Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

A novel representation of spin 1/2 combines in a single geometric object the roles of the standard Pauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists of three orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) is Hermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one-particle spin modulus 𝑆tot = (ℏ/2)√3; (4) constrains basis opposite spins to have same handedness; (5) allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs have precisely related gauge phases; (2) the dimensionality of the spin space doubles due to variation of handedness; (3) the four maximally entangled states are naturally defined by the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) and inversion (singlet). Cross-product terms in the expression for the squared total spin of two particles relate to experiment and they yield all standard expectation values after measurement. Here spin is directly defined and transformed in 3D orientation space, without use of eigen algebra and tensor product as done in the standard formulation. The formalism allows working with whole geometric objects instead of only components, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘on lab’ (uncontrolled gauge phase) spin transformations.

Spin-1/2 one- and two- particle systems in physical space without eigen-algebra or tensor product

A novel representation of spin 1/2 combines in a single geometric object the roles of the standardPauli spin vector operator and spin state. Under the spin-position decoupling approximation it consists ofthree orthonormal vectors comprising a gauge phase. In the one-particle case the representation: (1) isHermitian; (2) shows handedness; (3) reproduces all standard expectation values, including the total one particlespin modulus 𝑆tot = √3ℏ/2; (4) constrains basis opposite spins to have same handedness; (5)allows to formalize irreversibility in spin measurement. In the two-particle case: (1) entangled pairs haveprecisely related gauge phases and can be of same or opposite handedness; (2) the dimensionality of the spinspace doubles due to variation of handedness; (3) the four maximally entangled states are naturally definedby the four improper rotations in 3D: reflections onto the three orthogonal frame planes (triplets) andinversion (singlet). The cross-product terms in the expression for the squared total spin of two particlesrelates to experiment and they yield all standard expectation values after measurement. Here spin is directlydefined and transformed in 3D orientation space, without use of eigen algebra and tensor product as in thestandard formulation. The formalism allows working with whole geometric objects instead of onlycomponents, thereby helping keep a clear geometric picture of ‘on paper’ (controlled gauge phase) and ‘onlab’ (uncontrolled gauge phase) spin transformations.

Solving optimal reactive power problem by enhanced fruit fly optimization algorithm and status of material algorithm

<div data-canvas-width="397.2735184438622">This paper proposes enhanced fruit fly optimization algorithm (EFF) and status of material algorithm (SMA) to solve the optimal reactive power problem. Fruit fly optimization algorithm is based on the food finding behavior of the fruit fly. There are two steps in food finding procedure of fruit fly: At first it smells the food source by means of osphresis organ and it flies in that direction; afterwards, when it gets closer to the food site, through its sensitive vision it will find the food. At the beginning of the run by diminishing the inertia weight from a large value to a small value, will lead to enhance the global search capability and more local search ability will be in process the end of the run of the EFF algorithm. Then SMA is projected to solve the problem. Three state of material are solid, liquid, and gas. For evolution procedure direction vector operator assign a direction to every molecule consecutively to guide the particle progression. Collision operator imitates the collisions factor in which molecules are interacting to each other. Proposed enhanced EFF, SMA has been tested in standard IEEE 30 bus test system and simulation results show the projected algorithms reduced the real power loss considerably.</div>

Scattering of an elastic wave by a rigid sphere in a semi-bounded domain

The problem of scattering of plane elastic waves by a rigid sphere near a rigid boundary is considered. This leads to the appearance of multiply re-reflected dilatation and shear waves, which generate strong oscillations of the wave field. The problem for a vector operator of the shear waves is reduced to the definition of scalar functions as a consequence of symmetry. Approximate formulas for the far field and the long-wave Rayleigh approximation are presented. The construction of multiply re-reflected waves by the image method is presented and analyzed. Calculations of the scattered wave fields are plotted in the form of scattering diagrams.

Geometric phase of a spin-1 2 particle coupled to a quantum vector operator

We calculate Berry’s phase when the driving field, to which a spin-[Formula: see text] is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g. the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus “quantum driving field”, is first computed, and is then subdivided into the two subsystems, using the Schmidt decomposition of the total wave function — the resulting expression shows a marked, purely quantum effect, involving the commutator of the field components. We also compute the corresponding mean “classical” phase, involving a precessing magnetic field in the presence of noise, up to terms quadratic in the noise amplitude — the results are shown to be in excellent agreement with numerical simulations in the literature. Subtleties in the relation between the quantum and classical case are pointed out, while three concrete examples illustrate the scope and internal consistency of our treatment.

VECTOR BETA FUNCTION

We propose various properties of renormalization group beta functions for vector operators in relativistic quantum field theories. We argue that they must satisfy compensated gauge invariance, orthogonality with respect to scalar beta functions, Higgs-like relation among anomalous dimensions and a gradient property. We further conjecture that nonrenormalization holds if and only if the vector operator is conserved. The local renormalization group analysis guarantees the first three within power counting renormalization. We verify all the conjectures in conformal perturbation theories and holography in the weakly coupled gravity regime.

On the Kronecker Products and Their Applications

This paper studies the properties of the Kronecker product related to the mixed matrix products, the vector operator, and the vec-permutation matrix and gives several theorems and their proofs. In addition, we establish the relations between the singular values of two matrices and their Kronecker product and the relations between the determinant, the trace, the rank, and the polynomial matrix of the Kronecker products.

MATRIX ELEMENTS OF A HYPERBOLIC VECTOR OPERATOR UNDER SO(2,1)

We deal here with the use of Wigner–Eckart type arguments to calculate the matrix elements of a hyperbolic vector operator [Formula: see text] by expressing them in terms of reduced matrix elements. In particular, we focus on calculating the matrix elements of this vector operator within the basis of the hyperbolic angular momentum [Formula: see text] whose components [Formula: see text], [Formula: see text], [Formula: see text] satisfy an SO(2,1) Lie algebra. We show that the commutation rules between the components of [Formula: see text] and [Formula: see text] can be inferred from the algebra of ordinary angular momentum. We then show that, by analogy to the Wigner–Eckart theorem, we can calculate the matrix elements of [Formula: see text] within a representation where [Formula: see text] and [Formula: see text] are jointly diagonal.

On: “New prospects in shallow depth electrical surveying for archeological and pedological applications” by A. Hesse, A. Jolivet and A. Tabbagh (GEOPHYSICS, 51, 585–594, March 1986).

The forward solution given by Hesse et al. is incorrect. The error is a result of the erroneous governing differential equation [their equation (2)], which for the only nonzero component of the magnetic field has the following form: [Formula: see text]Unfortunately, the authors did not show how they arrived at this equation, but the mistake is so frequently encountered that its origin can be reconstructed quite easily. Indeed, by neglecting displacement currents in the fourth Maxwell equation and by applying the vector operator ∇× to both parts of the equation, we obtain [Formula: see text]Making use of the well known vector identity [Formula: see text]and of the first and third Maxwell equations, we obtain [Formula: see text]