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

Abstract

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...

The equations of motion for a classical color particle in background non-Abelian bosonic and fermionic fields

“…Obtaining the Wong equation for the usual color charge in [17] and the equations of motion for the Grassmann color charges in background gauge field in [18][19][20],…”

Hard thermal loops and beyond in the finite temperature world-line formulation of QED

“…At finite T , this was noted in [24] and later explored in [25]. Remarkably, a world-line approach, analogous to that used to obtain classical kinetic theory, can be applied to derive an effective action for small-x physics [26].…”

World-line approach to chiral kinetic theory in topological background gauge fields

“…In this talk, we will give an overview of our recent work [61,62] in developing a consistent Lorentz covariant transport theory within the world-line approach to quantum field theory [63,64,65,66,67,68,69,70,71]. A central ingredient, going back to seminal work by Berezin and Marinov [72] and related contemporaneous work [73,74,75,76,77,78,79,80,81,82] is the description of spin, color and other internal symmetries via Grassmann variables.…”