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Light-front computational methods
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<p>The <a href="/facts/Light_front_quantization/rJ5Ry83I">light-front quantization</a> of <a href="/facts/Quantum_field_theory/VoewsNJV">quantum field theories</a> provides a useful alternative to ordinary equal-time <a href="/facts/Quantization_(physics)/fvs2Jwpn">quantization</a>. In particular, it can lead to a <a href="/facts/Special_relativity/vsFFc63E">relativistic</a> description of <a href="/facts/Bound_state/NndW5JQl">bound systems</a> in terms of <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum-mechanical</a> <a href="/facts/Wave_function/KgM5ykjY">wave functions</a>. 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 <a href="/facts/Cartesian_coordinate_system/lkTqvMmB">Cartesian coordinate</a>, 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 <a href="/facts/Frame_of_reference/15yYfB9M">frame of reference</a> 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. </p><p>The solution of the LFQCD Hamiltonian eigenvalue equation will utilize the available mathematical methods of quantum mechanics and contribute to the development of advanced computing techniques for large quantum systems, including <a href="/facts/Atomic_nucleus/orpIm5Xl">nuclei</a>. For example, in the discretized light-cone quantization method (DLCQ), periodic conditions are introduced such that momenta are discretized and the size of the Fock space is limited without destroying Lorentz invariance. Solving a quantum field theory is then reduced to diagonalizing a large sparse <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrix</a>. The DLCQ method has been successfully used to obtain the complete spectrum and light-front wave functions in numerous model quantum field theories such as QCD with one or two space dimensions for any number of <a href="/facts/Flavour_(particle_physics)/RR7rsuTD">flavors</a> and quark masses. An extension of this method to <a href="/facts/Supersymmetry/hkWGrMet">supersymmetric theories</a>, SDLCQ, takes advantage of the fact that the light-front Hamiltonian can be factorized as a product of raising and lowering <a href="/facts/Ladder_operator/kyGdlRjQ">ladder operators</a>. SDLCQ has provided new insights into a number of supersymmetric theories including direct numerical evidence for a supergravity/super-Yang–Mills duality conjectured by Maldacena. </p><p>It is convenient to work in a Fock basis { | n : p i + , p → ⊥ i ⟩ } {\displaystyle \{|n:p_{i}^{+},{\vec {p}}_{\perp i}\rangle \}} where the light-front momenta P + {\displaystyle {\mathcal {P}}^{+}} and P → ⊥ {\displaystyle {\vec {\mathcal {P}}}_{\perp }} are diagonal. The state | P _ ⟩ {\displaystyle |{\underline {P}}\rangle } is given by an expansion </p> | P _ ⟩ = ∑ n ∫ [ d x ] n [ d 2 k ⊥ ] n ψ n ( x , k → ⊥ ) | n : x P + , x P → ⊥ + k → ⊥ ⟩ , {\displaystyle |{\underline {P}}\rangle =\sum _{n}\int [dx]_{n}\,[d^{2}k_{\perp }]_{n}\,\psi _{n}(x,{\vec {k}}_{\perp })|n:xP^{+},x{\vec {P}}_{\perp }+{\vec {k}}_{\perp }\rangle \,,} <p>with </p> [ d x ] n = 4 π δ ( 1 − ∑ i = 1 n x i ) ∏ i = 1 n d x i 4 π x i , [ d 2 k ⊥ ] n = 4 π 2 δ ( ∑ i = 1 n k → ⊥ i ) ∏ i = 1 n d 2 k ⊥ i 4 π 2 . {\displaystyle [dx]_{n}=4\pi \delta (1-\sum _{i=1}^{n}x_{i})\prod _{i=1}^{n}{\frac {dx_{i}}{4\pi {\sqrt {x_{i}}}}}\,,\;\;\;[d^{2}k_{\perp }]_{n}=4\pi ^{2}\delta (\sum _{i=1}^{n}{\vec {k}}_{\perp i})\prod _{i=1}^{n}{\frac {d^{2}k_{\perp i}}{4\pi ^{2}}}\,.} <p> ψ n {\displaystyle \psi _{n}} is interpreted as the wave function of the contribution from states with n {\displaystyle n} particles. The eigenvalue problem P − | P _ ⟩ = M 2 + P ⊥ 2 P + | P _ ⟩ {\displaystyle {\mathcal {P}}^{-}|{\underline {P}}\rangle ={\frac {M^{2}+P_{\perp }^{2}}{P^{+}}}|{\underline {P}}\rangle } is a set of coupled integral equations for these wave functions. Although the notation as presented supports only one particle type, the generalization to more than one is trivial. </p>