# Quantum Features of Macroscopic Fields: Entropy and Dynamics

## Abstract

**:**

## 1. Introduction

## 2. Jones and Mueller Formalism for Light Polarization

## 3. First- and Second-Quantization of Classical Field

- (1)
- $\left\{{f}_{k}\right\}$ form an orthonormal basis in $\mathcal{H}$; and
- (2)
- the classical energy of the field mode ${f}_{k}$ equals $\hslash {\omega}_{k}$.

## 4. Quantum Entropy of Macroscopic Field

## 5. Generalized Quasi-Free Dynamics

## 6. Examples

#### 6.1. Thermal Environment

#### 6.2. Polarization Optics Revisited

- (i)
- $\widehat{\omega}$ is a Hermitian $2\times 2$ matrix describing rotation of polarization vector;
- (ii)
- ${\widehat{\gamma}}_{\downarrow}$ is a positive $2\times 2$ matrix describing absorption of photons by the medium; and
- (iii)
- $\widehat{u}$ are $2\times 2$ unitaries describing depolarization of light by random scattering with the positive weight $\mu \left(du\right)$.

## 7. Concluding Remarks

## Funding

## Conflicts of Interest

## References

- Thorne, K.S.; Blandford, R.D. Modern Classical Physics; Princeton University Press: Oxford, UK, 2017. [Google Scholar]
- Alicki, R. The Theory of Open Systems in Application to Unstable Particles. Rep. Math. Phys.
**1978**, 14, 27–42. [Google Scholar] [CrossRef] - Evans, D.E. Completely positive quasifree maps on the CAR algebra. Commun. Math. Phys.
**1979**, 70, 53–68. [Google Scholar] [CrossRef] - Demoen, B.; Vanheuverzwijn, P.; Verbeure, A. Completely positive quasi-free maps of the CCR-algebra. Rep. Math. Phys.
**1979**, 15, 27–39. [Google Scholar] [CrossRef] - Alicki, R.; Lendi, K. Quantum Dynamical Semigroups and Applications, 2nd ed.; LNP 717; Springer: Berlin, Germany, 2007. [Google Scholar]
- Savenkov, S.N. Jones and Mueller matrices: Structure, symmetry relations and information content. Light Scatt. Rev.
**2009**, 4, 71–119. [Google Scholar] - Alicki, R. On von Neumann and Bell Theorems Applied to Quantumness Tests. Found. Phys.
**2009**, 39, 352–360. [Google Scholar] [CrossRef][Green Version] - Breuer, H.-P.; Petruccione, F. The Theory of Open Quantum Systems; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Rivas, A.; Huelga, S.F. Open Quantum Systems. An Introduction; Springer: Heidelberg, Germany, 2012. [Google Scholar]
- Ingarden, R.S.; Kossakowski, A.; Ohya, M. Information Dynamics and Open Systems; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997. [Google Scholar]
- Gorini, V.; Kossakowski, A.; Sudarshan, E.C.G. Completely positive dynamical semigroups of n-level systems. J. Math. Phys.
**1976**, 17, 821–825. [Google Scholar] [CrossRef] - Lindblad, G. On the generators of quantum dynamical semigroups. Commun. Math. Phys.
**1976**, 48, 119–130. [Google Scholar] [CrossRef] - Alicki, R.; Kosloff, R. Introduction to quantum thermodynamics: History and prospects. In Thermodynamics in the Quantum Regime; Volume 195 of Fundamental Theories of Physics; Springer: Cham, Swizerland, 2019. [Google Scholar]
- Alicki, R. The Markov master equation and the Fermi golden rule. Int. J. Theor. Phys.
**1977**, 16, 351–355. [Google Scholar] [CrossRef] - Alicki, R.; Jenkins, A. Interaction of a quantum field with a rotating heat bath. Ann. Phys.
**2018**, 395, 69–83. [Google Scholar] [CrossRef][Green Version] - Bekenstein, J.D.; Schiffer, M. The many faces of superradiance. Phys. Rev. D
**1998**, 58, 064014. [Google Scholar] [CrossRef][Green Version] - Brito, R.; Cardoso, V.; Pani, P. Superradiance: Energy Extraction, Black-Hole Bombs and Implications for Astrophysics and Particle Physics; Lect. Notes Phys. 906; Springer: Heidelberg, Germany, 2015. [Google Scholar]
- Benatti, F.; Floreanini, R. Tests of Complete Positivity in Fiber Optics. Open Syst. Inf. Dyn.
**2006**, 13, 229–238. [Google Scholar] [CrossRef][Green Version]

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Alicki, R.
Quantum Features of Macroscopic Fields: Entropy and Dynamics. *Entropy* **2019**, *21*, 705.
https://doi.org/10.3390/e21070705

**AMA Style**

Alicki R.
Quantum Features of Macroscopic Fields: Entropy and Dynamics. *Entropy*. 2019; 21(7):705.
https://doi.org/10.3390/e21070705

**Chicago/Turabian Style**

Alicki, Robert.
2019. "Quantum Features of Macroscopic Fields: Entropy and Dynamics" *Entropy* 21, no. 7: 705.
https://doi.org/10.3390/e21070705