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arXiv:1110.3903

The applications of the general and reduced Yangian algebras

Li‐Guo Qin, Li-Jun Tian, Jin, Yan-Ling, Guo-Hong Yang

Published Oct 18, 2011• 0 citations
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Abstract

The applications of the general and reduced Yangian Y(sl(2)) and Y(su(3)) algebras are discussed. By taking a special constraint, the representation of Y(sl(2)) and Y(su(3)) can be divided into two 2 \times 2 and three 3 \times 3 blocks diagonal respectively. The general and reduced Yangian Y(sl(2)) and Y(su(3)) are applied to the bi-qubit system and the mixed light pseudoscalar meson state, respectively. We can find that the general ones are not able to make the initial states disentangled by acting on the initial states, however the reduced ones are able to make the initial state disentangled. In addition, we show the effects of Y(su(3)) generators on the the decay channel.

DOI
10.48550/arxiv.1110.3903

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The applications of the general and reduced Yangian Y(sl(2)) and Y(su(3)) algebras are discussed. By taking a special constraint, the representation of Y(sl(2)) and Y(su(3)) can be divided into two 2 \times 2 and three 3 \times 3 blocks diagonal respectively. The general and reduced Yangian Y(sl(2)) and Y(su(3)) are applied to the bi-qubit system and the mixed light pseudoscalar meson state, respectively. We can find that the general ones are not able to make the initial states disentangled by acting on the initial states, however the reduced ones are able to make the initial state disentangled. In addition, we show the effects of Y(su(3)) generators on the the decay channel.
Yangian, as an algebra beyond the Lie algebra, is an infinite dimensional algebra and a powerful mathematical method for inVestigating the new symmetry of quantum systems which are nonlinear and integrable. Based on the su(3) symmetry of the quarK-flaVor in the meson states and the transition property of the generators in Yangian algebra, we study the applications of Yangian algebra Y(su(3)) in the decay of three mixed meson states(, K and Ki0) composed of the three positiVe and negatiVe meson states (, K, K0 and K0). As the transition operators, the eight generators (Ī, Ŭ, V, Ī3 and Ī8) of Yangian algebra Y(su(3)) are acting on the three mixed meson states, respectiVely. Then, the possible decay channels and the changes of the entanglement are studied. Results show: (i) Under the effects of Ī3 and Ī8 within the Lie algebra on the three mixed meson states, the compositions of the final states after decays of the three mixed meson states are not changed as compared with the initial state. The entanglement is not changed for the decay of the mixed meson state with the effect of Ī^8, and the others are changed. (ii) Under the effects of the other six generators (Ī, Ŭ and V) beyond the Lie algebra on the three mixed meson states, the compositions of the final states after the decay are changed compared with the initial state. In the six possible decay channels, the two final states become single states without entanglement; two decay channels are absent; and the entanglements of the final states in the remaining two decays are changed. In addition, the entanglement of the final meson states in the possible six decay channels of the two types K mixed meson states, the charged (K+, K-) and neutral (K0, K0) meson states, are the same two by two. (iii) The three mixed meson states can be circularly transferred by the operators Ī, Ŭ and V, implying the obVious symmetry. In this paper the Yangian method is presented to study the possible decay channels of the mixed meson states and may be used to present a possible interpretation of the new unKnown or Known particle in the decay of the mixed meson.
We now describe the properties of the SU( >K ) chain in more algebraic terms. The relevant symmetry is the SU( K ) Yangian, which is usually referred to as Y ( sl K ). On the other hand, the Yangian symmetry of the Sutherland model with spin 1/2 is given by Y ( gl 2 ). The difference between Y ( sl K ) and Y ( gl K ) is that the charge degrees of freedom U(1) is absent in Y ( sl K ), while the spin degrees of freedom is included in both Y ( sl K ) and Y ( sl K ). In Section 4.8.3, we have removed the U(1) component by means of the freezing trick, i.e., by taking the infinitely large repulsion parameter. In this chapter we take a more algebraic approach to study Y ( sl K ), which makes the relation between Y ( gl K ) and Y ( sl K ) clearer. In Section 9.1, we begin with the construction of the SU( K ) monodromy matrix as the product form. The commuting property of the Cherednik–Dunkl operators plays a crucial role here. The monodromy matrix is the basic building block of the Yangian algbebra, as discussed in Section 8.2. Because the Hamiltonian of the SU( K ) spin chain and the monodromy matrix commute, each eigenstate of the spin chain constitutes a basis for representations of the SU( K ) Yangian. In Section 9.2 and 9.3, we discuss the quantum determinant by comparing it with the ordinary determinant. It is demonstrated in Section 9.4 that the quantum determinant has a scalar nature that leads to a simple evaluation. Because of the scalar nature, however, the quantum determinant cannot distinguish different SU( K ) Yangian states.