R _ξ calibration

The R _ξ calibrations are a class of calibrations in quantum field theory . For gauge theories, there are an infinite number of mathematically different expressions that are physically equivalent. For a physical description of reality using path integrals , a calibration that has not yet been specified must be selected. The R _ξ calibrations are common calibrations because they are Lorentz-invariant , i.e. they have the same form in every inertial system, compatible with Einstein's special theory of relativity , and, moreover, in the equations of motion for the quantum fields of the gauge bosons in quantum electrodynamics, the equations of motion in Lorenz - Reproduce the calibration of classical electrodynamics .

The most important R _ξ calibrations are Feynman or Feynman-'t-Hooft calibration , Yennie or Fried-Yennie calibration and, in a broader sense, unitary calibration and Landau calibration .

Mathematical formulation

Massless case

The Lagrangian of a gauge field is in quantum electrodynamics ${\displaystyle {\mathcal {L}}}$ ${\displaystyle A}$

${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}(\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu })^{2}.}$

After a Fourier transformation , the Lagrangian can also be written as

${\displaystyle {\mathcal {L}}=-{\frac {1}{2}}A_{\mu }(k^{2}g^{\mu \nu }-k^{\mu }k^{\nu })A_{\nu }}$

to be written. The propagator , figuratively the propagation probability of a quantum field from one point to another, is the inverse of the term in brackets; In the case of the gauge fields, however, this has the determinant zero and therefore no inverse. ${\displaystyle G}$

The R _ξ calibrations lead in the Lagrangian the additional term, called the calibration fixation term,

${\displaystyle -{\frac {(\partial ^{\mu }A_{\mu })^{2}}{2\xi }},}$

or in Fourier space

${\displaystyle -{\frac {A_{\mu }k^{\mu }k^{\nu }A_{\nu }}{2\xi }},}$

a. There is a free calibration parameter that gives the R _ξ calibration its name. With the help of this additional term, the Lagrangian in Fourier space is ${\displaystyle \xi }$

${\displaystyle {\mathcal {L}}=-{\frac {1}{2}}A_{\mu }\left(k^{2}g^{\mu \nu }-\left(1-{\frac {1}{\xi }}\right)k^{\mu }k^{\nu }\right)A_{\nu },}$

which is the inverse

${\displaystyle G_{\mu \nu }={\frac {g_{\mu \nu }-(1-\xi ){\frac {k_{\mu }k_{\nu }}{k^{2}}}}{k^{2}}}}$

owns. Although quantum electrodynamics was used as the simplest example of a gauge theory implemented in the standard model of elementary particle physics for these explanations of the problem and its solution , these explanations can be generalized to non-Abelian gauge theories with massless gauge bosons, with Faddejew-Popow spirits generally occurring.

Massive fall

In the case of massive gauge theories with spontaneously broken symmetry, as in the electroweak interaction , Goldstone bosons always appear due to the Goldstone theorem, corresponding to the gauge bosons . These must be taken into account in the calibration fixation term. This then becomes ${\displaystyle A}$ ${\displaystyle \phi _{A}}$

${\displaystyle -{\frac {(\partial _{\mu }A^{\mu }+\xi m\phi _{A})^{2}}{2\xi }}}$

The mixed terms occurring in it proportionally cancel out the unphysical mixed terms resulting from the spontaneous symmetry breaking. Then is the propagator ${\displaystyle \partial _{\mu }A^{\mu }\phi _{A}}$

${\displaystyle G_{\mu \nu }={\frac {g_{\mu \nu }-(1-\xi ){\frac {k_{\mu }k_{\nu }}{k^{2}-\xi m^{2}}}}{k^{2}-m^{2}}}}$.

Compared to classical physics

The equations of motion (in spatial space) for the calibration field result in the case of quantum electrodynamics by means of the Lagrange equation from the Lagrangian

${\displaystyle \partial _{\mu }\partial ^{\mu }A^{\nu }-\left(1-{\frac {1}{\xi }}\right)\partial ^{\nu }\partial _{\mu }A^{\mu }=0}$.

In classical electrodynamics, the equation of motion for the vector potential , here noted as , ${\displaystyle A_{\text{klass}}}$

${\displaystyle \partial _{\mu }\partial ^{\mu }A_{\text{klass}}^{\nu }-\partial ^{\nu }\partial _{\mu }A_{\text{klass}}^{\mu }=0}$.

This arises in the Limes , which is synonymous with the disappearance of the legal term. On the other hand, in the (massless) quantum case, the propagator would then diverge. ${\displaystyle \xi \to \infty }$

A Lorentz-invariant calibration in classical physics is the Lorenz calibration , which requires that the equation of motion be compact ${\displaystyle \partial _{\mu }A_{\text{klass}}^{\mu }=0}$

${\displaystyle \partial _{\mu }\partial ^{\mu }A_{\text{klass}}^{\nu }=0}$

can be written as a wave equation . The enforcement of the quantum mechanical happens through the requirement that the entire calibration fixation term does not diverge. On the other hand, the classic wave equation can also be achieved by setting . ${\displaystyle \partial _{\mu }A^{\mu }=0}$ ${\displaystyle \xi =0}$ ${\displaystyle \xi =1}$

These three choices are called Landau, Feynman and Unitary Calibration. ${\displaystyle \xi =0,1,\infty }$

Special calibrations

Landau calibration

The Landau calibration , according to Lew Landau , sets . The Landau calibration has the advantage that the propagator is purely transverse, which means that it applies . In the Landau calibration, therefore, scalar and pseudoscalar bosons such as the Higgs boson and the Goldstone bosons do not interact with the Faddejew-Popow spirits , since their coupling strength is proportional to the calibration parameter . ${\displaystyle \xi =0}$ ${\displaystyle k^{\mu }G_{\mu \nu }=0}$ ${\displaystyle \xi }$

Another property of the Landau calibration is that the Goldstone bosons are massless in it, since their mass depends on the calibration used.

Feynman calibration

The Feynman or Feynman-'t-Hooft calibration , after Richard Feynman and Gerardus' t Hooft , sets . This gives the propagators their most compact form, since all terms that are not proportional to the metric are omitted. With regard to the Goldstone bosons, the Feynman calibration means that they have the same mass as their corresponding calibration bosons. In Feynman calibration the Goldstone-boson equivalence theorem becomes obvious. ${\displaystyle \xi =1}$ ${\displaystyle g_{\mu \nu }}$

Yennie calibration

The Yennie or Fried-Yennie calibration , after Herbert Fried and Donald Yennie , sets . This calibration does not seem to be of any advantage from the point of view of classical physics or for specific calculations. Their advantage lies in the simplicity of the results: Due to the Kinoshita-Lee-Nauenberg theorem , quantum field theories are infrared-safe, which means that different divergent contributions in the limit case of low energies cancel each other out, so that their total remains finite. The Yennie calibration means that many contributions do not even diverge. ${\displaystyle \xi =3}$

Unitary calibration

The unitary calibration sets . This leads to divergences in the propagator for massless gauge bosons, but not for the massive gauge bosons. In the unitary calibration, the Goldstone bosons decouple; they get an infinitely large mass and therefore do not propagate. Furthermore, no Faddejew-Popow ghosts occur in the unitary calibration. The unitary calibration is particularly suitable for calculations in theories with spontaneous symmetry breaking in the leading order of perturbation theory. ${\displaystyle \xi =\infty }$

literature

• Michael E. Peskin and Daniel V. Schroeder: An Introduction to Quantum Field Theory . Perseus Books, Reading 1995, ISBN 0-201-50397-2 (English).
• Mattew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).