Classical description of a particle interacting with a non-Abelian gauge field

Balachandran, A. P.; Salomonson, Per; Skagerstam, Bo-Sture; Winnberg, Jan-Olov

doi:10.1103/physrevd.15.2308

Cited by 154 publications

(177 citation statements)

References 5 publications

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“…Hence, extending the range of integration, 0 < τ < T −→ 0 < τ < ∞ should not modify the result. Performing this 7 The sum n k=1 e 2πik n ·(λ † λ+n/2−1) in the path integral in Eq. (39) is analogous to the well known GSO projection in string theory.…”

Section: The Many Body World Line Formalismmentioning

confidence: 99%

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Minding one’s P’s and Q’s: From the one loop effective action in quantum field theory to classical transport theory

et al. 2000

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The one loop effective action in quantum field theory can be expressed as a quantum mechanical path integral over world lines, with internal symmetries represented by Grassmanian variables. In this paper, we develop a real time, many body, world line formalism for the one loop effective action. In particular, we study hot QCD and obtain the classical transport equations which, as Litim and Manuel have shown, reduce in the appropriate limit to the non-Abelian Boltzmann-Langevin equation first obtained by Bödeker. In the Vlasov limit, the classical kinetic equations are those that correspond to the hard thermal loop effective action. We also discuss the imaginary time world line formalism for a hot φ 4 theory, and elucidate its relation to classical transport theory. IntroductionIn classical kinetic theory, a covariant formalism can be obtained in terms of phase space averages over the trajectories of particle world lines [1]. In quantum field theory, there are many instances at finite temperature and density where classical ideas are relevant, and where a classical kinetic picture would be useful. However, it is not immediately apparent how one recovers the classical world line picture directly from quantum field theory. This is especially problematic in theories with internal symmetries.Fortunately, in the last decade, there has been a considerable body of work relating the one loop effective action in quantum field theory to quantum mechanical path integrals over point particle Lagrangians 1 . For a survey of recent developments, see the review by Schubert [4]. These recent developments follow from the key insight by Berezin and Marinov [5] that internal symmetries such as color and spin had classical analogues in terms of Grassmanian variables. These obeyed classical commutation relations which, when quantized, gave the usual commutation relations for spin and color.Brink, DiVecchia, and Howe used these Grassmanian variables to construct a classical Lagrangian for spinning world lines in an Abelian background field [6]. Subsequently, Balachandran et al. [7] and Barducci et al. [8] wrote down the following Lagrangian for a classical colored particle in a non-Abelian background field 2 :Here the λ a (τ ) with a = 1, · · · , N are Grassmanian dynamical variables, and D ab λ b =λ a + igẋ µ A α µ T α ab λ b . The variables T α ab 's are N × N matrices in an irreducible representation of the Lie algebra of the group. The EulerLagrange equations of motion are deduced in the usual way from the above Lagrangian. It was shown by Balachandran et al. and by Barducci et al. that these equations are precisely the equations written down nearly thirty years ago by S.K. Wong [9].The connection of the work on point particle Lagrangians to quantum field theory was first made by Strassler [10]. He showed that the one loop effective action in quantum field theory could be expressed in terms of a quantum mechanical path integral over a point particle Lagrangian. For an Abelian gauge theory, Strassler showed that this Lag...

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Section: The Many Body World Line Formalismmentioning

confidence: 99%

“…Subsequently, Balachandran et al [7] and Barducci et al [8] wrote down the following Lagrangian for a classical colored particle in a non-Abelian background field 2 :…”

Section: Introductionmentioning

confidence: 99%

Minding one’s P’s and Q’s: From the one loop effective action in quantum field theory to classical transport theory

et al. 2000

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“…However, all those results were anticipated and an alternative formulation of relativistic wave equations [4,5,6] and quantum field theory can be obtained with the study of physics on the world line of the particle. Particles with spin N/2 could be described by an N − extented supersymmetry [6] on the world line, and gauge symmetries by the introduction of internal Grassmann variables [7]. All this was recently promoted into an alternative and efficient description of field theory using the world-line formalism [8], introducing 1D Feyman rules and appropriate one dimensional Green functions [9].…”

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confidence: 99%

Local Fractional Supersymmetry for Alternative Statistics

Fleury

Traubenberg

1996

Mod. Phys. Lett. A

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A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in 1D supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry i.e. a fractional supergravity which is then quantizedà la Dirac to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given, by means of a curved fractional superline.

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“…The method for incorporating internal symmetries at the first quantized level had been well known for quite some time, starting with [25] and further elaborated upon in [26,27]. However, their incorporation in worldline path-integrals for propagators [28,29] and effective actions [30] did not somehow seem appealing until only recently.…”

Section: Advances In High Energy Physicsmentioning

confidence: 99%

Worldline Path-Integral Representations for Standard Model Propagators and Effective Actions

Bhattacharya¹

2017

Advances in High Energy Physics

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We develop the formalism to do worldline calculations relevant for the Standard Model. For that, we first figure out the worldline representations for the free propagators of massless chiral fermions of a single generation and gauge bosons of the Standard Model. Then we extend the formalism to the massive and dressed cases for the fermions and compute the QED vertex. We then go over fermionic one-loop effective actions and anomalies. To our knowledge, in the places where there has been an attempt at deriving the gauge boson propagator, the derivation is somewhat contrived, and we believe our derivation is more straightforward. Moreover, our incorporation of internal degrees of freedom is novel and sports some new features. The derivation of the QED vertex is also new. The treatment of the fermionic one-loop effective actions leads to a particularly economical derivation of chiral anomalies and the gauge anomaly freedom in the Standard Model, improving upon the state of the art in the literature. The appropriate worldline formalism developed thus sets the stage for Standard Model calculations beyond the tree and one-loop cases that incorporate BernKosower type formulae for multiloop amplitudes, relevant for processes at the LHC.

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Classical description of a particle interacting with a non-Abelian gauge field

Cited by 154 publications

References 5 publications

Minding one’s P’s and Q’s: From the one loop effective action in quantum field theory to classical transport theory

Minding one’s P’s and Q’s: From the one loop effective action in quantum field theory to classical transport theory

Local Fractional Supersymmetry for Alternative Statistics

Worldline Path-Integral Representations for Standard Model Propagators and Effective Actions

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