# The Schwinger Representation of a Group: Concept and Applications

@article{Chaturvedi2006TheSR, title={The Schwinger Representation of a Group: Concept and Applications}, author={S. Chaturvedi and Giuseppe Marmo and N. Mukunda and R.Simon and Alessandro Zampini}, journal={Reviews in Mathematical Physics}, year={2006}, volume={18}, pages={887-912} }

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for SU(2), SO(3) and SU(n) for all n are constructed via specific carrier spaces and group actions. In the SU(2…

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## References

SHOWING 1-10 OF 19 REFERENCES

### The Schwinger SU(3) construction. I. Multiplicity problem and relation to induced representations

- Mathematics
- 2002

The Schwinger oscillator operator representation of SU(3) is analyzed with particular reference to the problem of multiplicity of irreducible representations. It is shown that with the use of an…

### Wigner–Weyl isomorphism for quantum mechanics on Lie groups

- Mathematics
- 2005

The Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group G is developed in detail. Several features are shown to arise which have no counterparts in the familiar Cartesian…

### Indistinguishability for quantum particles: spin, statistics and the geometric phase

- PhysicsProceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
- 1997

The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins…

### On the SU2 unit tensor

- Mathematics
- 1980

This paper deals with the SU2 ⊆ G unit tensor operators tkμα. In the case where the spinor point group G coincides with U1, then tkμα reduces (up to a constant) to the (Wigner–Racah–Schwinger) tensor…

### On the Representations of the Rotation Group

- Mathematics
- 1962

In a recent article (V. Bargmann, Comm. Pure Appl. Math., 14: 187(1981)) a family of Hilbert spaces were studied whose elements are entire analytic functions of n complex variables. The methods…

### SU(N) coherent states

- Physics
- 2002

We generalize Schwinger boson representation of SU(2) algebra to SU(N) and define coherent states of SU(N) using 2(2N−1−1) bosonic harmonic oscillator creation and annihilation operators. We give an…