Canonical Constraints, Dirac Observables, and Constraint Algebras

In canonical general relativity, lapse and shift are Lagrange multipliers enforcing Hamiltonian and spatial-diffeomorphism constraints. Their Poisson algebra has metric-dependent structure functions. A physical observable must commute with the constraints on the constraint surface, or be defined relationally with respect to dynamical clocks and rods.

Required background. Constraints, Dirac Brackets, and Symplectic Reduction supplies constrained Hamiltonian mechanics; Relational and Gauge-Invariant Gravitational Observables supplies observables.

Helpful background. Gauge Orbits, Gauss Constraints, and Stabilizers supplies gauge reduction; Covariant Symplectic Structure and Conserved Inner Products supplies the covariant comparison; Constraints, Conservation, and the Bianchi Identity supplies propagation.

ADM phase space

Write

$$ds^2=N^2dt^2-h_{ab}(dx^a+N^adt)(dx^b+N^bdt).$$

In units $16\pi G=1$, the action is

$$S=\int dt\,d^3x\, \left(\pi^{ab}\dot h_{ab}-N\mathcal H-N^a\mathcal H_a\right)+B,$$

with

$$\mathcal H= \frac1{\sqrt h}\left(\pi_{ab}\pi^{ab}-\frac12\pi^2\right) -\sqrt h\,{}^{(3)}R, \qquad \mathcal H_a=-2D_b\pi^b{}_a.$$

The boundary term $B$ and falloffs are required for differentiable generators.

First application: the hypersurface-deformation algebra

Smear the constraints as $H[N]=\int N\mathcal H$ and $D[\mathbf N]=\int N^a\mathcal H_a$. Direct use of $\{h_{ab}(x),\pi^{cd}(y)\}=\delta_{(a}^c\delta_{b)}^d\delta(x,y)$ gives

$$\{D[\mathbf N],D[\mathbf M]\}=D[[\mathbf N,\mathbf M]],$$ $$\{D[\mathbf N],H[M]\}=H[\mathcal L_{\mathbf N}M],$$ $$\{H[N],H[M]\} =D\!\left[h^{ab}(N\partial_bM-M\partial_bN)\right].$$

The last bracket contains $h^{ab}$, so this is not a Lie algebra with constant structure constants. It represents changes of spacetime slicing. The original ADM construction and constraint analysis are reviewed in Arnowitt, Deser, and Misner 1962.

For a background clock $T_0(t)$ and scalar $\phi_0(t)$, the perturbation

$$\Phi_{\rm rel} =\delta\phi-\frac{\dot\phi_0}{\dot T_0}\delta T$$

is invariant under an infinitesimal time shift. It is a perturbative relational observable: “the scalar when the clock reads $T$.”

Adversarial control: boundaries and anomalous brackets

Allow smearings that approach nonzero asymptotic values. Integrating by parts produces surface terms; omitting them makes the generators nondifferentiable and loses ADM charges. Change a regulator in the quantum theory and compute the commutator: closure only after discarding nonvanishing terms is not anomaly freedom.

The classical benchmark is the differentiable hypersurface-deformation algebra plus boundary charges. Coordinate components such as $h_{ab}(x)$ are not Dirac observables, and a quantum proposal must represent both the constraints and their physical inner product.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Arnowitt, Richard, Stanley Deser, and Charles W. Misner. “The Dynamics of General Relativity.” In Gravitation: An Introduction to Current Research, edited by Louis Witten, 227–265. Wiley, 1962. arXiv. Open PDF.