Light-front quantization applications

Main article: Light-front quantization

See also: Formalism Part 2, Computational Methods, and Applications

The light cone of special relativity. Light-front quantization uses light-front (or light-cone) coordinates to select an initial surface that is tangential to the light cone. Equal-time quantization uses an initial surface that is horizontal, labeled here as the "hypersurface of the present".

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 $x^{+}\equiv ct+z$ plays the role of time and the corresponding spatial coordinate is $x^{-}\equiv ct-z$. Here, $t$ is the ordinary time, $z$ is one Cartesian coordinate, and $c$ is the speed of light. The other two Cartesian coordinates, $x$ and $y$, are untouched and often called transverse or perpendicular, denoted by symbols of the type ${\vec {x}}_{\perp }=(x,y)$. The choice of the frame of reference where the time $t$ and $z$-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.

Fundamentals

Front-form relativistic quantum mechanics was introduced by Paul Dirac in a 1949 paper published in Reviews of Modern Physics and Bargmann 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, $[K^{j},P^{k}]=i\delta ^{jk}H$, where $P^{k}$ are components of the linear momentum and $K^{j}$ are components of rotation-less boost generators. If the Hamiltonian includes interactions, i.e. $H=H_{0}+V$, then the commutation relations cannot be satisfied unless at least three of the Poincare generators also include interactions.

Dirac's paperintroduced 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:

$x^{+}=x^{0}+{\vec {x}}\cdot {\hat {n}}=0,$ 1

with $x^{0}=ct$, where the usual convention is to choose ${\hat {n}}={\hat {z}}$. Coordinates of points on the light-front hyperplane are

$x^{-}=x^{0}-{\vec {x}}\cdot {\hat {n}}\quad {\mbox{and}}\quad {\vec {x}}_{\perp }:={\vec {x}}-{\hat {n}}({\hat {n}}\cdot {\vec {x}}).$ 2

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

$x\cdot y={\frac {1}{2}}(x^{-}y^{+}+x^{+}y^{-})-{\vec {x}}_{\perp }\cdot {\vec {y}}_{\perp }.$ 3

In a front-form relativistic quantum theory the three interacting generators of the Poincare group are $P^{-}:=H-{\vec {P}}\cdot {\hat {n}}$, the generator of translations normal to the light front, and ${\vec {J}}_{\perp }:={\vec {J}}-{\hat {n}}({\hat {n}}\cdot {\vec {J}})$, the generators of rotations transverse to the light-front. $P^{-}$ is called the "light-front" Hamiltonian.

The kinematic generators, which generate transformations tangent to the light front, are free of interaction. These include $P^{+}:=H+{\vec {P}}\cdot {\hat {n}}$ and ${\vec {P}}_{\perp }:={\vec {P}}-{\hat {n}}({\hat {n}}\cdot {\vec {P}})$, which generate translations tangent to the light front, $J_{3}:={\hat {n}}\cdot {\vec {J}}$ which generates rotations about the ${\hat {n}}$ axis, and the generators $K_{3}:={\hat {n}}\cdot {\vec {K}}$, $E_{1}$ and $E_{2}$ of light-front preserving boosts,

$(p^{+},{\vec {p}}_{\perp })\to (p^{+\prime },{\vec {p}}_{\perp }')=(v^{+}p^{+},{\vec {p}}_{\perp }+{\vec {v}}_{\perp }x^{+}),$ 4

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, $P^{+}$, 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 $S$-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

$j^{j}={\frac {1}{2}}\sum \epsilon _{jkl}\Lambda ^{-1}(p)^{k}{}_{\mu }\Lambda ^{-1}(p)^{l}{}_{\nu }J^{\mu \nu },$ 5

where $\Lambda ^{-1}(p)^{\mu }{}_{\nu }$ is a Lorentz boost that transforms $p^{\mu }$ to $(m,{\vec {0}})$.

The components of the resulting spin vector, ${\vec {j}}$, always satisfy $SU(2)$ commutation relations, but the individual components will depend on the choice of boost $\Lambda ^{-1}(P)^{\mu }{}_{\nu }$. The light-front components of the spin are obtained by choosing $\Lambda ^{-1}(P)^{k}{}_{\mu }$ 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 ${\hat {n}}$ 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 $P^{+}$ is non-negative and kinematic. The kinematic feature means that the generator $P^{+}$ is the sum of the non-negative single-particle $P_{i}^{+}$ generators, ($P^{+}=\sum _{i}P_{i}^{+})$. It follows that if $P^{+}$ is zero on a state, then each of the individual $P_{i}^{+}$ 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 $P^{+}$. The condition $P^{+}=0$ corresponds to infinite momentum $(-P^{3}\to H)$. 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 $P^{+}$ 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 $S$-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 $(P^{+}=0)$ divergences in light-front field theory. These have to be renormalized in a manner that recovers the full rotational covariance and maintains the $S$-matrix equivalence. The renormalization of light front field theories is discussed in Light_front_quantization_(formalism_part2)#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 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 $x^{+}$, one finds that the surfaces with $x^{+}\not =0$ are only invariant under a six parameter subgroup. This means that if one chooses a quantization surface with a fixed non-zero value of $x^{+}$, 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 Poincare commutation relations with $P^{-}$ 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, $U(R)$, the product $U_{0}(R)U^{\dagger }(R)$ of a kinematic rotation with the inverse of the corresponding dynamical rotation is a unitary operator that (1) preserves the $S$-matrix and (2) changes the kinematic subgroup to a kinematic subgroup with a rotated light front, ${\hat {n}}'=R{\hat {n}}$. Conversely, if the $S$-matrix is invariant with respect to changing the orientation of the light-front, then the dynamical unitary representation of rotations, $U(R)$, can be constructed using the generalized wave operators for different orientations of the light front and the kinematic representation of rotations

$U(R)=\Omega _{\pm }({\hat {n}})\Omega _{\pm }^{\dagger }(R{\hat {n}})U_{0}(R).$ 6

Because the dynamical input to the $S$-matrix is $P^{-}$, the invariance of the $S$-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 $SU(2)$.

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, ${\hat {n}}$, 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

${\vec {J}}_{\perp }={\frac {1}{P^{+}}}[{\frac {1}{2}}(P^{+}-P^{-})({\hat {n}}\times {\vec {E}}_{\perp })-({\hat {n}}\times {\vec {P}}_{\perp })({\vec {K}}\cdot {\hat {n}})+{\vec {P}}_{\perp }({\hat {n}}\cdot {\vec {j}})+M{\vec {j}}_{\perp }],$ 1

where here $P^{-}$ and $M$ contain interactions. The transverse components of the light-front spin, ${\vec {j}}_{\perp }$ 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_quantization_(formalism_part2)#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 $U_{0}(\Lambda ,a)$ 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 $U(\Lambda ,a)$ on this space where $U(g)=U_{0}(g)$ when $g$ 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 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 given by (1) and $P^{-}=({\vec {P}}_{\perp }^{2}+M^{2})/P^{+}$. 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 $P^{-}$ 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 ${\hat {n}}$ 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, 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 $x_{bj}={\frac {Q^{2}}{2M\nu }}$ measured in deep inelastic lepton-proton scattering $\ell p\to \ell ^{\prime }X$ in Feynman's parton model. (Here $Q^{2}=-q^{2}$ is the square of the spacelike momentum transfer imparted by the lepton and $\nu =E_{\ell }-E_{\ell ^{\prime }}$ 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, $P\to \infty$, in the negative ${\hat {z}}$ direction, then $x_{bj}$ can be interpreted as the longitudinal momentum fraction $x={\frac {k^{z}}{P^{z}}}$ 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 $t$ (the "instant form" ) where the observer is moving at infinite momentum and light-front Hamiltonian theory quantized at fixed light-front time $\tau =t+z/c$ (the "front form"). A typical energy denominator in the instant-form is ${1/[E_{initial}-E_{intermediate}+i\epsilon ]}$ where $E_{intermediate}=\sum _{j}E_{j}=\sum _{j}{\sqrt {m^{2}+{\vec {k}}_{j}^{2}}}$ is the sum of energies of the particles in the intermediate state. In the IMF, where the observer moves at high momentum $P$ in the negative ${\hat {z}}$ direction, the leading terms in $P$ cancel, and the energy denominator becomes $2P/[{\mathcal {M}}^{2}-\sum _{j}{\big [}{k_{\perp }^{2}+{\frac {m^{2}}{x_{i}}}}{\big ]}_{j}+i\epsilon ]$ where ${\mathcal {M}}^{2}$ is invariant mass squared of the initial state. Thus, by keeping the terms in ${\frac {1}{P}}$ 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 $t$ is acausal and infinitely complicated. For example, in quantum electrodynamics, bubble graphs of all orders, starting with the $e^{+}e^{-}\gamma$ 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 $1/P^{2}$ as the observer moves at $P\to \infty$. 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 $+$ 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 $P\to \infty$ is not a rigorous limit, and the need to boost the instant-form wave function introduces complexities.

Covariant formulation

In light-front coordinates, $x^{+}=ct+z$, $x^{-}=ct-z$, the spatial coordinates $x,y,z$ do not enter symmetrically: the coordinate $z$ is distinguished, whereas $x$ and $y$ 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 $z$ and $x,y$ 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 $x,y,z$.

To overcome this inconvenience, there was developed the explicitly covariant versionref name="karmanov:1982b"></ref> of light-front quantization (reviewed by Carbonell et al.), in which the state vector is defined on the light-front plane of general orientation: $\omega \cdot x=\omega _{0}ct-{\vec {\omega }}\cdot {\vec {x}}=\omega _{0}t-\omega _{x}x-\omega _{y}y-\omega _{z}z=0$ (instead of $ct+z=0$), where $x=(ct,{\vec {x}})$ is a four-dimensional vector in the four-dimensional space-time and $\omega =(\omega _{0},{\vec {\omega }})$ is also a four-dimensional vector with the property $\omega ^{2}=\omega _{0}^{2}-{\vec {\omega }}^{2}=0$. In the particular case $\omega =(1/c,0,0,-1/c)$ 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 $\omega$.

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 $\sigma =\omega \cdot x$ (in contrast to the evolution in the direction $x^{+}$ or $t$). 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 $k_{i}$ are on the corresponding mass shells $k_{i}^{2}=m_{i}^{2}$, 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 $x^{-}$ (or, in general, in the direction $\omega$). 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 $\omega$. 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 $\omega =(1/c,0,0,-1/c)$, the amplitudes found by the rules of covariant graph techniques, after replacement of variables, are reduced to the amplitudes given by the Weinberg rulesin 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 ${\vec {k}}_{\perp i},x_{i}$ taken separately but not in some particular combinations like the Mandelstam variables $s,t$.

On the energy shell, the amplitudes do not depend on the four-vector $\omega$ 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 $\omega$ 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 $\omega$ 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 ${\vec {k}}$ and with total angular momentum $l$, is proportional to the spherical function $Y_{lm}({\hat {\vec {k}}})$: $\psi _{lm}({\vec {k}})=f(k)Y_{lm}({\hat {k}})$, where ${\hat {k}}={\vec {k}}/k$ and $f(k)$ is a function depending on the modulus $k=|{\vec {k}}|$. The angular momentum operator reads: ${\vec {J}}=-i[{\vec {k}}\times \partial {\vec {k}}]$. Then the wave function of a relativistic system in the covariant formulation of light-front dynamics obtains the similar form:

$\psi _{lm}({\vec {k}},{\hat {n}})=f_{1}(k,{\vec {k}}\cdot {\hat {n}})Y_{lm}({\hat {k}})+f_{2}(k,{\vec {k}}\cdot {\hat {n}})Y_{lm}({\hat {n}}),$ 7

where ${\hat {n}}={\vec {\omega }}/|{\vec {\omega }}|$ and $f_{1,2}(k,{\vec {k}}\cdot {\hat {n}})$ are functions depending, in additional to $k$, on the scalar product ${\vec {k}}\cdot {\hat {n}}$. The variables $k$, ${\vec {k}}\cdot {\hat {n}}$ are invariant not only under rotations of the vectors ${\vec {k}}$, ${\hat {n}}$ but also under rotations and the Lorentz transformations of initial four-vectors $k$, $\omega$. The second contribution $\propto Y_{lm}({\hat {n}})$ means that the operator of the total angular momentum in explicitly covariant light-front dynamics obtains an additional term: ${\vec {J}}=-i[{\vec {k}}\times \partial {\vec {k}}]-i[{\hat {n}}\times \partial {\hat {n}}]$. For non-zero spin particles this operator obtains the contribution of the spin operators:

${\vec {J}}=-i[{\vec {k}}\times \partial {\vec {k}}]-i[{\hat {n}}\times \partial {\hat {n}}]+{\vec {s}}_{1}+{\vec {s}}_{2}.$

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 $f_{1,2}$ on the scalar product ${\vec {k}}\cdot {\hat {n}}$ varying when the orientation of the unit vector ${\hat {n}}$ changes (for fixed ${\vec {k}}$). This dependence (together with the dependence on $k$) is found from the dynamical equation for the wave function.

A peculiarity of this construction is in the fact that there exists the operator $A=({\hat {n}}\cdot {\vec {J}})^{2}$ which commutes both with the Hamiltonian and with ${\vec {J}}^{2},J_{z}$. Then the states are labeled also by the eigenvalue $a$ of the operator $A$: $\psi =\psi _{lma}({\vec {k}},{\hat {n}})$. For given angular momentum $l$, there are $l+1$ 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 an unique superposition of the states $\psi _{lma}({\vec {k}},{\hat {n}})$ with different eigenvalues $a$.

The extra contribution $-i[{\hat {n}}\times \partial {\hat {n}}]$ 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 ($S$- and $D$-waves). Whereas, the relativistic light-front deuteron wave function is determined by six components. These components were calculated in the one-boson exchange model.

See also

• Quantum field theories
• Quantum chromodynamics
• Quantum electrodynamics
• Light-front holography

References

External links

• ILCAC, Inc., the International Light-Cone Advisory Committee.

• Publications on light-front dynamics, maintained by A. Harindranath.