nLab Peierls bracket

Idea

The Peierls bracket (Peierls 52) is equivalently the canonical symplectic Poisson bracket on the covariant phase space of field theories such as of the scalar field (Dütsch-Fredenhagen 01, Khavkine 14, Collini 16, prop. 31 and below). Its construction requires the linearized field equations of motion satisfied by the gauge invariant fields to have unique retarded and advanced Green functions. Moreover, it extends “off shell” from the phase space to the space of all field configurations (Marolf 94, Dütsch-Fredenhagen 03, section 2.1, Fredenhagen-Rejzner 12, section 5), where however it is only Poisson bracket, no longer symplectic. For more on this aspect see at off-shell Poisson bracket.

The Peierls bracket of two suitably smooth functions $A$ and $B$ on field configuration space is the antisymmetrized influence on $B$ of an infinitesimal perturbation of a gauge-fixed action by a function that restricts to $A$ on the embedding of the space of solutions in the field configuration space. It is the construction of the influence of $A$ on $B$ that requires the existence of unique retarded and advanced Green's functions of the linearized field equations. One can avoid gauge fixing the action, as long as $A$ and $B$ are gauge invariant observables. In that case, unique retarded and advanced Green’s functions may not exist, but due to the gauge invariance of $A$ and $B$ any representative of the gauge equivalence class of Green’s functions with appropriate causal support. Since gauge invariant observables can be expressed in terms of gauge invariant fields (at least at the linearized level, which is all that matters in the construction) the existence and uniqueness of such equivalence classes of Green’s functions is equivalent to the existence and uniqueness of retarded and advanced Green’s functions for the linearized field equations satisfied by gauge invariant field combinations.

For example, in electrodynamics, advanced and retarded Green’s functions exist on globally hyperbolic spacetimes for Maxwell equations in terms of the field strength $F=\mathbf{d}A$ and correspond to unique gauge equivalence classes of Green’s functions for Maxwell equations in terms of the vector potential $A$.

The algebra of functions on the space of field configurations becomes a Poisson algebra in the following way. Pick a set of functions on the space of field configurations that restrict to a non-degeneratce coordinate system on the embedded covariant phase space. These functions, together with the equations of motion and gauge fixing conditions define a Poisson bivector by being declared canonical, such that the kernel of the bivector coincides with the ideal generated by the equations of motion and the gauge fixing conditions. Obviously the Poisson structure thus constructed on the algebra of functions on field configurations is not unique and depends on the above choice of coordinates; the same non-uniqueness may be parametrized instead by a choice of a connection on the space of field configurations. The embedded covariant phase space becomes a symplectic leaf of the symplectic foliation of the space of field configurations.

Definition

The Peirls bracket

under construction

Definition

(…)

…suitable PDE with advanced/retarded Green's function $\Delta_S^{A/R}$, then the causal Gree’s function is their difference

$$\Delta_S \coloneqq \Delta_S^{R} - \Delta_S^A$$

(…)

(#Khavkine 14, def. 3.9)

The off-shell Peierls bracket

The idea of (Marolf 93, section II) is this:

If $\{q(0), p(0)\}$ is the given (symplectic) Poisson bracket on the space of solutions, identified with the space of initial data, then requiring that everything Poisson-commutes with $EL(S)$, the Euler-Lagrange functional of the action, uniquely extends this to a bracket $\{q(t_1), p(t_2)\}$ on all hisories, because commutation with $EL(S)$ involves generation of time translation. Here $EL(S)$ generates a Poisson ideal and dividing that out reproduces the original symplectic bracket.

Now observe (Khavkine 1) that $EL(S)$ provides a foliation of history space by symplectic leaves. Because under the replacement $EL(S) \mapsto EL(S) - const$ the above still goes through. Observe also (Khavkine 2) that $EL(S) - const = 0$ is the equations of motion for the origial action with a source term $J q$ added. Hence the off-shell Peierls Poisson structure has symplectic leaves parameterized by the source $J$.

Observe finally (Khavkine 3) that with (Marolf 93, section III B) it follows that the Peierls bracket on the shifted leaves agrees with the original one. (…)

Properties

Proposition

(Peierls bracket is the canonical Poisson bracket in field theory)

Given a local Lagrangian field theory, assume that its gauge symmetries are globally recognizable (Khavkine 14, section 3.2.2). Then the Peierls bracket (def. ) is the Poisson bracket corresponding to the symplectic form on the reduced covariant phase space.

(Khavkine 14, theorem 3.2)

References

The definition of what now is called the Peierls bracket originates in

• R. Peierls, The commutation laws of relativistic field theory (1952) (jstor)

In this article the Peierls bracket on the covariant phase space of a non-gauge system is defined and the equivalence with the Hamiltonian phase space symplectic structure is (incompletely) demonstrated. Peierls also discusses how the definition extends to gauge theories and to fermionic theories.

An early review is in

• Bryce DeWitt, The Spacetime Approach to Quantum Field Theory, in Bryce DeWitt, Raymond Stora (eds.), Les Houches Session XL, Relativity, Groups and Topology II (North-Holland, 1983), pp. 382–738.

which is also the first to explicitly check the Jacobi identity for the Peirls bracket.

A streamlined general account is in

• Igor Khavkine, Covariant phase space, constraints, gauge and the Peierls formula, Int. J. Mod. Phys. A, 29, 1430009 (2014) (arXiv.1402.1282)

A traditional physics textbook account amplifying the Peierls bracket is

• Bryce DeWitt, The global approach to Quantum Field Theory (2 volumes), Oxford 2003

The fact that the Peirls bracket for the scalar field gives the Poisson bracket in its covariant phase space is discussed in

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic field theory, and deformation quantization, in Roberto Longo (ed.), Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001

  (arXiv:hep-th/0101079)

• Giovanni Collini, prop. 31 and below in Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

The off-shell generalization to a Poisson bracket on configuration space (history space) was first given in

• Don Marolf, Poisson Brackets on the Space of Histories Annals of Physics Volume 236, Issue 2, December 1994, Pages 374-391 (arXiv:hep-th/9308141)

• Don Marolf, The Generalized Peierls Bracket. Ann. Phys. (N.Y.) 236 (1994) 392–412 (arXiv:hep-th/9308150)

See also exercise 17.12 in

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems

A mathematically clean account of the (on- and off-shell) Peierls bracket (for scalar fields) is in section 2.1 of

• Michael Dütsch, Klaus Fredenhagen, The Master Ward Identity and Generalized Schwinger-Dyson Equation in Classical Field Theory, Commun.Math.Phys. 243 (2003) 275-314 (arXiv:hep-th/0211242)

and in section 2 of

• Ferdinand Brennecke, Michael Dütsch, Removal of violations of the Master Ward Identity in perturbative QFT, Rev.Math.Phys.20:119-172,2008 (arXiv:0705.3160)

there with an eye towards the renormalization program of perturbative Algebraic Quantum Field Theory (pAQFT) on flat and on curved spacetimes.

• Michael Dütsch and Klaus Fredenhagen, Causal perturbation theory in terms of retarded products, and a proof of the action Ward identity, Rev. Math. Phys. 16 (2004) 1291-1348 (arXiv:hep-th/0403213)

Further references to its use in the renormalization program of pAQFT can be found in:

• Klaus Fredenhagen, Katarzyna Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory. Commun.Math.Phys. 314 (2012) 93-127 (arXiv:1101.5112)

Functional analytic aspects of the definition and existence of the Peierls bracket (including its off-shell extension) are discussed in section 3.2 of

• Romeo Brunetti, Klaus Fredenhagen, Pedro Ribeiro, Algebraic Structure of Classical Field Theory I: Kinematics and Linearized Dynamics for Real Scalar Fields (arXiv:1209.2148)

The equivalence between the Peierls bracket and the symplectic Poisson bracket on the covariant phase space of classical field theory (by showing the equivalence of both to the canonical Poisson bracket in Hamiltonian formalism) was demonstrated in

• Glenn Barnich, Marc Henneaux, C. Schomblond, Covariant description of the canonical formalism. Phys.Rev. D44 (1991) R939-R941 (doi)

Further discussion of the manifestly covariant equivalence between the Peierls bracket and the symplectic Poisson bracket on the covariant phase space of classical field theory (avoiding traditional proof via the canonical Hamiltonian formalism) can be found in

• Michael Forger, Sandro Romero, Covariant Poisson Brackets in Geometric Field Theory, Commun.Math.Phys. 256 (2005) 375-410 (arXiv:math-ph/0408008)

• Igor Khavkine, Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory, section 5, (arXiv:1211.1914)