Path 10: Canonical Quantum Gravity & Quantum Cosmology

Apply standard quantization to general relativity’s canonical form (the Hamiltonian constraints), and study simplified universes quantum mechanically.

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Rationale: This “traditional” path goes back to the 1960s and earlier, with work by Bryce DeWitt, John Wheeler, and others. One starts with general relativity expressed in Hamiltonian form (the ADM formalism, which describes the dynamics of 3D spatial geometries evolving in time). Then one applies the quantization rules: momenta become operators, Poisson brackets become commutators, and the classical constraints (like the Hamiltonian constraint that classically equals zero for a closed universe) become conditions on a “wave functional” of the universe. The result is the famous Wheeler–DeWitt equation: $\hat{H}\Psi = 0$, an equation that in principle determines the quantum state of the entire universe. $\Psi$ here is a functional on superspace (the space of all 3-geometry configurations). Solutions of this equation, with appropriate boundary conditions (such as Hartle and Hawking’s no-boundary proposal, where $\Psi$ is summed over smooth compact Euclidean geometries) yield quantum amplitudes for different universes. Canonical quantum gravity doesn’t introduce new ingredients beyond GR and quantum mechanics – it’s a direct attempt to “guess the right wavefunction” obeying the right constraints. It has seen partial success in quantum cosmology: by symmetry-reducing the system (e.g. assume the universe is homogeneous – a few parameters describe the geometry), the Wheeler–DeWitt equation becomes a differential equation that can be studied. In simple cosmological models, one finds that at zero scale factor, the “wavefunction of the universe” can be finite, hinting that the big bang singularity might be resolved by quantum effects (the wavefunction can ‘bounce’). This approach also highlights deep conceptual issues like the problem of time: since the WDW equation has no explicit time parameter (it’s $H\Psi=0$), it’s unclear how to recover the flowing time experience of the universe from this timeless quantum state. Canonical quantization is the framework from which LQG (Path 2) emerged by choosing different variables (Ashtekar’s new variables) – but here we consider the broad canonical approach including older metric-variable quantization and quantum cosmology minisuperspace models.

Prerequisites: Hamiltonian mechanics and Dirac’s theory of constrained systems (because GR has gauge freedoms leading to constraints), differential geometry (to understand the superspace of 3-metrics and how constraints like diffeomorphism invariance work), and functional analysis (the wavefunctional lives on an infinite-dimensional space). A good grasp of quantum mechanics conceptual foundations is crucial due to the “time” issue (familiarity with different interpretations or the role of the observer, etc.). In quantum cosmology, one also needs ordinary differential equation skills and approximation methods to solve the minisuperspace WDW equations, plus some knowledge of cosmology (FRW models, inflation potential, etc.) to plug those into the quantum equation.

Dependencies: Canonical quantum gravity is essentially the root of Path 2 (LQG) – LQG is a canonical quantization that succeeded in a different set of variables where earlier attempts in metric variables had hit technical roadblocks (functional differential equations too ill-defined). So Path 10 overlaps with Path 2 at the foundation, but Path 2 goes much further with a specific strategy (discretizing via loops). Canonical QG also connects to Path 4 (CDT) conceptually: both start from the full GR Lagrangian/Hamiltonian; one then does Hamiltonian quantization (canonical) and the other does Lagrangian path integral (CDT). For consistency, if we could solve canonical QG exactly, its solutions should equal those obtained by summing over geometries in path integrals – a check we may one day do. Quantum cosmology, a part of this path, has points of contact with Path 7 (Emergent): some interpretations of the WDW equation use ideas like decoherence (from environment-induced emergence of classical behavior) to explain how classical spacetime arises from a timeless wavefunction – so notions of emergent time or thermodynamic time arrows creep in. Historically, Path 10 doesn’t unify other forces (it’s gravity-only); combining it with Path 1 or 9 (embedding QFT of matter into this canonical framework) is needed for a full Theory of Everything. The Wheeler–DeWitt approach also faces the issue that it’s hard to make it rigorously well-defined – one might need input from other approaches (like LQG’s polymer quantization or path integral methods) to properly define the operator $\hat{H}$.

Signs of Progress: For canonical QG in metric form, a sign of progress would be finding well-behaved solutions to the full WDW equation (perhaps via numerical methods or approximations) or showing that this quantization yields finite answers where expected (like computing quantum corrections to Newton’s law). Right now, solving the exact WDW for full GR is intractable – but progress has been made in understanding certain midisuperspace models (like one-dimensional inhomogeneities) quantized in this way. The resolution of the problem of time is a key sign: various ideas (the Wheeler–DeWitt equation might only describe stationary states and time emerges relationally by considering subsystems, or maybe adding fundamentally time-nonstationary terms is needed) have been debated. A convincing resolution (e.g. demonstrating in a toy model that yes, observers inside a WDW universe would experience time flowing thanks to semi-classical approximation) would be a breakthrough. Quantum cosmology, a subfield, has seen concrete progress: the no-boundary wavefunction proposed by Hartle–Hawking gave an elegant idea of the birth of inflationary perturbations, and loop quantum cosmology (though from Path 2) has shown robust singularity resolution. If such predictions could be connected to possible observations (say, a specific pattern in the cosmic microwave background resulting from a “bounce” rather than a bang), that would be huge – it hasn’t happened yet, but it’s being explored. In summary, Path 10’s progress is gauged by conceptual clarity (solving the time puzzle), mathematical rigor (does a well-defined Hilbert space and Hamiltonian operator exist for gravity?), and cosmological insights (does this approach give viable initial conditions for the universe or resolve the classical infinities?). Each partial success (like demonstrating how classical spacetime peaks in a WDW solution, or deriving the inflationary spectrum from quantum cosmology assumptions) keeps this approach relevant as a foundational piece of the quantum gravity puzzle.

What to Upload Next

This path is explored primarily through the other paths and dedicated research papers. The detailed reading lists and stepping stones from Path 1 (Superstring/M-Theory), Path 2 (LQG), and Path 11 (Experiment) provide complementary resources for deeper exploration.