Peierls Bracket Equivalence in Field Theory

• The Peierls bracket equivalence is a rigorous correspondence between causal, covariant constructions and canonical Poisson brackets in classical and quantum field theory.
• It provides a unifying framework linking deformation quantization, path-integral methods, and BRST techniques for gauge-invariant and non-Lagrangian systems.
• Its formulation relies on advanced-retarded Green’s functions, global hyperbolicity, and precise regularity conditions to ensure consistency across diverse field models.

The Peierls bracket equivalence refers to the rigorous correspondence between the Peierls bracket construction—a manifestly covariant, causal Poisson structure on the space of solutions of classical field equations—and the canonical Poisson bracket derived from the symplectic structure on the covariant phase space. This equivalence, established under suitable regularity and boundary conditions, provides a unifying framework for classical and quantum dynamics, ties together the deformation quantization, path-integral, and BRST perspectives, and generalizes to systems with gauge, constraints, and even non-Lagrangian dynamics.

1. Formulation of the Peierls Bracket

The Peierls bracket is defined on the space of gauge-invariant observables as follows. Given a field theory with action $S[\phi]$, one considers a deformation $S \to S + \epsilon F[\phi]$ and evaluates the response of another observable $G[\phi]$ by comparing advanced and retarded solutions to the perturbed Euler–Lagrange equations. The bracket is given by

$$\{F,G\}_{\text{Peierls}} = \int_{M\times M} \frac{\delta F}{\delta \phi^i(x)}\, \Delta^{ij}(x,y)\, \frac{\delta G}{\delta \phi^j(y)}\, d\mu(x)\, d\mu(y),$$

where $\Delta^{ij}(x,y)=G_+^{ij}(x,y) - G_-^{ij}(x,y)$ is the causal propagator defined via advanced and retarded Green’s functions of the linearized field operator about a background solution (Asorey et al., 2017, Khavkine, 2014, Berra-Montiel et al., 2014). The Peierls bracket is bilinear, antisymmetric, satisfies the Leibniz rule, and obeys the Jacobi identity whenever the Green’s functions exist and the system is sufficiently hyperbolic and globally hyperbolic.

2. Covariant Phase Space and Symplectic Structure

The covariant phase space $\text{Sol}$ is the space of all classical solutions of the field equations, equipped with a pre-symplectic (or symplectic, after factoring out gauge) two-form $\omega$, constructed from the Lagrangian via the presymplectic current. For a solution $\phi$ and linearized solutions $\eta_1,\eta_2$,

$$\omega_\phi(\eta_1, \eta_2) = \int_\Sigma [\eta_1 \cdot \pi(\eta_2) - \eta_2 \cdot \pi(\eta_1)]\,,$$

with $\Sigma$ a Cauchy surface and $\pi$ the presymplectic current from the second variation (Asorey et al., 2017, Khavkine, 2014). This symplectic form is closed and, under physical boundary/gauge conditions, non-degenerate on reduced phase space.

3. Equivalence Theorem and Proof

The core equivalence theorem, proved for globally hyperbolic and “hyperbolizable” systems with suitable cohomology conditions, asserts that the bivector field $\Lambda$ generated by the causal propagator $\Delta$ is the inverse of the symplectic form $\omega$, i.e., for $\alpha \in T^*_\phi\text{Sol}$,

$$\omega_\phi(G_\phi(\alpha), \eta_2) = \langle \alpha, \eta_2 \rangle,$$

where $G_\phi$ acts as the Green operator (Asorey et al., 2017, Khavkine, 2014). Explicit inversion is achieved by showing that:

• The symplectic pairing maps tangent vectors to dual densities (cotangent vectors).
• The Peierls kernel provides the unique solution inverting this map for compactly supported sources, modulo gauge/finiteness issues.
• On the space of gauge invariant observables, these two-sided inverses coincide, and the explicit Peierls bracket on observables matches the canonical bracket induced from the symplectic form.

This equivalence extends to systems with constraints and gauge symmetries, provided the vertical cohomologies (parametrizable constraints, recognizable gauge symmetries) are non-degenerate. The formalism accommodates gauge-fixing, ghosts, and extended BRST complexes (Khavkine, 2014, Sharapov, 2014).

4. Deformation Quantization Perspective

In deformation quantization, one replaces the classical pointwise product of functionals $F,G$ with the star-product

$$F \star G[\phi] = \exp\left(\frac{i\hbar}{2} \int dx\,dy\, \frac{\overleftarrow{\delta}}{\delta \phi^i(x)}\, \Delta^{ij}(x,y)\, \frac{\overrightarrow{\delta}}{\delta \phi^j(y)}\right) F[\phi] G[\phi],$$

(Berra-Montiel et al., 2014), with the first nontrivial order in $\hbar$ generating the Peierls bracket. The $\star$-commutator and the “classical limit” $\hbar \to 0$ recover the Poisson/Peierls bracket, demonstrating that deformation quantization provides a direct route to the covariant Poisson structure. This equivalence is exact in linear theories and holds to first order in nonlinear systems, reflecting the Peierls prescription.

5. Non-Lagrangian Extensions and Lagrange Structures

The Peierls bracket generalizes to non-Lagrangian systems through the concept of a “Lagrange anchor” $V$, a chain map between symmetries and equations of motion (Sharapov, 2014). For equations $T_a[\phi]=0$, the anchor $V$ ensures on-shell compatibility $J \circ V \simeq V^* \circ J^*$, allowing construction of advanced/retarded Green’s functions for $G=J \circ V$. In the Lagrangian case $V=1$, the anchor construction and resulting bracket reduce to the standard Peierls bracket. More generally, this framework yields covariant Poisson brackets—manifestly causal and gauge invariant—for both path-integral and deformation quantization approaches.

6. Illustrative Models

Several examples validate the Peierls bracket equivalence:

• Scalar field theory: For $(\Box + m^2)\phi=0$, the Peierls bracket coincides with the canonical Poisson bracket for fields and momenta, both with and without smearing (Berra-Montiel et al., 2014, Asorey et al., 2017).
• Bosonic string: The causal Peierls bracket generalizes the Virasoro algebra to two-time relations, collapsing to the canonical algebra in the equal-time limit (Berra-Montiel et al., 2014).
• Null initial data for gravity: A canonical Poisson bracket constructed from the Hilbert action symplectic form on free data of a double null sheet matches exactly the Peierls bracket on all sufficiently regular observables, handling caustics and generator crossings without additional constraints (0712.2541).
• Classical mechanics: For quadratic (and some time-dependent) systems, the equal-time Peierls bracket reduces to the Poisson bracket; full equivalence for general Hamiltonians remains contingent on explicit Green’s function construction (Sharan, 2010).

7. Constraints, Gauge Symmetries, and Technical Conditions

The Peierls bracket equivalence assumes:

• Global hyperbolicity for unique advanced/retarded Green’s operators.
• Field variations with compact support or appropriate boundary behavior.
• Hyperbolizability of equations of motion, ensuring the existence of the required Green’s functions (Khavkine, 2014).
• For gauge theories, proper factoring of gauge orbits and possibly inclusion of BRST–BV structures.

The equivalence extends under these conditions to theories with first-class constraints and gauge invariances, as well as those admitting generalized anchors (e.g., chiral bosons, first-order Maxwell forms) (Sharapov, 2014).

8. Significance and Applications

The Peierls bracket equivalence underpins a widely applicable causal and covariant formalism for both classical and quantum field theory. It provides a robust foundation for quantization (deformation, path integral, BRST), bridges canonical and covariant approaches, and facilitates gauge-invariant constructions for complex systems. Concrete benefits include constraint-free canonical gravity (0712.2541), causal algebraic generalizations for strings and field theories (Berra-Montiel et al., 2014), and fully covariant quantization of non-Lagrangian dynamics (Sharapov, 2014). A plausible implication is enhanced control over fundamental symmetry and causality in quantum gravity and effective field theories.

All claims, technical constructions, and explicit formulas appearing above are directly traceable to (Berra-Montiel et al., 2014, Asorey et al., 2017, Sharapov, 2014, Khavkine, 2014, 0712.2541), and (Sharan, 2010).