Functional Schrödinger-Equation for Fermions in External Gauge Fields

Abstract. These lecture notes review the foundations and some applications of lightcone quantization. First I explain how to choose a time in special relativity. Inclusion of Poincaré invariance naturally leads to Dirac's forms of relativistic dynamics. Among these, the front form, being the basis for light-cone quantization, is my main focus. I explain a few of its peculiar features such as boost and Galilei invariance or separation of relative and center-of-mass motion. Combining light-cone dynamics and field quantization results in light-cone quantum field theory. As the latter represents a first-order system, quantization is somewhat nonstandard. I address this issue using Schwinger's quantum action principle, the method of Faddeev and Jackiw, and the functional Schrödinger picture. A finite-volume formulation, discretized light-cone quantization, is analysed in detail. I point out some problems with causality, which are absent in infinite volume. Finally, the triviality of the light-cone vacuum is established. Coming to applications, I introduce the notion of light-cone wave functions as the solutions of the light-cone Schrödinger equation. I discuss some examples, among them nonrelativistic Coulomb systems and model field theories in two dimensions. Vacuum properties (like chiral condensates) are reconstructed from the particle spectrum obtained by solving the light-cone Schrödinger equation. In a last application, I make contact with phenomenology by calculating the pion wave function within the Nambu and Jona-Lasinio model. I am thus able to predict a number of observables like the pion charge and core radius, the r.m.s. transverse momentum, the pion structure function and the pion distribution amplitude. The latter turns out to be the asymptotic one.

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“…This expectation will indeed turn out to be true. The instant form fermionic field operators are given by (Kiefer and Wipf 1994) …”

Section: Fermions (Instant Form)mentioning

confidence: 99%

Light-Cone Quantization: Foundations and Applications

Heinzl

2001

Methods of Quantization

114

141

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“…While the massless Schwinger model is analytically solvable, no exact solution is known for massive fermions [1,2,[7][8][9][10][11][12][13][14]. The current quark masses of the light quarks are small compared to the QCD scale.…”

Section: Introductionmentioning

confidence: 99%

Connecting real-time properties of the massless Schwinger model to the massive case

Hebenstreit

Berges

2014

Phys. Rev. D

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Quantum electrodynamics in 1 + 1 space-time dimensions is analytically solvable for massless fermions, while no solution is known for massive fermions. Employing the classical-statistical approach, we simulate the real-time dynamics on a lattice using Wilson fermions with mass m at gauge coupling g. It is shown that quantitative properties of the massless Schwinger model are emerging in the limit of large g/m. We investigate two scenarios corresponding to opposite charges which are either held fixed or moving back-to-back along the light cone, as employed in effective descriptions for jet energy loss and photon production in the context of heavy-ion collisions. Remarkably, we find that the dynamics is rather well described by the massless limit for a wide range of mass values at fixed coupling. Moreover, our study shows that previous approximate scenarios with external charges on the light cone rather accurately capture the self-consistent dynamics of the energy conserving simulation.

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“…Several authors, inspired by the similarity of Q with a choice of Dirac sea, have taken Q ± to be (non-local) projections P ± onto +ve/-ve energy eigenstates. The resulting WF's are invariant under time-independent gauge transformations of the fields, but in general they do not satisfy Gauss' law [7,8]. However, if we take Q to satisfy the conditions above, Gauss' law is always automatically satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…Here S B and S F are the bosonic and fermionic parts of the Euclidean action, and the integral is evaluated with the following boundary conditions implied by (8):…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger Representation for Fermions and a Local Expansion of the Schwinger Model

Nolland

Mansfield

2000

Int. J. Mod. Phys. A

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We discuss the functional representation of fermions, and obtain exact expressions for wave-functionals of the Schwinger model. Known features of the model such as bosonization and the vacuum angle arise naturally. Contrary to expectations, the vacuum wave-functional does not simplify at large distances, but it may be reconstructed as a large time limit of the Schrödinger functional, which has an expansion in local terms. The functional Schrödinger equation reduces to a set of algebraic equations for the coefficients of these terms. These ideas generalize to a numerical approach to QCD in higher dimensions.

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Functional Schrödinger-Equation for Fermions in External Gauge Fields

Cited by 39 publications

References 21 publications

Light-Cone Quantization: Foundations and Applications

Light-Cone Quantization: Foundations and Applications

Connecting real-time properties of the massless Schwinger model to the massive case

The Schrödinger Representation for Fermions and a Local Expansion of the Schwinger Model

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