References analysis of the Wikipedia article "QED vacuum" in the English version

arxiv.org

For a review, see Paul Busch (2008). "Chapter 3: The Time–Energy Uncertainty Relation". In Muga, J. G.; Sala Mayato, R.; Egusquiza, Í. L. (eds.). Time in Quantum Mechanics (2nd ed.). Springer. p. 73ff. arXiv:quant-ph/0105049. Bibcode:2002tqm..conf...69B. doi:10.1007/978-3-540-73473-4_3. ISBN 978-3-540-73472-7. S2CID 14119708.

Heyl, Jeremy S.; Hernquist, Lars (1997). "Birefringence and Dichroism of the QED Vacuum". J. Phys. A30 (18): 6485–6492. arXiv:hep-ph/9705367. Bibcode:1997JPhA...30.6485H. doi:10.1088/0305-4470/30/18/022. S2CID 32306183.

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Cao, Tian Yu, ed. (2004). Conceptual Foundations of Quantum Field Theory. Cambridge University Press. p. 179. ISBN 978-0-521-60272-3. For each stationary classical background field there is a ground state of the associated quantized field. This is the vacuum for that background.

Mackay, Tom G.; Lakhtakia, Akhlesh (2010). Electromagnetic Anisotropy and Bianisotropy: A Field Guide. World Scientific. p. 201. ISBN 978-981-4289-61-0.

Shankar, Ramamurti (1994). Principles of Quantum Mechanics (2nd ed.). Springer. p. 507. ISBN 978-0-306-44790-7.

Silk, Joseph (2005). On the Shores of the Unknown: A Short History of the Universe. Cambridge University Press. p. 62. ISBN 978-0-521-83627-2.

A vaguer description is provided by Allday, Jonathan (2002). Quarks, Leptons and the Big Bang (2nd ed.). CRC Press. p. 224. ISBN 978-0-7503-0806-9. The interaction will last for a certain duration

Δ t

. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies

Δ E

This "borrowing" idea has led to proposals for using the zero-point energy of vacuum as an infinite reservoir and a variety of "camps" about this interpretation. See, for example, King, Moray B. (2001). Quest for Zero Point Energy: Engineering Principles for 'Free Energy' Inventions. Adventures Unlimited Press. p. 124ff. ISBN 978-0-932813-94-7.

Quantities satisfying a canonical commutation rule are said to be noncompatible observables, by which is meant that they can both be measured simultaneously only with limited precision. See Itô, Kiyosi, ed. (1993). "§ 351 (XX.23) C: Canonical commutation relations". Encyclopedic Dictionary of Mathematics (2nd ed.). MIT Press. p. 1303. ISBN 978-0-262-59020-4.

Schwabl, Franz (2007). "§ 3.1.3: The zero-point energy". Quantum Mechanics (4th ed.). Springer. p. 54. ISBN 978-3-540-71932-8.

Lambropoulos, Peter; Petrosyan, David (2007). Fundamentals of Quantum Optics and Quantum Information. Springer. p. 30. Bibcode:2007fqoq.book.....L. ISBN 978-3-540-34571-8.

Vogel, Werner; Welsch, Dirk-Gunnar (2006). "Chapter 2: Elements of quantum electrodynamics". Quantum Optics (3rd ed.). Wiley-VCH. p. 18. ISBN 978-3-527-40507-7.

This commutation relation is oversimplified, and a correct version replaces the

δ

product on the right by the transverse

δ

-tensor:

\delta _{\perp \ ij}(\mathbf {x} )={\frac {1}{8\pi ^{3}}}\int d^{3}\mathbf {k} \left(\delta _{ij}-{\hat {u}}_{i}{\hat {u}}_{j}\right)e^{i\mathbf {k\cdot x} }\ ,

where

û

is the unit vector of

k

û = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠k/k⁠

. For a discussion see, Compagno, G.; Passante, R.; Persico, F. (2005). "§2.1 Canonical quantization in the Coulomb gauge". Atom-Field Interactions and Dressed Atoms. Cambridge Studies in Modern Optics, vol. 17. Cambridge University Press. p. 31. ISBN 978-0-521-01972-9.

Vogel, Werner; Welsch, Dirk-Gunnar (2006). "§2.2.1 Canonical quantization: Eq. (2.50)". Quantum Optics (3rd ed.). Wiley-VCH. p. 21. ISBN 978-3-527-40507-7.

Grynberg, Gilbert; Aspect, Alain; Fabre, Claude (2010). "§5.2.2 Vacuum fluctuations and their physical consequences". Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light. Cambridge University Press. p. 351. ISBN 978-0-521-55112-0.

Parker, Ian (2003). Biophotonics, Volume 360, Part 1. Academic Press. p. 516. ISBN 978-0-12-182263-7.

Bregant, M.; et al. (2003). "Particle laser production at PVLAS: Recent developments". In Curwen Spooner, Neil John; Kudryavtsev, Vitaly (eds.). Proceedings of the Fourth International Workshop on the Identification of Dark Matter: York, UK, 2-6 September 2002. World Scientific. ISBN 9789812791313.

Gottfried, Kurt; Weisskopf, Victor Frederick (1986). Concepts of Particle Physics, Volume 2. Oxford University Press. p. 259. ISBN 978-0195033939.

Zeidler, Eberhard (2011). "§19.1.9 Vacuum polarization in quantum electrodynamics". Quantum Field Theory, Volume III: Gauge Theory: A Bridge Between Mathematicians and Physicists. Springer. p. 952. ISBN 978-3-642-22420-1.

Peskin, Michael Edward; Schroeder, Daniel V. (1995). "§7.5 Renormalization of the electric charge". An Introduction to Quantum Field Theory. Westview Press. p. 244. ISBN 978-0-201-50397-5.

Schweber, Silvan S. (2003). "Elementary particles". In Heilbron, J. L. (ed.). The Oxford Companion to the History of Modern Science. Oxford University Press. pp. 246–247. ISBN 978-0-19-511229-0. Thus in QED the presence of an electric charge

e o

polarizes the "vacuum" and the charge that is observed at a large distance differs from

e o

and is given by

e = ⁠ e o / ε ⁠

with

ε

the dielectric constant of the vacuum.

Donoghue, John F.; Golowich, Eugene; Holstein, Barry R. (1994). Dynamics of the Standard Model. Cambridge University Press. p. 47. ISBN 978-0-521-47652-2.

QCD vacuum is paramagnetic, while QED vacuum is diamagnetic. See Bertulani, Carlos A. (2007). Nuclear Physics in a Nutshell. Princeton University Press. p. 26. Bibcode:2007npn..book.....B. ISBN 978-0-691-12505-3.

Mészáros, Peter (1992). "§2.6 Quantum electrodynamics in strong fields". High-Energy Radiation from Magnetized Neutron Stars. University of Chicago Press. p. 56. ISBN 978-0-226-52094-0.

Hartemann, Frederic V. (2002). High-Field Electrodynamics. CRC Press. p. 428. ISBN 978-0-8493-2378-2.

Mendonça, José Tito; Eliezer, Shalom (2008). "Nuclear and particle physics with ultraintense lasers". In Eliezer, Shalom; Mima, Kunioki (eds.). Applications of Laser-Plasma Interactions. CRC Press. p. 145. ISBN 978-0-8493-7604-7.

Luciano Boi (2009). "Creating the physical world ex nihilo? On the quantum vacuum and its fluctuations". In Carafoli, Ernesto; Danieli, Gian Antonio; Longo, Giuseppe O. (eds.). The Two Cultures: Shared Problems. Springer. p. 55. ISBN 978-88-470-0868-7.

Dirac, P. A. M. (2001). Jong-Ping Hsu; Yuanzhong Zhang (eds.). Lorentz and Poincaré Invariance: 100 Years of Relativity. World Scientific. p. 440. ISBN 978-981-02-4721-8.

For example, see Gambini, Rodolfo; Pullin, Jorge (2010). "Chapter 1: Why quantize gravity?". A First Course in Loop Quantum Gravity. Oxford University Press. p. 1. ISBN 978-0-19-959075-9. and Rovelli, Carlo (2004). "§5.4.2 Much ado about nothing: the vacuum". Quantum Gravity. Cambridge University Press. p. 202ff. ISBN 978-0-521-83733-0. We use three distinct notions of vacuum in quantum gravity

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QED vacuum (English Wikipedia)

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