Light front quantization: Difference between revisions

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The [[Gauge theory|gauge]]-invariant [[meson]] and [[baryon]] distribution amplitudes which control hard exclusive and direct reactions are the [[Quark model|valence]] light-front wave functions integrated over transverse momentum at fixed <math>x_i= {k^+ _i/ P^+}</math>. The "ERBL" evolution<ref name="Lepage:1980fj" /><ref name="Efremov:1979qk" /> 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 rule in quantum mechanics|sum rules]].

The light-front wave functions contain information about novel features of QCD. These include effects suggested from other approaches, such as [[Quark#Strong interaction and color charge|colortransparency]] transparency, hidden color, intrinsic [[Charm (quantum number)|charm]], [[Quark#Sea quarks|sea-quark]] symmetries, dijet diffraction, direct hard processes, and hadronic [[Spin (physics)|spin]] dynamics.

[[File:DIS parton model.svg|thumbnail|right|alt=Deep inelastic scattering|Deep inelastic electron-proton scattering]]

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title=Chiral Perturbation Theory to One Loop |

journal=[[Annals of Physics]] | volume= 158 |issue=1 | pages= 142–210 | year=1984 |

doi= 10.1016/0003-4916(84)90242-2 |bibcode = 1984AnPhy.158..142G |url=https://cds.cern.ch/record/147992 |url-access=subscription }}</ref><ref name="condensates">{{ cite journal |

author=S. D. Glazek |

title=Light Front QCD in the Vacuum Background |

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doi= 10.2307/1968551|bibcode = 1939AnMat..40..149W |

jstor=1968551 |

s2cid=121773411 |}}</ref>

url=https://semanticscholar.org/paper/e7a88a5d0617b5adbefcedccb5e52a12725de6bf }}</ref>

and Bargmann<ref name="bargmann:1954">{{ cite journal |

author=V. Bargmann |

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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 <math>S</math>-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 Commutation Relations ===

[[Canonical commutation relations]] at equal time are the centerpiece of the [[canonical quantization]] method to quantized fields. In the standard quantization method (the "Instant Form" in Dirac's classification of relativistic dynamics<ref name="Dirac" />), the relations are, for example here for a spin-0 field <math> \phi </math> and its [[Conjugate_variables#Quantum_theory|canonical conjugate]] <math> \pi </math>:

<math display="block">{\rm Instant~Form:}~~[\phi(t, \vec x),\phi(t, \vec y)] = 0, \ \ [\pi(t, \vec x), \pi(t, \vec y)] = 0, \ \ [\phi(t, \vec x),\pi(t, \vec y)] = i\hbar \delta^3( \vec x- \vec y),</math>

where the relations are taken at equal time <math> t </math>, and <math> \vec x </math> and <math> \vec y </math> are the space variables. The equal-time requirement imposes that <math> \vec x - \vec y </math> is a [[Spacetime#Spacetime_interval|spacelike]] quantity. The non-zero value of the commutator <math>[\phi(t, \vec x),\pi(t, \vec y)]</math> expresses the fact that when <math> \phi </math> and <math> \pi </math> are separated by a spacelike distance, they cannot communicate with each other and thus commute, except when their separation <math> \vec x - \vec y \to 0</math>.<ref>{{cite book | last=Carroll | first=Sean | title=Spacetime and Geometry: An Introduction to General Relativity | publisher=Addison Wesley | year=2003 | isbn=0-8053-8732-3 | edition=Reprinted 2019 }}</ref>

In the Light-Front form however, fields at equal time <math> x^+ </math> are causally linked (i.e., they can communicate) since the Light-Front time <math> x^+ \equiv t-z </math> is along the light cone. Consequently, the Light-Front canonical commutation relations are different. For instance:<ref>{{cite book | last=Harindranath | first=A. | title=An Introduction to Light Front Dynamics for Pedestrians; In Light-Front Quantization and Non-Perturbative QCD | editor-last1=Vary | editor-first1=J.P. | editor-last2=Wolz | editor-first2=F. | publisher=International Institute of Theoretical and Applied Physics | ___location=Ames, IA | year=2000 | isbn=1-891815-00-8 | arxiv=hep-ph/9612244 }}</ref>

<math display="block">{\rm Light-Front~form:}~~[\phi(x^+, \vec x),\phi(x^+, \vec y)] = \frac{i}{4}\epsilon(x^- -y^-)\delta^2( \vec{x_\bot} - \vec{y_\bot}),</math>

where <math>\epsilon(x)=\theta(x)-\theta(-x)</math> is the antisymmetric [[Heaviside step function]].

On the other hand, the commutation relations for the [[creation and annihilation operators]] are similar for both the Instant and Light-Front forms:

<math display="block">{\rm Instant~Form:}~~[a(t, \vec k),a(t, \vec l)] = 0, \ \ [a^\dagger(t, \vec k),a^\dagger(t, \vec l)] = 0, \ \ [a(t, \vec k),a^\dagger(t, \vec l)]= \hbar \delta^3( \vec k- \vec l).</math>

<math display="block">{\rm Light-Front~form:}~~[a(x^+, \vec k),a(x^+, \vec l)] = 0, \ \ [a^\dagger(x^+, \vec k),a^\dagger(x^+, \vec l)] = 0, \ \ [a(x^+, \vec k),a^\dagger(x^+, \vec l)]= \hbar \delta(k^+-l^+) \delta^2( \vec{k_\bot}- \vec{l_\bot}).</math>

where <math> \vec k</math> and <math> \vec l</math> are the [[Wave vector|wavevectors]] of the fields, <math> k^+ = k_0 + k_3 </math> and <math> l^+ = l_0 + l_3 </math>.

=== Light-front boosts ===

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journal=[[Acta Physica Polonica B]] | volume= 43 |

issue=9 | page= 1843 | year=2012 |

doi= 10.5506/APhysPolB.43.1843|doi-access=free }}</ref><ref name="RGPEPrecent2">{{ cite journal |

author=S. D. Glazek |

title=Fermion mass mixing and vacuum triviality in the renormalization group procedure for effective particles |