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Problems and Study Projects in non-Abelian Gauge and String Theory
Published in Harish Parthasarathy, Supersymmetry and Superstring Theory with Engineering Applications, 2023
Note that ve is 4 × 1, e is also 4 × 1 and ϕ is 2 × 1. Thus, le is 8 × 1 and ϕ∇e is also 8 × 1 and hence L4 is a scalar as it should be. When the Higgs doublet ϕ falls to the ground state with vacuum expected value < ϕ >, then L4 contributes a term that is quadratic in the electron field e and it is precisely this mechanism that accounts for the electronic mass. This completes the story of the electroweak theory based on the gauge group SU(2) × U(1). If, in addition, we are to describe quarks out of which the protons and neutrons in the nuclei are built, we must introduce the gauge group SU (3) corresponding to the three flavours of quarks that are observed in nature. This describes the so called strong interactions along with the electro-weak interactions and the appropriate gauge group for doing this is SU (3) × SU(2) × U(1). The coupling constants for the Lie algebra of this group must be derived from a fundamental coupling constant. The strong interactions are described by means of a Hadronic current that interacts with the current in the electro-weak theory and these interactions break the symmetry leading to the quark masses. Specifically, if Q denotes a three component quark,
Representation Theory and Operational Calculus for SU(2) and SO(3)
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
Furthermore, the pair (A, B) ∈ SU(2) × SU(2) is redundant in the sense that (−A, −B) describes the same rotation as (A, B). In group-theoretic language, SU(2) × SU(2) is the double covering group of SO(4) in the same way that SU (2) is the double cover of SO(3). One writes (SU(2)×SU(2))/[−I,I]≅SO(4).
The Weak Interaction in the Framework of Grand Unification Theories
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
It is not as straightforward to satisfy point 2 above, as it was for example in the case of the weak SU(2)L group. Whilst in equation (5.88), only SU(2)L doublets and singlets arise, i.e. only fundamental and trivial representations, we now require so-called higher representations (see the Appendix (A.2)). A well known example of such a higher representation is a spin 1 state (possibly composed of two fermions). This forms a higher representation of SU(2) in the spinor space. But in addition the combination of two spin 12 particles also produces a spin 0 state, an SU(2) singlet. Formally, this state of affairs is described using the symbolic notation () 2⊗2=3⊕1
Phase space information in a non-linear quantum system containing a Kerr-like medium through Su(1, 1)-algebraic treatment
Published in Journal of Modern Optics, 2018
where , and are the Pauli matrices, and denotes the reduced density matrix of the Su(2)-system. As one of the important applications of Husimi function is the Wehrl theory [38] that introduces the Wehrl density and its entropy. They are powerful tools giving insight into the evolution some quantum effects of the phase space information. Based on the Husimi function, the Wehrl entropy of the Su(2)-system is given by: where represents the Wehrl density of the Su(2)-system, and it is given by
An optical channel modeling of a single mode fiber
Published in Journal of Modern Optics, 2018
Neda Nabavi, Peng Liu, Trevor James Hall
In a coherent detection scheme, the coherent receiver must be phase diverse to compensate for fluctuations in the overall phase (47). It follows from that one is at liberty to set when the state of polarization only of interest, i.e. may be chosen as a member of the special unitary group in two dimensions SU(2). Every member of SU(2) has the Caley–Klein representation (48):
Second quantisation for unrestricted references: formalism and quasi-spin-adaptation of excitation and spin-flip operators
Published in Molecular Physics, 2023
Second quantisation approaches in quantum chemistry are either spin-orbital or spatial orbital based. The first one is applied with success in many areas in chemistry, although it is not trivial to ensure spin-adaptation using it. Coupled cluster (CC) theory is especially complicated in this regard, due to the non-linearity of the ansatz. Szalay et al. solved this problem by introducing an additional set of spin-equations that leads to exact spin-eigenstates if all CSFs are included. While this is usually not possible, the spin equations can be solved to yield approximate eigenstates in a truncated CSF manifold. In this study, we will focus on spatial orbital or spin-free approaches that ultimately put constraints on the commutator of the ansatz operator with the total-spin-squared operator. Already in the 1960s Matsen called for a spin-free formulation of quantum chemistry [31] based on the observation that spin-adapting the spin part of the wavefunction using a two dimensional special unitary group, SU(2), approach is equivalent to constructing the spatial part using irreps of the n dimensional symmetric group, S describing permutation symmetry [32]. But since the Hamiltonian can also be expressed using the generators of the unitary group, it is also possible to give a spin-free representation using representations of the n dimensional unitary group, U(n) [32], of spatial orbitals which can be constructed using second quantisation. Unfortunately, this formulation leads to non-commuting cluster operators for open shells and linear dependencies in the more highly excited manifolds. How this problem can be overcome for CC theory is discussed elsewhere [33], here, we will limit ourselves to the simplest ansatz: configuration interaction singles (CIS) [34], which is a useful starting point for excited state calculations. Beyond restricted CIS (RCIS), CIS has also been extended to open shell systems [35] both in an unrestricted (UCIS) and restricted open shell (ROCIS) fashion. In fact, ROCIS has extended formulations that are not only spin-adapted, but spin-complete (span the entire spin space) [36], explore different multiplets using appropriately constructed flip operators [37, 38] and also ionised states [39]. For a deeper discussion of most spin adaptation related issues, see the book of Pauncz [40].