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Discrete symmetries and the reduction of the two-body Dirac equations with interactions
Published in C Constanda, J Saranen, S Seikkala, Integral methods in science and engineering, 2020
We consider two-particle relativistic equations with scalar, pseudoscalar, vector and oscillator-like interactions. These basic equations describing the steady states of the system are [1], [2] ()Liψ=[α1p1+α2p2+(β1+β2)m+Ui]ψ=Eiψ,i=1,2,3,4,
Expansion to 3D Computations
Published in Dietmar Hildenbrand, Introduction to Geometric Algebra Computing, 2020
Compared to Compass Ruler Algebra, Conformal Geometric Algebra consists of one additional basis vector e3 for 3D space. Table 15.1 lists all the 32 basis blades of CGA. The basis vectors e1,e2,e3,e0,e∈ are the five grade-1 blades of this algebra. There is one grade-0 blade (the scalar) and one grade-5 blade (the pseudoscalar). Linear combinations of the 10 grade-2 blades, the 10 grade-3 blades and the five grade-4 blades are called bivectors, trivectors and quadvectors. A linear combination of blades with different grades is called a multivector. Multivectors are the main algebraic elements of Conformal Geometric Algebra.
Classical Theory of the Weak Interaction and Foundations of Nuclear Beta Decay
Published in K Grotz, H V Klapdor, S S Wilson, The Weak Interaction in Nuclear, Particle and Astrophysics, 2020
K Grotz, H V Klapdor, S S Wilson
The matrix γ5 is responsible for an in some sense pathological behaviour of pseudoscalar and axial vector observables under a parity transformation. Whilst, e.g. a scalar current density, as usual for scalar quantities, is invariant under the parity transformation, and in general under all Lorentz transformations, the single-component quantity ψ¯2γ5ψ1 changes its sign. Axial vector currents are transformed, as shown in Table 2.3, also with the opposite sign compared to vector current densities.
Symmetry arguments and the totalitarian principle in the physics of liquid crystals and other condensed matter systems
Published in Liquid Crystals, 2023
Tianyi Guo, Xiaoyu Zheng, Peter Palffy-Muhoray
We note that vectors, formed by taking the cross-product of two proper vectors, such as torque and magnetic field, are pseudovectors. Their components do not change sign under parity, that is, under the inversion of the coordinate system. More interestingly, pseudotensors of rank 0, pseudoscalars, formed by taking the dot product of a proper vector with a pseudovector, do change sign under inversion. If three distinct non-planar proper vectors exist in a system, they may be used to form a Cartesian coordinate system with handedness. These vectors may also be used to construct a pseudoscalar. Pseudoscalars are therefore associated with handed chiral systems. If a system is chiral, there must be an associated nonzero pseudoscalar, and conversely, if there exists a pseudoscalar in a system, the system is necessarily chiral.
Surface anchoring energy of cholesteric liquid crystals
Published in Liquid Crystals, 2020
Tianyi Guo, Xiaoyu Zheng, Peter Palffy-Muhoray
Now we turn to the central point of our paper, surface anchoring of chiral nematic or cholesteric liquid crystals. The novel aspect of cholesterics is broken inversion symmetry due to the chirality of the constituent molecules. Since chiral objects lack inversion symmetry, it is possible to associate three non-collinear proper vectors with chiral systems. The three proper vectors , and can be combined to form the pseudoscalar . Since the converse is also true (a handed vector triad can be constructed if a pseudoscalar exists), pseudoscalars are indicators of chirality. In the case of cholesteric liquid crystals, a pseudoscalar is often used to represents the intrinsic twist in the orientation of molecules, with for right-handed chirality, and for left-handed chirality. The wavelength of the periodicity of the system is the pitch . With a pseudoscalar in the system, one can construct one additional proper scalar term, arising from chirality, in the surface anchoring energy density. The new term, in addition to the ones in Equation (5), is2
Magnetic dipolar modes in magnon-polariton condensates
Published in Journal of Modern Optics, 2021
It was shown [47] that together with power-flow vortices, in the near-field region adjacent to the MDM ferrite disk, there exists also another quadratic-form parameter determined by a scalar product between the electric and magnetic field components: The vector measures the rotation of the vector field . A scalar product of with is an indicator of how much the electric-field vector field rotates around itself. We name parameter F as the ME-field helicity density. In a case of ME helicity, this is a time-odd pseudoscalar. This is different from the known concepts of the magnetic, electric, and electromagnetic helicities, which are considered as time-even Lorentz pseudoscalars [77,78]. Pseudoscalar is a quantity that changes its sign when one changes RH coordinate system to LH coordinate system and vice versa. In the case of EM fields, helicity can be considered as the difference of the number of the RH and LH propagating photons. For ME fields, the parameter of helicity indicates characteristics of the near-field structure. It is related to the angle between the spinning electric and magnetic fields. We may say that at each MDM resonance, we observe time average near-field parameters: the power-flow vortex and the field helicity. An analysis shows that the ME near fields are characterized by unique symmetry properties, both with respect to time reversal and parity. The electric and magnetic fields outside a ferrite disk are rotating fields, which are not mutually perpendicular in vacuum. Depending on a direction of a bias magnetic field, we can distinguish the clockwise and counterclockwise topological-phase rotation of the fields. The mirror symmetry of the field vectors with respect to the plane of symmetry of the disk is very specific. As we can see from the analytical and numerical results [47,52,65,76], the magnetic field vectors appear to be polar relative to the plane of the disk. At the same time, electric field vectors behave like axial vectors with improper rotation. This means that in the case of the electric field vectors, we have a mirror inversion combined with the rotation of the mirror plane around the disk axis. These symmetry properties of the field vectors are illustrated in Figure 9.