nLab internal symmetry




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics



In physics by an “internal symmetry” one broadly means a symmetry other than spacetime symmetry, i.e. a symmetry not due to nor reflecting movement in space and time, but due to “internal degrees of freedom”.

In high energy physics

Notably gauge symmetry is counted as internal symmetry.

Or a spinor field, in the absence of any magnetic field or spin-orbit interaction or any other interaction involving the spin will have a dynamics which is invariant under transformations that just flips the spin, and this is then an internal symmetry.

In solid state physics

In condensed matter theory, internal symmetries are also referred to as “on-site” symmetries, this referring to the sites of the atomic nuclei in the crystal lattice.

For example, the Ising model has an on-site “spin flip”- / 2 \mathbb{Z}/2 symmetry |S|\vert \uparrow \rangle \xleftrightarrow{S} \vert \downarrow \rangle, while the Heisenberg model has the full Spin(3)=SU(2)-group acting as an on-site symmetry on the electron spins (cf. CGW11, p. 5).

(table from SS 22)


Explicit definition of internal symmetry condensed matter theory:

Beware that old historical articles (eg. Barlow 1883) used “internal symmetry” to refer to crystallographic symmetry which now is referred to as “spatial symmetry” and regarded as non-internal.

See also:

Last revised on May 16, 2022 at 13:32:52. See the history of this page for a list of all contributions to it.