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The Atiyah-Singer Index Theorem and Its Application to Anomalies in Quantum Field Theory
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
It is easy to see from the anticommutation relations for the flat space-time Gamma matrices and the above properties of the tetrad frame that {γμ(x),γv(x)}=gμv(x)
Some Aspects of Superstring Theory
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
Let ωmnµ denote the spinor connection of the gravitational field. Then if Γm are the Dirac matrices in four dimensions and emµ is the tetrad basis of space time being used, the covariant derivative of a spinor field is defined by Dμψ=(∂μ+(1/4)ωμmnΓmn)ψ where Γmn=[Γm,Γn]
Relativistic Quantum Mechanics and Quantum Field Theory
Published in Xavier Oriols, Jordi Mompart, Applied Bohmian Mechanics, 2019
Nevertheless, there is a way to introduce a matrix γµ that transforms as a true vector [13, 19]. At each point of space-time, one introduces the tetrad eα¯μ(x), which is a collection of four space-time vectors, one for each α¯=0,1,2,3. The tetrad is chosen so that () ηα¯β¯eα¯μ(x)eβ¯v(x)=gμv(x),
Berry phase of the linearly polarized light wave along an optical fiber and its electromagnetic curves via quasi adapted frame
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Rıdvan Cem Demirkol
The intrinsic geometric features of a curve in the 3D space is determined mostly by using the Serret–Frenet tetrad. However, when the curve is degenerate i.e. it has vanishing second derivative at some points along the curve, the Serret–Frenet frame do not exist. An ideal model of parallel transport frame or quasi frame has been improved by Soliman et al. [28] to deal with that problem. According to this approach, one can define a quasi frame by only parallel transporting to the tangent vector of the frame along the curve. The quasi frame of a regular space curve is given by where is a projection vector and can be selected as one of the following If the angle between the quasi normal vector and the normal vector is choosen as ψ, then following relation is obtained between the quasi and SF frame. Therefore, the quasi frame equations are expressed as where
Ideas and graphs: the Tetrad of activity
Published in International Journal of General Systems, 2018
One important sequence that is, however, implicitly discussed in this literature arrays the terms of the Tetrad in a hierarchical dimension with ground and goal at its limits and instrument and direction at intermediate points. This is shown in Figure 4 as an undirected graph which can be read going up from ground to goal (AB:BC:CD) or going down from goal to ground (DC:CB:BA). The undirected graph AB:BC:CD (where the individual relations are undirected and the order of relations is also arbitrary) can represent either direction or both. The zig-zag path in Figure 4 conveys an additional non-hierarchical idea: although direction is closer to goal and thus higher than instrument which is closer to ground, there is a secondary sense (in the idea of a motivational axis) in which direction and instrument are on the same level. The hierarchical sequence of Figure 4 is actually not explicitly given by Bennett, but is implicit in his discussion of the Tetrad, and features prominently in Blake’s work.
Micropolar continua as projective space of Skyrmions
Published in Philosophical Magazine, 2022
In [71], Skyrme used an explicit field configuration for (67) with the components of (75), but the field is given by the tetrad field -rotated hedgehog field representing the transformation of the spin-isospin system. The constraint immediately indicates that the tetrad field must be orthogonal matrices that rotate coordinate and isospin space. On the other hand, the tetrad field of (10) rotates coordinate and tangent space, and this will be reduced to the microrotation under the condition . This might explain the consistent results (81) and (80), in two distinct physical systems, which are again equivalent to the vanishing Riemann curvature tensor of the form (19) expressed by the Maurer–Cartan equation in terms of the contortion , Skyrmions are -dimensional field configurations for the quantised invariant number defined by the total charge of the integration of conserved current defined in (45) for d = 4 for a,b,c,d = 1,2,3,4 and of (63). This topological invariant number is regarded as a particle-like quantity and postulated to be a baryon number. In particular, under the configuration (82), we obtain the topological invariant charge Q = N = 1, one proton or neutron. The integer N = 1 comes entirely from the hedgehog field . In other words, if we use the general axial configuration such as (63) or (67), we will obtain a baryon number Q = N by the following integration of the topological density of the current (84), Using the relation with Nye's tensor (79), this can be rewritten by Furthermore, after a rather lengthy calculation using the relation with the contortion (17), this further becomes All three expressions (85), (86) and (87) will give identical topological invariant integer N satisfying the finite energy requirement we considered. The form of the integration (86) is noted in [73, 74] in the context of Cosserat elasticity without referring to the Skyrmions.