# A Compact Formula for Rotations as Spin Matrix Polynomials

## Abstract

Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. Here, the simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.

- Authors:

- Univ. of Miami, Coral Gables, FL (United States)
- Durham Univ., Durham (United Kingdom)
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- Argonne National Laboratory; USDOE Office of Science (SC)

- OSTI Identifier:
- 1395147

- Grant/Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications

- Additional Journal Information:
- Journal Volume: 10; Journal ID: ISSN 1815-0659

- Publisher:
- Institute of Mathematics, National Academy of Sciences Ukraine

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; spin matrices; matrix exponentials

### Citation Formats

```
Curtright, Thomas L., Fairlie, David B., and Zachos, Cosmas K.
```*A Compact Formula for Rotations as Spin Matrix Polynomials*. United States: N. p., 2014.
Web. doi:10.3842/SIGMA.2014.084.

```
Curtright, Thomas L., Fairlie, David B., & Zachos, Cosmas K.
```*A Compact Formula for Rotations as Spin Matrix Polynomials*. United States. doi:10.3842/SIGMA.2014.084.

```
Curtright, Thomas L., Fairlie, David B., and Zachos, Cosmas K. Tue .
"A Compact Formula for Rotations as Spin Matrix Polynomials". United States.
doi:10.3842/SIGMA.2014.084. https://www.osti.gov/servlets/purl/1395147.
```

```
@article{osti_1395147,
```

title = {A Compact Formula for Rotations as Spin Matrix Polynomials},

author = {Curtright, Thomas L. and Fairlie, David B. and Zachos, Cosmas K.},

abstractNote = {Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group. Here, the simple explicit result exhibits connections between group theory, combinatorics, and Fourier analysis, especially in the large spin limit. Salient intuitive features of the formula are illustrated and discussed.},

doi = {10.3842/SIGMA.2014.084},

journal = {Symmetry, Integrability and Geometry: Methods and Applications},

number = ,

volume = 10,

place = {United States},

year = {Tue Aug 12 00:00:00 EDT 2014},

month = {Tue Aug 12 00:00:00 EDT 2014}

}

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Cited by: 8 works

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