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This issue goes beyond understanding the role of symmetries within the framework of Continuous Group Theory, in the sense that it attempts to exploit such study to understand the behavior of both classical and quantum physical systems.
In this sense, the first two chapters begin by utilizing the background of existing symmetry beyond rotational symmetry, by studying simple classical and quantum systems, and conclude by finding the solution in a straightforward and, above all, elegant manner, without the need to solve differential equations.
For more complex problems, it is necessary to study more complex mathematical systems in necessary depth, such as Lie groups and the corresponding Lie algebras. Of particular interest is the study of the representations of the related operators of the algebra in the form of matrices, especially the irreducible representations, which can be identified with interesting physical variables.
This process leads to the study of both the classical groups and Lie algebras related to unitary, special unitary, orthogonal, and symplectic transformations, as well as the 5 special algebras G2, F4, E6, E7, and E8. This is done within the framework of the Cartan-Weyl theory through root vectors and root diagrams, particularly those in the Dynkin representation.
In this way, the representations are indicated by characteristic sets of integers known as weights. One of the weights is the so-called maximal weight, which is unambiguously defined for each representation of the algebra. All weights can arise from this through the action of appropriate lowering operators of the algebra as well as the corresponding states that characterize a given symmetry.
Alternatively, the irreducible representations can also be described by Young tableaux relating to the discrete symmetry Sn. These diagrams provide convenient formulas for the reduction of the Kronecker product of irreducible representations of a given semisimple algebra. The correlation between Young tableaux and the Dynkin basis is provided.
In this way, the calculation of the Clebsch-Gordan coefficients in the reduction of the Kronecker product becomes simple, and useful tables are provided. Often, for a clearer understanding of the subject, it is necessary to consider the possible subsymmetries of the maximal symmetry, i.e., subalgebras of the algebra. Of particular importance are the maximal subalgebras, which are studied in simple cases and recorded in tables.
Manufacturer
- Author
- Ioannis D. Vergados
- Publisher
- Symmetria
- Type
- Biology of Natural Sciences, Physics of Positive Sciences
- Language
- Greek
- Subtitle
- Continuous Groups and Applications in Physics
- Cover
- Soft
- Number of Pages
- 408
- Release Date
- 04/03/2022
- Publication Date
- 2022
- Dimensions
- 17x24 cm
- ISBN-13
- 9789602664834
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