Basis Of SU Research Articles

Unfolding a quantum master equation into a system of real-valued equations: Computationally effective expansion over the basis of SU(N) generators.

Dynamics of an open N-state quantum system is often modeled with a Markovian master equation describing the evolution of the system density operator. By using generators of SU(N) group as a basis, the density operator can be transformed into a real-valued "coherence-vector." A generator of the dissipative evolution, so-called "Lindbladian," can be expanded over the same basis and recast in the form of a real matrix. Together, these expansions result is a nonhomogeneous system of N^{2}-1 real-valued linear ordinary differential equations. Now one can, e.g., implement standard high-performance algorithms to integrate the system of equations forward in time while being sure in exact preservation of the trace (norm) and Hermiticity of the density operator. However, when performed in a straightforward way, the expansion turns to be an operation of the time complexity O(N^{10}). The complexity can be reduced when the number of dissipative operators is independent of N, which is often the case for physically meaningful models. Here we present an algorithm to transform quantum master equation into a system of real-valued differential equations and propagate it forward in time. By using a specific scalable model, we evaluate computational efficiency of the algorithm and demonstrate that it is possible to handle the model system with N=10^{3} states on a single node of a computer cluster.

Orbifold family unification on six dimensions

We study the possibility of family unification on the basis of SU(N) gauge theory on the 6-dimensional space-time, $M^4\times T^2/Z_N$. We obtain enormous numbers of models with three families of SU(5) matter multiplets and those with three families of the standard model multiplets, from a single massless Dirac fermion with a higher-dimensional representation of SU(N), through the orbifold breaking mechanism.

Controllability of spin 1 systems and realization of ternary SWAP gate in two spin 1 systems coupled with Ising interaction

In this paper, we investigate the controllability of spin 1 systems and the realization of ternary gates. Using dipole and quadrupole operators as the orthogonal basis of su(3) algebra, we discuss the controllability of one spin 1 systems and offer the concept of a complete set of control operators first. Then we present the controllability of two spin 1 systems coupled with Ising interaction and the transforming relations of the drift process of the system. Finally the specific realization of the ternary SWAP gate in these systems is discussed. It takes 9 drift processes and 25 basic control processes.

A quark model of $$\bar \Lambda \Lambda $$ production in $$\bar \Lambda \Lambda $$ interactionsinteractions

A quark model which includes both scalar and vector contributions to the reaction mechanism is used in a DWBA calculation of total and differential cross-sections, polarizations, and spin correlation coefficients for the reaction\(\bar pp \to \bar \Lambda \Lambda \) at laboratory momenta from threshold to 1.92GeV/c. The free parameters of the calculation include the scalar and vector strengths, a quark cluster size parameter, and six parameters in the unknown\(\bar \Lambda \Lambda \) interaction. Excellent agreement with experiment is found for a constructive interference of scalar and vector terms, and for a\(\bar \Lambda \Lambda \) potential which differs from that suggested by several authors on the basis ofSU(3) arguments. The fit to the data is better than that obtained by other quark models, which use only scalar or vector annihilation terms. The agreement with experiment is as good as that found in meson exchange models, which use more parameters than the present calculation.

A quark model of antilambda-lambda production in p̄p interactions

A quark model which includes both scalar and vector contributions to the reaction mechanism is used in a DWBA calculation of total and differential cross sections, polarizations, and spin-correlation coefficients for the reaction p p → Λ Λ at laboratory momenta from threshold to 1.92 GeV/ c. The free parameters of the calculation include the scalar and vector strengths, a quark-cluster size parameter, and six parameters in the unknown Λ Λ interaction. Excellent agreement with experiment is found for a constructive interference of scalar and vector terms, and for a Λ Λ potential which differs from that suggested by several authors on the basis of SU(3) arguments. The fit to the data is better than that obtained by other quark models, which use only scalar or vector annihilation terms. The agreement with experiment is as good as that found in meson-exchange models, which use more parameters than the present calculation.

Theoretical aspects of kaon condensation in neutron matter

Some theoretical aspects of kaon condensation in neutron matter are considered on the basis of SU(3) L×SU(3) R current algebra and PCAC. It is clarified that the critical condition is different from charged pion condensation and the weak interaction plays a significant role near the threshold. The critical density lies around (3–4) n 0 with the standard nuclear density n 0 irrespect ive of the ambiguities of the s̄s content inside the nucleon. For comparison, K +K − -pair condensation in symmetric nuclear matter is also discussed.

Normalized irreducible tensorial matrix expansions applied to an effective sp‐type Hamiltonian for the PES of water

AbstractAny matrix can be expanded on a basis of SU(2) normalized irreducible tensorial matrices, NITM, defined in terms of 3‐j symbols or coupling coefficients of SU(2). The NITM transform under rotations according to Wigner's matrices. If one dimension of an NITM is odd and the other even, the tensor has half‐integer rank. A simple NITM basis consists of all NITM having the same numbers of rows and columns as the expanded matrix. A compound NITM basis consists of two or more simple bases, each spanning a corresponding block in the expanded matrix. The choice of NITM basis for expanding an effective Hamiltonian matrix is a crucial step in formulating a model. To illustrate the use of a compound NITM basis, including nonsquare NITM, an effective sp‐type overlap‐free superposition Hamiltonian is constructed and applied to the photoelectron ionization potential spectrum of water.

Construction of coupling coefficients of SU(4) in a supermultiplet basis

Matrix elements of the unit tensor operators of ranks (1), (2) and (1,1) in the non-orthonormal Draayer basis of SU(4) are found. The evaluation algorithms for the SU(4) coupling coefficients are proposed for coupling one arbitrary and one four-, six- or ten-dimensional irreducible representation of SU(4) in the SU(2)*SU(2) chain, as well as in the case of the semistretched coupling. The construction of the overlaps of the Draayer basis states is considered.

Matrix elements of the labelling operators for SU(4) ⊃ SU(2) × SU(2)

A matrix representation of the labelling operators of Moshinsky and Nagel (1963) and of Partensky and Maguin (1978) in the non-orthogonal Draayer basis of SU(4) is derived; this allows one to solve explicitly the missing label problem for the spin-isospin multiplets in the arbitrary SU(4) supermultiplets. The simplification and degeneration of these representations are considered for two parametric irreducible representations of SU(4), as well as for some special cases of irreps typical of nuclear theory. The expansion of the linearly dependent states of the Draayer basis is discussed.

The Clebsch-Gordan coefficient for SU(m/n) Gel'fand basis

The outer-product reduction coefficients (ORC) of the graded state permutation group, which differ from the ORC of the ordinary permutation group only in signs, have been identified with the Clebsch-Gordan coefficients (CGC) for the special Gel'fand basis of the graded unitary group SU(m/n). The CGC for a general Gel'fand basis of SU(m/n) can be easily obtained from those for the special one. Tables of the SU(m/n) CGC are presented which are valid for arbitrary m and n.

The Casimir invariants and Gel'fand basis of the graded unitary group SU(m/n)

In the space of f-particle product states, the Casimir invariants of the graded unitary group SU(m/n) are functions of the class operators of the graded coordinate permutation group S(f), which are equal to the class operators of the graded state permutation group S(f). Formulae for the quadratic and cubic Casimir invariants are given. It is shown that in general the eigenvalue formulae for the U(m/n) Casimir invariants as functions of the partition can be obtained from those for U(m+n) by simply changing the argument of the functions from (m+n) to (m-n). The quasi-standard basis of S(f) is identified with the Gel'fand basis of SU(m/n), i.e. the irreducible basis classified according to the group chain SU(m/n) contains/implies SU(m/n-1) contains/implies ...SU(m) contains/implies ...SU(2) contains/implies U(1). The special Gel'fand basis of SU(m/n) (for which the weight is restricted to (1,1,...,1)) can be easily obtained from the Gel'fand basis of SU(m+n) with due allowances for modifying sign factors arising from the grading. The construction of a general SU(m/n) Gel'fand basis is also discussed.

The Clebsch-Gordan coefficients and isoscalar factors of the graded unitary group SU(m/n)

It is pointed out that the Clebsch-Gordan coefficients (CGC) of the permutation group are the coefficients which couple irreducible bases (IRB) of SU(m/n) and SU(p/q) to the IRB of SU(mp+nq/mp+np), and the outer-product reduction coefficients (ORC) of the permutation group are the coefficients which couple the IRB of SU(m/n) and SU(p/q) into the IRB of SU(m+p/n+q). The ORC of permutation groups with due modification in signs are the CGC for the special Gel'fand basis of SU(m/n). The isoscalar factor (ISF) for the permutation group chain S(f) contains/implies S(f1)*S(f2) is the ISF for the graded unitary group chain SU(mp+nq/mq+np) contains/implies SU(m/n)*SU(p/q), and the outer-product ISF for the group chain S(f) contains/implies S(f1)*S(f2) is the ISF for SU(mp+nq/mq+np) contains/implies SU(m/n)(X)SU(p/q). All these coefficients can be calculated easily and some are already available.

SU(4) Symmetric Duality Approach to Deep Inelastic Processes and Its Relation to Quark Parton Model

Deep inelastic lepton scattering off proton and neutron is investigated on the basis of SU(4) symmetry and duality, i.e., the absence of exotics in each channel. In this case, several relations among the structure functions, which can be easily compared with experiments, come out to be more restrictive than those obtained in the SU(2) and SU(3) cases. We also study the connection between the duality approach and the quark parton model, and give the answer to the question to what extent we can replace the quark parton model with the duality approach.

Parity non-conserving effects in the reaction [formula omitted

We have calculated the circular polarization and the asymmetry in the distribution of outgoing photons in the reaction n + 2 H → 3 H+γ for thermal incident neutrons. These effects become possible because of an admixture of odd-parity states in the three-body system by means of the weak, parity non-conserving component of the N-N interaction. We have in- vestigated several theoretical models for this interaction and have found that the parity non- conserving effects are generally of order 10 −6. With the exception of potentials derived on the basis of SU(6) considerations, our results do not show remarkable sensitivity to different models.

The nature of effective interaction in the 1f-2 p shell

The effective two-body interaction in the 1f-2p shell given by Kuo and Brown is expressed in the basis of SU 3 representations. The interaction is dominantly central in character and also contains some SU 3 breaking components. The SU 3 mixing caused by the interaction is such that it is greatly reduced when the single-particle energies have a l( l+1) sequence, i.e. the 2p state is below the 1f state. This implies that the two-body interaction is also dominantly quadrupolequadrupole type. To identify explicitly the SU 3 breaking components, the interaction is expressed in terms of radial integrals and the SU 3 symmetry properties of these radial integrals are studied.