[Paper Review] Peierls Brackets in Theoretical Physics

Giampiero Esposito, G. Marmo+1 more|ArXiv.org|Sep 18, 2002

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper introduces Peierls brackets as a covariant Poisson bracket formalism in quantum field theory and general relativity, constructed via retarded and advanced Green's functions of the field equations' linearized operator. It demonstrates that Peierls brackets are invariant under the full diffeomorphism group and recover standard canonical commutation relations in point mechanics and field theories, providing a manifestly covariant alternative to the canonical Hamiltonian formalism.

ABSTRACT

Peierls brackets are part of the space-time approach to quantum field theory, and provide a Poisson bracket which, being defined for pairs of observables which are group invariant, is group invariant by construction. It is therefore well suited for combining the use of Poisson brackets and the full diffeomorphism group in general relativity. The present paper provides an introduction to the topic, with applications to field theory and point Lagrangians.

Motivation & Objective

• To provide a pedagogical introduction to Peierls brackets as a covariant alternative to the canonical Hamiltonian formalism in field theory.
• To demonstrate how Peierls brackets preserve full diffeomorphism invariance in general relativity, overcoming the limitations of the standard Hamiltonian approach.
• To establish the equivalence of Peierls brackets with canonical Poisson brackets in both point mechanics and field theories.
• To clarify the mathematical structure of Peierls brackets using Green's functions and gauge-invariant perturbations.
• To unify the symplectic and covariant descriptions of dynamics in classical and quantum field theories.

Proposed method

• Construct the operator $ F = S_2 + ho R ilde{ ho}^{-1} R^t ho $, where $ S_2 $ is the second functional derivative of the action, and $ R $ generates gauge transformations.
• Define the advanced and retarded Green's functions $ G^lat $ as left and right inverses of $ -F $, satisfying $ G^lat F = - ext{1 lap{ aise0.15ex extdegree}{ extnormal{I}}}} $ and $ F G^lat = - ext{1 lap{ aise0.15ex extdegree}{ extnormal{I}}}} $.
• Introduce the antisymmetric combination $ ilde{G} = G^+ - G^- $, which defines the Peierls bracket via $ (A,B) = A_{,i} ilde{G}^{ij} B_{,j} $, with $ A_{,i} = rac{ ho A}{ ho ho^i} $.
• Use reciprocity relations $ ilde{G}^{ij} = - ilde{G}^{ji} $ and symmetry properties of $ F $ to ensure the bracket is antisymmetric and satisfies Jacobi identity.
• Apply the formalism to point mechanics by computing $ (q^i, ilde{q}^j) $, $ ( ilde{q}^i, ilde{q}^j) $, and $ (p_i, p_j) $, recovering canonical commutation relations.
• Establish correspondence between classical mechanics and field theory by mapping symplectic forms, Poisson brackets, and functionals to their Peierls counterparts.

Experimental results

Research questions

• RQ1 How can a Poisson bracket formalism be constructed that is manifestly invariant under the full diffeomorphism group in general relativity?
• RQ2 What is the relationship between Peierls brackets and the standard canonical Poisson bracket in point mechanics and field theories?
• RQ3 How do retarded and advanced Green's functions of the linearized field operator yield a consistent, antisymmetric bracket structure?
• RQ4 Can the Peierls bracket formalism recover the canonical commutation relations in both classical and quantum field theories?
• RQ5 What is the precise correspondence between the symplectic structure in finite-dimensional mechanics and the Peierls bracket in infinite-dimensional field configuration space?

Key findings

• Peierls brackets are constructed from the antisymmetrized difference of retarded and advanced Green's functions, $ ilde{G} = G^+ - G^- $, ensuring antisymmetry and covariance.
• The Peierls bracket $ (A,B) = A_{,i} ilde{G}^{ij} B_{,j} $ reproduces the canonical commutation relations $ (q^i, p_j) = ho^i_j $, $ (p_i, p_j) = 0 $, and $ (q^i, ilde{q}^j) = -C^{-1ij} $ in point mechanics.
• The bracket $ (p_i, p_j) $ vanishes identically, confirming consistency with the canonical structure in the absence of higher-order time derivatives.
• The formalism preserves full diffeomorphism invariance because the Green's functions are defined on spacetime without splitting into space and time, avoiding the $ ho imes ext{R} $ topology issue.
• The reciprocity relations $ ilde{G}^{ij} = - ilde{G}^{ji} $ and $ G^{lat ij} = G^{lat ji} $ ensure that the retarded effect of $ A $ on $ B $ equals the advanced effect of $ B $ on $ A $, reflecting causality.
• The Peierls bracket is invariant under the full invariance group of the theory, including gauge symmetries, due to the construction of $ F $ and $ ilde{G} $ from gauge-covariant operators.