An algebra of proper observables at null infinity: Dirac brackets, Memory and Goldstone probes

Abstract: We develop a rigorous evaluation of Dirac brackets for classical observables on the phase space of radiative gravitational modes at null infinity that naturally incorporates memory effects. Considering the Ashtekar-Streubel phase space, with boundary conditions in time given by vanishing {\it news} and purely electric {\it shear}, and taking into account the infinite dimensionality of the phase space, we identify the algebra of proper observables (understood as functions on phase space that can be associated with smooth symplectic flows). We show that the action of supertranslation charges generate the correct transformations on the shear. We also show that the conventional definition of the ``Goldstone mode'' adopted in the literature cannot be associated with a proper observable, but nevertheless there exists an infinite family of proper observables, which we call {\it Goldstone probes}, that are capable of measuring the Goldstone mode. We notice that there are no Goldstone probes constructed only out of the shear {\it or} the news, providing a possible explanation for why attempts to construct a (separable) Hilbert space with different memory states have failed so far. Finally, we derive formulas for distributional Dirac brackets between local shear and news, and show that they contain non-local corrections.

• The paper establishes a rigorous framework for the algebra of proper observables using Dirac brackets on the radiative phase space at null infinity.
• It extends Dirac’s method to infinite-dimensional systems, ensuring precise treatment of boundary conditions and the Ashtekar-Streubel phase space.
• The study shows gravitational memory as a non-central observable while the Goldstone mode fails to yield a regular symplectic flow, leading to the concept of Goldstone probes.

Algebra of Proper Observables at Null Infinity: Dirac Brackets, Memory, and Goldstone Probes

Overview

This work presents a mathematically rigorous analysis of the algebra of proper observables on the radiative phase space of gravity at future null infinity ($\mathscr{I}^+$), taking full account of gravitational memory and the infrared (IR) structure intrinsic to asymptotically flat spacetimes. The focus is on the Ashtekar-Streubel phase space with boundary data allowing for electric-type shear memory and implementing constraints via a technically precise extension of Dirac’s method to infinite-dimensional systems. A key theme is the distinction between observables that arise from regular (functional) symplectic flows—dubbed proper observables—and those that do not, particularly the so-called Goldstone mode. The analysis leads to a precise determination of the observable algebra, clarifies the status of gravitational memory as a non-central element, and provides detailed formulas for Dirac brackets, including crucial non-local corrections missed by previous treatments.

Phase Space Structure and Boundary Conditions

The radiative phase space at null infinity is defined in terms of the Bondi-Sachs framework with a focus on gravitational radiation (news tensor $N_{ab}$) and shear ($\sigma_{ab}$). The phase space is infinite-dimensional, with non-trivial boundary conditions:

• The news vanishes at early and late times: $\lim_{u\rightarrow \pm \infty} N_{ab} = 0$.
• The shear asymptotes to purely electric configurations: $\lim_{u\rightarrow \pm \infty}\sigma_{ab}$ electric.

The Ashtekar-Streubel symplectic form is adopted, which—upon constraint reduction via

$K_{ab}\equiv N_{ab}-2\mathcal{L}_n\sigma_{ab}=0$

—yields the reduced phase space used for gravitational radiative degrees of freedom.

Dirac Brackets and Infinite-Dimensional Constraint Analysis

Classical observables must generate (possibly nonlocal) symplectic flows that respect both the boundary conditions and the infinite-dimensional constraint structure. The authors extend the Dirac bracket construction by formulating a functional analytic framework that:

• Distinguishes regular (proper) observables as those associated with smooth Hamiltonian vector fields;
• Characterizes constraint smearings compatible with the symplectic structure and boundary data;
• Identifies the maximal set of regular constraint functionals and the subset of functions ("shear-news functionals") that admit tangential extensions, i.e., modified by constraints so their flows are tangent to the physical subspace.

The Dirac bracket is defined only on these proper observables, using tangential extension methods in analogy with finite-dimensional second-class constraints. The resulting Dirac bracket algebra is proven to coincide with that derived from direct reduction on the constrained phase space.

Algebra of Observables: Memory, Goldstone Mode, and Probes

The main results concerning the observable algebra are:

• Memory as a Non-central Observable: Gravitational memory, defined as the difference in asymptotic shear, is a well-defined non-central proper observable; its action reflects the nontrivial IR sector of gravity.
• Absence of Proper Goldstone Mode: The Goldstone mode, often imagined as conjugate to memory in the literature, does not correspond to any function admitting a regular symplectic flow. As a result, brackets involving the Goldstone mode are formally ill-defined in the proper observable algebra.
• Goldstone Probes: There exists an infinite set of Goldstone probes—proper observables built from combinations of shear and news that do not commute with the memory. These functionals can effectively "measure" the Goldstone sector, even though the Goldstone mode itself is not an observable.
• No Pure Shear or News Probes: Intriguingly, there do not exist Goldstone probes constructed solely from smearings of shear or news; probes require mixed functionals, providing a structural explanation for the failure of certain canonical quantizations to distinguish memory sectors.

The authors also demonstrate that the subalgebra generated by "shear-news" observables (combined linear functionals of $\sigma$ and $N$) forms a complete set separating points in the constrained phase space.

BMS Symmetry and Supertranslation Charges

The paper analyzes the action of the BMS group, particularly supertranslations, within this algebraic framework. It is shown:

• The Dirac bracket correctly generates supertranslation-induced transformations on the shear, resolving longstanding issues in the literature, such as the "factor-of-2" mismatch in earlier treatments.
• The soft and hard contributions to supertranslation charges are handled without reference to ill-defined local fields but via proper observable directional brackets.

Distributional Structure and Nonlocal Corrections

A significant technical development is the introduction of distributional Dirac brackets, which provide distributional objects encoding the algebraic relations between local fields. The explicit construction reveals:

• Standard expressions for, e.g., $\{N(u),N(u')\}$, miss essential non-local boundary terms required by the imposed memory-inclusive boundary conditions.
• These non-local corrections are vital for the correct IR structure and for symmetry transformations, e.g., supertranslation actions.

Implications and Prospects

Theoretical Implications

• Non-Separability of Memory Sectors: The failure to realize Goldstone modes as proper observables offers a conceptual resolution for prior observations that distinct memory sectors cannot be housed within a single separable Hilbert space when quantizing only the shear or news algebra.
• Goldstone Probes and IR Completion: The necessity of mixed functionals for accessing the Goldstone sector suggests new routes for IR-complete quantization schemes, circumventing the pitfalls of naive Fock space or soft mode extensions.
• Algebraic Quantization: Given that only the proper observable algebra is suitable for canonical quantization, the work identifies the minimal complete subalgebra suitable for defining a quantum theory—including both "hard" radiation and memory.

Practical Perspectives

The results provide a rigorous blueprint for quantizing gravity’s radiative sector including memory and for constructing gravitational S-matrix elements sensitive to IR physics. The precise observable algebra supports developing new representations of BMS symmetry and may guide future work on black hole information and holography at null infinity.

Future Directions

• Extension to enlarged symmetry algebras (e.g., extended BMS) or weaker boundary conditions;
• Investigation of Kähler structures for Hilbert space construction on the observable algebra;
• Analysis of the algebra’s implications for Hawking radiation, quantum memory, and gravitational dressing in scattering;
• Applying the method to "partial" radiative phase spaces between finite cross-sections.

Conclusion

This work supplies a definitive characterization of the proper observable algebra for radiative gravity at null infinity, including the rigorously derived Dirac bracket structure accommodating memory and its nonlocalities. The Goldstone mode cannot be promoted to a proper observable; however, an infinite family of Goldstone probes exists, clarifying previous puzzles regarding the IR sector's quantum representation. The results have direct implications for canonical quantum gravity, the existence and structure of a gravitational S-matrix, and the theoretical understanding of asymptotic symmetry dynamics in general relativity.