Example of a classical action changing by a nonzero boundary term under a continuous transformation

Question

Is there an example of a continuous transformation in classical field theory under which the classical action changes by a nonzero boundary term? I'd prefer an example from field theory in flat spacetime.

Answer 1

OP is asking for examples of a quasisymmetry of the action that is not a strict symmetry:

• Gaugesymmetries are often quasisymmetries.

• A global additive shift of the free Schroedinger field, cf. e.g. this Phys.SE post.

• A Galilean boost of the non-relativistic free particle, cf. e.g. this Phys.SE post.

• The quasisymmetry behind the conservation of Laplace-Runge-Lenz vector, cf. e.g. this Phys.SE post.

• Examples 1, 2 & 3 in the Wikipedia article for Noether's theorem.

Answer 2

A (Lorentz-noninvariant) Lagrange density in electromagnetism proportional to $\vec{A}\cdot\vec{B}$ changes by a boundary term under a gauge transformation.* Under the change of gauge $\vec{A}\rightarrow\vec{A}+\vec{\nabla}\Lambda$, a Lagrange density $${\cal L}=-\frac{1}{4}\left(\vec{E}^{2}-\vec{B}^{2}\right)+k\vec{A}\cdot\vec{B}$$ changes by $${\cal L}\rightarrow{\cal L}+k\left(\vec{\nabla}\Lambda\right)\cdot\vec{B}={\cal L}+\vec{\nabla}\cdot\left(k\Lambda\vec{B}\right)$$ (Here, $k$ is a constant coefficient.) To derive the transformation properties of ${\cal L}$, we have used the facts that the magnetic field is gauge invariant, $\vec{B}\rightarrow\vec{B}$ under the gauge transformation, and divergenceless, $\vec{\nabla}\cdot\vec{B}=0$.

If the Lagrange density changes by a total derivative, then the action changes by a surface term, $$S=\int d^{4}x\,{\cal L}\rightarrow S+\int d^{4}x\,\vec{\nabla}\cdot\left(k\Lambda\vec{B}\right)=S+\int d\vec{S}\cdot\left(k\Lambda\vec{B}\right),$$ where the last surface integral is over the boundary of four-dimensional Minkowski space.

*Such terms have been studied as models of CPT violation—for example in "Limits on a Lorentz- and parity-violating modification of electrodynamics," S. M. Carroll, G. B. Field, R. Jackiw, Phys. Rev. D 41, 1231 (1990).