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of hadrons in physical space-time. The mapping between the LF invariant variable ζ and the fifth-dimension AdS coordinate z was originally obtained by matching the expression for electromagnetic (EM) current matrix elements in AdS space with the corresponding expres- sion for the current matrix element, using LF theory in physical spacetime. [5] It has also been shown that one obtains the identical holographic mapping using the matrix elements of the energy-momentum tensor, [6] thus verifying the consistency of the holographic mapping from AdS to physical observables defined on the light front. LF holography thus provides a direct correspondence between an effective gravity theory defined in a fifth-dimensional warped space and a physical description of hadrons in 3 + 1 spacetime. 

Light-front quantization is the ideal framework for describing the structure of hadrons in terms of their quark and gluon degrees of freedom. LFWFs play the same role in hadron physics that Schr¨ odinger wave functions play in atomic physics. [26] The simple structure of the LF vacuum provides an unambiguous definition of the partonic content of a hadron in QCD. A physical hadron in four-dimensional Minkowski space has four-momentum P$_{µ}$ and invariant hadronic mass states, P$_{µ}$P µ = M $^{2}$, determined by the Lorentz-invariant Hamilto- nian equation for the relativistic bound-state system 

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The hadron four-momentum generator is P = ( P $^{+}$, P $^{−}$, P ⊥ ), P ± = P $^{0}$± P $^{3}$, and the hadronic state | ψ 〉 is an expansion in multiparticle Fock eigenstates | n 〉 of the free light-front Hamilto- nian: | ψ 〉 = ∑ $_{n}$ψ$_{n}$ | n 〉 . The internal partonic coordinates of the hadron are the momentum fractions x$_{i}$ = k + i /P + and the transverse momenta k ⊥ $_{i}$, i = 1 , 2 , . . . , n , where n is the num- ber of partons in a given Fock state. Momentum conservation requires ∑ n i $_{=1}$x$_{i}$ = 1 and ∑ n i $_{=1}$k ⊥ i = 0. It is useful to employ a mixed representation [27] in terms of n − 1 inde- pendent momentum fraction variables x$_{j}$ and position coordinates b ⊥ $_{j}$, j = 1 , 2 , . . . , n − 1, so that ∑ n i $_{=1}$b ⊥ i = 0. The relative transverse variables b ⊥ i are Fourier conjugates of the momentum variables k ⊥ $_{i}$. 

In AdS space the physical states are represented by normalizable modes Φ$_{P}$ ( x $^{µ}$, z ) = e − iP · $^{x}$Φ( z ), with plane waves along the Poincar´e coordinates and a profile function Φ( z ) along the holographic coordinate z . Each LF hadronic state | ψ ( P ) 〉 is dual to a normalizable string mode Φ$_{P}$ ( x $^{µ}$, z ). The hadronic mass M 2 is found by solving the eigenvalue problem for the corresponding wave equation in AdS space, which, as we discuss below, is equivalent 

