Covariance group

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 O(1,3) and is often referred to as Lorentz group.

For example, the Maxwell equation with sources,

$\partial_\mu F^{\mu\nu}=4\pi j^\nu$ ,

transforms as a four-vector, that is, under the (½,½) representation of the O(1,3) group.

The Dirac equation,

$(i\gamma^\mu\partial_\mu-m)\psi=0$ ,

transforms as a bispinor, that is, under the (½,0)⊕(0,½) representation of the O(1,3) 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

1. ^ Ryckman 2005, p. 22.

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