PlanetPhysics/General Theories and Axioms Metatheories

This is a topic on the applications of meta-theories, metalogic and metamathematics to axiomatics in [[../PhysicalMathematics2/|theoretical physics]]. The topic is of potential importance for areas such as [[../CosmologicalConstant2/|Axiomatic Quantum Field Theory]] (AX-QFT), local [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum field theories]] (AQFT), [[../SR/|general relativity]] theory, unified physical theories, general [[../NewtonianMechanics/|dynamics]] [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] theories and axiomatic mathematical biophysics or [[../SystemsBiology/|abstract relational biology]]. The axiomatic approach already has wide applications both in metalogic, mathematics, metamathematics and [[../LQG2/|quantum logics]].

Metatheory, Meta-Logic and Meta-Mathematics

A methatheory or meta-theory can be described as a higher level theory about theories belonging to a lower theory class $𝕄$, or first-level theories. With this meaning, a theory $𝒯$ of the [[../Bijective/|domain]] $𝒟$ is a meta-theory if $𝒟$ is a theory belonging to a class $𝕄$ of (lower-level, or first level) theories. A [[../GeneralTheory/|general theory]] is not a meta-theory because its domain $𝒟$ does not contain any other theories. Valid statements made in a meta-theory are called meta-theorems or metatheorems .

A metalogic is then a meta-theory of various [[../Bijective/|types]] of logic.

Meta-mathematics is concerned with the study of metatheories containing mathematical metatheorems.

As an example of a meta-theory is the theory of [[../SuperCategory6/|super-categories]] $𝒮$ concerned with metatheorems about [[../Cod/|categories]] of categories. On the other hand, an example of a metatheory of [[../SuperCategory6/|supercategories]] $\mathrm{\S }$, such as organismic supercategories $OS$, is the metatheory of the higher dimensional supercategory of supercategories. [[../HigherDimensionalAlgebra2/|Higher dimensional algebra]] ([[../2Groupoid2/|HDA]]) is a metatheory of [[../CategoryOfLogicAlgebras/|algebraic categories]] and other [[../TrivialGroupoid/|algebraic structures]]; good examples are [[../ThinEquivalence/|double groupoids]], [[../GeneralizedSuperalgebras/|double algebroids]] and their categories, as well as [[../HorizontalIdentities/|double categories]]. Further specific examples of HDA are 2-Lie [[../TrivialGroupoid/|groups]] and 2-Lie algebras, as well as their categories of 2-Lie groups and 2-Lie algebras.

In the perspective of the development of mathematics, advances in logic --and over the last century in logics and [[../GeneralTheory/|meta-logics]] -- have played, and are playing, very important roles both in the foundations of mathematics, as well as in related areas such as: categorical logics, [[../LM_nLogicAlgebra/|many-valued logic]] algebras, model theory and many specific [[../CosmologicalConstant/|fields]] of mathematics including, but not limited to, number theory/arithmetics.

Axiomatic Quantum Field Theory (AQFT or AX-QFT)

[[../CosmologicalConstant/|Algebraic Quantum Field Theory]] is the [[../CoIntersections/|algebraic]], geometric and [[../CoIntersections/|topological]] study of quantum field theories (QFT) and [[../NewtonianMechanics/|local quantum physics]] in relativistic space-times using tools from [[../CubicalHigherHomotopyGroupoid/|algebraic topology]], [[../TrivialGroupoid/|category theory]], and [[../Groupoid/|quantum operator algebras/]] algebraic topology ([[../QuantumOperatorAlgebra5/|QAT]]).

Whereas quantum field theory is the general framework for describing the physics of relativistic quantum systems (notably of elementary [[../Particle/|particles]]), algebraic quantum field theories are usually described as algebraic formulations (in terms of an algebraic system and/or physical-axiomatic frameworks) of quantum field theories. Thus, whereas QFT represents a synthesis of quantum theory (QT) and special relativity (SR), (which is supplemented by the principle of locality in space and time , and by the spectral condition in [[../CosmologicalConstant2/|energy]] and [[../Momentum/|momentum]]), \htmladdnormallink{algebraic QFTs {http://unith.desy.de/research/aqft/} study the role of algebraic relations among [[../QuantumSpinNetworkFunctor2/|observables]] that determine a physical system.}

An important example of AQFT is the \htmladdnormallink{Haag-Kastler axiomatic framework {http://planetphysics.us/encyclopedia/PureState.html} for quantum field theory} (thus named after Rudolf Haag and Daniel Kastler who introduced this axiomatic approach), which represents local quantum physics in terms of unital $C^{*}$-algebras. As in the standard formalism of quantum physics, [[../PureState/|pure states]] are described in AQFTs as "rays" in a [[../NormInducedByInnerProduct/|Hilbert space]] $\mathscr{H}$ --which are [[../PureState/|unit vectors]] up to a [[../PureState/|phase factor]] $\varphi$ -- and (quantum) observables defined by self-adjoint (quantum) [[../QuantumOperatorAlgebra4/|operators]] acting in $\mathscr{H}$. Let us recall that a state $\mathrm{\Psi }$ of a $C^{*}$-algebra is defined as a positive linear functional over the algebra equipped with unit [[../NormInducedByInnerProduct/|norm]]. With this definition, pure states correspond to [[../PureState/|irreducible representations]] of the unital $C^{*}$-algebras, and [[../PureState/|mixed states]] correspond to [[../HamiltonianAlgebroid3/|reducible representations]]; moreover, an irreducible representation (which is unique up to equivalence) is called a [[../PureState/|superselection sector]]. Furthermore, for each $C^{*}$-algebra state, one can associate a Hilbert space [[../CategoricalGroupRepresentation/|representation]] of a $C^{*}$-algebra corresponding to a specific choice of relativistic space-time (such as the Minkowski [[../Curved4DimensionalSpace/|4D-space]] in [[../SR/|SR]]).

The [[../TopologicalOrder2/|symmetry group]] of a classical Minkowski space-time $ℳ$ is the [[../PureState/|Poincar\'e group]], generated by translations and [[../CosmologicalConstant/|Lorentz transformations]]. The physical [[../PureState/|vacuum sector]] can be then shown to correspond to the pure state, and the Hilbert space associated with the vacuum sector can be regarded as a unitary representation of the Poincar\'e group; if one looks at the dual, Poincar\'e algebra then the energy-momentum [[../CohomologyTheoryOnCWComplexes/|spectrum]] corresponding to spacetime translations lies on--and also within --the positive light cone. In a more general, supersymmetric context, anti-deSitter vacuum sectors are also possible in principle, but they are not stable (viz. Weinberg, 2000).

A recent review of specific AQFT formulations presented in ref. provides several examples of AQFT approaches in sufficient mathematical detail to be able to evaluate their correctness from a mathematical viewpoint.

According to a recent monograph by Halvorson and Mueger (ref. ), an algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools: the theory of operator algebras, category theory, etc. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT.

Axioms of Quantum Logics

The axioms and derived [[../Formula/|theorems]] of quantum logics (QL) are being intensely studied. However, a metalogic approach to the class of axiomatic quantum logics and the [[../TrivialGroupoid/|classification]] problem of quantum logic is yet to be defined. An important question is that of finding or formulating [[../TheoryOfHilbertLattices/|quantum logic axioms]] that are compatible with General Relativity postulates and/or AQFT.