# Light front quantization

The **light-front quantization**^{[1]}
^{[2]}
^{[3]}
of quantum field theories
provides a useful alternative to ordinary equal-time
quantization. In
particular, it can lead to a relativistic description of bound systems
in terms of quantum-mechanical wave functions. The quantization is
based on the choice of light-front coordinates,^{[4]}
where plays the role of time and the corresponding spatial
coordinate is . Here, is the ordinary time,
is one Cartesian coordinate,
and is the speed of light. The other
two Cartesian coordinates, and , are untouched and often called
transverse or perpendicular, denoted by symbols of the type
. The choice of the
frame of reference where the time
and -axis are defined can be left unspecified in an exactly
soluble relativistic theory, but in practical calculations some choices may be more suitable than others.

## Overview

In practice, virtually all measurements are made at fixed light-front
time. For example, when an electron scatters on a proton as in the
famous SLAC experiments that discovered the quark structure of
hadrons, the interaction with
the constituents occurs at a single light-front time.
When one takes a flash photograph, the recorded image shows the object
as the front of the light wave from the flash crosses the object.
Thus Dirac used the terminology "light-front" and "front form" in
contrast to ordinary instant time and "instant form".^{[4]}
Light waves traveling in the negative direction
continue to propagate in at a single light-front time .

As emphasized by Dirac, Lorentz boosts of states at fixed light-front time are simple kinematic transformations. The description of physical systems in light-front coordinates is unchanged by light-front boosts to frames moving with respect to the one specified initially. This also means that there is a separation of external and internal coordinates (just as in nonrelativistic systems), and the internal wave functions are independent of the external coordinates, if there is no external force or field. In contrast, it is a difficult dynamical problem to calculate the effects of boosts of states defined at a fixed instant time .

The description of a bound state in a quantum field theory, such as an atom in quantum electrodynamics (QED) or a hadron in quantum chromodynamics (QCD), generally requires multiple wave functions, because quantum field theories include processes which create and annihilate particles. The state of the system then does not have a definite number of particles, but is instead a quantum-mechanical linear combination of Fock states, each with a definite particle number. Any single measurement of particle number will return a value with a probability determined by the amplitude of the Fock state with that number of particles. These amplitudes are the light-front wave functions. The light-front wave functions are each frame-independent and independent of the total momentum.

The wave functions are the solution of a field-theoretic analog of the Schrodinger equation of nonrelativistic quantum mechanics. In the nonrelativistic theory the Hamiltonian operator is just a kinetic piece and a potential piece . The wave function is a function of the coordinate , and is the energy. In light-front quantization, the formulation is usually written in terms of light-front momenta , with a particle index, , , and the particle mass, and light-front energies . They satisfy the mass-shell condition

The analog of the nonrelativistic Hamiltonian is the light-front
operator , which generates
translations in light-front time.
It is constructed from the Lagrangian for the chosen quantum field
theory. The total light-front momentum of the system,
, is the sum of the
single-particle light-front momenta. The total light-front energy
is fixed by the mass-shell condition to be
, where is the invariant mass of the system.
The Schrodinger-like equation of light-front quantization is then
. This provides a
foundation for a nonperturbative analysis of quantum field theories
that is quite distinct from the lattice
approach.^{[5]}
^{[6]}
^{[7]}

Quantization on the light-front provides the rigorous
field-theoretical realization of the intuitive ideas of the
parton model
which is formulated at fixed in the
infinite-momentum frame.^{[8]}
^{[9]}
(see #Infinite momentum frame )
The same results are obtained in the front
form for any frame; e.g., the structure functions and other
probabilistic parton distributions measured in deep inelastic scattering
are obtained from the squares of the boost-invariant light-front wave
functions,^{[10]}
the eigensolution of the light-front
Hamiltonian. The Bjorken kinematic variable of deep
inelastic scattering becomes identified with the light-front fraction at small
. The Balitsky-Fadin-Kuraev-Lipatov
(BFKL)^{[11]}
Regge behavior of structure functions can be
demonstrated from the behavior of light-front wave functions at small .
The Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP)
evolution^{[12]}
of structure functions and the
Efremov-Radyushkin-Brodsky-Lepage (ERBL)
evolution^{[13]}
^{[14]}
of distribution amplitudes
in are properties of the light-front wave functions at high
transverse momentum.

Computing hadronic matrix elements of currents is particularly simple
on the light-front, since they can be obtained rigorously as overlaps
of light-front wave functions as in the Drell-Yan-West
formula.^{[15]}
^{[16]}
^{[17]}

The gauge-invariant meson
and baryon distribution amplitudes which control hard exclusive and
direct reactions are the valence light-front wave functions integrated over transverse
momentum at fixed . The "ERBL"
evolution^{[13]}^{[14]} of distribution
amplitudes and the factorization theorems for hard exclusive processes
can be derived most easily using light-front methods. Given the
frame-independent light-front wave functions, one can compute a large range of hadronic
observables including generalized parton distributions, Wigner
distributions, etc. For example, the "handbag" contribution to the
generalized parton distributions for deeply virtual Compton scattering,
which can be computed from the overlap of light-front wave functions,
automatically satisfies the known sum rules.

The light-front wave functions contain information about novel features of QCD. These include effects suggested from other approaches, such as color transparency, hidden color, intrinsic charm, sea-quark symmetries, dijet diffraction, direct hard processes, and hadronic spin dynamics.

One can also prove fundamental theorems for relativistic quantum
field theories using the front form, including:
(a) the cluster decomposition theorem^{[18]}
and (b) the vanishing
of the anomalous gravitomagnetic moment for any Fock state of a
hadron;^{[19]}
one also can show that a nonzero
anomalous magnetic moment of a bound state requires nonzero
angular momentum of the constituents. The cluster
properties^{[20]}
of light-front time-ordered perturbation theory,
together with conservation, can be used
to elegantly derive the Parke-Taylor rules for multi-gluon scattering
amplitudes.^{[21]}
The counting-rule^{[22]}
behavior of structure functions
at large and Bloom-Gilman
duality^{[23]}
^{[24]}
have also been derived in light-front QCD (LFQCD).
The existence of "lensing effects" at leading twist, such as the
-odd "Sivers effect" in spin-dependent semi-inclusive deep-inelastic
scattering, was first demonstrated using light-front
methods.^{[25]}

Light-front quantization is thus the natural framework for the
description of the nonperturbative relativistic bound-state structure
of hadrons in quantum chromodynamics. The formalism is rigorous,
relativistic, and frame-independent. However, there exist subtle
problems in LFQCD that require thorough investigation. For example,
the complexities of the vacuum in the usual instant-time formulation,
such as the Higgs mechanism and
condensates in theory, have
their counterparts in zero modes or, possibly, in additional terms in
the LFQCD Hamiltonian that are allowed by power
counting.^{[26]}
Light-front considerations of the vacuum as well as
the problem of achieving full covariance in LFQCD require close
attention to the light-front singularities and zero-mode
contributions.^{[27]}
^{[28]}
^{[29]}
^{[30]}
^{[31]}
^{[32]}
^{[33]}
^{[34]}
^{[35]}
^{[36]}
^{[37]}
The truncation of the light-front
Fock-space calls for the introduction of effective quark and gluon
degrees of freedom to overcome truncation effects. Introduction of
such effective degrees of freedom is what one desires in seeking the
dynamical connection between canonical (or current) quarks and
effective (or constituent) quarks that Melosh sought, and Gell-Mann
advocated, as a method for truncating QCD.

The light-front Hamiltonian formulation thus opens access to QCD at the amplitude level and is poised to become the foundation for a common treatment of spectroscopy and the parton structure of hadrons in a single covariant formalism, providing a unifying connection between low-energy and high-energy experimental data that so far remain largely disconnected.

## Fundamentals

Front-form relativistic quantum mechanics was introduced by Paul Dirac
in a 1949 paper published in Reviews of Modern Physics^{[38]}
and Bargmann^{[39]}
showed that this symmetry must be realized by a unitary representation of the
connected component of the Poincare group on the Hilbert space of
the quantum theory. The Poincare symmetry is a dynamical symmetry
because Poincare transformations mix both space and time variables.
The dynamical nature of this symmetry is most easily seen by noting
that the Hamiltonian appears on the right-hand side of three of the
commutators of the Poincare generators,
, where are
components of the linear momentum and
are components of rotation-less boost generators. If the
Hamiltonian includes interactions, i.e. , then the
commutation relations cannot be satisfied unless at least three of the
Poincare generators also include interactions.

Dirac's paper^{[4]} introduced three distinct ways to minimally
include interactions in the Poincare Lie algebra. He referred to
the different minimal choices as the "instant-form", "point-form"
and "front-from" of the dynamics. Each "form of dynamics" is
characterized by a different interaction-free (kinematic) subgroup of
the Poincare group. In Dirac's instant-form dynamics the kinematic
subgroup is the three-dimensional Euclidean subgroup generated by
spatial translations and rotations, in Dirac's point-form dynamics
the kinematic subgroup is the Lorentz group and in Dirac's
"light-front dynamics" the kinematic subgroup
is the group of transformations that leave a three-dimensional
hyperplane tangent to the light cone invariant.

A light front is a three-dimensional hyperplane defined by the condition:

with , where the usual convention is to choose . Coordinates of points on the light-front hyperplane are

The Lorentz invariant inner product of two four-vectors, and , can be expressed in terms of their light-front components as

In a front-form relativistic quantum theory the three interacting generators of the Poincare group are , the generator of translations normal to the light front, and , the generators of rotations transverse to the light-front. is called the "light-front" Hamiltonian.

The kinematic generators, which generate transformations tangent to the light front, are free of interaction. These include and , which generate translations tangent to the light front, which generates rotations about the axis, and the generators , and of light-front preserving boosts,

which form a closed subalgebra.

Light-front quantum theories have the following distinguishing properties:

- Only three Poincare generators include interactions. All of Dirac's other forms of the dynamics require four or more interacting generators.

- The light-front boosts are a three-parameter subgroup of the Lorentz group that leave the light front invariant.

- The spectrum of the kinematic generator, , is the positive real line.

These properties have consequences that are useful in applications.

There is no loss of generality in using light-front relativistic quantum theories. For systems of a finite number of degrees of freedom there are explicit -matrix-preserving unitary transformations that transform theories with light-front kinematic subgroups to equivalent theories with instant-form or point-form kinematic subgroups. One expects that this is true in quantum field theory, although establishing the equivalence requires a nonperturbative definition of the theories in different forms of dynamics.

### light-front boosts

In general if one multiplies a Lorentz boost on the right by a momentum-dependent rotation, which leaves the rest vector unchanged, the result is a different type of boost. In principle there are as many different kinds of boosts as there are momentum-dependent rotations. The most common choices are rotation-less boosts, helicity boosts, and light-front boosts. The light-front boost (Template:EquationNote) is a Lorentz boost that leaves the light front invariant.

The light-front boosts are not only members of the light-front kinematic subgroup, but they also form a closed three-parameter subgroup. This has two consequences. First, because the boosts do not involve interactions, the unitary representations of light-front boosts of an interacting system of particles are tensor products of single-particle representations of light-front boosts. Second, because these boosts form a subgroup, arbitrary sequences of light-front boosts that return to the starting frame do not generate Wigner rotations.

The spin of a particle in a relativistic quantum theory is the angular momentum of the particle in its rest frame. Spin observables are defined by boosting the particle's angular momentum tensor to the particle's rest frame

where is a Lorentz boost that transforms to .

The components of the resulting spin vector, , always satisfy commutation relations, but the individual components will depend on the choice of boost . The light-front components of the spin are obtained by choosing to be the inverse of the light-front preserving boost, (Template:EquationNote).

The light-front components of the spin are the components of the spin measured in the particle's rest frame after transforming the particle to its rest frame with the light-front preserving boost (Template:EquationNote). The light-front spin is invariant with respect to light-front preserving-boosts because these boosts do not generate Wigner rotations. The component of this spin along the direction is called the light-front helicity. In addition to being invariant, it is also a kinematic observable, i.e. free of interactions. It is called a helicity because the spin quantization axis is determined by the orientation of the light front. It differs from the Jacob-Wick helicity, where the quantization axis is determined by the direction of the momentum.

These properties simplify the computation of current matrix elements because (1) initial and final states in different frames are related by kinematic Lorentz transformations, (2) the one-body contributions to the current matrix, which are important for hard scattering, do not mix with the interaction-dependent parts of the current under light front boosts and (3) the light-front helicities remain invariant with respect to the light-front boosts. Thus, light-front helicity is conserved by every interaction at every vertex.

Because of these properties, front-form quantum theory is the only form of relativistic dynamics that has true "frame-independent" impulse approximations, in the sense that one-body current operators remain one-body operators in all frames related by light-front boosts and the momentum transferred to the system is identical to the momentum transferred to the constituent particles. Dynamical constraints, which follow from rotational covariance and current covariance, relate matrix elements with different magnetic quantum numbers. This means that consistent impulse approximations can only be applied to linearly independent current matrix elements.

### spectral condition

A second unique feature of light-front quantum theory follows because the operator is non-negative and kinematic. The kinematic feature means that the generator is the sum of the non-negative single-particle generators, (. It follows that if is zero on a state, then each of the individual must also vanish on the state.

In perturbative light-front quantum field theory this property leads
to a suppression of a large class of diagrams, including all vacuum
diagrams, which have zero internal . The condition
corresponds to infinite momentum . Many of the
simplifications of light-front quantum field theory are realized in
the infinite momentum
limit^{[40]}
^{[41]}
of ordinary canonical field theory (see #Infinite momentum frame).

An important consequence of the spectral condition on and the subsequent suppression of the vacuum diagrams in perturbative field theory is that the perturbative vacuum is the same as the free-field vacuum. This results in one of the great simplifications of light-front quantum field theory, but it also leads to some puzzles with regard the formulation of theories with spontaneously broken symmetries.

### equivalence of forms of dynamics

Sokolov^{[42]}
^{[43]}
demonstrated that
relativistic quantum theories based on different forms of dynamics are
related by -matrix-preserving unitary transformations. The
equivalence in field theories is more complicated because the
definition of the field theory requires a redefinition of the
ill-defined local operator products that appear in the dynamical
generators. This is achieved through renormalization. At the
perturbative level, the ultraviolet divergences of a canonical field
theory are replaced by a mixture of ultraviolet and infrared
divergences in light-front field theory. These have to be
renormalized in a manner that recovers the full rotational covariance and
maintains the -matrix equivalence. The renormalization of light
front field theories is discussed in Light-front computational methods#Renormalization group.

### classical vs quantum

One of the properties of the classical wave equation is that the light-front is a characteristic surface for the initial value problem. This means the data on the light front is insufficient to generate a unique evolution off of the light front. If one thinks in purely classical terms one might anticipate that this problem could lead to an ill-defined quantum theory upon quantization.

In the quantum case the problem is to find a set of ten self-adjoint
operators that satisfy the Poincare Lie algebra. In the absence of
interactions, Stone's theorem applied to tensor products of known
unitary irreducible representations of the Poincare group gives a
set of self-adjoint light-front generators with all of the required
properties. The problem of adding interactions is no
different^{[44]}
than it is in non-relativistic quantum
mechanics, except that the added interactions also need to preserve
the commutation relations.

There are, however, some related observations. One is that if one takes seriously the classical picture of evolution off of surfaces with different values of , one finds that the surfaces with are only invariant under a six parameter subgroup. This means that if one chooses a quantization surface with a fixed non-zero value of , the resulting quantum theory would require a fourth interacting generator. This does not happen in light-front quantum mechanics; all seven kinematic generators remain kinematic. The reason is that the choice of light front is more closely related to the choice of kinematic subgroup, than the choice of an initial value surface.

In quantum field theory, the vacuum expectation value of two fields
restricted to the light front are not well-defined distributions on
test functions restricted to the light front. They only become
well defined distributions on functions of four space time
variables.^{[45]}
^{[46]}

### rotational invariance

The dynamical nature of rotations in light-front quantum theory means that preserving full rotational invariance is non-trivial. In field theory, Noether's theorem provides explicit expressions for the rotation generators, but truncations to a finite number of degrees of freedom can lead to violations of rotational invariance. The general problem is how to construct dynamical rotation generators that satisfy Poincare commutation relations with and the rest of the kinematic generators. A related problem is that, given that the choice of orientation of the light front manifestly breaks the rotational symmetry of the theory, how is the rotational symmetry of the theory recovered?

Given a dynamical unitary representation of rotations, , the
product of a kinematic rotation with the
inverse of the corresponding dynamical rotation is a unitary operator
that (1) preserves the -matrix and (2) changes the kinematic
subgroup to a kinematic subgroup with a rotated light front,
. Conversely, if the -matrix
is invariant with respect to changing the orientation of the
light-front, then the dynamical unitary representation of rotations,
, can be constructed using the generalized wave operators for
different orientations of the light
front^{[47]}
^{[48]}
^{[49]}
^{[50]}
^{[51]}
and the kinematic representation of rotations

Because the dynamical input to the -matrix is , the invariance of the -matrix with respect to changing the orientation of the light front implies the existence of a consistent dynamical rotation generator without the need to explicitly construct that generator. The success or failure of this approach is related to ensuring the correct rotational properties of the asymptotic states used to construct the wave operators, which in turn requires that the subsystem bound states transform irreducibly with respect to .

These observations make it clear that the rotational covariance of the
theory is encoded in the choice of light-front Hamiltonian.
Karmanov^{[52]}
^{[53]}
^{[54]}
introduced a
covariant formulation of light-front quantum theory, where the
orientation of the light front is treated as a degree of freedom.
This formalism can be used to identify observables that do not depend
on the orientation, , of the light front (see
#Covariant formulation).

While the light-front components of the spin are invariant under
light-front boosts, they Wigner rotate under rotation-less boosts and
ordinary rotations. Under rotations the light-front components of the
single-particle spins of different particles experience different
Wigner rotations. This means that the light-front spin components
cannot be directly coupled using the standard rules of angular
momentum addition. Instead, they must first be transformed to the
more standard canonical spin components, which have the property that
the Wigner rotation of a rotation is the rotation. The spins can then
be added using the standard rules of angular momentum addition and the
resulting composite canonical spin components can be transformed back
to the light-front composite spin components. The transformations
between the different types of spin components are called Melosh
rotations.^{[55]}
^{[56]}
They are the momentum-dependent
rotations constructed by multiplying a light-front boost
followed by the inverse
of the corresponding rotation-less boost. In order to also add the
relative orbital angular momenta, the relative orbital
angular momenta of each particle must also be converted to a
representation where they Wigner rotate with the spins.

While the problem of adding spins and internal orbital angular momenta
is more complicated,^{[57]}
it is only total angular
momentum that requires interactions; the total spin does not
necessarily require an interaction dependence. Where the interaction
dependence explicitly appears is in the relation between the total spin
and the total angular
momentum^{[56]}
^{[58]}

where here and contain interactions. The transverse
components of the
light-front spin, may or may not have an
interaction dependence; however, if one also demands cluster
properties,^{[59]}
then the transverse components of
total spin necessarily have an interaction dependence. The result is
that by choosing the light front components of the spin to be
kinematic it is possible to realize full rotational invariance at the
expense of cluster properties. Alternatively it is easy to realize
cluster properties at the expense of full rotational symmetry. For
models of a finite number of degrees of freedom there are
constructions that realize both full rotational covariance and cluster
properties;^{[60]}
these realizations all have additional
many-body interactions in the generators that are functions of
fewer-body interactions.

The dynamical nature of the rotation generators means that tensor and spinor operators, whose commutation relations with the rotation generators are linear in the components of these operators, impose dynamical constraints that relate different components of these operators.

### nonperturbative dynamics

The strategy for performing nonperturbative calculations in light-front field theory is similar to the strategy used in lattice calculations. In both cases a nonperturbative regularization and renormalization are used to try to construct effective theories of a finite number of degrees of freedom that are insensitive to the eliminated degrees of freedom. In both cases the success of the renormalization program requires that the theory has a fixed point of the renormalization group; however, the details of the two approaches differ. The renormalization methods used in light-front field theory are discussed in Light-front computational methods#Renormalization group. In the lattice case the computation of observables in the effective theory involves the evaluation of large-dimensional integrals, while in the case of light-front field theory solutions of the effective theory involve solving large systems of linear equations. In both cases multi-dimensional integrals and linear systems are sufficiently well understood to formally estimate numerical errors. In practice such calculations can only be performed for the simplest systems. Light-front calculations have the special advantage that the calculations are all in Minkowski space and the results are wave functions and scattering amplitudes.

## Relativistic quantum mechanics

While most applications of light-front quantum mechanics are to the light-front formulation of quantum field theory, it is also possible to formulate relativistic quantum mechanics of finite systems of directly interacting particles with a light-front kinematic subgroup. Light-front relativistic quantum mechanics is formulated on the direct sum of tensor products of single-particle Hilbert spaces. The kinematic representation of the Poincar\'e group on this space is the direct sum of tensor products of the single-particle unitary irreducible representations of the Poincar\'e group. A front-form dynamics on this space is defined by a dynamical representation of the Poincar\'e group on this space where when is in the kinematic subgroup of the Poincare group.

One of the advantages of light-front quantum mechanics is that it is
possible to realize exact rotational covariance for system of a finite
number of degrees of freedom. The way that this is done is to start
with the non-interacting generators of the full Poincar\'e group,
which are sums of single-particle generators, construct the kinematic invariant
mass operator, the three kinematic generators of translations tangent
to the light-front, the three kinematic light-front boost generators
and the three components of the light-front spin operator.
The generators are well-defined functions of these
operators^{[58]}
^{[61]}
given by (Template:EquationNote)
and . Interactions
that commute with all of these operators except the kinematic mass are
added to the kinematic mass operator to construct a dynamical mass
operator. Using this mass operator in (Template:EquationNote) and the expression
for gives a set of dynamical Poincare generators with a
light-front kinematic subgroup.^{[60]}

A complete set of irreducible eigenstates can be found by diagonalizing the interacting mass operator in a basis of simultaneous eigenstates of the light-front components of the kinematic momenta, the kinematic mass, the kinematic spin and the projection of the kinematic spin on the axis. This is equivalent to solving the center-of-mass Schrodinger equation in non-relativistic quantum mechanics. The resulting mass eigenstates transform irreducibly under the action of the Poincare group. These irreducible representations define the dynamical representation of the Poincare group on the Hilbert space.

This representation fails to satisfy cluster
properties,^{[59]} but this can be restored using a
front-form generalization^{[56]}
^{[60]} of the
recursive construction given by Sokolov.^{[42]}

## Infinite momentum frame

The "infinite momentum frame" (IMF) was originally
introduced^{[62]} to provide a physical interpretation
of the Bjorken variable measured in deep
inelastic lepton-proton scattering in
Feynman's parton model. (Here is the square of the
spacelike momentum transfer imparted by the lepton and
is the energy transferred in the proton's rest
frame.) If one considers a hypothetical Lorentz frame where the
observer is moving at infinite momentum, , in the
negative direction, then can be interpreted as the
longitudinal momentum fraction carried by the
struck quark (or "parton") in the incoming fast moving proton. The
structure function of the proton measured in the experiment is then
given by the square of its instant-form wave function boosted to
infinite momentum.

Formally, there is a simple connection between the Hamiltonian
formulation of quantum field theories quantized at fixed time (the
"instant form" ) where the observer is moving at infinite momentum
and light-front Hamiltonian theory quantized at fixed light-front time
(the "front form"). A typical energy denominator in
the instant-form is
where
is the sum of energies of the particles in the
intermediate state. In the IMF, where the observer moves at high
momentum in the negative direction, the leading terms in
cancel, and the energy denominator becomes where
is invariant mass squared of the initial state. Thus, by
keeping the terms in in the instant form, one recovers the
energy denominator which appears in light-front Hamiltonian theory.
This correspondence has a physical meaning: measurements made by an
observer moving at infinite momentum is analogous to making
observations approaching the speed of light—thus matching to the
front form where measurements are made along the front of a
light wave. An example of an application to quantum electrodynamics
can be found in the work of Brodsky, Roskies and
Suaya.^{[63]}

The vacuum state in the instant form defined at fixed is acausal
and infinitely complicated. For example, in quantum electrodynamics,
bubble graphs of all orders, starting with the
intermediate state, appear in the ground state vacuum; however, as
shown by Weinberg,^{[41]} such vacuum graphs are
frame-dependent and formally vanish by powers of as the
observer moves at . Thus, one can again match the
instant form to the front-form formulation where such vacuum loop
diagrams do not appear in the QED ground state. This is because the
momentum of each constituent is positive, but must sum to zero in
the vacuum state since the momenta are conserved. However, unlike
the instant form, no dynamical boosts are required, and the front form
formulation is causal and frame-independent. The infinite momentum
frame formalism is useful as an intuitive tool; however, the limit
is not a rigorous limit, and the need to boost the
instant-form wave function introduces complexities.

## Covariant formulation

In light-front coordinates, , , the spatial coordinates do not enter symmetrically: the coordinate is distinguished, whereas and do not appear at all. This non-covariant definition destroys the spatial symmetry that, in its turn, results in a few difficulties related to the fact that some transformation of the reference frame may change the orientation of the light-front plane. That is, the transformations of the reference frame and variation of orientation of the light-front plane are not decoupled from each other. Since the wave function depends dynamically on the orientation of the plane where it is defined, under these transformations the light-front wave function is transformed by dynamical operators (depending on the interaction). Therefore, in general, one should know the interaction to go from given reference frame to the new one. The loss of symmetry between the coordinates and complicates also the construction of the states with definite angular momentum since the latter is just a property of the wave function relative to the rotations which affects all the coordinates .

To overcome this inconvenience, there was developed the explicitly
covariant version^{[64]}^{[54]} of
light-front quantization (reviewed by Carbonell
et al.^{[65]}),
in which the state vector is defined on the light-front plane of
general orientation:
(instead of ),
where
is a four-dimensional vector in the four-dimensional space-time and
is also a four-dimensional vector with the property . In the particular case
we come back to the standard construction. In the explicitly covariant formulation the
transformation of the reference frame and the change of orientation of the light-front plane
are decoupled. All the rotations and the Lorentz transformations are purely
kinematical (they do not require knowledge of the interaction), whereas the
(dynamical) dependence on the orientation of the light-front plane is covariantly parametrized
by the wave function dependence on the four-vector .

There were formulated the rules of graph techniques which, for a given Lagrangian,
allow to calculate the perturbative decomposition of the state vector evolving in the
light-front time (in contrast to the evolution in the
direction or ). For the instant form of dynamics,
these rules were firstl developed by
Kadyshevsky.^{[66]}
^{[67]}
By these rules, the light-front amplitudes are represented as the
integrals over the momenta of particles in intermediate states. These
integrals are three-dimensional, and all the four-momenta
are on the corresponding mass shells ,
in contrast to the Feynman rules containing four-dimensional integrals
over the off-mass-shell momenta. However, the calculated light-front amplitudes, being
on the mass shell, are
in general the off-energy-shell amplitudes. This means that the on-mass-shell four-momenta,
which these amplitudes depend on, are not conserved in the direction
(or, in general, in the direction ).
The off-energy shell amplitudes do not coincide with the Feynman amplitudes, and they depend on
the orientation of the light-front plane. In the covariant formulation, this dependence is explicit:
the amplitudes are functions of . This allows one to apply to them in
full measure the well-known techniques developed for the covariant Feynman
amplitudes (constructing the invariant variables, similar to the Mandelstam variables,
on which the amplitudes depend;
the decompositions, in the case of particles with spins, in invariant amplitudes;
extracting electromagnetic form factors; etc.). The irreducible off-energy-shell
amplitudes serve as the kernels of equations for the light-front wave functions.
The latter ones are found from these equations and used to analyze hadrons
and nuclei.

For spinless particles, and in the particular case of ,
the amplitudes found by the rules of covariant graph techniques, after replacement
of variables, are reduced to the amplitudes given by the Weinberg
rules^{[41]} in the
infinite momentum frame. The dependence on orientation of the
light-front plane manifests itself in the dependence of the off-energy-shell Weinberg
amplitudes on the variables taken separately but not
in some particular combinations like the Mandelstam variables .

On the energy shell, the amplitudes do not depend on the four-vector determining orientation of the corresponding light-front plane. These on-energy-shell amplitudes coincide with the on-mass-shell amplitudes given by the Feynman rules. However, the dependence on can survive because of approximations.

## Angular momentum

The covariant formulation is especially useful for constructing the states with definite angular momentum. In this construction, the four-vector participates on equal footing with other four-momenta, and, therefore, the main part of this problem is reduced to the well-know one. For example, as is well known, the wave function of a non-relativistic system, consisting of two spinless particles with the relative momentum and with total angular momentum , is proportional to the spherical function : , where and is a function depending on the modulus . The angular momentum operator reads: . Then the wave function of a relativistic system in the covariant formulation of light-front dynamics obtains the similar form:

where
and are functions depending, in additional
to , on the scalar product .
The variables , are invariant not only under rotations
of the vectors , but also under rotations and the Lorentz
transformations of initial four-vectors , .
The second contribution
means that the operator of the total angular momentum in explicitly covariant
light-front dynamics obtains an additional
term: .
For non-zero spin particles this operator obtains the contribution of the spin
operators:^{[47]}
^{[48]}
^{[49]}^{[50]}
^{[68]}^{[69]}

The fact that the transformations changing the orientation of the light-front
plane are dynamical (the corresponding generators of the Poincare group contain
interaction) manifests itself in the dependence
of the coefficients on the scalar product varying
when the orientation of the unit vector changes (for fixed ).
This dependence (together with the dependence on ) is found from the dynamical
equation for the wave function.

A peculiarity of this construction is in the fact
that there exists the operator which commutes both
with the Hamiltonian and with . Then the states are labeled also
by the eigenvalue of the operator : .
For given angular momentum , there are such the states. All of them are
degenerate, i.e. belong to the same mass (if we do not make an approximation).
However, the wave function should also satisfy the so-called angular
condition^{[53]}^{[54]}
^{[70]}
^{[71]}
^{[72]}
After satisfying it, the solution obtains the form of an unique superposition of
the states with different eigenvalues
.^{[54]}^{[65]}

The extra contribution in the light-front angular
momentum operator increases the number of spin components
in the light-front wave function. For example, the non-relativistic deuteron wave function
is determined by two components (- and -waves).
Whereas, the relativistic light-front deuteron wave function is determined by six
components.^{[68]}^{[69]}
These components were calculated in the one-boson exchange
model.^{[73]}

## Goals and Prospects

The central issue for light-front quantization is the rigorous description of hadrons, nuclei, and systems thereof from first principles in QCD. The main goals of the research using light-front dynamics are

- Evaluation of masses and wave functions of hadrons using the light-front Hamiltonian of QCD.

- The analysis of hadronic and nuclear phenomenology based on fundamental quark and gluon dynamics, taking advantage of the connections between quark-gluon and nuclear many-body methods.

- Understanding of the properties of QCD at finite temperatures and densities, which is relevant for

understanding the early universe as well as compact stellar objects.

- Developing predictions for tests at the new and upgraded hadron experimental facilities -- JLAB, LHC, RHIC, J-PARC, GSI(FAIR).

- Analyzing the physics of intense laser fields, including a nonperturbative approach to strong-field QED.

- Providing bottom-up fitness tests for model theories as exemplified in the case of Standard Model.

The nonperturbative analysis of light-front QCD requires the following:

- Continue testing the light-front Hamiltonian approach in simple theories in order to improve our understanding of its peculiarities and treacherous points vis a vis manifestly-covariant quantization methods.

This will include work on theories such as Yukawa theory and QED and on theories with unbroken supersymmetry, in order to understand the strengths and limitations of different methods. Much progress has already been made along these lines.

- Construct symmetry-preserving regularization and renormalization schemes for light-front QCD, to include the Pauli-Villars-based method of the St. Petersburg group,
^{[74]}^{[75]}Glazek-Wilson similarity renormalization-group procedure for Hamiltonians,^{[76]}^{[77]}^{[78]}Mathiot-Grange test functions,^{[79]}Karmanov-Mathiot-Smirnov^{[80]}realization of sector-dependent renormalization, and determine how to incorporate symmetry breaking in light-front quantization;^{[81]}^{[82]}^{[83]}^{[84]}^{[85]}^{[86]}this is likely to require an analysis of zero modes and in-hadron condensates.^{[5]}^{[27]}^{[28]}^{[29]}^{[30]}^{[31]}^{[32]}^{[33]}^{[34]}^{[35]}^{[36]}^{[37]}

- Develop computer codes which implement the regularization and renormalization schemes.

Provide a platform-independent, well-documented
core of routines that allow investigators to
implement different numerical approximations to
field-theoretic eigenvalue problems, including the
light-front coupled-cluster
method^{[87]}
^{[88]}
finite elements, function
expansions,^{[89]}
and the complete orthonormal wave functions obtained from
AdS/QCD. This will build on
the Lanczos-based MPI code developed for
nonrelativistic nuclear physics applications and
similar codes for Yukawa theory and
lower-dimensional supersymmetric Yang—Mills
theories.

- Address the problem of computing rigorous bounds on truncation errors, particularly for energy scales where QCD is strongly coupled.

Understand the role of renormalization group methods, asymptotic freedom and spectral properties of in quantifying truncation errors.

- Solve for hadronic masses and wave functions.

Use these wave functions to compute form factors, generalized parton distributions, scattering amplitudes, and decay rates. Compare with perturbation theory, lattice QCD, and model calculations, using insights from AdS/QCD, where possible. Study the transition to nuclear degrees of freedom, beginning with light nuclei.

- Classify the spectrum with respect to total angular momentum.

In equal-time quantization, the three generators of rotations are kinematic, and the analysis of total angular momentum is relatively simple. In light-front quantization, only the generator of rotations around the -axis is kinematic; the other two, of rotations about axes and , are dynamical. To solve the angular momentum classification problem, the eigenstates and spectra of the sum of squares of these generators must be constructed. This is the price to pay for having more kinematical generators than in equal-time quantization, where all three boosts are dynamical. In light-front quantization, the boost along is kinematic, and this greatly simplifies the calculation of matrix elements that involve boosts, such as the ones needed to calculate form factors. The relation to covariant Bethe-Salpeter approaches projected on the light-front may help in understanding the angular momentum issue and its relationship to the Fock-space truncation of the light-front Hamiltonian. Model-independent constraints from the general angular condition, which must be satisfied by the light-front helicity amplitudes, should also be explored. The contribution from the zero mode appears necessary for the hadron form factors to satisfy angular momentum conservation, as expressed by the angular condition. The relation to light-front quantum mechanics, where it is possible to exactly realize full rotational covariance and construct explicit representations of the dynamical rotation generators, should also be investigated.

- Explore the AdS/QCD correspondence and light-front holography.
^{[90]}^{[91]}^{[92]}^{[93]}^{[94]}^{[95]}

The approximate duality in the limit of massless
quarks motivates few-body analyses of meson and
baryon spectra based on a one-dimensional
light-front Schrodinger equation in terms of the
modified transverse coordinate . Models
that extend the approach to massive quarks have
been proposed, but a more fundamental
understanding within QCD is needed. The nonzero
quark masses introduce a non-trivial dependence on
the longitudinal momentum, and thereby highlight
the need to understand the representation of
rotational symmetry within the formalism.
Exploring AdS/QCD wave functions as part of a
physically motivated Fock-space basis set to
diagonalize the LFQCD Hamiltonian should shed
light on both issues. The complementary Ehrenfest
interpretation^{[96]}
can be used to introduce effective
degrees of freedom such as diquarks in
baryons.

- Develop numerical methods/computer codes to directly evaluate the partition function (viz. thermodynamic potential) as the basic thermodynamic quantity.

Compare to lattice QCD, where applicable, and focus on a finite chemical potential, where reliable lattice QCD results are presently available only at very small (net) quark densities. There is also an opportunity for use of light-front AdS/QCD to explore non-equilibrium phenomena such as transport properties during the very early state of a heavy ion collision. Light-front AdS/QCD opens the possibility to investigate hadron formation in such a non-equilibrated strongly coupled quark-gluon plasma.

- Develop a light-front approach to the neutrino oscillation experiments possible at Fermilab and elsewhere, with the goal of reducing the energy spread of the neutrino-generating hadronic sources, so that the three-energy-slits interference picture of the oscillation pattern
^{[97]}can be resolved and the front form of Hamiltonian dynamics utilized in providing the foundation for qualitatively new (treating the vacuum differently) studies of neutrino mass generation mechanisms.

- If the renormalization group procedure for effective particles (RGPEP)
^{[98]}^{[99]}does allow one to study intrinsic charm, bottom, and glue in a systematically renormalized and convergent light-front Fock-space expansion, one might consider a host of new experimental studies of production processes using the intrinsic components that are not included in the calculations based on gluon and quark splitting functions.

## See also

- Light-front computational methods
- Light-front quantization applications
- Quantum field theories
- Quantum chromodynamics
- Quantum electrodynamics
- Light-front holography

## References

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## External links

- ILCAC, Inc., the International Light-Cone Advisory Committee.
- Publications on light-front dynamics, maintained by A. Harindranath.