Poisson bracket and $L_\infty$ algebras

Abstract: We describe the Poisson bracket of a Lagrangian field theory expressed in the framework of $L_\infty$ algebras. We show that the recently proposed symplectic structure implies that the associated Poisson bracket can be computed through the Peierls formula. We consider Poisson brackets in $p$-adic string theory, where interesting complications arise. In addition we give an elegant interpretation of the inverse relation between the Poisson bracket and symplectic structure in the language of homological algebra, extending some ideas in the mathematical physics literature.

• The paper proves that the $L_\infty$ symplectic form is inverted by the causal propagator, yielding the Peierls bracket and establishing antisymmetry, gauge invariance, derivation, and Jacobi identities for on-shell observables.
• The paper constructs an explicit contracting homotopy for the Peierls complex, showing homologically that the causal propagator and the sigmoid commutator provide inverse maps between localized observables and on-shell symmetries.
• The paper demonstrates that $p$-adic string theory lacks a consistent full Poisson bracket because infinitely many higher-derivative poles obstruct advanced and retarded propagators, although a tachyon-only truncation remains an open possibility.

The paper develops the Poisson bracket within a program that formulates classical field theory using $L_\infty$ algebras, building on prior work by Bernardes, Erler, and Fırat on covariant phase space constructions and conserved charges. The central result is that the symplectic structure previously proposed for this framework is inverted by the causal propagator, so that the Poisson bracket is computed by the Peierls formula. Beyond establishing this equivalence, the authors give an explicit contracting homotopy — the "Peierls complex" — realizing the inverse relation between Poisson bracket and symplectic structure as exactness of a homological algebra construction, and they probe the limits of the formalism in $p$-adic string theory, where higher-derivative instabilities obstruct a consistent definition of advanced and retarded propagators.

Setup: $L_\infty$ data and the covariant phase space

The formalism starts from a graded vector space $H$ equipped with a nondegenerate, graded-antisymmetric BV inner product $\omega$ at grade $-1$. Classical fields $\Phi$ are grade-zero elements; dynamics is encoded in an anticommuting Euler-Lagrange state $q_\Phi$ at grade 1 (with equations of motion $q_\Phi = 0$), whose variation defines the cyclic kinetic operator $Q_\Phi$ satisfying the Noether identity $p$0. Gauge transformations take the form $p$1, while general symmetries obey $p$2. Observables are written via characteristic states $p$3 at grade 1 with $p$4.

The key structural element is the symplectic form

$p$5

where $p$6 is a "sigmoid" operator interpolating from 0 in the infinite past to 1 in the infinite future. Nontriviality of $p$7 relies entirely on temporal boundary contributions to cyclicity of $p$8: since $p$9 on-shell, the commutator with $L_\infty$0 is the only source of a nonzero two-form. Conserved charges associated with action symmetries were constructed in earlier work as $L_\infty$1, with a useful total-derivative form under tau regularization. The present paper asks whether this $L_\infty$2 is the correct symplectic structure by checking whether its inverse reproduces the standard Poisson bracket.

From observables to symmetries: the Peierls formula

The first main claim is that every on-shell observable $L_\infty$3 with characteristic state $L_\infty$4 defines a Hamiltonian vector field

$L_\infty$5

satisfying $L_\infty$6, where $L_\infty$7 is built from retarded and advanced propagators satisfying $L_\infty$8 on time-localized, $L_\infty$9-invariant inputs. The proof proceeds by splitting $H$0 into advanced and retarded pieces; each piece vanishes after opening the $H$1 commutator and invoking cyclicity, provided the propagator-sigmoid combinations $H$2 and $H$3 are sufficiently localized in time that tau regularization can be dropped. This localization requirement is stated explicitly as a condition on the definition of the propagators, and the authors note it can fail — e.g., a point charge on a torus cannot be consistently accommodated by the electric field. The remaining term yields exactly $H$4.

Two consequences follow immediately. First, nondegeneracy of $H$5 holds: contracting $H$6 against arbitrary Peierls vector fields recovers Lie derivatives of arbitrary observables, hence determines $H$7 up to gauge. Second, defining $H$8, one obtains the Peierls formula

$H$9

This agreement with the covariant phase space result of Forger–Romero and Khavkine indirectly validates the $\omega$0 symplectic structure, at least for finite-derivative theories. The bracket is shown to be antisymmetric, bilinear, gauge invariant (via Cartan calculus: $\omega$1 again generates a gauge transformation), a derivation, and Jacobi, so on-shell observables form a Lie algebra. A perturbative formula relating the propagator around a background to the vacuum propagator, $\omega$2, is also given via geometric series.

From symmetries to observables and central charges

Conversely, every symmetry of the symplectic structure — not merely of the action, so including cases like electromagnetic duality — arises locally as a Hamiltonian vector field with conserved charge $\omega$3, where the interpolation must run through solution space. The authors concede this has limited practical utility since finding such paths is difficult; conserved charges are usually obtained from action symmetries.

For two action symmetries $\omega$4 with charges $\omega$5, the Poisson bracket algebra closes as

$\omega$6

The central charge is a Lie algebra 2-cocycle, antisymmetric and Chevalley–Eilenberg closed; it is removable precisely when it is a 2-coboundary, i.e., linear in $\omega$7. Notably, the direct computation never invokes the causal propagator, because known symmetry generators supply the needed vector fields directly.

Examples

For a nonrelativistic particle with kinetic operator $\omega$8, the causal propagator is $\omega$9, giving $-1$0. The canonical bracket $-1$1 is reproduced, with the sigmoid boundary condition $-1$2 supplying the unit and total derivatives dropping out because $-1$3 decays faster than $-1$4. Computing $-1$5 yields $-1$6: the mass appears as a central extension, recovering the Bargmann algebra. This confirms both the Peierls formula and the central-charge formula in a concrete setting.

The $-1$7-adic string example is more consequential. Around the unstable vacuum $-1$8, the kinetic operator $-1$9 produces a momentum-space denominator $\Phi$0 with infinitely many poles on two hyperbola branches extending to unbounded imaginary part, beyond the tachyon at $\Phi$1. No contour parallel to the real axis lies above all poles, so no acceptable advanced or retarded propagator exists in the usual sense. The authors analyze what a partial contour definition would require: for the proof of Claim 1 to go through, the pole nearest the advanced contour must satisfy $\Phi$2, where solutions grow at most as $\Phi$3. Since the poles themselves imply $\Phi$4 is unbounded, no consistent Poisson bracket exists that accounts for all solutions. One may restrict the solution space to discard higher-derivative modes, keeping only the tachyon, but the authors state plainly that consistency of this truncation in the interacting theory is unclear, even though elimination of higher derivatives may be possible via equations of motion or field redefinitions. They note the parallel situation in open string field theory at the tachyon vacuum, where discarding poles collapses phase space to zero dimension — suggestive of Sen's conjecture, but contingent on whether the unphysical poles are removed by cohomology at higher level.

The Peierls complex

The final section reframes the inverse relation homologically. Working on-shell, define subspaces $\Phi$5 (vanishing in the past), $\Phi$6 (vanishing in the future), and their intersection $\Phi$7 (localized). Assuming retarded and advanced propagators exist as full contracting homotopies, $\Phi$8 on their respective domains — an assumption the authors flag as established for local theories but merely assumed in general — the causal propagator satisfies $\Phi$9 and induces mutually inverse maps on cohomology:

$q_\Phi$0

Both compositions equal the identity up to $q_\Phi$1-exact terms, with the sigmoid ensuring all operators act on their proper domains. This upgrades the heuristic that $q_\Phi$2 "inverts" the causal propagator into a precise statement, and extends the symmetry–observable correspondence to all grades $q_\Phi$3, connecting $q_\Phi$4 and $q_\Phi$5 — a cohomological isomorphism the authors note has not received much attention, with potential bearing on rigid symmetries, conserved currents, and anomalies.

To descend from cohomology to actual field configurations, the authors construct the Peierls complex: the mapping cone of $q_\Phi$6 on $q_\Phi$7 (with shifted grading), whose differential combines $q_\Phi$8 diagonally with $q_\Phi$9 off-diagonally. In the absence of gauge symmetry this reduces to the standard exact sequence familiar from algebraic QFT in curved spacetime (Benini–Dappiaggi, Wald, Bär–Ginoux–Pflaumle, Khavkine); the novelty is the treatment of gauge symmetries together with an explicit contracting homotopy

$q_\Phi = 0$0

satisfying $q_\Phi = 0$1. The upper-right block is precisely the operator appearing in the symplectic form, making the connection between the homological inverse and $q_\Phi = 0$2 manifest. The most nontrivial component identity involves the peculiar difference of cubes of advanced and retarded propagators, which the authors verify explicitly. By the standard homological-algebra theorem (Spanier, Thm. 4.2.10), existence of this contracting homotopy is equivalent to the causal propagator having a two-sided inverse up to homotopy. The authors attribute the late recognition of this simple structure to the fact that the symplectic form, usually written as an integral over an initial-value surface, does not naturally present itself as an operator on $q_\Phi = 0$3; the sigmoid softens the time slice and makes the inverse visible.

Limitations and open questions

Several assumptions carry the main results and are acknowledged as such. The existence of advanced/retarded propagators as genuine contracting homotopies on the full retarded/advanced subspaces is proven for local theories with finite propagation speed but assumed otherwise. The proof of Claim 1 requires sufficient time localization of propagator-sigmoid combinations relative to the growth of solutions — a condition that fails in $q_\Phi = 0$4-adic string theory and potentially other nonlocal theories. Whether discarding higher-derivative modes in $q_\Phi = 0$5-adic string theory is consistent in the interacting theory remains unresolved, as does the fate of the analogous unphysical poles in open string field theory at higher levels. Finally, the physical interpretation of the all-grade cohomological isomorphism induced by the Peierls complex is left unexplored.

Conclusion

The paper establishes that the $q_\Phi = 0$6-based symplectic structure $q_\Phi = 0$7 is inverted by the causal propagator through the Peierls formula, thereby validating the framework against the standard covariant phase space formalism and extending Hamiltonian mechanics to settings where canonical methods fail. The explicit contracting homotopy for the Peierls complex provides a compact, gauge-inclusive generalization of exact-sequence arguments from algebraic QFT, with the symplectic-form operator appearing as a component of the homotopy itself. The $q_\Phi = 0$8-adic analysis delineates sharply where the construction breaks down: unbounded higher-derivative instabilities destroy the propagator structure underlying the bracket, leaving truncation-dependent remedies of uncertain consistency.