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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Peierls bracket canonical | 2 |
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Peierls bracket Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Rudolf Peierls