Relational and Deparametrized Evolution

An internal clock can turn a constrained system into an ordinary-looking quantum evolution, but only on a domain where the clock labels gauge orbits uniquely and the reduced generator is self-adjoint. Relational evolution is therefore conditional: it describes correlations between a clock and the remaining degrees of freedom. This page derives that construction, identifies its unitarity conditions, and shows how turning points and quantum clock fluctuations delimit it.

Required background. Variational and Symplectic Geometry supplies constraint flows, gauge fixing, and reduced symplectic structure.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical operators; Einstein Equations supplies the gravitational dynamics to recover; Quantum Information supplies conditional states and reduced density operators; Formulations of General Relativity supplies canonical constraints; and Quantum Gravity supplies the scale and scope distinctions.

Complete observables from a clock

Let one first-class constraint $C$ generate a gauge flow $\alpha_C^s$. Choose a clock function $T$ and another quantity $A$. Locally, solve

$$\alpha_C^{s_*}(T)=\tau$$

and define the complete observable

$$F_{A,T}(\tau)=\alpha_C^{s_*}(A).$$

Although $A$ and $T$ separately need not be gauge invariant, their correlation is: $\{F_{A,T}(\tau),C\}=0$ wherever the construction is regular. The essential condition is $\{T,C\}\ne0$ on the clock patch. A complete observable can fail globally if a level surface $T=\tau$ misses an orbit or intersects it more than once. Dittrich develops the local series and its multi-constraint extension in Dittrich 2006, §§2–3.

For many constraints $C_I$, one needs as many clock variables $T^I$ as independent gauge directions and an invertible matrix $M^I{}_J=\{T^I,C_J\}$. Its determinant is the classical gauge-fixing, or Faddeev–Popov, factor. Vanishing determinant marks a clock horizon; it cannot be repaired by merely reparametrizing the evolution equation.

Exact deparametrization

The cleanest case has canonical pair $(T,P_T)$ and constraint

$$C=P_T+H(q,p)=0,$$

with $H$ independent of $T$. Quantization with $\widehat P_T=-i\partial_T$ gives

$$i\frac{\partial}{\partial T}\Psi(q,T)=\widehat H\Psi(q,T).$$

If $\widehat H$ is self-adjoint on a fixed dense domain, then $U(T_2,T_1)=\exp[-i\widehat H(T_2-T_1)]$ is unitary and the inner product on any constant-$T$ slice is conserved. This is not an approximation: within the deparametrized model it is exact.

Three changes spoil the simple conclusion. If $H$ depends explicitly on $T$, unitary evolution still exists only when a suitable time-dependent self-adjoint domain is controlled. If the constraint is quadratic in $P_T$, selecting a positive-frequency square root is an additional branch choice. If $T$ ceases to be monotonic, no single constant-$T$ surface represents every orbit once.

Conditional quantum states

The same logic can be written without first reducing the classical system. In a clock–system tensor product, take a physical state satisfying

$$(\widehat P_T+\widehat H_S)\lvert\Psi\rangle=0.$$

Conditioning on an idealized clock reading gives $\lvert\psi_S(T)\rangle\propto\langle T\vert\Psi\rangle$, which obeys the Schrödinger equation. Page and Wootters give this stationary-global-state construction and its conditional probabilities in Page and Wootters 1983, pp. 2887–2889.

An actual clock has finite energy, resolution, and coherence. A finite-resolution reading can mix several ideal times, so the conditional evolution need not be an exact one-parameter unitary group. For example, model the uncertainty in an ideal time $t$ at reported reading $T$ by a normalized Gaussian of width $\sigma$. If the ideal system has energy eigenvalues $E_m$, averaging $U(t)\rho_0U(t)^\dagger$ gives

$$\rho_{mn}(T)=e^{-i(E_m-E_n)T} e^{-\sigma^2(E_m-E_n)^2/2}(\rho_0)_{mn}.$$

This elementary Gaussian integral shows how energy coherences can be blurred by clock resolution. For fixed $\sigma$ the suppression is fixed; it is not a decay rate growing universally with time. More generally the correction depends on the clock Hamiltonian, measurement model, and correlations in the physical state, so it must not be promoted to a universal fundamental decoherence law.

Clock dependence and gravitational use

In canonical gravity, possible clocks include a scalar field, dust, spatial volume, extrinsic curvature, or asymptotic boundary time. Each selects a different patch and may produce a different reduced Hamiltonian. In homogeneous minimally coupled massless-scalar cosmology, nonzero conserved $p_\varphi$ gives $\dot\varphi=Np_\varphi/(V_0a^3)$, so for positive lapse the scalar remains monotonic even through a recollapse of $a$. The scale factor itself fails as a clock there. A scalar potential or inhomogeneous dynamics can remove this simple guarantee, while a geometric clock can fluctuate strongly in a quantum regime.

Rovelli’s formulation clarifies that relational predictions need gauge-invariant correlations, not a preferred fundamental time Rovelli 2002, §§2–4. The stronger assertion that all admissible clocks produce equivalent quantum theories remains unproved. An independent check is therefore to quantize the same finite model with two clocks, match their overlapping relational observables, and test whether the physical inner products and transition amplitudes agree.

This page does not claim a clock for full quantum gravity, recover quantum field theory on a generic spacetime, or settle semiclassical thermodynamics. It supplies the conditions a proposed deparametrization must satisfy: transversality, adequate global coverage, controlled branch choice, a positive physical inner product, self-adjoint dynamics, and clock fluctuations small on the resolution scale of the observable.

Exercises

For $C=P_T+(P^2+\omega^2Q^2)/2=0$, compute the complete observable $F_{Q,T}(\tau)$ and verify that it commutes with $C$. What is the quantum condition for the corresponding $T$ evolution to be unitary?

Solution

The constraint flow advances $T$ by $s$ and evolves $(Q,P)$ by the harmonic-oscillator Hamiltonian. Setting $s=\tau-T$ gives

$$F_{Q,T}(\tau)=Q\cos[\omega(\tau-T)]+\frac{P}{\omega}\sin[\omega(\tau-T)].$$

Differentiating shows that the explicit $T$ dependence cancels its Poisson bracket with the oscillator Hamiltonian, so $\{F_{Q,T},C\}=0$. Quantum mechanically, $\widehat H=(\widehat P^2+\omega^2\widehat Q^2)/2$ must be self-adjoint on a specified domain. Stone’s theorem then makes $e^{-i\widehat H(\tau-T)}$ unitary and preserves the physical inner product. Merely writing the formal exponential is insufficient if the operator is only symmetric or its domain changes with the clock value.

References

• Dittrich, Bianca. “Partial and Complete Observables for Canonical General Relativity.” Classical and Quantum Gravity 23 (2006): 6155–6184. DOI.
• Page, Don N., and William K. Wootters. “Evolution without Evolution: Dynamics Described by Stationary Observables.” Physical Review D 27 (1983): 2885–2892. DOI.
• Rovelli, Carlo. “Partial Observables.” Physical Review D 65 (2002): 124013. DOI.