# Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea

## Abstract

The connection between real-time quantum field theory (RTQFT) [see, e.g., A. Kamenev and A. Levchenko, Adv. Phys. 58 (2009) 197] and phase-space techniques [E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1882]. Each rotation is a phase-space mapping, generalising the corresponding ordering from free to interacting fields. In particular, response transformation [L.I. Plimak, S. Stenholm, Ann. Phys. (N.Y.) 323 (2008) 1989] extends the normal ordering of free-field operators to the time-normal ordering of Heisenberg operators. Structural properties of the response transformation, such as its association with the nonlinear quantum response problem and the related causality properties, hold for all generalised Keldysh rotations. Furthermore, we argue that response transformation is especially suited for RTQFT formulation of spatial, in particular, relativistic, problems, because it extends cancellation of zero-point fluctuations, characteristic of the normal ordering, to interacting fields. As an example, we considermore »

- Authors:

- Institut fuer Quantenphysik, Universitaet Ulm, 89069 Ulm (Germany)
- (Sweden)
- (Finland)

- Publication Date:

- OSTI Identifier:
- 22157008

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 327; Journal Issue: 11; Other Information: Copyright (c) 2012 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; CAUSALITY; COMPARATIVE EVALUATIONS; ELECTROMAGNETIC FIELDS; FIELD OPERATORS; FLUCTUATIONS; MAPPING; PHASE SPACE; QUANTUM ELECTRODYNAMICS; RELATIVISTIC RANGE; ROTATION; SYMMETRY; TRANSFORMATIONS

### Citation Formats

```
Plimak, L.I., E-mail: lev.plimak@uni-ulm.de, Stenholm, S., Physics Department, Royal Institute of Technology, KTH, Stockholm, and Laboratory of Computational Engineering, HUT, Espoo.
```*Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea*. United States: N. p., 2012.
Web. doi:10.1016/J.AOP.2012.06.007.

```
Plimak, L.I., E-mail: lev.plimak@uni-ulm.de, Stenholm, S., Physics Department, Royal Institute of Technology, KTH, Stockholm, & Laboratory of Computational Engineering, HUT, Espoo.
```*Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea*. United States. doi:10.1016/J.AOP.2012.06.007.

```
Plimak, L.I., E-mail: lev.plimak@uni-ulm.de, Stenholm, S., Physics Department, Royal Institute of Technology, KTH, Stockholm, and Laboratory of Computational Engineering, HUT, Espoo. Thu .
"Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea". United States. doi:10.1016/J.AOP.2012.06.007.
```

```
@article{osti_22157008,
```

title = {Causal signal transmission by quantum fields. V: Generalised Keldysh rotations and electromagnetic response of the Dirac sea},

author = {Plimak, L.I., E-mail: lev.plimak@uni-ulm.de and Stenholm, S. and Physics Department, Royal Institute of Technology, KTH, Stockholm and Laboratory of Computational Engineering, HUT, Espoo},

abstractNote = {The connection between real-time quantum field theory (RTQFT) [see, e.g., A. Kamenev and A. Levchenko, Adv. Phys. 58 (2009) 197] and phase-space techniques [E. Wolf and L. Mandel, Optical Coherence and Quantum Optics (Cambridge, 1995)] is investigated. The Keldysh rotation that forms the basis of RTQFT is shown to be a phase-space mapping of the quantum system based on the symmetric (Weyl) ordering. Following this observation, we define generalised Keldysh rotations based on the class of operator orderings introduced by Cahill and Glauber [K.E. Cahill, R.J. Glauber, Phys. Rev. 177 (1969) 1882]. Each rotation is a phase-space mapping, generalising the corresponding ordering from free to interacting fields. In particular, response transformation [L.I. Plimak, S. Stenholm, Ann. Phys. (N.Y.) 323 (2008) 1989] extends the normal ordering of free-field operators to the time-normal ordering of Heisenberg operators. Structural properties of the response transformation, such as its association with the nonlinear quantum response problem and the related causality properties, hold for all generalised Keldysh rotations. Furthermore, we argue that response transformation is especially suited for RTQFT formulation of spatial, in particular, relativistic, problems, because it extends cancellation of zero-point fluctuations, characteristic of the normal ordering, to interacting fields. As an example, we consider quantised electromagnetic field in the Dirac sea. In the time-normally-ordered representation, dynamics of the field looks essentially classical (fields radiated by currents), without any contribution from zero-point fluctuations. For comparison, we calculate zero-point fluctuations of the interacting electromagnetic field under orderings other than time-normal. The resulting expression is physically inconsistent: it does not obey the Lorentz condition, nor Maxwell's equations. - Highlights: Black-Right-Pointing-Pointer The Keldysh rotation is a phase-space mapping based on Weyl's operator ordering. Black-Right-Pointing-Pointer Generalised Keldysh rotations (GKRs) based on other orderings are introduced. Black-Right-Pointing-Pointer Special properties of the GKR based on the normal ordering are elucidated. Black-Right-Pointing-Pointer In relativistic QED, other rotations are shown to be physically inconsistent.},

doi = {10.1016/J.AOP.2012.06.007},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 11,

volume = 327,

place = {United States},

year = {2012},

month = {11}

}