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# Contents

## Idea

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”-$\mathbb{Z}/2$symmetry $\vert \uparrow \rangle \xleftrightarrow{S} \vert \downarrow \rangle$, 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)

## References

Explicit definition of internal symmetry condensed matter theory:

• Charles Zhaoxi Xiong, p. 13 of: Classification and Construction of Topological Phases of Quantum Matter $[$arXiv:1906.02892$]$

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.