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In mathematical physics, constructive quantum field theory is the field devoted to showing that quantum field theory can be defined in terms of precise mathematical structures. This demonstration requires new mathematics, in a sense analogous to classical real analysis, putting calculus on a mathematically rigorous foundation.
- Quantum field theory
This line of study is called constructive quantum field...
- Quantum field theory
- Rationale
- Axioms
- Consequences of The Axioms
- Relation to Other Frameworks and Concepts in Quantum Field Theory
- Existence of Theories That Satisfy The Axioms
- See Also
- Further Reading
One basic idea of the Wightman axioms is that there is a Hilbert space, upon which the Poincaré group acts unitarily. In this way, the concepts of energy, momentum, angular momentum and center of mass (corresponding to boosts) are implemented. There is also a stability assumption, which restricts the spectrum of the four-momentum to the positive li...
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Quantum mechanics is described according to von Neumann; in particular, the pure states are given by the rays, i.e. the one-dimensional subspaces, of some separable complex Hilbert space. In the following, the scalar product of Hilbert space vectors Ψ and Φ is denoted by ⟨ Ψ , Φ ⟩ {\displaystyle \langle \Psi ,\Phi \rangle } , and the norm of Ψ is denoted by ‖ Ψ ‖ {\displaystyle \lVert \Psi \rVert } . The transition probability between two pure states [Ψ] and [Φ] can be defined in terms of non...
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For each test function f, i.e. for a function with a compact support and continuous derivatives of any order, there exists a set of operators A 1 ( f ) , … , A n ( f ) {\displaystyle A_{1}(f),\ldots ,A_{n}(f)} which, together with their adjoints, are defined on a dense subset of the Hilbert state space, containing the vacuum. The fields A are operator-valued tempered distributions. The Hilbert state space is spanned by the field polynomials acting on the vacuum (cyclicity condition).
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The fields are covariant under the action of Poincaré group and transform according to some representation S of the Lorentz group, or SL(2, C) if the spin is not integer: 1. U ( a , L ) † A ( x ) U ( a , L ) = S ( L ) A ( L − 1 ( x − a ) ) . {\displaystyle U(a,L)^{\dagger }A(x)U(a,L)=S(L)A{\big (}L^{-1}(x-a){\big )}.}
From these axioms, certain general theorems follow: 1. CPT theorem— there is general symmetry under change of parity, particle–antiparticle reversal and time inversion (none of these symmetries alone exists in nature, as it turns out). 2. Connection between spin and statistic — fields that transform according to half integer spin anticommute, while...
The Wightman framework does not cover infinite-energy states like finite-temperature states. Unlike local quantum field theory, the Wightman axioms restrict the causal structure of the theory explicitly by imposing either commutativity or anticommutativity between spacelike separated fields, instead of deriving the causal structure as a theorem. If...
One can generalize the Wightman axioms to dimensions other than 4. In dimension 2 and 3, interacting (i.e. non-free) theories that satisfy the axioms have been constructed. Currently, there is no proof that the Wightman axioms can be satisfied for interacting theories in dimension 4. In particular, the Standard Model of particle physics has no math...
7 de nov. de 2017 · Constructive quantum field theory is a synthesis of ideas and methods of axiomatic field theory and renormalization theory with modern mathematical methods. The concept of a relativistic quantum field itself admits various equivalent mathematical interpretations, enabling one to use methods from different areas of mathematics.
the goal of this paper is to provide a perspective on “constructive quantum field theory” (CQFT), the subfield of mathematical physics concerned with establishing the existence of concrete models of relativistic quantum field theory in a very precise mathematical sense and then studying their properties from the point of
Wegive a pedagogical introduction toalgebraic quantum fieldtheory (AQFT),with the aim of explaining its key structures and features. Topics covered include: algebraic formulations of quantum theory and the GNS representation theorem, the appearance of unitarily inequivalent
14 de jul. de 2014 · The authors present a rigorous treatment of the first principles of the algebraic and analytic core of quantum field theory. Their aim is to correlate modern mathematical theory with the explanation of the observed process of particle production and of particle-wave duality that heuristic quantum field theory provides.
