Stochastic Quantization of Gauge Fields

  • Daniel Zwanziger
Part of the NATO ASI Series book series (NSSB, volume 141)


Stochastic quantization of gauge fields is reviewed, with particular attention to the mechanism by which stochastic gauge-fixing resolves the Gribov ambiguity, and the light which this casts on the geometry of gauge orbits. Some geometric properties of gauge orbits are derived, and the difficulty which they present for the Faddeev-Popov method is reviewed. Stochastic quantization is introduced, and it is shown how the Euclidean probability distribution may be determined by a diffusion process with a drift force K=−S, where S=S(A) is the classical Yang-Mills action. The equivalence of the diffusion and Langevin equations is demonstrated. Stochastic gauge-fixing is explained, and it is demonstrated that an additional drift force, called the gauge-fixing force, may be introduced which does not affect expectation values of gauge-invariant quantities because it is everywhere tangent to the gauge orbits. The gauge-fixing force is non-conservative and thus cannot be accommodated in an action formalism. Properties of the gauge-fixing force are explained and it is shown how, by a mechanism of “instability stabilizes”, it concentrates the probability near the interior Ω of the first Gribov horizon. As a new result, the diffusion equation is solved for large values of the gauge-fixing force, the solution being a modified Faddeev-Popov formula, whereby the integration extends only over Ω. Another new result is the ℏ or loop expansion of the effective action for stochastic gauge-fixing, and it is shown that all primitive divergences arize in the iterative calculation of an effective drift force Keff.


Diffusion Equation Gauge Field Langevin Equation Orbit Space Gluon Propagator 
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Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Daniel Zwanziger
    • 1
  1. 1.Physics DepartmentNew York UniversityNew YorkUSA

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