QED vacuum

The QED vacuum or quantum electrodynamic vacuum is the field-theoretic vacuum of quantum electrodynamics. It is the lowest energy state (the ground state) of the electromagnetic field when the fields are quantized.^[1] When the Planck constant is hypothetically allowed to approach zero, QED vacuum is converted to classical vacuum, which is to say, the vacuum of classical electromagnetism.^[2]^[3]

Another field-theoretic vacuum is the QCD vacuum of the Standard Model.

Fluctuations

The video of an experiment showing vacuum fluctuations (in the red ring) amplified by spontaneous parametric down-conversion.

The QED vacuum is subject to fluctuations about a dormant zero average-field condition;^[4] Here is a description of the quantum vacuum:

The quantum theory asserts that a vacuum, even the most perfect vacuum devoid of any matter, is not really empty. Rather the quantum vacuum can be depicted as a sea of continuously appearing and disappearing [pairs of] particles that manifest themselves in the apparent jostling of particles that is quite distinct from their thermal motions. These particles are ‘virtual’, as opposed to real, particles. ...At any given instant, the vacuum is full of such virtual pairs, which leave their signature behind, by affecting the energy levels of atoms.
— Joseph Silk On the Shores of the Unknown, p. 62^[5]

Virtual particles

It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg energy-time uncertainty principle: $\Delta E\Delta t\geq {\frac {\hbar }{2}}\,,$ (where $Δ E$ and $Δ t$ are energy and time variations, and $ħ$ the Planck constant divided by 2 $π$ ) arguing along the lines that the short lifetime of virtual particles allows the "borrowing" of large energies from the vacuum and thus permits particle generation for short times.^[6]

This interpretation of the energy-time uncertainty relation is not universally accepted, however.^[7]^[8] One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty $Δ t$ determines a "budget" for borrowing energy $Δ E$ . Another issue is the meaning of "time" in this relation, because energy and time (unlike position $q$ and momentum $p$ , for example) do not satisfy a canonical commutation relation (such as $[q, p] = iħ$ ).^[9] Various schemes have been advanced to construct an observable that has some kind of time interpretation, and yet does satisfy a canonical commutation relation with energy.^[10]^[11] The many approaches to the energy-time uncertainty principle are a continuing subject of study.^[11]

Quantization of the fields

The Heisenberg uncertainty principle does not allow a particle to exist in a state in which the particle is simultaneously at a fixed location, say the origin of coordinates, and has also zero momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations; if confined, it has a zero-point energy.^[12]

An uncertainty principle applies to all quantum mechanical operators that do not commute.^[13] In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.^[14]

The standard approach to the quantization of the electromagnetic field begins by introducing a vector potential

A

and a scalar potential

V

to represent the basic electromagnetic electric field

E

and magnetic field

B

using the relations:^[14]

{\begin{aligned}\mathbf {B} &=\mathbf {\nabla \times A} \,,\\\mathbf {E} &=-{\frac {\partial }{\partial t}}\mathbf {A} -\mathbf {\nabla } V\,.\end{aligned}}

The vector potential is not completely determined by these relations, leaving open a so-called gauge freedom. Resolving this ambiguity using the Coulomb gauge leads to a description of the electromagnetic fields in the absence of charges in terms of the vector potential and the momentum field

Π

, given by:

\mathbf {\Pi } =\varepsilon _{0}{\frac {\partial }{\partial t}}\mathbf {A} \,,

where

ε 0

is the electric constant of the SI units. Quantization is achieved by insisting that the momentum field and the vector potential do not commute. That is, the equal-time commutator is:^[15]

{\bigl [}\Pi _{i}(\mathbf {r} ,t),\ A_{j}(\mathbf {r} ',t){\bigr ]}=-i\hbar \delta _{ij}\delta (\mathbf {r} -\mathbf {r} ')\,,

where

r

,

r'

are spatial locations,

ħ

is the reduced Planck constant,

δ ij

is the Kronecker delta and

δ (r - r')

is the Dirac delta function. The notation

[ , ]

denotes the commutator.

Quantization can be achieved without introducing the vector potential, in terms of the underlying fields themselves:^[16]

\left[{\hat {E}}_{k}({\boldsymbol {r}}),{\hat {B}}_{k'}({\boldsymbol {r}}')\right]=-\epsilon _{kk'm}{\frac {i\hbar }{\varepsilon _{0}}}{\frac {\partial }{\partial x_{m}}}\delta ({\boldsymbol {r-r'}})\,,

where the circumflex denotes a Schrödinger time-independent field operator, and

ε ijk

is the antisymmetric Levi-Civita tensor.

Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero.^[17] The electromagnetic field has therefore a zero-point energy, and a lowest quantum state. The interaction of an excited atom with this lowest quantum state of the electromagnetic field is what leads to spontaneous emission, the transition of an excited atom to a state of lower energy by emission of a photon even when no external perturbation of the atom is present.^[18]

Electromagnetic properties

The polarization of the observed light in the extremely strong magnetic field suggests that the empty space around the neutron star RX J1856.5−3754 is subject to the vacuum birefringence.^[19]

As a result of quantization, the quantum electrodynamic vacuum can be considered as a material medium.^[20] It is capable of vacuum polarization.^[21]^[22] In particular, the force law between charged particles is affected.^[23]^[24] The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple $ε 0$ of the classical vacuum. Likewise, its permeability can be calculated and differs slightly from $μ 0$ . This medium is a dielectric with relative dielectric constant > 1, and is diamagnetic, with relative magnetic permeability < 1.^[25]^[26] Under some extreme circumstances in which the field exceeds the Schwinger limit (for example, in the very high fields found in the exterior regions of pulsars^[27]), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields.^[28] Calculations also indicate birefringence and dichroism at high fields.^[29] Many of electromagnetic effects of the vacuum are small, and only recently have experiments been designed to enable the observation of nonlinear effects.^[30] PVLAS and other teams are working towards the needed sensitivity to detect QED effects.

Attainability

A perfect vacuum is itself only attainable in principle.^[31]^[32] It is an idealization, like absolute zero for temperature, that can be approached, but never actually realized:

One reason [a vacuum is not empty] is that the walls of a vacuum chamber emit light in the form of black-body radiation...If this soup of photons is in thermodynamic equilibrium with the walls, it can be said to have a particular temperature, as well as a pressure. Another reason that perfect vacuum is impossible is the Heisenberg uncertainty principle which states that no particles can ever have an exact position ...Each atom exists as a probability function of space, which has a certain nonzero value everywhere in a given volume. ...More fundamentally, quantum mechanics predicts ...a correction to the energy called the zero-point energy [that] consists of energies of virtual particles that have a brief existence. This is called vacuum fluctuation.
— Luciano Boi, "Creating the physical world ex nihilo?" p. 55^[31]

Virtual particles make a perfect vacuum unrealizable, but leave open the question of attainability of a quantum electrodynamic vacuum or QED vacuum. Predictions of QED vacuum such as spontaneous emission, the Casimir effect and the Lamb shift have been experimentally verified, suggesting QED vacuum is a good model for a high quality realizable vacuum. There are competing theoretical models for vacuum, however. For example, quantum chromodynamic vacuum includes many virtual particles not treated in quantum electrodynamics. The vacuum of quantum gravity treats gravitational effects not included in the Standard Model.^[33] It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.

References

^ 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.
^ Classical vacuum is not a material medium, but a reference state used to define the SI units. Its permittivity is the electric constant and its permeability is the magnetic constant, both of which are exactly known by definition, and are not measured properties. See Mackay & Lakhtakia, p. 20, footnote 6.
^ 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.
^ For an example, see Davies, P. C. W. (1982). The Accidental Universe. Cambridge University Press. p. 106. ISBN 978-0-521-28692-3.
^ 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.
^ Busch, Paul; Grabowski, Marian; Lahti, Pekka J. (1995). "§III.4: Energy and time". Operational Quantum Physics. Springer. p. 77. ISBN 978-3-540-59358-4.
^ ^a ^b 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.
^ 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.
^ ^a ^b 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.
^ "First Signs of Weird Quantum Property of Empty Space? – VLT observations of neutron star may confirm 80-year-old prediction about the vacuum". www.eso.org. Retrieved 5 December 2016.
^ 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.
^ 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.
^ 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.
^ ^a ^b 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

This article incorporates material from the Citizendium article "Vacuum (quantum electrodynamic)", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.

[Cao-1] 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-2] Mackay, Tom G.; Lakhtakia, Akhlesh (2010). Electromagnetic Anisotropy and Bianisotropy: A Field Guide. World Scientific. p. 201. ISBN 978-981-4289-61-0.

[classical-3] Classical vacuum is not a material medium, but a reference state used to define the SI units. Its permittivity is the electric constant and its permeability is the magnetic constant, both of which are exactly known by definition, and are not measured properties. See Mackay & Lakhtakia, p. 20, footnote 6.

[Shankar-4] Shankar, Ramamurti (1994). Principles of Quantum Mechanics (2nd ed.). Springer. p. 507. ISBN 978-0-306-44790-7.

[Silk-5] 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.

[Davies-6] For an example, see Davies, P. C. W. (1982). The Accidental Universe. Cambridge University Press. p. 106. ISBN 978-0-521-28692-3.

[Allday-7] 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 $Δ E$ .

[King-8] 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.

[commutation-9] 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.

[Busch0-10] Busch, Paul; Grabowski, Marian; Lahti, Pekka J. (1995). "§III.4: Energy and time". Operational Quantum Physics. Springer. p. 77. ISBN 978-3-540-59358-4.

[Busch-11] 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.

[Schwabl-12] Schwabl, Franz (2007). "§ 3.1.3: The zero-point energy". Quantum Mechanics (4th ed.). Springer. p. 54. ISBN 978-3-540-71932-8.

[Petrosyan-13] 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.

[Vogel2-14] 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.

[Compagno-15] 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-16] 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-17] 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-18] Parker, Ian (2003). Biophotonics, Volume 360, Part 1. Academic Press. p. 516. ISBN 978-0-12-182263-7.

[19] "First Signs of Weird Quantum Property of Empty Space? – VLT observations of neutron star may confirm 80-year-old prediction about the vacuum". www.eso.org. Retrieved 5 December 2016.

[medium-20] 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.

[Weisskopf-21] Gottfried, Kurt; Weisskopf, Victor Frederick (1986). Concepts of Particle Physics, Volume 2. Oxford University Press. p. 259. ISBN 978-0195033939.

[Zeidler-22] 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.

[Schroeder-23] 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-24] 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-25] 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.

[Bertulani-26] 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.

[Meszaros-27] 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-28] Hartemann, Frederic V. (2002). High-Field Electrodynamics. CRC Press. p. 428. ISBN 978-0-8493-2378-2.

[Heyl-29] 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.

[nonlinear-30] 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.

[Longo-31] 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-32] 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.

[gravity-33] 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

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

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[33]