The decomposition of Spinc(4) gauge potential in terms of the Dirac 4-spinor is investigated, where an important characterizing equation ∆Aμ = −λAμ has been discovered. Here λ is the vacuum expectation value of the spinor field, λ = ‖Φ‖, and Aμ the twisting U(1) potential. It is found that when λ takes constant values, the characterizing equation becomes an eigenvalue problem of the Laplacian o...

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G,σ) (S = (G,μ)), where G = (V, E) is a graph called the underlying graph of S and σ : E → (e1, e2, · · · , ek) (μ : V → (e1, e2, · · · , ek)) is a function, where each ei ∈ {+,−}. Particularly, a Smarandachely 2-singed graph or 2-marked graph is called abbreviated to a singed graph or a marked graph. We chara...

We consider the loop equation in four-dimensional N = 4 SYM, which is a functional differential equation for the Wilson loop W (C) and expresses the propagation and the interaction of the string C. Our W (C) consists of the scalar and the gaugino fields as well as the gauge field. The loop C is specified by six bosonic coordinates y(s) and two fermionic coordinates ζ(s) and η(s) besides the fou...

symbol or equation Corresponding expression for the Yang-Mills field Corresponding expression for the gravitational field p=0, 1,2,3 n=1 ~ ~ n Ppv=gfsv 'QIbv j P&v =Op1p2p3b Ppy= Pv F P @vs7 S— = g'" &'&m~,

∗October 4, 2009: The body of this document is an almost literal transcription of the original manuscript, with the above date, entitled “Notes on the Euclidean φ functional field equation.” The table of contents was added in April, 1994, and the introduction in March, 2008. G. A. Baker and A. S. Wightman, “Trying to Violate Coupling Constant Bounds in φ ν Quantum Field Theory,” in Progress in ...

The work contains a detailed investigation of free neutral (Hermitian) or charged (non-Hermitian) scalar fields and the describing them (system of) Klein-Gordon equation(s) in momentum picture of motion. A form of the field equation(s) in terms of creation and annihilation operators is derived. An analysis of the (anti-)commutation relations on its base is presented. The concept of the vacuum a...

While the equation above is correct, it is not very useful for incremental updates since the equation above updates the checksum C, rather than the 1’s complement of the checksum, ̃C, which is the value stored in the checksum field. In addition, it suffers because the notation does not clearly specify that all arithmetic, including the unary negation, must be performed one’s complement, and so i...

Using a formalism of minitwistors, we derive infinitely many conserved charges for the sl(∞)-Toda equation which accounts for gravitational instantons with a rotational Killing symmetry. These charges are shown to form an infinite dimensional algebra through the Poisson bracket which is isomorphic to two dimensional area preserving diffeomorphism with central extentions. E-mail address; be10@dk...