BRST-Invariant Quantization of Gravity

• BRST-invariant quantization is a method that augments the Einstein–Hilbert action with gauge-fixing and Faddeev–Popov ghost terms to rigorously implement diffeomorphism invariance.
• The approach uses canonical ADM variables and auxiliary fields to resolve traditional constraint problems, enabling a consistent Hamiltonian evolution.
• The framework recovers classical general relativity from quantum expectation values via BRST-invariant coherent states and highlights the role of spontaneously-broken supersymmetry in vacuum stability.

A BRST-invariant quantization of general relativity rigorously constructs a quantum theory of gravity in which diffeomorphism gauge symmetry is implemented at the quantum level through nilpotent BRST symmetry. This procedure facilitates a consistent Hamiltonian formulation, resolves the canonical constraint problem via the introduction of auxiliary and ghost fields, and provides a framework for extracting the classical physics of Einstein gravity from quantum expectation values. In the most recent developments, classical backgrounds—including cosmological spacetimes—are not fundamental but arise as BRST-invariant coherent states over a unique, Poincaré-invariant Minkowski vacuum, with deep implications for both quantum gravity and its consistency requirements, such as spontaneous supersymmetry breaking (Berezhiani et al., 27 Sep 2024).

1. Formulation of the BRST-Invariant Canonical Quantum Gravity

In the canonical approach, the Lagrangian density of general relativity is made BRST-invariant by supplementing the Einstein–Hilbert term, $\mathcal{L}_\mathrm{EH} \propto \sqrt{-g} R(g)$, with both a gauge-fixing term and a Faddeev–Popov ghost term: $$\mathcal{L} = \mathcal{L}_\mathrm{EH} + \mathcal{L}_\mathrm{GF} + \mathcal{L}_\mathrm{FP}$$ The gauge-fixing sector is introduced, for example, in de Donder gauge as

$$\mathcal{L}_\mathrm{GF} = b_\nu\, \partial_\mu(\sqrt{-g} g^{\mu\nu}) - \frac{\alpha}{2} \eta^{\mu\nu} b_\mu b_\nu$$

where $b_\mu$ are Nakanishi–Lautrup auxiliary fields implementing the gauge condition, and ghost fields $c^\mu$ and antighosts $\bar{c}_\mu$ are included in $\mathcal{L}_\mathrm{FP}$. For canonical quantization, a formulation in ADM variables is chosen with the spatial metric $\gamma_{ij}$ and $A^\mu \equiv \sqrt{-g} g^{0\mu}$ as configuration variables. Canonical momenta are defined consistently with the BRST extension; e.g.,

$$\Pi^{ij} = -\frac{1}{2\kappa} \sqrt{\gamma}\,(K^{ij} - \gamma^{ij} K) \qquad,\qquad \Pi_\nu = b_\nu - i (\partial_\mu \bar{c}_\nu) c^\mu$$

The quantum theory is constructed by imposing equal-time (anti)commutation relations on fields and their conjugate momenta, including the ghost sector. The total Hamiltonian,

$$\widehat{H} = \widehat{H}_\mathrm{EH} + \widehat{H}_\mathrm{FP+GF}$$

is thus well defined on the extended BRST-invariant phase space.

2. Auxiliary Fields and Resolution of the Constraint Problem

The canonical Einstein–Hilbert action manifests local constraints (notably, vanishing canonical momenta for temporal components of the metric) which obstruct quantization. In BRST-invariant quantization, auxiliary fields $b_\mu$ render all metric components dynamical by promoting gauge constraints to equations of motion. For instance,

$$\Pi_\nu = b_\nu - i (\partial_\mu \bar{c}_\nu) c^\mu$$

ensures $A^\mu$ has well-defined momenta, and the resulting Hamiltonian contains explicit terms involving $b_\nu$. The auxiliary, ghost, and antighost fields thereby enable a Hamiltonian evolution that preserves the BRST constraint, with all gauge degrees of freedom dynamically and consistently implemented.

3. Recovery of Classical General Relativity

Physical states $|\text{phys}\rangle$ in the quantum theory are defined by the nilpotent BRST charge via $\widehat{Q} |\text{phys}\rangle = 0$. The quantum-corrected Einstein equations emerge as expectation values in such states: $$\langle \text{phys} | \widehat{EOM}(h, \Pi, A^\mu, \dots) | \text{phys}\rangle = EOM_\text{classical} + \mathcal{O}(\hbar)$$ For the Hamiltonian constraint,

$$\mathcal{H} \equiv \frac{1}{2\kappa}\Bigl[(\gamma_{ik}\gamma_{jl} + \gamma_{il}\gamma_{jk} - \gamma_{ij}\gamma_{kl}) \Pi^{ij}\Pi^{kl} - (\text{3-curvature})\Bigr] = 0$$

one finds, as $\hbar \to 0$,

$$\lim_{\hbar \to 0} \langle \text{phys} | \widehat{H} | \text{phys} \rangle = 0$$

showing that the vanishing Hamiltonian and constraints are realized as expectation values, not imposed strongly on all states. The nonlinear classical Einstein equations are likewise recovered for one-point functions of fields, with quantum corrections systematically accessible.

4. Physicality, Dynamics, and the Resolution of the ‘Frozen Time’ Problem

Unlike in conventional Wheeler–DeWitt quantization (where the Hamiltonian annihilates physical states, seemingly freezing evolution), in the BRST formalism physicality is distributed between bras and kets through the BRST charge. The Hamiltonian generates genuine nontrivial time evolution for correlation functions,

$$\partial_t \widehat{A}^\mu = i [\widehat{H}, \widehat{A}^\mu]$$

and expectation values over BRST-invariant states yield the classical gauge-fixing equations as operator relations. Thus, while the expectation value of the bulk Hamiltonian vanishes in the classical limit, the theory describes time-dependent quantum gravity as required for causality and dynamics.

5. BRST-Invariant Coherent States and Emergence of Classical Spacetimes

The quantum theory is quantized around the Minkowski vacuum. Classical backgrounds, including all cosmologically relevant spacetimes, are realized as BRST-invariant coherent states: $$|g_{\text{cl}}\rangle \text{ such that } \langle g_{\text{cl}} | \widehat{h}_{ij} | g_{\text{cl}}\rangle = g^{\text{(cl)}}_{ij} - \delta_{ij}$$ This generalizes the standard approach of quantizing about a fixed metric; instead, the unique quantum theory supports an entire orbit of classical metrics via appropriate coherent excitations. The construction

$$|f\rangle = \exp\left[ -i \int d^3 x\, f^\nu_c(x) \widehat{b}_\nu(x) \right] |\Omega\rangle$$

demonstrates how “pure-gauge” coherent states can be constructed, ensuring gauge-equivalent slicings correspond to physically indistinguishable quantum states. In applications to cosmological backgrounds, this unifies the treatment of quantum fluctuations and classical gravitational fields.

6. Poincaré Invariance and the Necessity of Spontaneously-Broken Supersymmetry

The entire construction depends on the selection of a unique, exact Poincaré-invariant Minkowski vacuum, which serves as the foundation for the canonical quantization and S-matrix definition. Nonperturbative effects, notably from gravitational instantons such as Eguchi–Hanson, threaten to break this invariance by mediating transitions to de Sitter-like vacua. A key result is that spontaneous breaking of supersymmetry is required: zero modes deposited by gravitino fields suppress these instanton processes,

$$\text{Fermionic zero modes from the spin-}\tfrac{3}{2}\text{ gravitino restore Poincaré invariance}$$

and stabilize the Minkowski vacuum. This provides a dynamical explanation for the necessity of (spontaneously-broken) local supersymmetry in any consistent low-energy limit of quantum gravity, fully compatible with the BRST-invariant quantization (Berezhiani et al., 27 Sep 2024).

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Table 1: Key Features of BRST-Invariant Canonical Quantization

Feature	Description	Role in Quantization
BRST symmetry	Nilpotent fermionic symmetry encoding gauge invariance	Selects physical states
Auxiliary fields ($b_\mu$)	Promote Lagrangian constraints to dynamical variables	Ensures Hamiltonian flow
Coherent states	Realize classical backgrounds as quantum states over Minkowski vacuum	Link to semiclassical GR
Poincaré invariance	Maintained by construction; underlying vacuum is exactly Poincaré-invariant	S-matrix, consistency
Spontaneously-broken supersymmetry	Removes instanton-induced Poincaré breaking, ensures vacuum stability	Theoretical requirement

7. Implications and Novel Insights

• The BRST-invariant canonical quantization resolves the historical obstacle of non-dynamical constraints and clarifies that “physicality” does not preclude real-time evolution in quantum gravity.
• The treatment of cosmological and black hole backgrounds as coherent states inherently incorporates quantum back-reaction, enabling analysis of quantum break-times and nonperturbative effects absent in semiclassical approaches.
• The necessity of supersymmetry, in any consistent quantum gravity, arises from the demand for unbroken Poincaré invariance of the vacuum—a consequence enforced by BRST symmetry and quantum consistency rather than classical arguments.

This approach provides a compelling, technically rigorous architecture for quantum gravity, consistent with the emergence of classical general relativity, the preservation of gauge and spacetime symmetries, and the fundamental role of BRST invariance in organizing quantum gravitational dynamics (Berezhiani et al., 27 Sep 2024).