparing the forms of the Hamiltonian density constructed bytwo di erent

metho ds, namely, the one based on light-frontpower counting alone and the

one based on the canonical pro cedure starting from the Lagrangian density,it

app ears that the rst metho d has to o many arbitrary parameters compared to

the very few parameters resulting from the second metho d. This should cause no

surprise since the rst metho d has relied purely on p ower counting whereas the

second metho d has already implemented the consequences of Lorentz symmetries

by virtue of starting from a manifestly invariant Lagrangian density.We can hop e

to reduce the numb er of free parameters by studying the implications of various

symmetries in the theory. In the next section we provide an example of this idea.

5.4 Implementing Symmetries: A Simple Example

Wehave seen that the most general form of the canonical Hamiltonian density

can b e constructed using the p ower counting rules. However, the Hamiltonian

density so constructed su ers from an apparent proliferation of free parameters

in comparison with that obtained starting from the manifestly Lorentz invariant

Lagrangian density. In this section we provide an example of how implementing

symmetries implies relationship among the parameters and thus reduces the

numb er of free parameters in the theory.

Two of the most imp ortant symmetries in light-front theory are the longitu-

dinal and the transverse b o ost symmetries. As wehave already observed, lon-

gitudinal b o ost symmetry is a scale symmetry which is already implemented in

constructing the p ower counting rules for the canonical Hamiltonian P should

scale as x . Transverse b o ost symmetry implies that interaction vertices in the

theory in momentum space are indep endent of the total transverse momen-

tum in the problem. Let us consider the consequence of this symmetry for the

Hamiltonian for the Yukawa mo del wehave constructed from p ower counting.

24 Avaroth Harindranath

We consider the tree level matrix element for transition from a single fermion

state to a fermion - b oson state. Let us denote momenta of the initial fermion,

nal fermion and the b oson by P , k , and q resp ectively. The relevant terms of

interest are those involving the transverse derivative. A simple calculation shows

that, apart from common factors, the matrix element

? ? ? ?

 :P  :k

Mc c : 120

9 10

+ +

P k

+ + ? ? ?

Intro duce the internal momenta k = xP , k =  + xP . In terms of the

internal variables the matrix element

? ? ? ? ?

 :P  : + xP 

Mc c : 121

9 10

+ +

P xP

?

Requiring that the matrix element is indep endentofP immediately yields c =

9

c .Thus the implementation of transverse b o ost symmetry on the transition

10

matrix element results in the reduction of numb er of free parameters in the tree

level Hamiltonian by one.

Discussion. By relying on the p ower counting rules rather than app ealing to

a manifestly Lorentz invariant Lagrangian wehave a starting bare Hamiltonian

that do not have the symmetries of the real world. However, demanding that the

physical observables ob ey the symmetries we can hop efully correct our mistakes!

An analysis in QED along these lines can b e found in the b eautiful work of French

and Weisskopf 1949. An application of this idea to the problem of sp ontaneous

symmetry breaking in sigma mo del on the light-frontisworked out in App endix

A of Wilson et al. 1994.

The examples cited so far deals with the theory at the tree level. At this stage

it lo oks likewe are solving a simple problem in a complicated way.Fortunately,

for the light-front theory matters are not so simple. As we stated in the b eginning,

we need to construct the most general form of the Hamiltonian i.e., the canonical

terms plus the counterterms. The p ower counting rules wehave cited are for the

canonical terms. Light-front symmetries imply a far richer counterterm structure

than is familiar in the equal time theory. A discussion of this structure, however,

is b eyond the scop e of these p edagogical lectures and is the sub ject of active

research. For a study in the context of b ound state dynamics in the Yukawa

mo del see Glazek et al. 1993. A preliminary analysis is carried out in Wilson

et al. 1994. For a discussion of the reduction of free parameters in the context

of light-front renormalization group see the work of Perry and Wilson 1993

and Perry 1994.

A Notation, Conventions, and Useful Relations



We denote the four-vector x by

 0 3 1 2 0 3 ?

x =x ;x ;x ;x =x ;x ;x : 122

An Intro duction to Light-Front Dynamics for Pedestrians 25

Scalar pro duct

0 0 3 3 ? ?

x
y = x y x y x
y : 123

De ne light-frontvariables

+ 0 3 0 3

x = x + x ; x = x x : 124



Let us denote the four-vector x by

 + ?

x =x ;x ;x : 125

Scalar pro duct

1 1

+ + ? ?

x
y = x y + x y x
y : 126

2 2

The metric tensor is

0 1

0 2 0 0

2 0 0 0

B C



g = : 127

@ A

0 0 1 0

0 0 0 1

0 1

1

0 0 0

2

1

0 0 0

B C

2

g = : 128

@ A



0 0 1 0

0 0 0 1

Thus

1 1

+

x = x ;x = x: 129

+

2 2

Partial derivatives:

@

+

@ =2@ =2 : 130

@x

@

: 131 @ =2@ =2

+

+

@x

Four-dimensional volume element:

1

4 0 2 ? 3 + 2 ?

d x = dx d x dx = dx dx d x : 132

2

Three dimensional volume element:

1

2 ?

[dx]= dx d x 133 2

26 Avaroth Harindranath

Lorentz invariantvolume element in momentum space:

+ 2 ?

dk d k

3

[d k ]= : 134

3 +

22  k

The step function

x=0; x< 0

=1; x> 0: 135

The antisymmetric step function

x= x x: 136

@

=2 x 137

@x

where  x is the Dirac delta function.

j x j = xx: 138

We de ne the integral op erators

Z

1 1

f x = dy x y  f y ; 139

+

@ 4

Z

1 1

2

  f x = dy j x y j f y : 140

+

@ 8

Unless otherwise sp eci ed, wecho ose the Bjorken and Drell convention for

gamma matrices:

0 1

1 0 0 0

0 1 0 0

B C

0

: 141 = =

@ A

0 0 1 0

0 0 0 1

0 

= : 142

 0

0 1

 = : 143

x

1 0

0 i

 = : 144

y

i 0

An Intro duction to Light-Front Dynamics for Pedestrians 27

1 0

 = : 145

z

0 1

0 1

0 0 1 0

0 0 0 1

B C

5 0 1 2 3

= i = : 146

@ A

1 0 0 0

0 1 0 0

0

= : 147

 0 3

=  : 148

Explicitly,

0 1

1 0 1 0

0 1 0 1

B C



= : 149

@ A

1 0 1 0

0 1 0 1

1 1 1

  0  3 

= = I  : 150  =

4 2 2

Explicitly,

0 1

1 0 1 0

1

0 1 0 1

B C

+

: 151  =

@ A

1 0 1 0

2

0 1 0 1

0 1

1 0 1 0

1

0 1 0 1

B C

 = 152

@ A

1 0 1 0

2

0 1 0 1

 2 

  =  : 153

 y 

  =  : 154

+

 +  = I: 155

?   ?

 =  : 156

0   0

 =  : 157

28 Avaroth Harindranath

?   ?

 =  : 158

5   5

 =  : 159

  0  

=2 =  : 160

1 1

i ij j 5 i 

 i  : 161  =

2 2

i

j i + ij + 5

 =  : 162

2

Dirac spinors

r

 

1

+ ? ? +

u k = m  +k + :k   : 163

 F 

+

m k

F

0 1

1

p

0

B C

: 164 2 m =

@ A

F "

0

0

0 1

0

p

1

B C

= : 165 2 m

@ A

 F

0

0

0 1

+

k +m

F

1 2

1

k +ik

B C

p

u k = : 166

@ A

"

+

+

k m

F

2k

1 2

k + ik

0 1

1 2

k +ik

+

1

k + m

B C

F

p

: 167 u k =

@ A



1 2

+

k ik

2k

+

k + m

F

0 1

1

r

+

k

0

B C

+

+

u k = : 168

@ A

"

1

2 0

An Intro duction to Light-Front Dynamics for Pedestrians 29

0 1

0

r

+

k

1

B C

+

+

u k = : 169

@ A



0

2

1

T

v k = C u k 170

 

2 0

where C = i is the charge conjugation op erator.

r

 

1

+ ? ? +

: 171 v k = m  +k + :k  

  F

+

m k

F

0 1

0

p

0

B C

= 2 m : 172

@ A

" F

0

1

0 1

0

p

0

B C

2 m : 173 =

@ A

F 

1

0

0 1

1 2

k ik

+

1

k + m

B C

F

p

v k = : 174

@ A

"

1 2

+

k + ik

2k

+

k + m

F

0 1

+

k +m

F

1 2

1

k ik

B C

p

v k = : 175

@ A



+

+

k m

F

2k

1 2

k ik

1 0

0

r

+

k

1

C B

+

: 176 k = v

A @

"

0

2

1

0 1

1

r

+

k

0

B C

+

v k = : 177

@ A



1

2 0

30 Avaroth Harindranath

B Survey of Light-Front Related Reviews, Bo oks

B.1 Review Articles on Light-Front

Several review articles have app eared touching up on various asp ects of light-front

dynamics. An almost complete list till the end of 1995 follows.

The article by Rohrlich 1971 discusses quantization on the light-front to-

gether with a careful examination of the asso ciated b oundary value problem.

Topics covered also include scale invariance and conformal invariance. A nice in-

tro duction to the initial value problem on the light-front is also given by Domokos

1971. Susskind 1969 and Kogut and Susskind 1973 provide the rationale

for considering eld theories in in nite momentum frame IMF with particular

emphasis on high energy pro cesses. They also discuss the nonrelativistic anal-

ogy, i.e, the corresp ondence b etween IMF physics and two-dimensional Galilean

mechanics. Jackiw 1972 compares and contrasts the derivation of sum rules in

deep inelastic scattering using a equal time quantization together with in nite

momentum techniques and b light-cone quantization. Melosh transformation

and its connection with the more familiar Pryce-Tani-Foldy-Wouthuysen trans-

formation are reviewed by Bell 1974. Bell and Ruegg 1975 discusses the rela-

tion b etween relativistic parton mo del, non-relativistic quark mo del, and various

SU6 and SU6 broken symmetry schemes. Relativistic Hamiltonian quan-

W

tum theories of nitely many degrees of freedom are reviewed by Leutwyler and

Stern 1978. Phenomenological use of light-cone wavefunctions can b e found in

the review articles of Frankfurt and Strikman 1981 and Frankfurt and Strik-

man 1988. Light-cone p erturbation theory and its application to various elds

are reviewed by Namyslowski 1985. For applications to p erturbative QCD see

the review articles of Lepage, Bro dsky, Huang, and Mackenzie 1983, Bro dsky

and Lepage 1989 and Ji 1989. An approach to hadron sp ectroscopy and form

factors utilizing a null plane approximation to Bethe-Salp eter equation is re-

viewed in Chakrabarty, Gupta, Singh and Mitra 1989. Null plane dynamics

of particles and elds is reviewed in Co ester 1991 and Keister and Polyzou

1991. Two review articles on null plane dynamics with emphasis on covariance

are Karmanov 1988 and Fuda 1991. The discretized light-cone quantization

program of Bro dsky and Pauli and collab orators is reviewed in Bro dsky and

Pauli 1991 and Bro dsky, McCartor, Pauli, and Pinsky 1992. Bro dsky, Mc-

Cartor, Pauli, and Pinsky 1992 also has an account of the so-called zero-mo de

problem. An overview of the whole sub ject is given by Ji 1992. Reviews of

light-front dynamics with emphasis on renormalization problem are given by

Glazek 1993 and byPerry 1994. A detailed review with emphasis on QCD

and phenomenology of hadron structure is given by Zhang 1994. For review of

light-front dynamics with detailed discussion of the asp ects of zero mo de prob-

lem, see, Burkardt 1995.

An Intro duction to Light-Front Dynamics for Pedestrians 31

B.2 Light-Front in Bo oks

Light-front dynamics has made its entry into a few b o oks. In the following,

wehave omitted standard textb o oks that intro duce light-frontvariables in the

context of deep inelastic scattering.

Avery brief treatment app ears in The Theory of Photons and Electrons:

The Relativistic Quantum Field Theory of Charged Particles with Spin One-

Half, Expanded Second Edition, J.M. Jauch and F. Rohrlich, Springer-Verlag,

New York, 1976.

In the context of current algebra and deep inelastic scattering, light-front

dynamics app ears in Currents in Hadron Physics, V. de Alfaro, S. Fubini, G. Fur-

lan, and C. Rossetti, North-Holland Publishing Company, Amsterdam, 1973.

This b o ok also provides an excellent discussion of the in nite-momentum limit.

Also, see, Theory of Lepton-Hadron Processes at High Energies: Partons, Scale

Invariance and Light-Cone Physics,P.Roy, Clarendon Press, Oxford, 1975.

Sp eaking of deep inelastic scattering, one should not forget partons. The clas-

sic reference is Photon-Hadron Interactions, R.P.Feynman, Benjamin, Reading,

MA 1972.

For the utility of light-frontvariables in high energy scattering in the context

of high orders of Feynman diagrams, see, Expanding Protons: Scattering at High

Energies, H. Cheng and T.T. Wu, The M.I.T. Press, Cambridge, Massachusetts,

1987.

In the context of Poincare Group and relativistic harmonic oscillator, see,

Theory and Applications of the PoincareGroup, Y.S. Kim and M.E. Noz, D.

Reidel Publishing, Dordrecht, Holland, 1988.

For the application of light-front formalism to relativistic nuclear physics, see,

Relativistic Nuclear Physics in the Light-Front Formalism, V.R. Garsevanishvili

and Z.R. Menteshashvili, Nova Science Publishers Inc., New York, 11725, 1993.

The following workshop pro ceedings deal with light-front dynamics.

1. Nuclear and Particle Physics on the Light Cone, edited by M.B. Johnson

and L.S. Kisslinger, World Scienti c, Singap ore, 1989.

2. Theory of Hadrons and Light-Front QCD, edited by St. D. Glazek, World

Scienti c, Singap ore, 1995.

32 Avaroth Harindranath

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