Dirac and Relational Observables

A field value such as $\phi(x)$ is not a gravitational observable merely because $\phi$ is a spacetime scalar: the coordinate label $x$ moves under a diffeomorphism. A relational observable replaces that coordinate label by readings of physical clocks, rods, matter profiles, or asymptotic anchors. The result can commute with the constraints, but only within the domain where the chosen reference system is well defined.

Required background. Variational and Symplectic Geometry supplies first-class constraints and Hamiltonian flows. Equivalence Principles and Gravitational Measurement supplies operational reference frames.

Helpful background. Flat-Spacetime Quantum Field Theory supplies operator-valued fields, Einstein Equations supplies gravitational dynamics, Quantum Information supplies conditional states, and Quantum Gravity supplies the approximation hierarchy used here.

Partial and complete observables

Let first-class constraints $C_A\approx0$ generate gauge flows on a constraint surface. A Dirac observable $O$ obeys

$$\{O,C_A\}\approx0$$

for every proper gauge generator. Individual clock readings $T^A$ and a field $f$ need not satisfy this equation. They are partial observables: quantities that can participate in a measurement but whose separate values do not label a gauge-invariant state. A complete observable asks for the value of $f$ when the clocks read $T^A=\tau^A$. Rovelli develops this distinction and its predictive role in generally covariant systems Rovelli 2002, §§II–V.

For one constraint $C$ and a clock satisfying $\{T,C\}=1$, let $\alpha_C^s$ denote Hamiltonian flow by parameter $s$. Along an orbit, $\alpha_C^s(T)=T+s$. The relational observable is

$$O_{f,T}(\tau)=\left.\alpha_C^s(f)\right|_{s=\tau-T}.$$

Moving the starting point along the same orbit changes both $T$ and the required flow parameter by compensating amounts, so $O_{f,T}(\tau)$ is invariant. Locally one may expand

$$O_{f,T}(\tau)=\sum_{n=0}^{\infty} \frac{(\tau-T)^n}{n!}\underbrace{\{\cdots\{f,C\},C\}\cdots,C\}}_{n\ \text{brackets}}.$$

The flow definition is local in phase space. The displayed series is a formal Taylor expansion unless the analytic flow and its convergence domain are established; smoothness alone does not guarantee convergence. The construction fails if the clock stops, takes the same value repeatedly on an orbit, or does not provide an invertible gauge fixing.

Parameterized-particle derivation

Promote Newtonian time $t$ to a canonical coordinate with momentum $p_t$. A free particle has the constraint

$$C=p_t+\frac{p^2}{2m}\approx0.$$

The coordinate $q$ is not Dirac because $\{q,C\}=p/m$. Choose $t$ as the clock. Since $\{t,C\}=1$, the complete position at clock reading $\tau$ is

$$Q(\tau)=q+\frac{p}{m}(\tau-t).$$

Direct calculation gives

$$\{Q(\tau),C\}=\frac{p}{m}-\frac{p}{m}=0.$$

Solving the equations first gives the same result: $dq/dt=p/m$, so the position when the physical clock reaches $\tau$ is $q+p(\tau-t)/m$. Agreement between the Poisson-bracket and reduced-solution routes checks the sign and normalization.

Two limits discriminate the interpretation. At $\tau=t$, $Q=q$ on that clock slice, recovering the gauge-fixed variable. If $p=0$, $Q=q$ for all $\tau$, as a stationary trajectory requires. If a proposed clock has $\{T,C\}=0$, however, its reading is constant along the orbit and it cannot locate an event; the construction fails rather than producing a new observable.

Four scalar reference fields in gravity

Suppose four scalar fields $X^A(x)$ have a nonsingular Jacobian in a spacetime patch. They can provide physical coordinates. If $x_\tau$ is the unique event satisfying $X^A(x_\tau)=\tau^A$, then

$$\Phi(\tau)=\phi(x_\tau)$$

is invariant under diffeomorphisms that move $\phi$, $X^A$, and $x_\tau$ together. An equivalent distributional expression is

$$\Phi(\tau)=\int d^4x\,\sqrt{-g}\,\phi(x) \prod_{A=0}^{3}\delta\!\left(X^A(x)-\tau^A\right) \frac{\left|\det(\partial_\mu X^A)\right|}{\sqrt{-g}},$$

when the map $x\mapsto X^A$ is one-to-one and orientations are fixed. The determinant compensates the delta functions, leaving evaluation at the relationally specified event.

This formula states the assumptions that the compact notation $\phi(X=\tau)$ hides: a reference-field configuration, an invertible patch, a choice among multiple roots, and suitable smearing in quantum theory. Reference fields gravitate and fluctuate. Treating them as infinitely precise classical rods while quantizing everything else can conceal backreaction and uncertainty.

Quantum clocks and the problem of time

Dirac quantization imposes $\widehat C_A\lvert\Psi\rangle=0$. A naive external-time evolution may then appear frozen, although correlations between partial observables remain nontrivial. Isham’s analysis distinguishes this constraint problem from the separate choices of internal time, Hilbert-space inner product, and global foliation Isham 1993, §§3–6.

One can define conditional amplitudes or evolving constants relative to a clock operator, but different clocks need not yield unitarily equivalent quantum descriptions globally. The clock has a spectrum, dispersion, and interaction with the system. Ordering choices can modify the constraint commutator, while an anomalous quantum constraint algebra would invalidate the Dirac construction.

Relational observables also need not be compactly supported in the underlying variables. Solving for $x_\tau$ uses reference fields over a region, and an asymptotically anchored observable carries a dressing to the boundary. Relational localization is therefore operational and state-dependent, not identical to coordinate support.

Practical construction test

Before calling a quantity relationally local, report: the constraint set; clock and rod variables; the domain in which their Jacobian is invertible; the operator ordering and state; the smearing resolution; boundary conditions; and the residual gauge group. Then verify the constraint bracket or commutator through the retained order and compare with a gauge-fixed calculation of the same observable.

Changing clocks is an adverse check. Robust predictions should agree where both reference systems are semiclassical and overlap. Disagreement can signal clock fluctuations, a multiple-choice problem, a bad gauge patch, or omitted backreaction rather than new physics.

Exercises

Exercise. For the parameterized free particle, define $Q(\tau)$ as above and $P(\tau)=p$. Show that both are Dirac observables and that their reduced Poisson bracket is canonical. Then replace the clock by $T=t+\beta q$ and identify the condition under which it is locally admissible.

Solution

We already found $\{Q,C\}=0$, and $\{p,C\}=0$, so both are Dirac. At fixed $\tau$,

$$\{Q(\tau),P(\tau)\} =\left\{q+\frac{p}{m}(\tau-t),p\right\}=1.$$

For $T=t+\beta q$,

$$\{T,C\}=1+\beta\frac{p}{m}.$$

The clock is locally admissible only where this expression is nonzero, because the gauge condition $T=\tau$ must intersect nearby gauge orbits transversely. At $p=-m/\beta$ the clock is constant along the orbit and the relational construction becomes singular. This failure depends on the reference choice, not on the particle trajectory itself.

References

• Isham, Christopher J. “Canonical Quantum Gravity and the Problem of Time.” In Integrable Systems, Quantum Groups, and Quantum Field Theories, edited by L. A. Ibort and M. A. Rodríguez, 157–287. Dordrecht: Kluwer, 1993. arXiv:gr-qc/9210011.
• Rovelli, Carlo. “Partial Observables.” Physical Review D 65 (2002): 124013. DOI.