Classical Observables and Poisson Factorization

Classical observables form a Poisson-compatible factorization structure when their supports, equations-of-motion resolution, and bracket kernel are all controlled. In Euclidean BV theory the local bracket is naturally a degree-$+1$ $P_0$ bracket. In Lorentzian theory the Peierls bracket is degree zero on physical observables but is generally multilocal rather than local. These are related constructions, not interchangeable formulas.

Required background. Prefactorization and factorization algebras supplies disjoint multiplication, while microcausal functionals and Peierls brackets supplies the Lorentzian domain on which functional derivatives can be contracted safely.

Helpful background. Elliptic complexes and factorization observables gives the Euclidean BV model, and wavefront-set products gives the distributional criterion behind the Lorentzian bracket.

Functions on the derived solution space

For a field complex $\mathcal E(U)$ with differential $Q$, polynomial off-shell observables are functions on $\mathcal E(U)$. Equations of motion and gauge symmetry should not be imposed by an ordinary quotient that loses stabilizers and relations; the BV differential resolves the derived critical locus. With a degree-$-1$ local pairing, compactly supported linear observables generate

$$\operatorname{Obs}^{\mathrm{cl}}(U) =\widehat{\operatorname{Sym}}\bigl(\mathcal E_c(U)[1]\bigr),$$

equipped with $d_{\mathrm{cl}}$ and an odd Poisson bracket $\{-,-\}_{\mathrm{BV}}$ of degree $+1$. It satisfies graded antisymmetry, the Jacobi identity, and the biderivation rule

$$\{F,GH\}_{\mathrm{BV}} =\{F,G\}_{\mathrm{BV}}H +(-1)^{(|F|+1)|G|}G\{F,H\}_{\mathrm{BV}}.$$

Locality of the pairing implies that the bracket vanishes for disjoint supports. It therefore fits the factorization maps rather than competing with them. The construction and the homotopy-$P_0$ qualification are developed in Costello and Gwilliam 2021, Chs. 4–5.

The Lorentzian Peierls bracket

Let $S$ be a hyperbolic action and $S^{(2)}(\phi)$ its linearized Euler–Lagrange operator at a background $\phi$. When retarded and advanced Green operators exist, their difference $\Delta_S=\Delta_S^{\mathrm R}-\Delta_S^{\mathrm A}$ defines

$$\{F,G\}_{S}(\phi) =\left\langle F^{(1)}(\phi), \Delta_S(\phi)G^{(1)}(\phi)\right\rangle.$$

Compact spacetime support makes the pairing meaningful at infinity; wavefront restrictions on the functional derivatives make it meaningful at coincident singular directions. Antisymmetry follows from formal skew-adjointness of $\Delta_S$, and the Jacobi identity uses the variation of the Green operators together with the equations of motion. If the supports of $F$ and $G$ are spacelike separated, causal support forces the bracket to vanish. Timelike-separated disjoint supports can have a nonzero bracket, so “disjoint” in a factorization product must not be silently replaced by “Poisson commuting.”

The Peierls bracket obeys the ordinary Leibniz rule and descends to the equation-of-motion quotient. For gauge theories it is defined on gauge-invariant cohomology only after a gauge-fixed Green-hyperbolic resolution and proof of gauge-fixing independence. The free-field comparison, including how the bracket is recovered from time-ordered factorization data, is proved in Gwilliam and Rejzner 2020, §§3–4.

Interacting scalar example

On Minkowski space with the inherited (+---) metric, take

$$S[\phi]=\int \mathrm d^dx\, \left(\frac12\partial_\mu\phi\,\partial^\mu\phi -\frac12m^2\phi^2-\frac{\lambda}{4!}\phi^4\right),$$

with the interaction multiplied by a compactly supported switching function when global infrared behavior is not controlled. For local functionals $F_f=\int f(x)P(j\phi(x))\,\mathrm d^dx$, the first functional derivative is supported in $\operatorname{supp}f$. The causal propagator of $S^{(2)}(\phi)=-(\Box+m^2+\lambda\phi^2/2)$ gives the bracket above. Differentiating a product directly verifies the Leibniz rule; placing $f$ and $g$ in spacelike-separated regions verifies causal commutativity.

The physical meaning and limitations of specifying an interaction are developed at What an Interacting Lagrangian Does and Does Not Specify. The construction here is classical and local in field space. It neither supplies a quantum state nor proves that the nonlinear equation has global solutions for every background.

Failure test: local functionals are not bracket-closed

A common overstatement is that local functionals alone form a Poisson algebra. Even if $F$ and $G$ are local, $\Delta_S(x,y)$ propagates between two points, so $\{F,G\}_S$ is generally bilocal. The correct Lorentzian domain is a multilocal or suitably microcausal class closed under the bracket. Recent work also shows that pointwise wavefront restrictions alone need not ensure smooth dependence on the background; an equicausal refinement restores closure Brouder, Dang, and Hélein 2024, Theorem 6.1. Dropping that functional-analytic hypothesis invalidates the bracket as a smooth operation even if every fixed-background contraction looks legal.

Exercises

Show that spacelike-separated supports give a vanishing Peierls bracket.

Solution

$\Delta_S G^{(1)}$ is supported in the causal hull of $\operatorname{supp}G^{(1)}$. If $\operatorname{supp}F^{(1)}$ is spacelike separated from that hull, the supports in the pairing are disjoint, so the pairing and hence the bracket vanish.

Why does the BV bracket have a different degree from the physical Peierls bracket?

Solution

The BV symplectic form has cohomological degree $-1$, so its inverse induces a bracket of degree $+1$ on the shifted resolution. The degree-zero physical bracket appears after passing to the appropriately shifted, gauge-invariant on-shell observables. Erasing the shifts conflates resolution data with physical ghost number.

References

• Brouder, Christian, Nguyen Viet Dang, and Frédéric Hélein. “A Novel Class of Functionals for Perturbative Algebraic Quantum Field Theory.” 2024. arXiv:2312.15203.
• Costello, Kevin, and Owen Gwilliam. Factorization Algebras in Quantum Field Theory, Volume 2. Cambridge University Press, 2021. doi:10.1017/9781316678664.
• Gwilliam, Owen, and Katarzyna Rejzner. “Relating Nets and Factorization Algebras of Observables: Free Field Theories.” Communications in Mathematical Physics 373 (2020): 107–174. doi:10.1007/s00220-019-03652-9.