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We formulated a field theory in space-time elements, which are obtained by dividing the space-time continuum into arbitrary-shaped space-time elements, such as hyper-octahedra (orthoplexes), which are aligned locally periodically, without long-range order. Each field is expressed in terms of scalar expansion functions with rotationally covariant coefficients. The cosmological constant was related to the cut-off energy of the quadratic Higgs self-energy. From the literature, examinations find an experimental break in the cosmic ray energy spectrum called 'knee' at about 3 [PeV], which agrees with this theoretical ultraviolet cut-off energy. Moreover, the expansion functions are updated to plane waves using the Lorentz/Poincar\'e covariant/invariant inner product of vectors with each four-momentum within the cut-off energy. Due to the long-range disordered alignment of space-time elements, this study newly introduces an effective field, which is not in the standard model and lies within the Planck energy. Particles are then rarely excited to the energy region between the cut-off energy and the Planck energy via extremely weak interaction solely with the effective field. The theoretical ultraviolet cut-off energy is consistent with the experimental phenomena such as the stability of our Universe, inflation in the early Universe and no detection of neutrinos above PeV-order energy., Comment: v1: 7 pages; v2: Sec. 3 revised; v3: Secs. 1 and 3 revised; v4: Secs. 3 and 4 revised; v5: Abstract slightly modified; v6: Sec. 3 modified; v7: language editing performed; v8: Secs. 2-3 revised; revision summary added to Sec. 1; v9: Secs. 2-3 and Appendix modified; related parts modified; v10: Secs. 2 and 3 modified; related parts modified; v11: last part of Sec. 2 modified; typo corrected

This article is a review by the authors concerning the construction of a Poincar${\rm \acute{e}}$ covariant (owing to spacetime continuum) field-theoretic formalism in terms of step-function-type basis functions without ultraviolet divergences. This formalism analytically derives confinement/deconfinement, mass-gap and Regge trajectory for non-Abelian gauge fields, and gives solutions for self-interacting scalar fields. Fields propagate in spacetime continuum and fields with finite degrees of freedom toward continuum limit have no ultraviolet divergence. Basis functions defined in a parameter spacetime are mapped to real spacetime. The authors derive a new solution comprised of classical fields as a vacuum and quantum fluctuations, leading to the linear potential between the particle and antiparticle from the Wilson loop. The Polyakov line gives finite binding energies and reveals the deconfining property at high temperatures. The quantum action yields positive mass from the classical fields and quantum fluctuations produces the Coulomb potential. Pure Yang-Mills fields show the same mass-gap owing to the particle-antiparticle pair creation. The Dirac equation under linear potential is analytically solved in this formalism, reproducing the principal properties of Regge trajectories at a quantum level. Further outlook mentions a possibility of the difference between conventional continuum and present wave functions responsible for the cosmological constant., Comment: v6: 42 pages; typos corrected; to appear in International Journal of Modern Physics A (title shortened)

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product between the zeta and gamma functions. It is known that two integral lines, expressing the completed zeta function, rotated from the real axis in the opposite directions, can be shifted without affecting the completed zeta function owing to the residue theorem. The completed zeta function is regular in the region of the complex plane under consideration. For convenience in the subsequent singularity analysis of the above integral, we first deform and shift the integral contours. We then investigate the singularities of the composite elements (caused by polynomial integrals in opposite directions), which appear only in the case for which the distance between the contours and the origin of the coordinates approaches zero. The real part of the zeros of the zeta function is determined to be 1/2 along a symmetry line from the singularity removal condition. (In the other points, the singularities are adequately cancelled as a whole to lead to a finite value.), Comment: Comments: v2: grammar/spell corrected; typos corrected including a typo above Eq. (33); v3: descriptions including title amended to more mathematical ones; v4: Sec. 2 revised; v5: Sec. 2 and related descriptions revised including Eq. (38); description below Eq. (51) modified; v6: Sec. 3 revised and Lemmas 2, 3 added; v7: 19 pages: Sec. 2 and related parts (mainly those in Sec. 3) revised

This article reports an explicit function form for confining classical Yang-Mills vector potentials and quantum fluctuations around the classical field. The classical vector potential, which is composed of a confining localized function and an unlocalized function, satisfies the classical Yang-Mills equation. The confining localized function contributes to the Wilson loop, while the unlocalized function makes no contribution to this loop. The confining linear potential between a heavy fermion and antifermion is due to (1) the Lie algebra and (2) the form of the confining localized function which has opposite signs at the positions of the particle and antiparticle along the Wilson loop in the time direction. Some classical confining parts of vector potentials also change sign on inversion of the coordinates of the axis perpendicular to the axis joining the two particles. The localized parts of the vector potentials are squeezed around the axis connecting the two particles, and the string tension of the confining linear potential is derived. Quantum fluctuations are formulated using a field expression in terms of local basis functions in real spacetime. The quantum path integral gives the Coulomb potential between the two particles in addition to the linear potential due to the classical fields., Comment: 14 pages; v2:missing factor inserted (Eqs.82,87,89,90);spell/grammar corrected; v3:15 pages; Sect.1, Subsect.3.3 and related descriptions in Subsect.3.1 revised with citations; v4:16 pages;Sect.2 (Subsections 2.1,2.2,2.3), Sect.3, Sect.4 (Subsections 4.1,4.2), and related Sect.1 and References revised; v5:a few typos corrected; to appear in Bulg. J. Phys. 41 (2014) 142-171 (title in BJP changed)