Peierls bracket

E136243

bracket operation mathematical concept structure in classical field theory structure in quantum field theory

The Peierls bracket is a covariant generalization of the Poisson bracket used in quantum field theory and classical field theory to define commutation relations in a way that respects spacetime causality.

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All labels observed (3)

Label	Occurrences
Peierls bracket canonical	4
Peierls bracket satisfies the Jacobi identity under suitable conditions	1
Peierls–DeWitt bracket formalism	1

How this entity was disambiguated

This entity first appeared as the object of triple T1190367 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Peierls bracket
Context triple: [Rudolf Peierls, notableIdea, Peierls bracket]

A. Weyl quantization
Weyl quantization is a mathematical procedure in quantum mechanics that systematically associates classical observables with quantum operators in a symmetric and coordinate-independent way.
B. Tomonaga–Schwinger equation
The Tomonaga–Schwinger equation is a relativistic generalization of the Schrödinger equation that formulates quantum field evolution on arbitrary spacelike hypersurfaces, forming a key part of covariant quantum field theory.
C. Osterwalder–Schrader axioms
The Osterwalder–Schrader axioms are a set of mathematical conditions that characterize Euclidean quantum field theories in a way that allows them to be rigorously continued to physically meaningful relativistic quantum field theories.
D. Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
E. Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Peierls bracket
Target entity description: The Peierls bracket is a covariant generalization of the Poisson bracket used in quantum field theory and classical field theory to define commutation relations in a way that respects spacetime causality.

A. Weyl quantization
Weyl quantization is a mathematical procedure in quantum mechanics that systematically associates classical observables with quantum operators in a symmetric and coordinate-independent way.
B. Tomonaga–Schwinger equation
The Tomonaga–Schwinger equation is a relativistic generalization of the Schrödinger equation that formulates quantum field evolution on arbitrary spacelike hypersurfaces, forming a key part of covariant quantum field theory.
C. Osterwalder–Schrader axioms
The Osterwalder–Schrader axioms are a set of mathematical conditions that characterize Euclidean quantum field theories in a way that allows them to be rigorously continued to physically meaningful relativistic quantum field theories.
D. Wick’s theorem
Wick’s theorem is a fundamental result in quantum field theory that expresses time-ordered products of field operators as sums of normal-ordered products with all possible contractions, forming the basis for deriving Feynman rules and diagrammatic expansions.
E. Weyl algebra
The Weyl algebra is a fundamental noncommutative algebra generated by position and momentum operators satisfying canonical commutation relations, central in quantum mechanics and representation theory.
F. None of above. chosen

Statements (40)

Predicate	Object
instanceOf	bracket operation ⓘ mathematical concept ⓘ structure in classical field theory ⓘ structure in quantum field theory ⓘ
actsOn	functionals of fields ⓘ observables in classical field theory ⓘ
alternativeTo	canonical Hamiltonian formalism in non-covariant coordinates ⓘ
basedOn	action functional of a field theory ⓘ
clarifies	relation between classical causality and quantum commutators ⓘ
compatibleWith	relativistic invariance ⓘ
definedInTermsOf	difference between advanced and retarded effects of perturbations ⓘ
dependsOn	choice of action and boundary conditions ⓘ
ensures	microcausality in field theory formulations ⓘ
field	classical field theory ⓘ covariant Hamiltonian formalism ⓘ mathematical physics ⓘ quantum field theory ⓘ theoretical physics ⓘ
formalismType	Lagrangian-based bracket construction ⓘ
historicalPublication	Rudolf Peierls' 1952 paper on commutation laws of relativistic field theories ⓘ
inspired	covariant phase space approaches to field theory ⓘ
namedAfter	Rudolf Peierls ⓘ
property	antisymmetric bilinear operation ⓘ covariant with respect to spacetime transformations ⓘ reduces to the canonical Poisson bracket in suitable limits ⓘ respects spacetime causality ⓘ satisfies the Jacobi identity under appropriate conditions ⓘ
quantizationRole	provides a route from classical to quantum commutators ⓘ
relatedTo	Green's functions ⓘ Poisson bracket ⓘ advanced Green's function ⓘ canonical commutation relations ⓘ covariant phase space ⓘ retarded Green's function ⓘ
usedFor	constructing quantum commutators from classical field theory ⓘ defining commutation relations in field theory ⓘ formulating dynamics in a manifestly covariant way ⓘ generalizing the Poisson bracket in a covariant way ⓘ
usedIn	algebraic quantum field theory ⓘ covariant canonical quantization ⓘ

How these facts were elicited

Referenced by (6)

Full triples — surface form annotated when it differs from this entity's canonical label.

Rudolf Peierls → notableIdea → Peierls bracket ⓘ

Rudolf Peierls → notableConcept → Peierls bracket ⓘ

subject linked to: Peierls

Rudolf Peierls' 1952 paper on commutation laws of relativistic field theories → mainTopic → Peierls bracket ⓘ

Rudolf Peierls' 1952 paper on commutation laws of relativistic field theories → introducedConcept → Peierls bracket ⓘ

Rudolf Peierls' 1952 paper on commutation laws of relativistic field theories → propertyShown → Peierls bracket satisfies the Jacobi identity under suitable conditions ⓘ

linked to: Peierls bracket

Rudolf Peierls' 1952 paper on commutation laws of relativistic field theories → influenceOn → Peierls–DeWitt bracket formalism ⓘ