Canonical Quantum Gravity and the Problem of Time

Canonical quantum gravity begins with a sharp classical fact: in a generally covariant theory, the lapse and shift multiply constraints rather than supplying an external Hamiltonian. Quantization therefore asks simultaneously what the physical states are, how their inner product is defined, which quantities are observable, and how the time used in ordinary quantum theory can reappear. This chapter develops those questions from the ADM phase space through relational and semiclassical evolution, without claiming that any one construction already gives a complete microscopic theory.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical quantization and operator-valued fields; Einstein Equations supplies the classical dynamics being constrained; Quantum Information supplies Hilbert spaces, conditional states, and unitarity; Formulations of General Relativity supplies the $3+1$ decomposition; and Variational and Symplectic Geometry supplies constrained phase space and gauge degeneracies.

Enter the canonical theory

A spatial slice $\Sigma$ carries a Riemannian metric $h_{ij}$ and its momentum density $\pi^{ij}$. The lapse $N$ and shift $N^i$ specify how a foliation is threaded through spacetime, but their canonical momenta vanish. Varying them imposes the Hamiltonian and momentum constraints. Up to a boundary charge, the canonical generator is

$$H[N,N^i]=\int_\Sigma d^3x\left(N\mathcal H_\perp+N^i\mathcal H_i\right)+H_{\partial\Sigma}.$$

The constraints do more than remove redundant coordinates. Their Poisson brackets realize the hypersurface-deformation algebra, whose structure functions depend on $h_{ij}$. This is why quantizing gravity is not equivalent to imposing a fixed Lie algebra of internal symmetries. The canonical decomposition and its boundary term are developed in Arnowitt, Deser, and Misner 2008, §§3–7.

Dirac quantization promotes the constraints to operators and seeks states annihilated by them. The formal equation $\widehat{\mathcal H}_\perp\Psi=0$ is only a starting point: products of functional derivatives at one point require regularization, different factor orderings can change the operator, and a solution space is not yet a Hilbert space. DeWitt 1967, §§4–7 gives the foundational superspace formulation while making visible several of these unresolved choices.

Six steps through the chapter

The pages form a dependency chain, and their order matters.

1. ADM Phase Space and Constraints derives $h_{ij}$, $\pi^{ij}$, both constraint families, their algebra, and the boundary charge.
2. The Wheeler–DeWitt Equation explains what the operator constraint says and why ordering, regularization, and domain data cannot be suppressed.
3. Dirac Quantization and Physical Hilbert Space compares quantize-then-constrain, reduced quantization, and group averaging.
4. The Problem of Time separates the frozen equation, global-clock obstruction, multiple-choice problem, observables, and foliation questions.
5. Relational and Deparametrized Evolution asks when a material or geometric clock produces a controlled conditional dynamics.
6. Semiclassical WKB Time and Recovery of QFT derives the Born–Oppenheimer hierarchy in which a WKB geometry supplies approximate Schrödinger time to lighter fields.

For a mathematical route, take the first three pages before studying the constraint algebra and inner product. For a conceptual route, read the first page, then the fourth and fifth. For a bridge to quantum fields on an emergent background, read the Wheeler–DeWitt and WKB pages together. Classical formulations are used as input; generic flat-space field theory and semiclassical thermodynamics remain in their dedicated treatments rather than being redeveloped here.

Time is several problems, not one

The phrase problem of time collects inequivalent issues. A Hamiltonian constraint gives the frozen formalism; an internal variable may fail to be monotonic, producing a global-time problem; two admissible clocks can lead to inequivalent quantum descriptions, producing the multiple-choice problem; and observables must commute with constraints while still encoding change. Isham’s canonical taxonomy remains a useful guide precisely because it keeps these questions separate Isham 1992, §§5.1 and 6.2–6.4.

Relational observables answer “what is $A$ when $T=\tau$?” rather than “what is $A$ at an external time?” In a deparametrizable model with constraint $C=p_T+H(q,p)$, quantization can yield $i\partial_T\psi=\widehat H\psi$. But this result depends on a clock whose level surfaces intersect the relevant gauge orbits once, a self-adjoint reduced Hamiltonian, and a physical inner product compatible with the reduction. A quadratic clock constraint, turning point, or clock with large quantum fluctuations changes the conclusion. Rovelli’s partial-observable framework emphasizes correlations without presuming that every generally covariant system admits a global preferred clock Rovelli 2002, §§2–4.

The WKB route is different. It does not pick a fundamental clock; it expands the total wave functional around a rapidly varying gravitational phase. A time derivative is then defined along a classical Hamilton–Jacobi trajectory. The resulting matter Schrödinger equation is branch-dependent and approximate. At a WKB turning point, between interfering branches, or when backreaction is not hierarchically small, that recovered time can fail.

Results and open questions as of 2026-08-20

The canonical framework is mathematically informative, but the physical Hilbert space and controlled dynamics of full four-dimensional gravity remain unresolved. The strongest positions are genuinely different rather than stylistic variants.

Formulation or position	Controlled result or evidence	Scope and limitation
Dirac constraint quantization	Gauge-invariant states and group averaging are explicit in finite-dimensional systems and selected field-theory models	A regulator, anomaly-free constraint representation, and positive physical inner product for full gravity are not established
Reduced or deparametrized quantization	A good clock converts some constrained models into ordinary unitary evolution	Global clocks may not exist, and different clock choices can yield inequivalent quantum theories
Semiclassical WKB recovery	A Born–Oppenheimer expansion yields QFT evolution along a classical gravitational branch	It assumes a narrow WKB branch and controlled backreaction; it does not define fundamental time
Relational or timeless interpretation	Complete observables encode gauge-invariant correlations without external time	Construction is often local in phase space, and operational completeness for full gravity remains open

The adverse test common to all four positions is a clock turning point. If $\{T,C\}=0$ on part of a gauge orbit, $T=\tau$ is not a global gauge fixing: the Faddeev–Popov factor vanishes, conditional evolution becomes multivalued, and a reduced Hamiltonian based on that clock cannot cover the orbit. This does not refute relational physics; it prevents a local clock construction from being promoted to a universal solution.

The conclusions could change after an anomaly-free regulated constraint algebra with a controlled continuum limit and physical inner product is independently constructed, or after a theorem establishes equivalence—or a concrete inequivalence—between broad classes of clock choices beyond symmetry-reduced models.

Check your preparation

Consider the parametrized nonrelativistic particle with extended phase space $(q,p;t,p_t)$ and constraint $C=p_t+p^2/(2m)=0$. Explain why its vanishing canonical Hamiltonian does not imply an absence of change, and identify what fails if $t$ is replaced by a clock $T(q,p)$ whose Poisson bracket with $C$ vanishes somewhere.

Solution

The constraint generates simultaneous shifts along the particle trajectory and in the label $t$. Choosing the gauge $t=\tau$ intersects each relevant orbit once because $\{t,C\}=1$; solving $C=0$ gives $p_t=-p^2/(2m)$ and the reduced generator $H=p^2/(2m)$. Thus $dq/dt=p/m$ describes relational change even though the extended Hamiltonian is constrained. For a proposed clock $T$, the gauge condition is locally valid only where $\{T,C\}\ne0$. At a zero of that bracket, neighboring gauge orbits are tangent to a clock surface, the gauge-fixed measure becomes singular, and a single-valued reduced evolution in $T$ cannot be continued without changing clock or patch.

References

• Arnowitt, Richard, Stanley Deser, and Charles Misner. “The Dynamics of General Relativity.” Reprint of the 1962 chapter, 2008. arXiv:gr-qc/0405109.
• DeWitt, Bryce S. “Quantum Theory of Gravity. I. The Canonical Theory.” Physical Review 160 (1967): 1113–1148. DOI.
• Isham, Chris J. “Canonical Quantum Gravity and the Problem of Time.” In Integrable Systems, Quantum Groups, and Quantum Field Theories, 1992. arXiv:gr-qc/9210011.
• Rovelli, Carlo. “Partial Observables.” Physical Review D 65 (2002): 124013. DOI.