Faddeev Popov ghosts

Spirit fields or also Faddejew Popow spirits (after Ludwig Faddejew and Wiktor Popow ) are unphysical fields that occur in the quantization of gauge theories in the path integral formalism. They were first used by Richard Feynman .

The ghosts are relics of the mathematical treatment of non-Abelian gauge theories ( Yang-Mills theory ). In the path integral, all vector potentials are integrated, including those that are equivalent due to the freedom from calibration. This is compensated for in the Faddejew-Popow formalism by introducing new scalar fields, the ghosts . These appear only in closed loops of the Feynman diagrams , where they cancel out the contribution of the redundant degrees of freedom of the vector potentials. So that the loop diagram shows an amount that is opposite to the contribution of the vector fields (according to the statistics, bosonic ) , the spirit fields must obey a fermionic statistics. So they formally violate the spin statistics theorem , but this is irrelevant because they do not correspond to any physical particles. In the case of quantum chromodynamics , they are complex scalar fields (the index a denotes their degrees of color freedom, i.e. values in adjoint representations of the calibration group , such as that of the vector potential fields ). In the weak interaction there are three different spirit fields which correspond to the three weak gauge bosons. ${\ displaystyle c ^ {a}}$ ${\ displaystyle c ^ {\ pm}, c ^ {Z}}$

For Yang-Mills fields like quantum chromodynamics , their contribution to the Lagrangian is :

{\ displaystyle {\ mathcal {L}} _ {\ mathrm {ghost}} = \ partial _ {\ mu} {\ bar {c}} ^ {a} \ partial ^ {\ mu} c ^ {a} + gf ^ {abc} (\ partial ^ {\ mu} {\ bar {c}} ^ {a}) A _ {\ mu} ^ {b} c ^ {c}}

For purely Abelian gauge theories such as quantum electrodynamics , where is, the spirit fields decouple and make no contribution to the Lagrangian. In theories with residual abelian gauge theory, such as the electro-weak of the standard model, after which a refractive with present, the ghost decouple only by their corresponding bosons, but not of the calibration and Goldstone bosons the fractional part. ${\ displaystyle f ^ {abc} = 0}$ ${\ displaystyle SU (2) \ times U (1)}$ ${\ displaystyle U (1)}$ ${\ displaystyle f ^ {abc} = 0}$

Just like the propagators of the calibration bosons, the propagators and the couplings of the Faddejew-Popow spirits depend on the chosen calibration . In unitary calibration the weak spirits get an infinitely large mass and thereby decouple from the other particles. In quantum chromodynamics, the choice of the Arnowitt-Fickler calibration can be used to decouple the spirits.

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↑ LD Faddeev, VN Popov: Feynman diagrams for the Yang-Mills field . In: Phys. Lett. B . tape 25 , no. 1 , 1967, p. 29 , doi : 10.1016 / 0370-2693 (67) 90067-6 .

[1] LD Faddeev, VN Popov: Feynman diagrams for the Yang-Mills field . In: Phys. Lett. B . tape 25 , no. 1 , 1967, p. 29 , doi : 10.1016 / 0370-2693 (67) 90067-6 .