Symplectic Reduction and Boundary Degrees of Freedom

Symplectic reduction turns a constrained description with redundant directions into a phase space of distinct states. In its cleanest finite-dimensional form, one restricts to a regular moment-map level and then divides by a free, proper group action. Field theories with boundaries expose every hypothesis in this sentence: admissible transformations depend on boundary conditions, the presymplectic form can acquire boundary flux, and the quotient can be singular. Boundary degrees of freedom are therefore a conclusion of a specified variational problem, not an automatic consequence of drawing a boundary.

Required background. Exterior Algebra and Differential Forms supplies pullbacks and form algebra, while the generalized Stokes theorem supplies the boundary identity. Tensor Fields and Natural Operations supplies smooth tensors and natural maps; Symmetry Algebras, Orbits, and Fixed Sets supplies group actions and orbits, and this page states the extra regularity needed for a smooth quotient.

Helpful background. Calculus and Linear Algebra supplies regular values, kernels, and dimension counts; Variational Mechanics and Symmetries supplies constrained Hamiltonian intuition.

Regular symplectic reduction

Let a Lie group $G$ act on $(P,\Omega)$ in a Hamiltonian fashion with equivariant moment map $\mu:P\to\mathfrak g^*$. At the zero level set $\mathcal C=\mu^{-1}(0)$, the pullback $i^*\Omega$ has the orbit directions in its kernel. Suppose that zero is a regular value, the action on $\mathcal C$ is free and proper, and the quotient map $\pi:\mathcal C\to\mathcal C/G$ is therefore a smooth principal bundle. Then there is a unique two-form $\Omega_{\rm red}$ satisfying

$$\pi^*\Omega_{\rm red}=i^*\Omega.$$

Horizontality follows because orbit generators lie in the kernel; invariance follows from the symplectic group action. These properties make the pulled-back form basic, so it descends. Closedness descends from $d(i^*\Omega)=0$, and after quotienting precisely the kernel directions, $\Omega_{\rm red}$ is nondegenerate. This is the Marsden–Weinstein–Meyer construction presented in Cannas da Silva 2006, Lectures 23–24, pp. 141–154.

When $G$ has dimension $r$ and all hypotheses hold, regular restriction removes $r$ dimensions and quotienting removes another $r$:

$$\dim(P\!\sslash G)=\dim P-2r.$$

The count fails at nonregular values or where stabilizers change. In those cases the quotient may have symplectic strata of different dimensions rather than one manifold. For a connected Hamiltonian manifold with a compact Lie group and equivariant moment map, Sjamaar and Lerman 1991, Theorem 2.1 and §6, pp. 9 and 36–44 of the preprint, PDF constructs the zero-level orbit-type pieces and their stratification. This cited version uses compactness of the group; it should not be read as a theorem about every nonproper action. Properness of the moment map is a separate hypothesis used for stronger global connectedness conclusions.

A direct reduction check

Consider $T^*\mathbb R^2$ with canonical form and the rotation moment map $J=xp_y-yp_x$. On the open set $r=\sqrt{x^2+y^2}>0$, use polar canonical coordinates $(r,\theta,p_r,p_\theta)$, for which

$$\Omega=dr\wedge dp_r+d\theta\wedge dp_\theta, \qquad J=p_\theta.$$

Restriction to $J=0$ removes variations in $p_\theta$, leaving $i^*\Omega=dr\wedge dp_r$. Its kernel is generated by $\partial_\theta$, exactly the rotation orbit. Quotienting by $SO(2)$ yields coordinates $(r,p_r)$ and

$$\Omega_{\rm red}=dr\wedge dp_r.$$

This calculation independently verifies both descent and the dimension count. It excludes $r=0$ because polar coordinates on the base fail there. That coordinate failure is not itself an enhanced stabilizer: at $x=y=0$ with nonzero momentum, rotations still act freely on the full phase-space point. The actual fixed point is $x=y=p_x=p_y=0$, where the whole $SO(2)$ action fixes both position and momentum and the zero-level quotient has a singular orbit type.

Boundaries change the degeneracy test

For a field theory, let $\Omega_\Sigma$ be obtained by integrating a presymplectic current over a hypersurface $\Sigma$. A gauge variation $\delta_\lambda\phi$ is redundant only if

$$\Omega_\Sigma(\delta\phi,\delta_\lambda\phi)=0$$

for every admissible tangent variation. Integration by parts often turns this pairing into a constraint term plus an integral over $\partial\Sigma$. A parameter that vanishes at the boundary may kill the surface term. A parameter allowed to act nontrivially there may instead generate a symmetry with boundary data. Which case occurs depends on the action, boundary terms, falloff, allowed variations, and flux through the boundary.

This kernel test is distinct from asking whether $\Omega_\Sigma$ is independent of $\Sigma$. The latter is controlled by flux through the intervening boundary. A boundary symmetry can have a nonzero pairing even when that flux vanishes, as explained in Presymplectic Potentials and Currents.

Harlow and Wu derive covariant phase space from a well-posed boundary variational principle and show why the boundary contribution must be retained in the symplectic structure Harlow and Wu 2020, §§2.1–2.2, pp. 6–15. Their construction also illustrates that different boundary conditions define different phase spaces; there is no boundary-condition-independent list of gauge directions.

Edge variables are a choice, not a theorem

One way to obtain a gauge-covariant factorization across a subregion boundary is to enlarge phase space with boundary variables and then reduce an extended symmetry. Such variables are often called edge modes. They can be extremely useful, but neither their field content nor their polarization is unique. In other formulations the same physics may be described by restricting gauge transformations, choosing boundary conditions, retaining surface symmetries, or working directly with an algebra of observables.

Thus a boundary does not compel one universal collection of physical modes. One must first identify the variational principle and admissible transformations, then compute the kernel and surface pairing, and only then decide whether extension and reduction are useful. This page does not define gravitational charges, choose a gravitational boundary action, or perform quantization.

Exercise: reduction by translations

Let $P=T^*\mathbb R^2$ with coordinates $(x,y,p_x,p_y)$, and let $\mathbb R$ translate $x$. Reduce at moment-map value $p_x=0$. Verify the descended form and explain what changes if $x$ is restricted to an interval whose endpoints must be fixed.

Solution

On $p_x=0$, the pullback of $dx\wedge dp_x+dy\wedge dp_y$ is $dy\wedge dp_y$. Its kernel is $\partial_x$, the translation generator. Quotienting gives $T^*\mathbb R$ with coordinates $(y,p_y)$ and symplectic form $dy\wedge dp_y$.

If $x$ lies in an interval and the boundary locations are fixed, finite translations do not preserve the admissible configuration space. They are not gauge transformations of that boundary problem, so quotienting by them is unjustified. The local bulk generator has not changed, but its global status has.

References

• Cannas da Silva, Ana. Lectures on Symplectic Geometry. Lecture Notes in Mathematics 1764, revised edition, Springer, 2006. Lectures 23–25, pp. 141–162.
• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 10 (2020), 146. DOI; Open PDF.
• Sjamaar, Reyer, and Eugene Lerman. “Stratified Symplectic Spaces and Reduction.” Annals of Mathematics 134, no. 2 (1991): 375–422. DOI; Open PDF, November 1990 preprint.