Gödel axiom mappings in special relativity and quantum-electromagnetic theory

Exponential mappings into an imaginary space or number field for the axioms of a theory, which are in the form of propositional constants and variables, make possible: (a) an understanding of the meaning and differences between the Lorentz transformation constants, such that their product is still equal to one, but the axioms at each end of the transformations are logically inverse and separately consistent; (b) an interpretation of the psi function phase factor which is part of the axiomE=hf; (c) the unification of the quantum-mechanical psi function and the electromagnetic wave function. Thus, those statements whose mechanisms are unknown (the axioms of the theory) are to be assigned the axiom propositional number symbol and are to be associated with the complex probability eⁱ, which is a uniform factor of the energy equations expressing the physical state. Such probabilistic axiom functions can be associated with both the special theory of relativity and the quantum-electromagnetic theory.     

Twisted parafermions

A new type of nonlocal currents (quasi-particles), which we call twisted parafermions, and its corresponding twisted Z-algebra are found. The system consists of one spin-1 bosonic field and six nonlocal fields of fractional spins. Jacobi-type identities for the twisted parafermions are derived, and a new conformal field theory is constructed from these currents. As an application, a parafermionic representation of the twisted affine current algebra A⁽²⁾₂ is given.     

Form factors in the presence of integrable defects

Form factor axioms are derived in two dimensional integrable defect theories for matrix elements of operators localized both in the bulk and on the defect. The form factors of bulk operators are expressed in terms of the bulk form factors and the transmission factor. The structure of the form factors of defect operators is established in general, and explicitly calculated in particular, for the free boson and for some operator of the Lee–Yang model. Fusion method is also presented to generate boundary form factor solutions for a fused boundary from the known unfused ones.     

Noncommutative supergeometry, duality and deformations

Following Lett. Math. Phys. 50 (1999) 309, we introduce a notion of Q-algebra that can be considered as a generalization of the notion of Q-manifold (a supermanifold equipped with an odd vector field obeying {Q,Q}=0). We develop the theory of connections on modules over Q-algebras and prove a general duality theorem for gauge theories on such modules. This theorem containing as a simplest case SO(d,d,Z)-duality of gauge theories on noncommutative tori can be applied also in more complicated situations. We show that Q-algebras appear naturally in Fedosov construction of formal deformation of commutative algebras of functions and that similar Q-algebras can be constructed also in the case when the deformation parameter is not formal.     

The Sudakov form factor in QCD

The quark electromagnetic form factor in the Sudakov region is calculated with the use of the renormalization properties of the contour functional <0 | T P exp(ig∫_cdz^μA_μ(z)) | 0>. It is shown that the nonleading logarithmic corrections to the Sudakov form factor are summed to give a decreasing exponential and they do not destroy the leading double logarithmic result.     

Charged Pion Form Factor Determination in the Range of Q^2= 0.6-1.6 （GeV／c）^2

Using the most recent differential cross section data for e-p quasi-elastic scattering, the charged pion formation and its form factor Fπ is calculated in the energy range of 2.4-4 GeV at Q^2 = 0.6-1.6 (GeV/c)^2. The functional dependence of the charged pion form factor to the separated cross section aL is investigated and compared to the previously determined result.     

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.     

Functional Integral Construction of the Massive Thirring model: Verification of Axioms and Massless Limit

We present a complete construction of a Quantum Field Theory for the Massive Thirring model by following a functional integral approach. This is done by introducing an ultraviolet and an infrared cutoff and by proving that, if the “bare” parameters are suitably chosen, the Schwinger functions have a well defined limit satisfying the Osterwalder-Schrader axioms, when the cutoffs are removed. Our results, which are restricted to weak coupling, are uniform in the value of the mass. The control of the effective coupling (which is the main ingredient of the proof) is achieved by using the Ward Identities of the massless model, in the approximated form they take in the presence of the cutoffs. As a byproduct, we show that, when the cutoffs are removed, theWard Identities have anomalies which are not linear in the bare coupling. Moreover, we find for the interacting propagator of the massless theory a closed equation which is different from that usually stated in the physical literature.     

Meaningfulness and Order-Invariance: Two Fundamental Principles for Scientific Laws

The first invariance principle, called meaningfulness, is germane to the common practice requiring that the form of a scientific law must not be altered by a change of the units of the measurement scales. By itself, meaningfulness does not put any constraint on the possible data. The second principle requires that the output variable is order-invariant with respect to any transformation (of one of the input variables) belonging to a particular family or class of such transformations which are characteristic of the law. These principles are formulated as axioms of a theory. Taken together, meaningfulness and order-invariance axioms have strong consequences on the feasible theories. Three applications of our results are discussed in details, involving the Lorentz–FitzGerald contraction, Beer's law, and the Monomial laws, each of which is derived from three axioms implementing meaningfulness and order-invariance concepts. (An initial condition axiom is also used.) Not all scientific laws are order-invariant in the sense of this paper. An example is van der Waals' equation.     

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.     

A new theory of elementary matter

This is the first of a series of articles that reviews and expands upon a new theory of elementary matter. This paper presents an exposition of the philosophical approach and its general implications. The ensuing explicit form of the mathematical expression of the theory and several applications in the atomic and elementary particle domains will be developed in the succeeding parts of this series.The theory is based on three axioms: the principle of general relativity, a generalized Mach principle, and a correspondence principle. The approach is basically a deterministic, relativistic field theory which fully incorporates the idea that any realistic physical system is in facta closed system, without separable parts. It is shown that the most primitive mathematical expression of this theory, following as anecessary consequence of its axioms, is in terms of a set of coupled nonlinear spinor field equations. Nevertheless, the exact formalism is constructed to asymptotically approach the quantum mechanical formalism for a many-particle system, in the limit of sufficiently small energy-momentum transfer among the components of the considered closed system. Thus, all of the mathematical predictions of nonrelativistic quantum mechanics are contained in this theory, as a mathematical approximation. However, predictions follow from the exact form of this theory (where energy-momentum transfer can be arbitrarily large) that are not contained in the quantum theory.     

A new theory of elementary matter part III: A self-consistent field theory of electrodynamics and correspondence with quantum mechanics

This paper exploits the axioms and general mathematical structure of a new theory of elementary matter, thus far developed in two earlier papers (Sachs, 1971b, c). It is shown here, in an explicit fashion, how the exact form of this theory approaches that of quantum mechanics of a many-particle system that interacts electromagnetically. The form of the mathematical expression of quantum mechanics of a many-particle system is found to be a linear approximation for the nonlinear (deterministic) field theory of this author's approach. The latter approximation is valid only when the components of the (asserted) closed system are sufficiently weakly coupled so that it appears as a many-particle system. The physical equivalent of the Pauli exclusion principle is derived in this paper as anexact feature of the theory, which is, in fact, sensitive to its closed and nonlinear features. It is then shown how the Fermi-Dirac statistics in particle physics follows from the present nonlinear theory only in a linear approximation.     

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.     

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.     

Neutron magnetic form factor in strongly correlated materials

We introduce a formalism to compute the neutron magnetic form factor FM(q) within a first-principles density functional theory and dynamical mean field theory. The approach treats spin and orbital interactions on the same footing and reduces to earlier methods in the fully localized or the fully itinerant limit. We test the method on various actinides of current interest NpCoGa5, PuSb and PuCoGa5, and we show that PuCoGa5 is in mixed valent state, which naturally explains the measured magnetic form factor.     

Simple example of quantum causal structure

The axioms of quantum causal structure and the definition of AlexandrovT-structure are presented in an improved form and an illustrative example is shown.     

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.     

Theory of ordered spaces

This is the first of a planned series of investigations on the theory of ordered spaces based upon four axioms. Two of these, the order (I.1.1) and the local structure (II.5.1) axioms provide the structure of the theory, and the other two [the identification (I.1.11) and cone (I.2.7) axioms] eliminate pathologies or excessive generality. In the present paper the axioms are supplemented by the nontriviality conditions (I.1.9) and a regularity property (II.4.2).The starting point is a nonempty setM and a family of distinguished subsets, calledlight rays, which are totally ordered. The order axiom provides the properties of this order. Positive and negative cones at a point are defined in terms of increasing and decreasing subsets and are used to extend the total order on the light rays to a partial order over all ofM. The first significant result is thepolygon lemma (I.2.3) which provides an essential constructive tool. A non-topological definition is found for the interiors of the cones; it leads to a more homogeneous partial order relation onM.In Sect. II, subsets calledD-sets (Def. II.2.2), possessing certain desirable properties, are studied. The key concept ofperpendicularity of light rays is isolated (Def. II. 3.1) and used to derive the basic separation properties, provided that the interiors of cones are nonempty. It is shown that, in aD-set, good properties of one cone can be transported along light rays, so that the structure of aD-set is homogeneous. In particular, if one cone has nomempty interior, so have all others. However, the existence of even one cone with nomepty interior does not follow from the axioms, but has to be imposed as an additionalregularity condition. The local structure axiom now states that every point lies in a regularD-set. It is proved that the family of regularD-sets is closed under finite intersections. Theorder topology is defined as the topology which has this family as a base. This topology is Hausdorff, and coincides with the usual topology for Minkowski spaces.     

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.