Local and Microcausal Functionals with Peierls Brackets

The Peierls bracket of a given pair is well defined when the linearized equation is Green hyperbolic and every distributional contraction satisfies Hörmander’s wavefront-set criterion. The traditional microcausal cone excludes functional-derivative covectors that are all future directed or all past directed. That pointwise condition does not, by itself, make the whole microcausal class closed as smooth functionals; a uniform equicausal refinement supplies the missing control. A formal functional derivative without its topology and wavefront behavior is not an admissible observable.

Required background. Causal factorization and time-ordered products motivate the functional setting, while support and regularity domains supply distributional discipline. Helpful background. Time-slice and relative Cauchy evolution explain dynamical quotients, and states, GNS representations, and folia distinguish this classical Poisson algebra from a representation.

Smooth, local, and microcausal functionals

For a real scalar field take configuration space $\mathcal E(M)=C^\infty(M)$. A Bastiani-smooth functional $F:\mathcal E(M)\to\mathbb C$ has derivatives

$$F^{(n)}[\varphi]\in\mathcal E'(M^n),$$

compactly supported distributions depending smoothly on $\varphi$. Its spacetime support is the closure of points where an arbitrarily small field variation can change $F$. A local functional has the form $F(\varphi)=\int f(x)j_x(\varphi)\,\mathrm d\mu_g(x)$ for a finite-jet density; its higher derivatives are supported on the thin diagonal.

Let $\overline V_\pm$ be the closed future and past causal covector cones. The microcausal condition is

$$\operatorname{WF}(F^{(n)}[\varphi]) \cap\left(\overline V_+^{,n}\cup \overline V_-^{,n}\right)=\varnothing \quad\text{for every }n\geq1.$$

It forbids precisely the all-same-time-orientation configurations that would collide with propagator wavefront sets in contractions. The product theorem says $uv$ exists if no $(x,k)\in\operatorname{WF}(u)$ has $(x,-k)\in\operatorname{WF}(v)$ Hörmander 1990, Theorem 8.2.10, pp. 267–268.

There is an important topology qualification. Microcausality is tested separately at each configuration $\varphi$. Smoothness of a bracket also requires uniform control of the derivative distributions as $\varphi$ ranges over compact subsets of configuration space. An equicausal functional imposes precisely the corresponding equicontinuity condition in the distribution spaces with open wavefront cones. Local functionals satisfy this stronger condition Hawkins, Rejzner, and Visser 2026, §5.1, Theorem 5.8, pp. 27–30 of the open manuscript.

Peierls bracket and its domain

Let $S$ be a nonlinear action and suppose its linearization $P_\varphi=S^{(2)}[\varphi]$ is normally hyperbolic on the configurations considered. Let $\Delta_\varphi^{R,A}$ be its retarded and advanced Green operators and $\Delta_\varphi=\Delta_\varphi^R-\Delta_\varphi^A$. Define

$$\{F,G\}_S(\varphi) =\left\langle F^{(1)}[\varphi], \Delta_\varphi G^{(1)}[\varphi]\right\rangle.$$

Compact support makes the retarded/advanced pairing proper, and the wavefront condition makes the displayed contraction defined. Antisymmetry follows from formal skew-adjointness of $\Delta_\varphi$; the Jacobi identity follows, on a functional class where all differentiated contractions are smooth, by differentiating $P_\varphi\Delta_\varphi=0$ and canceling the cyclic terms. On-shell, functionals in the ideal generated by the equations of motion have zero bracket with equivalence classes, so the bracket descends to the quotient. Brunetti, Fredenhagen, and Ribeiro develop the smooth functional and Green-hyperbolic framework in Brunetti, Fredenhagen, and Ribeiro 2019, §§3.2–4.2.

Closure requires more than the displayed first-derivative pairing. Differentiating the bracket produces terms containing $F^{(r)}$, $G^{(s)}$, derivatives of $\Delta_\varphi$, and compact diagonal contractions. The variation identity

$$\delta\Delta_\varphi =-\Delta_\varphi^R(\delta P_\varphi)\Delta_\varphi -\Delta_\varphi(\delta P_\varphi)\Delta_\varphi^A$$

expresses every new propagator derivative through the same retarded and advanced kernels and a local variation of $P_\varphi$. Wavefront composition controls each candidate contraction, while equicontinuity controls its dependence on $\varphi$. Equicausal functionals are closed under the resulting star product and Peierls bracket Hawkins, Rejzner, and Visser 2026, §7, Propositions 7.1 and 7.3 and Theorem 7.4, pp. 32–39 of the open manuscript. Without this extra condition, even the bracket of two regular functionals can fail to be smooth Hawkins, Rejzner, and Visser 2026, §4, Proposition 4.5 and Theorem 4.6, pp. 21–25.

Quadratic observables on a nonlinear scalar background

Choose $f,g\in C_0^\infty(M)$ and

$$F(\varphi)=\frac12\int f\varphi^2\,\mathrm d\mu_g, \qquad G(\varphi)=\frac12\int g\varphi^2\,\mathrm d\mu_g.$$

Then $F^{(1)}[\varphi]=f\varphi$, $G^{(1)}[\varphi]=g\varphi$, and the first QFT application is

$$\{F,G\}_S(\varphi) =\int_{M^2}f(x)\varphi(x)\Delta_\varphi(x,y) g(y)\varphi(y)\,\mathrm d\mu_g(x)\,\mathrm d\mu_g(y).$$

The smearings and background field are smooth, so their wavefront sets are empty; the only singular factor is $\Delta_\varphi$. Its wavefront set consists of null-related covectors transported by the bicharacteristic flow, with opposite covectors in the two slots. Pairing against compact smooth factors is therefore allowed. This is the rigorous version of the calculation based on curved-spacetime Green functions and causal propagators.

An independent check exchanges $F$ and $G$. Formal skew-adjointness gives $\int g\varphi\Delta(f\varphi)=-\int f\varphi\Delta(g\varphi)$, verifying antisymmetry directly. If the supports are causally disjoint, $\Delta_\varphi$ has no support on their product and the bracket vanishes.

For the quadratic example, differentiating once more gives kernels supported on the diagonal from the explicit factors of $\varphi$, plus the field variation of $\Delta_\varphi$. The former have smooth compact coefficients $f,g$, and locality makes their derivative families equicontinuous on compact configuration sets. The latter has the factorization above. Thus this local example lies in the equicausal class and its bracket is again equicausal, not merely a finite pairing at one background.

Characteristic smearing failure

Adversarial test. Replace $f$ by a delta distribution $\delta_\Sigma$ on a characteristic hypersurface $\Sigma$. Its wavefront set is the conormal bundle $N^*\Sigma\setminus0$, which contains null covectors because $\Sigma$ is characteristic. The propagator carries matching oppositely oriented null covectors. Their sum can be zero at the contraction point, violating Hörmander’s condition. The displayed bracket is then not defined by the stated theorem.

This failure does not prove that every characteristic observable is impossible; one may choose a different class of boundary data or a separately defined pullback theorem. It does prove that the microcausal bulk bracket cannot be extended by merely substituting $f=\delta_\Sigma$.

Exercises

1. Linear observables. For $F_f(\varphi)=\int f\varphi$ and $F_g(\varphi)=\int g\varphi$, compute the bracket.

Solution

The first derivatives are $f$ and $g$, and all higher derivatives vanish. Hence $\{F_f,F_g\}=\langle f,\Delta g\rangle$, independent of $\varphi$ for a linear theory. It vanishes for causally disjoint supports by the support property of $\Delta$.

2. Equation-of-motion ideal. Let $F(\varphi)=\langle P\varphi,h\rangle$ in the linear theory. Show that its bracket with $G$ vanishes on the quotient.

Solution

$F^{(1)}=P^*h=Ph$. Thus $\{F,G\}=\langle Ph,\Delta G^{(1)}\rangle=\langle h,P\Delta G^{(1)}\rangle=0$. Therefore changing a functional by an equation-of-motion term does not change its on-shell class.

References

• Brunetti, Romeo, Klaus Fredenhagen, and Pedro Lauridsen Ribeiro. “Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields.” Communications in Mathematical Physics 368 (2019): 519–584. DOI; Open preprint.
• Hawkins, Eli, Kasia Rejzner, and Berend Visser. “A Novel Class of Functionals for Perturbative Algebraic Quantum Field Theory.” Revised 2026. Open manuscript, arXiv:2312.15203.
• Hörmander, Lars. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis. 2nd ed. Berlin: Springer, 1990. DOI.