Hierarchy Of Symmetries Research Articles

Degrees of symmetry in quantum field theories

A hierarchy of possible symmetries in quantum field theory is defined, which reaches from a purely mathematical invariance to the conventional physical invariance, including the commonly discussed type of spontaneously broken symmetry (SBS). It is shown that one type of SBS, which is usually not considered, naturally leads to theories with an algebra of non-conserved currents and a non-linearly transforming phenomenological Lagrangian. An exactly solvable model is given and some general remarks are made.

Theory of carrier mobility in polar-semiconductors : B. S. Krishnamurthy and K. P. Sinha, J. Phys. Chem. Solids, 26 (1965), pp. 1949–1959

We lay down a nonlinear elastic constitutive framework for the modeling of some 2D crystals of current interest. The 2D crystals we treat are graphene, hexagonal boron nitride and some metal dichalcogenides: molybdenium disulfide (MoS2), tungsten selenium (WSe2), and niobium diselenide (NbSe2). We first find their arithmetic symmetries by using the theory of monoatomic and diatomic 2-nets. Then, by confinement to weak transformation neighborhoods and by applying the Cauchy–Born rule we are able to use the symmetries continuum mechanics utilizes: geometric symmetries. We give the complete and irreducible representation for energies depending on an in-plane measure, the curvature tensor and the shift vector. This is done for the symmetry hierarchies that describe how symmetry changes at the continuum level: C6ν→C2ν→C1 for monoatomic 2-nets, and C6ν→C1ν→C1 for diatomic two nets. We stress that we do not take into account how energy behaves at the transition regime. Having these energies at hand we are able to evaluate stresses and couple stresses for each symmetry regime. These quantities participate to the field equations: the momentum equation, the moment of momentum equation and the equation ruling the shift vector. By making specific assumption for the loading histories that correspond to in-plane biaxial tension/compression as well as in-plane simple shear, we obtain necessary expressions for the shift vector components in order all field equations to be satisfied. When the algebraic equations ruling the shift vector render infinite solutions, we use the Cauchy–Kowalevski theorem-when it applies-for the momentum equation viewed as a quasilinear system in order to single out the unique solution.

Higher Symmetries and the Neutron-Proton Magnetic-Moment Ratio

A quark model of $\mathrm{SU}(4)$ is developed in which the quarks possess charge $\frac{2}{3}$ and $\ensuremath{-}\frac{1}{3}$ and baryon number $N=\frac{1}{3}$. The new particles predicted are characterized by a quantum number $W\ensuremath{\ne}0$ (superstrangeness); they possess integer charge but fractional hypercharge $\frac{1}{3}$ or $\frac{2}{3}$, and are called hyperquarks. A spin extension of $\mathrm{SU}(4)$ is formulated and leads to the study of the group $\mathrm{SU}(8)$ and the subalgebra $\mathrm{SU}(4)\ensuremath{\bigotimes}\mathrm{SU}(2)$. The baryons and isobars are grouped in the representation 120 and the mesons in the adjoint representation 63. It is found that the $\frac{F}{D}$ ratio in $\mathrm{SU}(8)$ is uniquely specified by the scheme. The ratio of the magnetic moments of the neutron and proton is uniquely determined by assuming that the magnetic-moment operator transforms as the adjoint representation of $\mathrm{SU}(8)$, and by specifying the extended Gell-Mann-Nishijima relation for the quarks. The value $\ensuremath{-}\frac{2}{3}$ is found for the neutron-proton magnetic-moment ratio in agreement with the result found in $\mathrm{SU}(6)$. The selection rules forbidding processes like $\ensuremath{\varphi}\ensuremath{\rightarrow}\ensuremath{\rho}\ensuremath{\pi}$ and $\ensuremath{\pi}+N\ensuremath{\rightarrow}\ensuremath{\pi}+N+\ensuremath{\varphi}$ are obtained from the conservation of the four quark spins. These results strongly indicate that the physical predictions of a symmetry scheme like $\mathrm{SU}(6)$ are not unique and that there exists a hierarchy of symmetries all possessing equally good (or bad) physical predictions, but with new quantum numbers associated with superstrange particles.