| induced representations | |
|---|---|
| Name | Induced Representations |
| Field | Representation theory and Quantum Physics |
| Introduced by | Frobenius and Wigner |
induced representations
Induced representations is a fundamental concept in representation theory and Quantum Physics, which describes how a group representation of a subgroup can be extended to a representation of the larger group. This concept is crucial in understanding the symmetry properties of physical systems and has numerous applications in Quantum Mechanics and Quantum Field Theory. The theory of induced representations was developed by Frobenius and Wigner, and has since been extensively used in various areas of physics and mathematics, including the work of Harish-Chandra and Gelfand.
Induced Representations Induced representations is a technique used to construct a representation of a group from a representation of one of its subgroups. This is particularly useful in Quantum Physics, where the symmetry group of a physical system can be used to determine its energy levels and transition probabilities. The concept of induced representations is closely related to the work of Wigner, who used it to develop the theory of group representations in Quantum Mechanics. Other notable physicists, such as Dirac and Heisenberg, have also contributed to the development of induced representations in Quantum Physics. The University of Göttingen and the Institute for Advanced Study have been centers of research in this area, with notable researchers including Von Neumann and Dyson.
The mathematical background for induced representations involves the theory of group representations and module theory. A group representation is a homomorphism from a group to the general linear group of a vector space, and induced representations provide a way to construct new representations from existing ones. The Peter-Weyl theorem and the Frobenius reciprocity theorem are key results in the theory of induced representations, and have been used by researchers such as Bott and Atiyah to study the properties of Lie groups and their representations. The American Mathematical Society and the London Mathematical Society have published numerous papers on this topic, including work by Langlands and Arthur.
in Group Theory In group theory, induced representations are used to study the properties of group representations and their relationships to subgroups. The Mackey theory of induced representations provides a framework for constructing representations of a group from representations of its subgroups, and has been used by researchers such as Mackey and Blattner to study the properties of induced representations. The theory of characters is also closely related to induced representations, and has been used by researchers such as Brauer and Feit to study the properties of group representations. The Institute for Advanced Study and the University of Chicago have been centers of research in this area, with notable researchers including Gelfand and Kirillov.
in Quantum Mechanics Induced representations have numerous applications in Quantum Mechanics, particularly in the study of symmetry properties of physical systems. The Wigner-Eckart theorem provides a framework for using induced representations to study the properties of physical systems with symmetry, and has been used by researchers such as Wigner and Eckart to study the properties of atomic spectra and molecular spectra. The theory of angular momentum is also closely related to induced representations, and has been used by researchers such as Dirac and Heisenberg to study the properties of particles with spin. The Los Alamos National Laboratory and the European Organization for Nuclear Research have been centers of research in this area, with notable researchers including Feynman and Schwinger.
Induced representations are closely related to the concept of symmetry in physics, and provide a framework for understanding the relationships between symmetry and conservation laws. The Noether's theorem provides a framework for using symmetry to derive conservation laws, and has been used by researchers such as Noether and Hilbert to study the properties of physical systems with symmetry. The theory of Lie groups is also closely related to induced representations, and has been used by researchers such as Lie and Cartan to study the properties of symmetry groups and their representations. The University of California, Berkeley and the Massachusetts Institute of Technology have been centers of research in this area, with notable researchers including Chern and Simons.
in Quantum Field Theory Induced representations also have applications in Quantum Field Theory, particularly in the study of symmetry properties of field theories. The theory of currents is closely related to induced representations, and has been used by researchers such as Gell-Mann and Lehmann to study the properties of field theories with symmetry. The theory of anomalies is also closely related to induced representations, and has been used by researchers such as Adler and Bell to study the properties of field theories with anomalies. The Stanford Linear Accelerator Center and the CERN have been centers of research in this area, with notable researchers including Weinberg and Glashow.
There are many examples and case studies of induced representations in Quantum Physics, including the study of atomic spectra and molecular spectra. The hydrogen atom is a classic example of a physical system with symmetry, and has been studied using induced representations by researchers such as Schrodinger and Dirac. The theory of nuclear reactions is also closely related to induced representations, and has been used by researchers such as Feynman and Bethe to study the properties of nuclear reactions. The Lawrence Berkeley National Laboratory and the Argonne National Laboratory have been centers of research in this area, with notable researchers including Seaborg and Teller.