The light-front quantization
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,
where
x
+
≡
c
t
+
z
{\displaystyle x^{+}\equiv ct+z}
plays the role of time and the corresponding
spatial
coordinate is
x
−
≡
c
t
−
z
{\displaystyle x^{-}\equiv ct-z}
. Here,
t
{\displaystyle t}
is the ordinary time,
z
{\displaystyle z}
is one Cartesian coordinate,
and
c
{\displaystyle c}
is the speed of light. The other
two Cartesian coordinates,
x
{\displaystyle x}
and
y
{\displaystyle y}
, are untouched and often called
transverse or perpendicular, denoted by symbols
of the type
x
→
⊥
=
(
x
,
y
)
{\displaystyle {\vec {x}}_{\perp }=(x,y)}
. The choice of the
frame of reference where the time
t
{\displaystyle t}
and
z
{\displaystyle z}
-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".
Light waves traveling in the negative
z
{\displaystyle z}
direction
continue to propagate in
x
−
{\displaystyle x^{-}}
at a single light-front time
x
+
{\displaystyle x^{+}}
.
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
t
{\displaystyle t}
.
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
Schrödinger equation
H
ψ
=
E
ψ
{\displaystyle H\psi =E\psi }
of nonrelativistic quantum
mechanics. In the nonrelativistic theory the
Hamiltonian operator
H
{\displaystyle H}
is just a kinetic
piece
−
ℏ
2
2
m
∇
2
{\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla
^{2}}
and
a potential piece
V
(
r
→
)
{\displaystyle V({\vec {r}})}
.
The wave function
ψ
{\displaystyle \psi }
is a function of the coordinate
r
→
{\displaystyle {\vec {r}}}
, and
E
{\displaystyle E}
is the energy. In light-front quantization,
the formulation is
usually written in terms of light-front momenta
p
_
i
=
(
p
i
+
,
p
→
⊥
i
)
{\displaystyle {\underline {p}}_{i}=(p_{i}^{+},{\vec
{p}}_{\perp i})}
, with
i
{\displaystyle i}
a particle index,
p
i
+
≡
p
i
2
+
m
i
2
+
p
i
z
{\displaystyle p_{i}^{+}\equiv {\sqrt {p_{i}^{2}+m_{i}^{2}}}+p_{iz}}
,
p
→
⊥
i
=
(
p
i
x
,
p
i
y
)
{\displaystyle {\vec {p}}_{\perp i}=(p_{ix},p_{iy})}
, and
m
i
{\displaystyle m_{i}}
the particle mass, and light-front
energies
p
i
−
≡
p
i
2
+
m
i
2
−
p
i
z
{\displaystyle p_{i}^{-}\equiv {\sqrt {p_{i}^{2}+m_{i}^{2}}}-p_{iz}}
. They satisfy the
mass-shell
condition
m
i
2
=
p
i
+
p
i
−
−
p
→
⊥
i
2
{\displaystyle m_{i}^{2}=p_{i}^{+}p_{i}^{-}-{\vec
{p}}_{\perp i}^{2}}
The analog of the nonrelativistic Hamiltonian
H
{\displaystyle H}
is the light-front
operator
P
−
{\displaystyle {\mathcal {P}}^{-}}
, 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,
P
_
≡
(
P
+
,
P
→
⊥
)
{\displaystyle {\underline {P}}\equiv (P^{+},{\vec
{P}}_{\perp })}
, is the sum of the
single-particle light-front momenta. The total
light-front energy
P
−
{\displaystyle P^{-}}
is fixed by the mass-shell condition to be
(
M
2
+
P
⊥
2
)
/
P
+
{\displaystyle (M^{2}+P_{\perp }^{2})/P^{+}}
, where
M
{\displaystyle M}
is the invariant mass of the system.
The Schrödinger-like equation of light-front
quantization is then
P
−
ψ
=
M
2
+
P
⊥
2
P
+
ψ
{\displaystyle {\mathcal {P}}^{-}\psi ={\frac
{M^{2}+P_{\perp }^{2}}{P^{+}}}\psi }
. This provides a
foundation for a nonperturbative analysis
of quantum field theories
that is quite distinct from the lattice
approach.Quantization on the light-front provides
the rigorous
field-theoretical realization of the intuitive
ideas of the
parton model
which is formulated at fixed
t
{\displaystyle t}
in the
infinite-momentum frame.
(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,
the eigensolution of the light-front
Hamiltonian. The Bjorken kinematic variable
x
b
j
{\displaystyle x_{bj}}
of deep
inelastic scattering becomes identified with
the light-front fraction at small
x
{\displaystyle x}
. The Balitsky–Fadin–Kuraev–Lipatov
(BFKL)
Regge behavior of structure functions can
be
demonstrated from the behavior of light-front
wave functions at small
x
{\displaystyle x}
.
The Dokshitzer–Gribov–Lipatov–Altarelli–Parisi
(DGLAP)
evolution
of structure functions and the
Efremov–Radyushkin–Brodsky–Lepage (ERBL)
evolution
of 
distribution amplitudes
in
log
⁡
Q
2
{\displaystyle \log Q^{2}}
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.
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
x
i
=
k
i
+
/
P
+
{\displaystyle x_{i}={k_{i}^{+}/P^{+}}}
. The "ERBL"
evolution 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
and (b) the vanishing
of the anomalous gravitomagnetic moment for
any Fock state of a
hadron;
one also can show that a nonzero
anomalous magnetic moment of a bound state
requires nonzero
angular momentum of the constituents. The
cluster
properties
of light-front time-ordered perturbation theory,
together with
J
z
{\displaystyle J^{z}}
conservation, can be used
to elegantly derive the Parke-Taylor rules
for multi-gluon scattering
amplitudes.
The counting-rule
behavior of structure functions
at large
x
{\displaystyle x}
and Bloom-Gilman
duality
have also been derived in light-front QCD
(LFQCD).
The existence of "lensing effects" at leading
twist, such as the
T
{\displaystyle T}
-odd "Sivers effect" in spin-dependent semi-inclusive
deep-inelastic
scattering, was first demonstrated using light-front
methods.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
ϕ
4
{\displaystyle \phi ^{4}}
theory, have
their counterparts in zero modes or, possibly,
in additional terms in
the LFQCD Hamiltonian that are allowed by
power
counting.
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.
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.
Light-front quantum field theory is the front-form
representation of
local relativistic quantum field theory.
The relativistic invariance of a quantum theory
means that the
observables (probabilities, expectation values
and ensemble averages) have the same values
in all inertial coordinate systems. Since
different inertial coordinate systems are
related by inhomogeneous
Lorentz transformations (Poincaré transformations),
this requires
that the Poincaré group is a symmetry group
of the theory.
Wigner
and Bargmann
showed that this symmetry must be realized
by a unitary representation of the
connected component of the Poincaré group
on the Hilbert space of
the quantum theory. The Poincaré symmetry
is a dynamical symmetry
because Poincaré 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 Poincaré generators,
[
K
j
,
P
k
]
=
i
δ
j
k
H
{\displaystyle [K^{j},P^{k}]=i\delta ^{jk}H}
, where
P
k
{\displaystyle P^{k}}
are
components of the linear momentum and
K
j
{\displaystyle K^{j}}
are components of rotation-less boost generators.
If the
Hamiltonian includes interactions, i.e.
H
=
H
0
+
V
{\displaystyle H=H_{0}+V}
, then the
commutation relations cannot be satisfied
unless at least three of the
Poincaré generators also include interactions.
Dirac's paper introduced three distinct ways
to minimally
include interactions in the Poincaré 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 Poincaré 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
x
0
=
c
t
{\displaystyle x^{0}=ct}
, where the usual convention is to choose
n
^
=
z
^
{\displaystyle {\hat {n}}={\hat {z}}}
.
Coordinates of points on the light-front hyperplane
are
The Lorentz invariant inner product of two
four-vectors,
x
{\displaystyle x}
and
y
{\displaystyle y}
,
can be expressed in terms of their light-front
components as
In a front-form relativistic quantum theory
the three interacting
generators of the Poincaré group are
P
−
:=
H
−
P
→
⋅
n
^
{\displaystyle P^{-}:=H-{\vec {P}}\cdot {\hat
{n}}}
,
the generator of translations normal to the
light front, and
J
→
⊥
:=
J
→
−
n
^
(
n
^
⋅
J
→
)
{\displaystyle {\vec {J}}_{\perp }:={\vec
{J}}-{\hat {n}}({\hat {n}}\cdot {\vec {J}})}
,
the generators of rotations
transverse to the light-front.
P
−
{\displaystyle P^{-}}
is called the "light-front"
Hamiltonian.
The kinematic generators, which generate transformations
tangent to
the light front, are free of interaction.
These include
P
+
:=
H
+
P
→
⋅
n
^
{\displaystyle P^{+}:=H+{\vec {P}}\cdot {\hat
{n}}}
and
P
→
⊥
:=
P
→
−
n
^
(
n
^
⋅
P
→
)
{\displaystyle {\vec {P}}_{\perp }:={\vec
{P}}-{\hat {n}}({\hat {n}}\cdot {\vec {P}})}
,
which generate translations tangent to the
light front,
J
3
:=
n
^
⋅
J
→
{\displaystyle J_{3}:={\hat {n}}\cdot {\vec
{J}}}
which generates rotations
about the
n
^
{\displaystyle {\hat {n}}}
axis, and the generators
K
3
:=
n
^
⋅
K
→
{\displaystyle K_{3}:={\hat {n}}\cdot {\vec
{K}}}
,
E
1
{\displaystyle E_{1}}
and
E
2
{\displaystyle E_{2}}
of
light-front preserving boosts,
which form a closed subalgebra.
Light-front quantum theories have the following
distinguishing properties:
Only three Poincaré 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,
P
+
{\displaystyle P^{+}}
, 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
S
{\displaystyle S}
-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
(4)
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
Λ
−
1
(
p
)
μ
ν
{\displaystyle \Lambda ^{-1}(p)^{\mu }{}_{\nu
}}
is a Lorentz boost that
transforms
p
μ
{\displaystyle p^{\mu }}
to
(
m
,
0
→
)
{\displaystyle (m,{\vec {0}})}
.
The components of the resulting spin vector,
j
→
{\displaystyle {\vec {j}}}
, always
satisfy
S
U
(
2
)
{\displaystyle SU(2)}
commutation relations, but the individual
components will
depend on the choice of boost
Λ
−
1
(
P
)
μ
ν
{\displaystyle \Lambda ^{-1}(P)^{\mu }{}_{\nu
}}
.
The light-front components of the spin are
obtained by choosing
Λ
−
1
(
P
)
k
μ
{\displaystyle \Lambda ^{-1}(P)^{k}{}_{\mu
}}
to be the inverse of the light-front
preserving boost, (4).
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 (4).
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
n
^
{\displaystyle {\hat {n}}}
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
P
+
{\displaystyle P^{+}}
is non-negative and kinematic. The kinematic
feature means that the generator
P
+
{\displaystyle P^{+}}
is the sum of the non-negative
single-particle
P
i
+
{\displaystyle P_{i}^{+}}
generators, (
P
+
=
∑
i
P
i
+
)
{\displaystyle P^{+}=\sum _{i}P_{i}^{+})}
. It follows
that if
P
+
{\displaystyle P^{+}}
is zero on a state, then each of the individual
P
i
+
{\displaystyle P_{i}^{+}}
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
P
+
{\displaystyle P^{+}}
. The condition
P
+
=
0
{\displaystyle P^{+}=0}
corresponds to infinite momentum
(
−
P
3
→
H
)
{\displaystyle (-P^{3}\to H)}
. Many of the
simplifications of light-front quantum field
theory are realized in
the infinite momentum
limit
of ordinary canonical field theory (see #Infinite
momentum frame).
An important consequence of the spectral condition
on
P
+
{\displaystyle P^{+}}
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
demonstrated that
relativistic quantum theories based on different
forms of dynamics are
related by
S
{\displaystyle S}
-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
(
P
+
=
0
)
{\displaystyle (P^{+}=0)}
divergences in light-front field theory. These
have to be
renormalized in a manner that recovers the
full rotational covariance and
maintains the
S
{\displaystyle S}
-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 Poincaré Lie algebra.
In the absence of
interactions, Stone's theorem applied to tensor
products of known
unitary irreducible representations of the
Poincaré group gives a
set of self-adjoint light-front generators
with all of the required
properties. The problem of adding interactions
is no
different
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
x
+
{\displaystyle x^{+}}
, one finds that the surfaces with
x
+
≠
0
{\displaystyle x^{+}\not =0}
are only invariant under a six parameter subgroup.
This means
that if one chooses a quantization surface
with a fixed non-zero
value of
x
+
{\displaystyle x^{+}}
, 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.
=== 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
Poincaré commutation relations with
P
−
{\displaystyle P^{-}}
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,
U
(
R
)
{\displaystyle U(R)}
, the
product
U
0
(
R
)
U
†
(
R
)
{\displaystyle U_{0}(R)U^{\dagger }(R)}
of a kinematic rotation with the
inverse of the corresponding dynamical rotation
is a unitary operator
that (1) preserves the
S
{\displaystyle S}
-matrix and (2) changes the kinematic
subgroup to a kinematic subgroup with a rotated
light front,
n
^
′
=
R
n
^
{\displaystyle {\hat {n}}'=R{\hat {n}}}
. Conversely, if the
S
{\displaystyle S}
-matrix
is invariant with respect to changing the
orientation of the
light-front, then the dynamical unitary representation
of rotations,
U
(
R
)
{\displaystyle U(R)}
, can be constructed using the generalized
wave operators for
different orientations of the light
front
and the kinematic representation of rotations
Because the dynamical input to the
S
{\displaystyle S}
-matrix is
P
−
{\displaystyle P^{-}}
, the invariance
of the
S
{\displaystyle S}
-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
S
U
(
2
)
{\displaystyle SU(2)}
.
These observations make it clear that the
rotational covariance of the
theory is encoded in the choice of light-front
Hamiltonian.
Karmanov
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,
n
^
{\displaystyle {\hat {n}}}
, 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.
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,
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
where here
P
−
{\displaystyle P^{-}}
and
M
{\displaystyle M}
contain interactions. The transverse
components of the
light-front spin,
j
→
⊥
{\displaystyle {\vec {j}}_{\perp }}
may or may not have an
interaction dependence; however, if one also
demands cluster
properties,
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;
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
U
0
(
Λ
,
a
)
{\displaystyle U_{0}(\Lambda ,a)}
of the Poincaré group on
this space is the direct sum of tensor products
of the single-particle
unitary irreducible representations of the
Poincaré group. A
front-form dynamics on this space is defined
by a dynamical
representation of the Poincaré group
U
(
Λ
,
a
)
{\displaystyle U(\Lambda ,a)}
on this space
where
U
(
g
)
=
U
0
(
g
)
{\displaystyle U(g)=U_{0}(g)}
when
g
{\displaystyle g}
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 Poincaré 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
given by (1)
and
P
−
=
(
P
→
⊥
2
+
M
2
)
/
P
+
{\displaystyle P^{-}=({\vec {P}}_{\perp }^{2}+M^{2})/P^{+}}
. 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 (1)
and the expression
for
P
−
{\displaystyle P^{-}}
gives a set of dynamical Poincare generators
with a
light-front kinematic subgroup.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
n
^
{\displaystyle {\hat {n}}}
axis. This is equivalent to
solving the center-of-mass Schrödinger 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, but this can be restored using
a
front-form generalization of the
recursive construction given by Sokolov.
== Infinite momentum frame ==
The infinite momentum frame (IMF) was originally
introduced to provide a physical interpretation
of the Bjorken variable
x
b
j
=
Q
2
2
M
ν
{\displaystyle x_{bj}={\frac {Q^{2}}{2M\nu
}}}
measured in deep
inelastic lepton-proton scattering
ℓ
p
→
ℓ
′
X
{\displaystyle \ell p\to \ell ^{\prime }X}
in
Feynman's parton model. (Here
Q
2
=
−
q
2
{\displaystyle Q^{2}=-q^{2}}
is the square of the
spacelike momentum transfer imparted by the
lepton and
ν
=
E
ℓ
−
E
ℓ
′
{\displaystyle \nu =E_{\ell }-E_{\ell ^{\prime
}}}
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,
P
→
∞
{\displaystyle P\to \infty }
, in the
negative
z
^
{\displaystyle {\hat {z}}}
direction, then
x
b
j
{\displaystyle x_{bj}}
can be interpreted as the
longitudinal momentum fraction
x
=
k
z
P
z
{\displaystyle x={\frac {k^{z}}{P^{z}}}}
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
t
{\displaystyle t}
(the
"instant form" ) where the observer is moving
at infinite momentum
and light-front Hamiltonian theory quantized
at fixed light-front time
τ
=
t
+
z
/
c
{\displaystyle \tau =t+z/c}
(the "front form"). A typical energy denominator
in
the instant-form is
1
/
[
E
i
n
i
t
i
a
l
−
E
i
n
t
e
r
m
e
d
i
a
t
e
+
i
ϵ
]
{\displaystyle {1/[E_{initial}-E_{intermediate}+i\epsilon
]}}
where
E
i
n
t
e
r
m
e
d
i
a
t
e
=
∑
j
E
j
=
∑
j
m
2
+
k
→
j
2
{\displaystyle E_{intermediate}=\sum _{j}E_{j}=\sum
_{j}{\sqrt {m^{2}+{\vec {k}}_{j}^{2}}}}
is the sum of energies of the particles in
the
intermediate state. In the IMF, where the
observer moves at high
momentum
P
{\displaystyle P}
in the negative
z
^
{\displaystyle {\hat {z}}}
direction, the leading terms in
P
{\displaystyle P}
cancel, and the energy denominator becomes
2
P
/
[
M
2
−
∑
j
[
k
⊥
2
+
m
2
x
i
]
j
+
i
ϵ
]
{\displaystyle 2P/[{\mathcal {M}}^{2}-\sum
_{j}{\big [}{k_{\perp }^{2}+{\frac {m^{2}}{x_{i}}}}{\big
]}_{j}+i\epsilon ]}
where
M
2
{\displaystyle {\mathcal {M}}^{2}}
is invariant mass squared of the initial state.
Thus, by
keeping the terms in
1
P
{\displaystyle {\frac {1}{P}}}
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.The vacuum state in the instant form
defined at fixed
t
{\displaystyle t}
is acausal
and infinitely complicated. For example, in
quantum electrodynamics,
bubble graphs of all orders, starting with
the
e
+
e
−
γ
{\displaystyle e^{+}e^{-}\gamma }
intermediate state, appear in the ground state
vacuum; however, as
shown by Weinberg, such vacuum graphs are
frame-dependent and formally vanish by powers
of
1
/
P
2
{\displaystyle 1/P^{2}}
as the
observer moves at
P
→
∞
{\displaystyle P\to \infty }
. 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
+
{\displaystyle +}
momentum of each constituent is positive,
but must sum to zero in
the vacuum state since the
+
{\displaystyle +}
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
P
→
∞
{\displaystyle P\to \infty }
is not a rigorous limit, and the need to boost
the
instant-form wave function introduces complexities.
== Covariant formulation ==
In light-front coordinates,
x
+
=
c
t
+
z
{\displaystyle x^{+}=ct+z}
,
x
−
=
c
t
−
z
{\displaystyle x^{-}=ct-z}
, the spatial coordinates
x
,
y
,
z
{\displaystyle x,y,z}
do not enter symmetrically: the coordinate
z
{\displaystyle z}
is distinguished,
whereas
x
{\displaystyle x}
and
y
{\displaystyle y}
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
z
{\displaystyle z}
and
x
,
y
{\displaystyle x,y}
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
x
,
y
,
z
{\displaystyle x,y,z}
.
To overcome this inconvenience, there was
developed the explicitly
covariant version of
light-front quantization (reviewed by Carbonell
et al.),
in which the state vector is defined on the
light-front plane of
general orientation:
ω
⋅
x
=
ω
0
c
t
−
ω
→
⋅
x
→
=
ω
0
t
−
ω
x
x
−
ω
y
y
−
ω
z
z
=
0
{\displaystyle \omega \cdot x=\omega _{0}ct-{\vec
{\omega }}\cdot {\vec {x}}=\omega _{0}t-\omega
_{x}x-\omega _{y}y-\omega _{z}z=0}
(instead of
c
t
+
z
=
0
{\displaystyle ct+z=0}
),
where
x
=
(
c
t
,
x
→
)
{\displaystyle x=(ct,{\vec {x}})}
is a four-dimensional vector in the four-dimensional
space-time 
and
ω
=
(
ω
0
,
ω
→
)
{\displaystyle \omega =(\omega _{0},{\vec
{\omega }})}
is also a four-dimensional vector with the
property
ω
2
=
ω
0
2
−
ω
→
2
=
0
{\displaystyle \omega ^{2}=\omega _{0}^{2}-{\vec
{\omega }}^{2}=0}
. In the particular case
ω
=
(
1
/
c
,
0
,
0
,
−
1
/
c
)
{\displaystyle \omega =(1/c,0,0,-1/c)}
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
ω
{\displaystyle \omega }
.
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
σ
=
ω
⋅
x
{\displaystyle \sigma =\omega \cdot x}
(in contrast to the evolution in the
direction
x
+
{\displaystyle x^{+}}
or
t
{\displaystyle t}
). For the instant form of dynamics,
these rules were firstl developed by
Kadyshevsky.
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
k
i
{\displaystyle k_{i}}
are on the corresponding mass shells
k
i
2
=
m
i
2
{\displaystyle k_{i}^{2}=m_{i}^{2}}
,
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
x
−
{\displaystyle x^{-}}
(or, in general, in the direction
ω
{\displaystyle \omega }
).
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
ω
{\displaystyle \omega }
. 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
ω
=
(
1
/
c
,
0
,
0
,
−
1
/
c
)
{\displaystyle \omega =(1/c,0,0,-1/c)}
,
the amplitudes found by the rules of covariant
graph techniques, after replacement of variables,
are reduced to the amplitudes given by the
Weinberg
rules 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
k
→
⊥
i
,
x
i
{\displaystyle {\vec {k}}_{\perp i},x_{i}}
taken separately but not
in some particular combinations like the Mandelstam
variables
s
,
t
{\displaystyle s,t}
.
On the energy shell, the amplitudes do not
depend
on the four-vector
ω
{\displaystyle \omega }
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
ω
{\displaystyle \omega }
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
ω
{\displaystyle \omega }
participates on equal footing
with other four-momenta, and, therefore, the
main part of this problem is reduced to the
well known one.
For example, as is well known, the wave function
of a non-relativistic system,
consisting of two spinless particles with
the relative momentum
k
→
{\displaystyle {\vec {k}}}
and with total angular momentum
l
{\displaystyle l}
, is proportional to the spherical
function
Y
l
m
(
k
→
^
)
{\displaystyle Y_{lm}({\hat {\vec {k}}})}
:
ψ
l
m
(
k
→
)
=
f
(
k
)
Y
l
m
(
k
^
)
{\displaystyle \psi _{lm}({\vec {k}})=f(k)Y_{lm}({\hat
{k}})}
,
where
k
^
=
k
→
/
k
{\displaystyle {\hat {k}}={\vec {k}}/k}
and
f
(
k
)
{\displaystyle f(k)}
is a function depending on the
modulus
k
=
|
k
→
|
{\displaystyle k=|{\vec {k}}|}
.
The angular momentum operator reads:
J
→
=
−
i
[
k
→
×
∂
k
→
]
{\displaystyle {\vec {J}}=-i[{\vec {k}}\times
\partial {\vec {k}}]}
.
Then the wave function of a relativistic system
in the covariant formulation of
light-front dynamics obtains the similar form:
where
n
^
=
ω
→
/
|
ω
→
|
{\displaystyle {\hat {n}}={\vec {\omega }}/|{\vec
{\omega }}|}
and
f
1
,
2
(
k
,
k
→
⋅
n
^
)
{\displaystyle f_{1,2}(k,{\vec {k}}\cdot {\hat
{n}})}
are functions depending, in addition
to
k
{\displaystyle k}
, on the scalar product
k
→
⋅
n
^
{\displaystyle {\vec {k}}\cdot {\hat {n}}}
.
The variables
k
{\displaystyle k}
,
k
→
⋅
n
^
{\displaystyle {\vec {k}}\cdot {\hat {n}}}
are invariant not only under rotations
of the vectors
k
→
{\displaystyle {\vec {k}}}
,
n
^
{\displaystyle {\hat {n}}}
but also under rotations and the Lorentz
transformations of initial four-vectors
k
{\displaystyle k}
,
ω
{\displaystyle \omega }
.
The second contribution
∝
Y
l
m
(
n
^
)
{\displaystyle \propto Y_{lm}({\hat {n}})}
means that the operator of the total angular
momentum in explicitly covariant
light-front dynamics obtains an additional
term:
J
→
=
−
i
[
k
→
×
∂
k
→
]
−
i
[
n
^
×
∂
n
^
]
{\displaystyle {\vec {J}}=-i[{\vec {k}}\times
\partial {\vec {k}}]-i[{\hat {n}}\times \partial
{\hat {n}}]}
.
For non-zero spin particles this operator
obtains the contribution of the spin
operators:
J
→
=
−
i
[
k
→
×
∂
k
→
]
−
i
[
n
^
×
∂
n
^
]
+
s
→
1
+
s
→
2
.
{\displaystyle {\vec {J}}=-i[{\vec {k}}\times
\partial {\vec {k}}]-i[{\hat {n}}\times \partial
{\hat {n}}]+{\vec {s}}_{1}+{\vec {s}}_{2}.}
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
f
1
,
2
{\displaystyle f_{1,2}}
on 
the scalar product
k
→
⋅
n
^
{\displaystyle {\vec {k}}\cdot {\hat {n}}}
varying
when the orientation of the unit vector
n
^
{\displaystyle {\hat {n}}}
changes (for fixed
k
→
{\displaystyle {\vec {k}}}
).
This dependence (together with the dependence
on
k
{\displaystyle k}
) is found from the dynamical
equation for the wave function.
A peculiarity of this construction is in the
fact
that there exists the operator
A
=
(
n
^
⋅
J
→
)
2
{\displaystyle A=({\hat {n}}\cdot {\vec {J}})^{2}}
which commutes both
with the Hamiltonian and with
J
→
2
,
J
z
{\displaystyle {\vec {J}}^{2},J_{z}}
. Then the states are labeled also
by the eigenvalue
a
{\displaystyle a}
of the operator
A
{\displaystyle A}
:
ψ
=
ψ
l
m
a
(
k
→
,
n
^
)
{\displaystyle \psi =\psi _{lma}({\vec {k}},{\hat
{n}})}
.
For given angular momentum
l
{\displaystyle l}
, there are
l
+
1
{\displaystyle l+1}
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
After satisfying it, the solution obtains
the form of a unique superposition of
the states
ψ
l
m
a
(
k
→
,
n
^
)
{\displaystyle \psi _{lma}({\vec {k}},{\hat
{n}})}
with different eigenvalues
a
{\displaystyle a}
.The extra contribution
−
i
[
n
^
×
∂
n
^
]
{\displaystyle -i[{\hat {n}}\times \partial
{\hat {n}}]}
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 (
S
{\displaystyle S}
- and
D
{\displaystyle D}
-waves).
Whereas, the relativistic light-front deuteron
wave function is determined by six
components.
These components were calculated in the one-boson
exchange
model.
== 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, Glazek-Wilson
similarity renormalization-group procedure
for Hamiltonians, Mathiot-Grange test functions,
Karmanov-Mathiot-Smirnov realization of sector-dependent
renormalization, and determine how to incorporate
symmetry breaking in light-front quantization;
this is likely to require an analysis of zero
modes and in-hadron condensates.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
finite elements, function
expansions,
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
P
+
{\displaystyle P^{+}}
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
z
{\displaystyle z}
-axis is
kinematic; the other two, of rotations about
axes
x
{\displaystyle x}
and
y
{\displaystyle y}
, 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
z
{\displaystyle z}
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.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 Schrödinger equation in terms
of the
modified transverse coordinate
ζ
{\displaystyle \zeta }
. 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
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 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) 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
