Upper entropy axioms and lower entropy axioms

The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak upper entropy axioms and weak lower entropy axioms. Their conditions are weaker than those of Shannon–Khinchin axioms and Tsallis axioms, while these conditions are stronger than those of the axiomatics based on the first three Shannon–Khinchin axioms. The subadditivity and strong subadditivity of entropy are obtained in the new axiomatics. Tsallis statistics is a special case of satisfying our axioms. Moreover, different forms of information measures, such as Shannon entropy, Daroczy entropy, Tsallis entropy and other entropies, can be unified under the same axiomatics.     

Nonstandard Methods in Quantum Field Theory I: A Hyperfinite Formalism of Scalar Fields

A nonstandard approach to axiomatic quantum field theory is given. Nonstandard axioms for a Hermitian scalar field is proposed, where the field operators act on a hyperfinite-dimensional Hilbert space. The axioms are shown to be equivalent to the Gårding–Wightman axioms. An example of a model of the nonstandard axioms is examined.     

Remarks on the Definition of a Courant Algebroid

The notion of a Courant algebroid was introduced by Liu, Weinstein, and Xu in 1997. Its definition consists of five axioms and a defining relation for a derivation. It is shown that two of the axioms and the relation (assuming only the Leibniz rule) follow from the rest of the axioms.     

A note on misunderstandings of Piron's axioms for quantum mechanics

Piron's axioms for a realistically interpreted quantum mechanics are analyzed in detail within the context of a formal mathematical structure expressed in the conventional set-theoretic idiom of mathematics. As a result, some of the serious misconceptions that have encouraged recent criticisms of Piron's axioms are exposed.     

An algebraic approach to form factors

An associative *-algebra is introduced (containing a TTR-algebra as a subalgebra) that implements the form factor axioms, and hence indirectly the Wightman axioms, in the following sense: Each T-invariant linear functional over the algebra automatically satisfies all the form factor axioms. It is argued that this answers the question (posed in the functional Bethe ansatz) how to select the dynamically correct representations of the TTR-algebra. Applied to the case of integrable QFTs with diagonal factorized scattering theory a universal formula for the eigenvalues of the conserved charges emerges.     

Quantum probability zero-one law for sequential terminal events

On the basis of the Jauch-Piron quantum probability calculus a zero-one law for sequential terminal events is proven, and the significance of certain crucial axioms in the quantum probability calculus is discussed. The result shows that the Jauch-Piron set of axioms is appropriate for the non-Boolean algebra of sequential events.     

Axioms for nonrelativistic quantum mechanics

On the basis of the axioms assumed it is proved that the logic of propositions concerning any quantum-mechanical system may be endowed with the structure of an orthomodular atomistic complete lattice satisfying the covering postulate, and hence, as a consequence of these axioms, the Piron-MacLaren representation theorem for the logic is obtained.     

Euclidean formulation of quantum field theory without positivity

General properties of local quantum field theories (QFT) without positivity are discussed in connection with their euclidean formulation. Modified euclidean axioms for local QFT's without positivity are presented, which allow us to recover by analytic continuation Wightman functions satisfying the modified Wightman axioms for indefinite metric QFT's.     

Axioms for renormalization in Euclidean quantum field theory

A set of axioms which fix Euclidean renormalizations up to a finite renormalization is proposed. There exists a one to one correspondence between Euclidean renormalizations and renormalizations in Minkowski space-time satisfying Hepp's axioms. No restrictions on masses are imposed.     

Axioms of an Experimental System

In this paper a system of axioms is presented todefine the notion of an experimental system. The primaryfeature of these axioms is that they are based solely onthe mathematical notion of a direct product decomposition of a set. Properties ofexperimental systems are then developed. This includesdefining negation, implication, conjunction, anddisjunction on the set of all binaryexperiments of the system and showing that the resulting structure is aregular orthomodular poset. The theory of observables ofexperimental systems is also developed. Finally, theusual models of experiments from classical as well as quantum physics are shown to satisfythe axioms of an experimental system, and a mechanism tocreate new models of the axioms is given.     

Axiomatic approach to the cosmological constant

A theory of the cosmological constant Λ is currently out of reach. Still, one can start from a set of axioms that describe the most desirable properties a cosmological constant should have. This can be seen in certain analogy to the Khinchin axioms in information theory, which fix the most desirable properties an information measure should have and that ultimately lead to the Shannon entropy as the fundamental information measure on which statistical mechanics is based. Here we formulate a set of axioms for the cosmological constant in close analogy to the Khinchin axioms, formally replacing the dependence of the information measure on probabilities of events by a dependence of the cosmological constant on the fundamental constants of nature. Evaluating this set of axioms one finally arrives at a formula for the cosmological constant given by , where G is the gravitational constant, m_e the electron mass, and α_el the low-energy limit of the fine structure constant. This formula is in perfect agreement with current WMAP data. Our approach gives physical meaning to the Eddington-Dirac large-number hypothesis and suggests that the observed value of the cosmological constant is not at all unnatural.     

Spin Axioms in Different Geometries of Relativistic Continuum Physics

The 24 components of the relativistic spin tensor consist of 3 + 3 basic spin fields and 9 + 9 constitutive fields. Empirically only three basic spin fields and nine constitutive fields are known. This empirem can be expressed by two spin axioms, one of them denying purely relativistic spin fields, and the other one relating the three additional basic fields and the nine additional constitutive fields to the known (and measurable) ones. This identification by the spin axioms is material-independent and does not mix basic spin fields with constitutive properties. The approaches to the Weyssenhoff fluid and the Dirac-electron fluid found in literature are discussed with regard to these spin axioms. The conjecture is formulated, that another reduction from six to three basic spin fields which does not obey the spin axioms introduces special material properties by not allowed mixing of constitutive and basic fields.     

Gödel's theorem and information

Gödel's theorem may be demonstrated using arguments having an informationtheoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.     

The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case

We show how averaging defines an Ehresmann connection whose holonomy is the classical adiabatic angles which Hannay defined for families of completely integrable systems. The averaging formula we obtain for the connection only requires that the family of Hamiltonians has a continuous symmetry group. This allows us to extend the notion of the Hannay angles to families of non-integrable systems with symmetry. We state three geometric axioms satisfied by the connection. These axioms uniquely determine the connection, thus enabling us to find new formulas for the connection and its curvature. Two examples are given.     

The Free Will Theorem

On the basis of three physical axioms, we prove that if the choice of a particular type of spin 1 experiment is not a function of the information accessible to the experimenters, then its outcome is equally not a function of the information accessible to the particles. We show that this result is robust, and deduce that neither hidden variable theories nor mechanisms of the GRW type for wave function collapse can be made relativistic and causal. We also establish the consistency of our axioms and discuss the philosophical implications.     

A Lie group structure underlying the triplectic geometry

We consider the pair of degenerate compatible antibrackets satisfying a generalization of the axioms imposed in the triplectic quantization of gauge theories. We show that these data encode a Lie group structure such that the antibrackets are related to the left- and right- invariant vector fields on the group. Conversely, every Lie group admits a “triplectic bundle.” The standard triplectic quantization axioms then correspond to Abelian Lie groups.     

Equivalence of Euclidean and Wightman field theories

A new inversion formula for the Laplace transformation of tempered distributions with supports in the closed positive semiaxis is obtained. The inverse Laplace transform of a tempered distribution is defined by means of a limit of a special distribution constructed from this distribution. The weak spectral condition on the Euclidean Green's functions implies that some of the limits needed for the inversion formula exist for any Euclidean Green's function with an even number of variables. We then prove that the initial Osterwalder-Schrader axioms [1] and the weak spectral condition are equivalent with the Wightman axioms.The research described in this publication was made possible in part by Grant No. 93-011-147 from the Russian Foundation for Basic Research     

Axiomatic Conformal Theory in Dimensions >2 and AdS/CT Correspondence

We formulate axioms of conformal theory (CT) in dimensions >2 modifying Segal’s axioms for two-dimensional CFT. (In the definition of higher-dimensional CFT, one includes also a condition of existence of energy-momentum tensor.) We use these axioms to derive the AdS/CT correspondence for local theories on AdS. We introduce a notion of weakly local quantum field theory and construct a bijective correspondence between conformal theories on the sphere S^d and weakly local quantum field theories on \({H^{d+1}}\) that are invariant with respect to isometries. (Here \({H^{d+1}}\) denotes hyperbolic space = Euclidean AdS space.) We give an expression of AdS correlation functions in terms of CT correlation functions. The conformal theory has conserved energy-momentum tensor iff the AdS theory has graviton in its spectrum.     

Foundation of quasilinear irreversible thermodynamics

In the set of axioms of the linear thermodynamics of irreversible processes the linearity law is replaced by a more general relation (8) coupling fluxes and forces and their time derivatives. It is shown that the remaining axioms of the classical irreversible thermodynamics (OnsagerCasimir relations and the entropy production expression) can be left unchanged. It is derived that in the frame of this theory it is possible to picture the Maxwell model of viscoelastic effects and the Debye model of the dielectric relaxation.