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{{Short description|Lowest energy state in quantum electrodynamics}}
{{For|articles related to the QED vacuum|Quantum vacuum (disambiguation)}}

The '''QED vacuum''' or '''quantum electrodynamic vacuum''' is the [[Vacuum state|field-theoretic vacuum]] of [[quantum electrodynamics]]. It is the lowest energy state (the [[ground state]]) of the electromagnetic field when the [[Quantization of the electromagnetic field|fields are quantized]].<ref name=Cao/> When the [[Planck constant]] is hypothetically allowed to approach zero, QED vacuum is converted to [[Vacuum#In electromagnetism|classical vacuum]], which is to say, the vacuum of classical electromagnetism.<ref name=Mackay/><ref name=classical/>

Another field-theoretic vacuum is the [[QCD vacuum]] of the [[Standard Model]].
[[File:Photon-photon scattering.svg|thumb|A [[Feynman diagram]] (''box diagram'') for photon-photon scattering, one photon scatters from the transient [[quantum fluctuation|vacuum charge fluctuation]]s of the other]]

== Fluctuations ==
{{Main|Quantum fluctuation}}

[[File:Vacuum fluctuations revealed through spontaneous parametric down-conversion.ogv|thumb|right|150px| 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;<ref name=Shankar/> Here is a description of the quantum vacuum:
{{quote|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<ref name= Silk/>}}

=== Virtual particles === 
{{Main|Virtual particles|Quantum fluctuation}}
It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg [[Uncertainty principle#Energy.E2.80.93time uncertainty principle|energy-time uncertainty principle]]:
<math display="block">\Delta E \Delta t \ge \frac{\hbar}{2} \, , </math>
(where {{math|Δ''E''}} and {{math|Δ''t''}} are [[energy]] and [[time]] variations, and {{mvar|ħ}} the [[Planck constant]] divided by 2{{pi}}) 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.<ref name=Davies/>

This interpretation of the energy-time uncertainty relation is not universally accepted, however.<ref name=Allday/><ref name=King/> One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty {{math|Δ''t''}} determines a "budget" for borrowing energy {{math|Δ''E''}}.  Another issue is the meaning of "time" in this relation, because energy and time (unlike position {{mvar|q}} and momentum {{mvar|p}}, for example) do not satisfy a [[canonical commutation relation]] (such as {{math|1=[''q'', ''p''] = ''iħ''}}).<ref name=commutation/> 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.<ref name=Busch0/><ref name=Busch/> The many approaches to the energy-time uncertainty principle are a continuing subject of study.<ref name=Busch/>

=== Quantization of the fields ===
{{Main|Quantization of the electromagnetic field}}
The [[Heisenberg Uncertainty Principle|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]].<ref name=Schwabl/>

An uncertainty principle applies to all quantum mechanical operators that do not [[commutator|commute]].<ref name=Petrosyan/> In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.<ref name=Vogel2/>
:The standard approach to the quantization of the electromagnetic field begins by introducing a ''vector'' potential {{math|'''A'''}} and a ''scalar'' potential {{mvar|V}} to represent the basic electromagnetic electric field {{math|'''E'''}} and magnetic field {{math|'''B'''}} using the relations:<ref name=Vogel2/> <math display="block">\begin{align}
\mathbf B &= \mathbf {\nabla  \times A}\,, \\ 
\mathbf E &= -\frac{\partial}{\partial t} \mathbf{A} - \mathbf{\nabla}V \, . 
\end{align}</math> 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'' {{math|'''Π'''}}, given by: <math display="block"> \mathbf \Pi = \varepsilon_0 \frac{ \partial }{\partial t} \mathbf A \, , </math> where {{math|''ε''<sub>0</sub>}} 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:<ref name=Compagno/> <math display="block">\bigl[\Pi_i(\mathbf{r}, t),\ A_j(\mathbf{r}', t)\bigr] = -i\hbar \delta_{ij}\delta (\mathbf{r}-\mathbf{r}')\, , </math> where {{math|'''r'''}}, {{math|'''r'''′}} are spatial locations, {{mvar|ħ}} is the [[reduced Planck constant]], {{mvar|δ<sub>ij</sub>}} is the [[Kronecker delta]] and {{math|''δ''('''r''' − '''r'''′)}} is the [[Dirac delta function]]. The notation {{math|[ , ]}} denotes the [[commutator]].
:Quantization can be achieved without introducing the vector potential, in terms of the underlying fields themselves:<ref name=Vogel/> <math display="block">\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'})  \, , </math> where the [[circumflex]] denotes a Schrödinger time-independent field operator, and {{mvar|ε<sub>ijk</sub>}} 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.<ref name=Grynberg/> 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.<ref name=Parker/>

== Electromagnetic properties ==
{{See also|Lamb shift|Casimir effect|Spontaneous emission}}
[[File:The polarisation of light emitted by a neutron star.jpg|thumb|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.<ref>{{cite web|title=First Signs of Weird Quantum Property of Empty Space? – VLT observations of neutron star may confirm 80-year-old prediction about the vacuum|url=https://www.eso.org/public/news/eso1641/|website=www.eso.org|access-date=5 December 2016}}</ref>]]

As a result of quantization, the quantum electrodynamic vacuum can be considered as a material medium.<ref name=medium/> It is capable of [[vacuum polarization]].<ref name=Weisskopf/><ref name=Zeidler/> In particular, the [[Coulomb's law|force law between charged particles]] is affected.<ref name=Schroeder/><ref name=Schweber/> The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple {{math|[[electric constant|''ε''<sub>0</sub>]]}} of the [[Vacuum#In electromagnetism|classical vacuum]]. Likewise, its permeability can be calculated and differs slightly from {{math|[[magnetic constant|''μ''<sub>0</sub>]]}}. This medium is a dielectric with relative dielectric constant > 1, and is diamagnetic, with relative magnetic permeability < 1.<ref name= Donoghue/><ref name=Bertulani/> 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]]<ref name=Meszaros/>), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields.<ref name=Hartemann/> Calculations also indicate birefringence and dichroism at high fields.<ref name=Heyl/>  Many of electromagnetic effects of the vacuum are small, and only recently have experiments been designed to enable the observation of nonlinear effects.<ref name=nonlinear/> [[PVLAS]] and other teams are working towards the needed sensitivity to detect QED effects.

== Attainability ==
A perfect vacuum is itself only attainable in principle.<ref name=Longo/><ref name=Dirac/> It is an idealization, like [[absolute zero]] for temperature, that can be approached, but never actually realized:
{{quote|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<ref name=Longo/>}}

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, [[QCD vacuum|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.<ref name=gravity/> It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.

== See also ==
* [[Feynman diagram]]
* [[History of quantum field theory]]
* [[Precision tests of QED]]

== References ==
{{reflist|refs=
<ref name=Compagno>
This commutation relation is oversimplified, and a correct version replaces the {{mvar|δ}} product on the right by the transverse {{mvar|δ}}-''tensor'':
: <math>\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}} \ , </math>
where {{math|'''û'''}} is the unit vector of {{math|'''k'''}}, {{math|'''û''' {{=}} {{sfrac|'''k'''|''k''}}}}. For a discussion see, {{cite book |title=Atom-Field Interactions and Dressed Atoms|series=Cambridge Studies in Modern Optics, vol. 17 |first1=G. |last1=Compagno |first2=R. |last2=Passante |first3=F. |last3=Persico |chapter-url=https://books.google.com/books?id=R2GUQlBEeekC&pg=PA31 |chapter=§2.1 Canonical quantization in the Coulomb gauge |page=31 |year=2005 |publisher=Cambridge University Press |isbn=978-0-521-01972-9 }}
</ref>

<ref name=Cao>
{{cite book |title=Conceptual Foundations of Quantum Field Theory |editor-first=Tian Yu |editor-last=Cao |url=https://books.google.com/books?id=d0wS0EJHZ3MC&pg=PA179 |page=179 |quote=For each stationary classical background field there is a ground state of the associated quantized field. This is the vacuum ''for that background''. |isbn=978-0-521-60272-3 |publisher=Cambridge University Press |year=2004}}
</ref>

<ref name=classical>
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.
</ref>

<ref name=Mackay>
{{cite book |first1=Tom G. |last1=Mackay |first2=Akhlesh |last2=Lakhtakia |title=Electromagnetic Anisotropy and Bianisotropy: A Field Guide |url=https://books.google.com/books?id=AcMyp7wxWpsC&pg=PA201 |page=201 |isbn=978-981-4289-61-0 |publisher=World Scientific |year=2010}}
</ref>

<ref name=Allday>
A vaguer description is provided by {{cite book |title=Quarks, Leptons and the Big Bang |first=Jonathan |last=Allday |url=https://books.google.com/books?id=kgsBbv3-9xwC&pg=PA224 |page=224 |quote=The interaction will last for a certain duration {{math|Δ''t''}}. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies {{math|Δ''E''}}. |isbn=978-0-7503-0806-9 |edition=2nd |publisher=CRC Press |year=2002}}
</ref>

<ref name=Bertulani>
[[QCD vacuum]] is [[Paramagnetism|paramagnetic]], while QED vacuum is [[Diamagnetism|diamagnetic]]. See {{cite book |title=Nuclear Physics in a Nutshell |first=Carlos A. |last=Bertulani |url=https://books.google.com/books?id=n51yJr4b_oQC&pg=PA26 |page=26 |isbn=978-0-691-12505-3 |year=2007 |publisher=Princeton University Press|bibcode=2007npn..book.....B }}
</ref>

<ref name=Busch>
For a review, see {{cite book |title=Time in Quantum Mechanics |editor1-first=J. G. |editor1-last=Muga |editor2-first=R. |editor2-last=Sala Mayato |editor3-first=Í. L. |editor3-last=Egusquiza |page=73ff |chapter=Chapter 3: The Time–Energy Uncertainty Relation |author=Paul Busch |author-link=Paul Busch (physicist) |arxiv=quant-ph/0105049 |isbn=978-3-540-73472-7 |year=2008 |edition=2nd |publisher=Springer|bibcode=2002tqm..conf...69B |doi=10.1007/978-3-540-73473-4_3 |s2cid=14119708 }} 
</ref>

<ref name=Busch0>
{{cite book |title=Operational Quantum Physics |url=https://archive.org/details/operationalquant00busc |url-access=limited |first1=Paul |last1=Busch |author-link1=Paul Busch (physicist) |first2=Marian |last2=Grabowski |first3=Pekka J. |last3=Lahti |chapter=§III.4: Energy and time |page=[https://archive.org/details/operationalquant00busc/page/n86 77] |isbn=978-3-540-59358-4 |year=1995 |publisher=Springer}}
</ref>

<ref name=commutation>
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 {{cite book |title=Encyclopedic Dictionary of Mathematics |editor-first=Kiyosi |editor-last=Itô |chapter-url=https://books.google.com/books?id=azS2ktxrz3EC&pg=PA1303 |page=1303 |chapter=§ 351 (XX.23) C: Canonical commutation relations |isbn=978-0-262-59020-4 |year=1993 |edition=2nd |publisher=MIT Press}}
</ref>

<ref name=Davies>
For an example, see {{cite book |title=The Accidental Universe |first=P. C. W. |last=Davies |url=https://archive.org/details/accidentaluniver0000davi |url-access=registration |page=[https://archive.org/details/accidentaluniver0000davi/page/106 106] |isbn=978-0-521-28692-3 |year=1982 |publisher=Cambridge University Press}}
</ref>

<ref name=Dirac>
{{cite book |first=P. A. M. |last=Dirac |author-link=Paul Dirac |title=Lorentz and Poincaré Invariance: 100 Years of Relativity |editor1=Jong-Ping Hsu |editor2=Yuanzhong Zhang |year=2001 |publisher=World Scientific |isbn=978-981-02-4721-8 |url=https://books.google.com/books?id=jryk42J8oQIC&pg=PA440 |page=440}}
</ref>

<ref name= Donoghue>
{{cite book |title=Dynamics of the Standard Model |first1=John F. |last1=Donoghue |first2=Eugene |last2=Golowich |first3=Barry R. |last3=Holstein |page=47 |url=https://books.google.com/books?id=hFasRlkBbpYC&pg=PA47 |isbn=978-0-521-47652-2 |publisher =Cambridge University Press |year=1994}}
</ref>

<ref name=gravity>
For example, see {{cite book |title=A First Course in Loop Quantum Gravity |chapter-url=https://books.google.com/books?id=sSzPlPP8JEwC&pg=PA1 |page=1 |chapter=Chapter 1: Why quantize gravity? |publisher=Oxford University Press |year=2010 |first1=Rodolfo |last1=Gambini |first2=Jorge |last2=Pullin |isbn=978-0-19-959075-9}} and {{cite book |title=Quantum Gravity |first=Carlo |last=Rovelli |chapter-url=https://books.google.com/books?id=HrAzTmXdssQC&pg=PA202 |page=202ff |chapter=§5.4.2 Much ado about nothing: the vacuum |publisher=Cambridge University Press |quote=We use three distinct notions of vacuum in quantum gravity |year=2004 |isbn=978-0-521-83733-0}}
</ref>

<ref name=Grynberg>
{{cite book |title=Introduction to Quantum Optics: From the Semi-Classical Approach to Quantized Light |first1=Gilbert |last1=Grynberg |first2=Alain |last2=Aspect |first3=Claude |last3=Fabre |chapter-url=https://books.google.com/books?id=l-l0L8YInA0C&pg=PA351 |page=351 |chapter=§5.2.2 Vacuum fluctuations and their physical consequences |year=2010 |isbn=978-0-521-55112-0 |publisher=Cambridge University Press}}
</ref>

<ref name=Hartemann>
{{cite book |title=High-Field Electrodynamics |first=Frederic V. |last=Hartemann |url=https://books.google.com/books?id=tIkflVrfkG0C&pg=PA428 |page=428 |isbn=978-0-8493-2378-2 |year=2002 |publisher=CRC Press}}
</ref>

<ref name=Heyl>
{{cite journal |title=Birefringence and Dichroism of the QED Vacuum |first1=Jeremy S. |last1=Heyl |first2=Lars |last2=Hernquist |journal=J. Phys. |volume=A30 |issue=18 |pages=6485–6492 |doi=10.1088/0305-4470/30/18/022 |year=1997|arxiv = hep-ph/9705367 |bibcode = 1997JPhA...30.6485H |s2cid=32306183 }}
</ref>

<ref name=King>
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, {{cite book |title=Quest for Zero Point Energy: Engineering Principles for 'Free Energy' Inventions |first=Moray B. |last=King |url=https://books.google.com/books?id=0RmkmrFxHM0C&pg=PA124 |page=124ff |isbn=978-0-932813-94-7 |year=2001 |publisher=Adventures Unlimited Press}}
</ref>

<ref name=Longo>
{{cite book |author=Luciano Boi |title=The Two Cultures: Shared Problems |contribution=Creating the physical world ''ex nihilo?'' On the quantum vacuum and its fluctuations |editor1-first=Ernesto |editor1-last=Carafoli |editor2-first=Gian Antonio |editor2-last=Danieli |editor3-first=Giuseppe O. |editor3-last=Longo |url=https://books.google.com/books?id=Kz38u2qT36kC&pg=PA55 |page=55 |isbn=978-88-470-0868-7 |year=2009 |publisher=Springer}} 
</ref>

<ref name=medium>
{{cite book |title=Proceedings of the Fourth International Workshop on the Identification of Dark Matter: York, UK, 2-6 September 2002|first=M. |last=Bregant |display-authors=etal |editor1-first=Neil John |editor1-last=Curwen Spooner |editor2-first=Vitaly |editor2-last=Kudryavtsev |contribution=Particle laser production at PVLAS: Recent developments |url=https://books.google.com/books?id=D82vKLeFd4oC&pg=PA414 |publisher=World Scientific |year=2003|isbn=9789812791313 }}
</ref>

<ref name=Meszaros>
{{cite book |title=High-Energy Radiation from Magnetized Neutron Stars |first=Peter |last=Mészáros |chapter-url=https://books.google.com/books?id=L6DfR49OWd8C&pg=PA56 |page=56 |chapter=§2.6 Quantum electrodynamics in strong fields |isbn=978-0-226-52094-0 |year=1992  |publisher=University of Chicago Press}}
</ref>

<ref name=nonlinear>
{{cite book |title=Applications of Laser-Plasma Interactions |contribution=Nuclear and particle physics with ultraintense lasers |first1=José Tito |last1=Mendonça |first2=Shalom |last2=Eliezer |url=https://books.google.com/books?id=inCD7lqR0K8C&pg=PA145 |page=145 |isbn=978-0-8493-7604-7 |editor1-first=Shalom |editor1-last=Eliezer |editor2-first=Kunioki |editor2-last=Mima |year=2008 |publisher=CRC Press}}
</ref>

<ref name=Parker>
{{cite book |title=Biophotonics, Volume 360, Part 1 |first=Ian |last=Parker |page=516 |url=https://books.google.com/books?id=ACcfLIRrLLIC&pg=PA516 |isbn=978-0-12-182263-7 |year=2003 |publisher=Academic Press}}
</ref>

<ref name=Petrosyan>
{{cite book |title=Fundamentals of Quantum Optics and Quantum Information |first1=Peter |last1=Lambropoulos |first2=David |last2=Petrosyan |url=https://books.google.com/books?id=53bpU-41U8gC&pg=PA30 |page=30 |isbn=978-3-540-34571-8 |year=2007 |publisher=Springer|bibcode=2007fqoq.book.....L }}
</ref>

<ref name=Schroeder>
{{cite book |title=An Introduction to Quantum Field Theory |first1=Michael Edward |last1=Peskin |first2=Daniel V. |last2=Schroeder |chapter-url=https://books.google.com/books?id=i35LALN0GosC&pg=PA244 |page=[https://archive.org/details/introductiontoqu0000pesk/page/244 244] |chapter=§7.5 Renormalization of the electric charge |isbn=978-0-201-50397-5 |publisher=Westview Press |year=1995 |url=https://archive.org/details/introductiontoqu0000pesk/page/244 }} 
</ref>

<ref name=Schwabl>
{{cite book |title=Quantum Mechanics |first=Franz |last=Schwabl |chapter=§ 3.1.3: The zero-point energy |page=54 |isbn=978-3-540-71932-8 |edition =4th |publisher=Springer |year=2007 |chapter-url=https://books.google.com/books?id=pTHb4NK2eZcC&pg=PA54}}
</ref>

<ref name=Schweber>
{{cite book |first=Silvan S. |last=Schweber |title=The Oxford Companion to the History of Modern Science |editor-first=J. L. |editor-last=Heilbron |chapter-url=https://books.google.com/books?id=abqjP-_KfzkC&pg=PA246 |pages=246–247 |quote=Thus in QED the presence of an electric charge {{mvar|e<sub>o</sub>}} polarizes the "vacuum" and the charge that is observed at a large distance differs from {{mvar|e<sub>o</sub>}} and is given by {{math|''e'' {{=}} {{sfrac|''e<sub>o</sub>''|''ε''}}}} with {{mvar|ε}} the dielectric constant of the vacuum. |chapter=Elementary particles |isbn=978-0-19-511229-0 |year=2003 |publisher=Oxford University Press}}
</ref>

<ref name=Shankar>
{{cite book|title=Principles of Quantum Mechanics |first=Ramamurti |last=Shankar |url=https://books.google.com/books?id=2zypV5EbKuIC&pg=PA507 |page=507 |isbn=978-0-306-44790-7 |year=1994 |edition=2nd |publisher=Springer}}
</ref>

<ref name= Silk>
{{cite book |title=On the Shores of the Unknown: A Short History of the Universe |first=Joseph |last=Silk |page=62 |url=https://books.google.com/books?id=JowbGgYiSDYC&pg=PA62 |isbn=978-0-521-83627-2 |publisher=Cambridge University Press |year=2005}}
</ref>

<ref name=Vogel>
{{cite book |first1=Werner |last1=Vogel |first2=Dirk-Gunnar |last2=Welsch |title=Quantum Optics |chapter=§2.2.1 Canonical quantization: Eq. (2.50) |page=21 |chapter-url=https://books.google.com/books?id=AFdY_zNZciAC&pg=PA21 |isbn=978-3-527-40507-7 |publisher=Wiley-VCH |year=2006 |edition=3rd}}
</ref>

<ref name=Vogel2>
{{cite book |title=Quantum Optics |first1=Werner |last1=Vogel |first2=Dirk-Gunnar |last2=Welsch |page=18 |chapter= Chapter 2: Elements of quantum electrodynamics |isbn=978-3-527-40507-7 |year=2006 |publisher=Wiley-VCH |edition =3rd  |chapter-url=https://books.google.com/books?id=eouwERvRrJEC&pg=PA18 }}
</ref>

<ref name=Weisskopf>
{{cite book |title=Concepts of Particle Physics, Volume 2 |first1=Kurt |last1=Gottfried |first2=Victor Frederick |last2=Weisskopf |url=https://books.google.com/books?id=KXvoI-m9-9MC&pg=PA259 |page=259 |isbn= 978-0195033939 |year=1986 |publisher=Oxford University Press}}
</ref>

<ref name=Zeidler>
{{cite book |chapter=§19.1.9 Vacuum polarization in quantum electrodynamics |title=Quantum Field Theory, Volume III: Gauge Theory: A Bridge Between Mathematicians and Physicists |first=Eberhard |last=Zeidler |chapter-url=https://books.google.com/books?id=miwuxaEXvOsC&pg=PA952 |page=952 |isbn=978-3-642-22420-1 |publisher=Springer |year=2011}}
</ref>
}}

{{QED}}
{{Quantum mechanics topics}}
{{citizendium|title=Vacuum (quantum electrodynamic)}}

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