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In [[physics]], '''covariance group''' is a [[group (mathematics)|group]] of [[Coordinate_system#Transformations|coordinate transformations]] between admissible [[Frame of reference|frames of reference]] (see for example<ref>Ryckman 2005, p. 22.</ref>). The frames are assumed to provide equivalent description of physical phenomena. The [[covariance principle]] suggests the equations, describing the laws of physics, should transform from one admitted frame to another covariantly, that is, according to a [[Group representation|representation]] of the covariance group. | |||
In [[special relativity]] only [[Inertial frame of reference|inertial frames]] are admitted and the covariance group consists of [[Rotation group SO(3)|rotations]], [[Lorentz transformation|velocity boosts]], and the [[Parity (physics)|parity transformation]]. It is denoted as [[Generalized_orthogonal_group|<math>O(1,3)</math>]] and is often referred to as [[Lorentz group]]. | |||
For example, the [[Maxwell_equation#Covariant_formulation_of_Maxwell.27s_equations|Maxwell equation]] with sources, | |||
:<math>\partial_\mu F^{\mu\nu}=4\pi j^\nu</math> , | |||
transforms as a [[four-vector]], that is, under the [[Representations_of_the_Lorentz_group#Common_reps|(½,½)]] representation of the <math>O(1,3)</math> group. | |||
The [[Dirac_equation#Covariant form and relativistic invariance|Dirac equation]], | |||
:<math>(i\gamma^\mu\partial_\mu-m)\psi=0</math> , | |||
transforms as a [[bispinor]], that is, under the (½,0)⊕(0,½) representation of the <math>O(1,3)</math> group. | |||
The covariance principle, unlike the [[Principle of relativity|relativity principle]], does not imply that the equations are [[Invariant_(physics)|invariant]] under transformations from the covariance group. In practice the equations for [[Electromagnetic force|electromagnetic]] and [[strong interaction|strong]] interactions ''are'' invariant, while the [[weak interaction]] is not invariant under the parity transformation. For example, the Maxwell equation ''is'' invariant, while the corresponding equation for the [[Standard_model_(basic_details)#The_Gauge_Field_Lagrangian|weak field]] explicitly contains [[Standard_model_(basic_details)#The_charged_and_neutral_current_couplings|left currents]] and thus is not invariant under the parity transformation. | |||
In [[general relativity]] the covariance group consists of all arbitrary (invertible and suitably differentiable) coordinate transformations. | |||
== Notes == | |||
{{reflist|2}} | |||
== References == | |||
* Thomas Ryckman, The Reign of Relativity: Philosophy in Physics 1915-1925, Oxford University Press US, 2005, ISBN 0-19-517717-7, ISBN 978-0-19-517717-6 | |||
[[Category:Concepts in physics]] | |||
Latest revision as of 21:09, 27 December 2012
In physics, covariance group is a group of coordinate transformations between admissible frames of reference (see for example[1]). The frames are assumed to provide equivalent description of physical phenomena. The covariance principle suggests the equations, describing the laws of physics, should transform from one admitted frame to another covariantly, that is, according to a representation of the covariance group.
In special relativity only inertial frames are admitted and the covariance group consists of rotations, velocity boosts, and the parity transformation. It is denoted as and is often referred to as Lorentz group.
For example, the Maxwell equation with sources,
transforms as a four-vector, that is, under the (½,½) representation of the group.
The Dirac equation,
transforms as a bispinor, that is, under the (½,0)⊕(0,½) representation of the group.
The covariance principle, unlike the relativity principle, does not imply that the equations are invariant under transformations from the covariance group. In practice the equations for electromagnetic and strong interactions are invariant, while the weak interaction is not invariant under the parity transformation. For example, the Maxwell equation is invariant, while the corresponding equation for the weak field explicitly contains left currents and thus is not invariant under the parity transformation.
In general relativity the covariance group consists of all arbitrary (invertible and suitably differentiable) coordinate transformations.
Notes
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References
- Thomas Ryckman, The Reign of Relativity: Philosophy in Physics 1915-1925, Oxford University Press US, 2005, ISBN 0-19-517717-7, ISBN 978-0-19-517717-6
- ↑ Ryckman 2005, p. 22.