## Constructing phase space distributions with internal symmetries

## Abstract

We discuss an ab initio world-line approach to constructing phase space distributions in systems with internal symmetries. Starting from the Schwinger-Keldysh real-time path integral in quantum field theory, we derive the most general extension of the Wigner phase space distribution to include color and spin degrees of freedom in terms of dynamical Grassmann variables. The corresponding Liouville distribution for colored particles, which obey Wong’s equation, has only singlet and octet components, while higher moments are fully constrained by the Grassmann algebra. The extension of phase space dynamics to spin is represented by a generalization of the Pauli-Lubanski vector; its time evolution via the Bargmann-Michel-Telegdi equation also follows from the phase space trajectories of the underlying Grassmann coordinates. Our results for the Liouville phase space distribution in systems with both spin and color are of interest in fields as diverse as chiral fluids, finite temperature field theory and polarized parton distribution functions. We also comment on the role of the chiral anomaly in the phase space dynamics of spinning particles.

- Authors:

- Brookhaven National Lab. (BNL), Upton, NY (United States). Physics Dept.

- Publication Date:

- Research Org.:
- Brookhaven National Lab. (BNL), Upton, NY (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Nuclear Physics (NP) (SC-26); German Research Foundation (DFG)

- OSTI Identifier:
- 1498591

- Alternate Identifier(s):
- OSTI ID: 1505106

- Report Number(s):
- BNL-211515-2019-JAAM

Journal ID: ISSN 2470-0010

- Grant/Contract Number:
- SC0012704; 404640738

- Resource Type:
- Published Article

- Journal Name:
- Physical Review D

- Additional Journal Information:
- Journal Volume: 99; Journal Issue: 5; Journal ID: ISSN 2470-0010

- Publisher:
- American Physical Society (APS)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; anomalies; many-body techniques; nonequilibrium statistical mechanics; path integrals; quantum field theory; quantum kinetic theory; relativistic heavy-ion collisions; relativistic kinetic theory; topology

### Citation Formats

```
Mueller, Niklas, and Venugopalan, Raju. Constructing phase space distributions with internal symmetries. United States: N. p., 2019.
Web. doi:10.1103/PhysRevD.99.056003.
```

```
Mueller, Niklas, & Venugopalan, Raju. Constructing phase space distributions with internal symmetries. United States. doi:10.1103/PhysRevD.99.056003.
```

```
Mueller, Niklas, and Venugopalan, Raju. Fri .
"Constructing phase space distributions with internal symmetries". United States. doi:10.1103/PhysRevD.99.056003.
```

```
@article{osti_1498591,
```

title = {Constructing phase space distributions with internal symmetries},

author = {Mueller, Niklas and Venugopalan, Raju},

abstractNote = {We discuss an ab initio world-line approach to constructing phase space distributions in systems with internal symmetries. Starting from the Schwinger-Keldysh real-time path integral in quantum field theory, we derive the most general extension of the Wigner phase space distribution to include color and spin degrees of freedom in terms of dynamical Grassmann variables. The corresponding Liouville distribution for colored particles, which obey Wong’s equation, has only singlet and octet components, while higher moments are fully constrained by the Grassmann algebra. The extension of phase space dynamics to spin is represented by a generalization of the Pauli-Lubanski vector; its time evolution via the Bargmann-Michel-Telegdi equation also follows from the phase space trajectories of the underlying Grassmann coordinates. Our results for the Liouville phase space distribution in systems with both spin and color are of interest in fields as diverse as chiral fluids, finite temperature field theory and polarized parton distribution functions. We also comment on the role of the chiral anomaly in the phase space dynamics of spinning particles.},

doi = {10.1103/PhysRevD.99.056003},

journal = {Physical Review D},

number = 5,

volume = 99,

place = {United States},

year = {2019},

month = {3}

}

DOI: 10.1103/PhysRevD.99.056003