Quantum consistency of the canonical gravitational field: well-defined energy and boost generators and the anomaly-free closure of Schwinger's 1963 tetrad constraint algebra

When Julian Schwinger in 1963 constructed the canonical quantum theory of the gravitational field in tetrad variables under the time gauge, the generators of spatial translations and rotations emerged as explicit functionals of the canonical pair with correct commutation properties, but the energy density $\theta^0$ was available only implicitly, through an elliptic constraint equation. Consequently the total energy $P^0=\int\theta^0$ and the boost generators $J^{0k}=\int x^k\theta^0$ — the Lorentz sector of the Poincaré algebra — could not be exhibited as explicit operators, and the paper explicitly left undecided whether the quantum formalism is Lorentz invariant, that is, internally consistent. Schwinger's own sequel formally verified the Lorentz commutators using nonlocal 'extended operators' and a factor-ordering choice fixed by consistency, but only formally: it covered integer-spin matter, conceded that the construction 'denies physical significance to the detailed specification of energy distributions', and acknowledged that 'a much more careful examination will be required to test whether the loosely stated physical boundary conditions can be maintained as assertions about operators in relation to a class of physical states'. The rigorous question has never been settled: later work established that the formal constraint commutators are not well defined until regularized, with different formal computations yielding different results. This problem asks for a mathematically well-defined, anomaly-free quantum realization of the canonical gravitational constraint and Poincaré generator algebra — or a proof that none exists.

Background

Canonical quantization of general relativity begins by foliating spacetime into spatial hypersurfaces and rewriting Einstein's theory as a Hamiltonian system on each surface. Because the foliation and the coordinates within each surface are arbitrary, the theory is constrained: certain phase-space functions (the constraints) must vanish on physical configurations, and the total energy-momentum and Lorentz generators of an isolated system are not volume integrals but surface integrals at spatial infinity. In a quantum theory built on canonical commutation relations, the corresponding operators must satisfy the same algebra — the four constraint equations and the ten Poincaré commutation relations — as operator identities on the physical states. If quantization produces extra, non-classical terms in these commutators, the theory has an anomaly and is inconsistent (or must be modified). This consistency question is the oldest open problem of canonical quantum gravity.

Schwinger's two-part paper 'Quantized Gravitational Field' (1963) pursued this program in the tetrad (vierbein) formulation, in which the spacetime metric is written through local orthonormal frames, $g_{\mu\nu}=e_\mu{}^a e_\nu{}^b\eta_{ab}$, with the tetrad (frame) fields $e_\mu{}^a$ and the spin connection $\omega_{\mu ab}$, the gauge field of local Lorentz rotations. Imposing the time gauge, which locks the local time axes to the normal of the spatial hypersurfaces, removes the boost part of the local Lorentz freedom and leaves the symmetric spatial tensor density $q^{kl}=g\,{}^{(3)}g^{kl}$ (with $g=\det{}^{(3)}g_{kl}$ and ${}^{(3)}g^{kl}$ the inverse spatial metric) and its conjugate momentum $\Pi_{kl}$, related to the extrinsic curvature describing how the hypersurface sits in spacetime. The constraints take the form

$$2(\partial_l\Pi_{kl}-\partial_k\Pi_{ll})=\theta_k,\qquad -\partial_k\partial_l q^{kl}=2\kappa\,\theta^0,$$

where $\kappa$ is the gravitational coupling constant and $\theta^\mu$ the total energy-momentum density, and where these hold as operator equations on the physical states in the quantum theory. The momentum constraints $\theta_k$ determine the longitudinal parts of $\Pi_{kl}$, while the second constraint is an elliptic (biharmonic-type) equation that determines the scalar part of $q^{kl}$ in terms of the two physical degrees of freedom — the transverse-traceless part (divergence-free and trace-free components) of the spatial metric — plus matter. The generators of spatial translations $P^k$ and rotations $J^{kl}$ are explicit functionals of the canonical variables, and Schwinger verified their commutators. But $\theta^0$ is given only implicitly: solving the constraint equation requires inverting a fourth-order differential operator, so $P^0=\int d^3x\,\theta^0$ and the boost generators $J^{0k}=\int d^3x\,x^k\theta^0$ are nonlocal functionals of the physical variables, and the Lorentz commutators could not be checked. Part I closed by acknowledging that this 'could also be a formidable barrier to verifying the consistency of the formalism' and deferring the issue to a separate paper.

That sequel, Part II (Phys. Rev. 132, 1317 (1963)), introduced 'extended operators' — nonlocal composite operators built with the Green's functions of $-\nabla^2$ — fixed an otherwise-free factor-ordering ambiguity (the power of $q=\det q^{kl}$ in the Hamiltonian density) by requiring the commutator algebra to have the correct structure, and thereby completed a formal verification of Lorentz invariance. Three explicit limitations remained. First, the matter fields were restricted to integer spin. Second, since the reduction of extended operators to localized ones is nonlocal, the construction 'effectively denies physical significance to the detailed specification of energy distributions by means of the operator $\theta^0(x)$'. Third, and most importantly, the surface-integral boundary conditions defining a physically closed system were imposed as classical assumptions, and Schwinger wrote that 'a much more careful examination will be required to test whether the loosely stated physical boundary conditions can be maintained as assertions about operators in relation to a class of physical states'.

The subsequent literature settled the classical side but not the quantum one. DeWitt's 1967 metric-variable quantization (leading to the Wheeler-DeWitt equation, the functional differential equation obeyed by quantum states) engaged Schwinger's ordering proposal but remained formal. Regge and Teitelboim (1974) clarified the surface-integral structure of the Poincaré generators in the ADM (Arnowitt-Deser-Misner) canonical formulation, where the spatial metric itself is the configuration variable. In the 1980s it was shown that the formal manipulations underlying all such verifications are not well defined: the constraint operators contain singular products of fields at the same point, so no factor-ordering question has a meaning until a regulator is chosen (a cutoff or smearing prescription that renders the singular operator products finite, for instance by separating the field points that collide), and different formal computations of the same commutator yield different results, and the regulator must respect general covariance — a requirement that is itself only formally implementable. Ashtekar's new canonical variables (1987) — a densitized triad (the spatial frame field scaled by the determinant of the spatial metric) and a self-dual spin connection (a connection for local frame rotations that equals its own Hodge dual with complex coefficients, making the constraint equations polynomial) — reformulated the problem without resolving it, and the modern effective approaches and loop quantum gravity (the program built on these variables) treat anomaly-freedom of the constraint algebra as the central open consistency condition of canonical quantum gravity. What has never been produced is a regulated, well-defined operator realization in which the gravitational energy density, the boosts, and the constraint algebra are simultaneously controlled — precisely the rigorous core of Schwinger's 1963 question.

Problem Statement

Decide the quantum consistency of the canonical formulation of general relativity that Schwinger (1963) left undecided: in the time-gauge tetrad canonical formulation whose fundamental variables are the symmetric spatial tensor density $q^{kl}=g\,{}^{(3)}g^{kl}$ ($g=\det{}^{(3)}g_{kl}$) and its conjugate momentum $\Pi_{kl}$, with constraints

$$2(\partial_l\Pi_{kl}-\partial_k\Pi_{ll})=\theta_k,\qquad -\partial_k\partial_l q^{kl}=2\kappa\,\theta^0,$$

for the gravitational field (optionally in interaction with integer-spin matter, as in the source), construct — or prove impossible to construct — a quantum realization in which:

(1) the energy density $\theta^0$, determined classically only implicitly by solving the elliptic constraint equation, is promoted to a well-defined operator, so that the total energy $P^0=\int d^3x\,\theta^0$ and the boost generators $J^{0k}=\int d^3x\,x^k\theta^0$ are well-defined operators on explicitly stated domains of a state space carrying the canonical commutation relations of $(q^{kl},\Pi_{kl})$;

(2) the physical state space is specified as the joint kernel of the four constraint operators, and the asymptotic boundary conditions at spatial infinity that define a physically closed system are maintained as assertions about operators and the class of physical states (the 'loosely stated physical boundary conditions' of the source), not merely as classical assumptions;

(3) the integrated Poincaré commutation relations — including the Lorentz-sector relations $[J^{0k},P^l]=i\hbar\,\delta^{kl}P^0$ and $[J^{0k},J^{0l}]=i\hbar\,J^{kl}$, together with the spatial translation and rotation relations — hold as operator identities on the physical states; and

(4) the constraint algebra closes without anomaly: within the chosen regularization of singular operator products and factor ordering, no non-classical commutator terms arise, or those that arise are shown to be removable consistently across the whole algebra.

A proof that no such anomaly-free quantum realization exists in this canonical framework must cover the admissible class of realizations described in (1)-(4) and be independent of the admissible choices of regulator and ordering. Perturbative checks in a truncated or symmetry-reduced sector, and purely classical verifications of the constraint algebra, do not by themselves resolve the question.

References

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