View source for Vacuum polarization

{{Short description|Gauge boson self-energy due to interactions with virtual particles}}
{{Use American English|date = February 2019}}
In [[quantum field theory]], and specifically [[quantum electrodynamics]], '''vacuum polarization''' describes a process in which a background [[electromagnetic field]] produces [[virtual particle|virtual]] [[electron]]&ndash;[[positron]] pairs that change the distribution of charges and currents that generated the original electromagnetic field.  It is also sometimes referred to as the '''[[self-energy]]''' of the [[gauge boson]] ([[photon]]).

After developments in radar equipment for [[World War II]] resulted in higher accuracy for measuring the energy levels of the hydrogen atom, [[Isidor Rabi]] made measurements of the [[Lamb shift]] and the [[anomalous magnetic dipole moment]] of the electron. These effects corresponded to the deviation from the value −2 for the spectroscopic electron [[g-factor (physics)|''g''-factor]] that are predicted by the [[Dirac equation]]. Later, [[Hans Bethe]]<ref>{{harvnb|Bethe|1947}}</ref> theoretically calculated those shifts in the [[hydrogen]] energy levels due to vacuum polarization on his return train ride from the [[Shelter Island Conference]] to Cornell.

The effects of vacuum polarization have been routinely observed experimentally since then as very well-understood background effects. Vacuum polarization, referred to below as the one loop contribution, occurs with leptons (electron–positron pairs) or quarks. The former (leptons) was first observed in 1940s but also more recently observed in 1997 using the [[KEK|TRISTAN]] particle accelerator in Japan,<ref>{{harvnb|Levine|1997}}</ref> the latter (quarks) was observed along with multiple quark–gluon loop contributions from the early 1970s to mid-1990s using the VEPP-2M particle accelerator at the [[Budker Institute of Nuclear Physics]] in [[Siberia]], [[Russia]] and many other accelerator laboratories worldwide.<ref>{{harvnb|Brown|Worstell|1996|pp=3237&ndash;3249}}</ref>

== History ==
Vacuum polarization was first discussed in papers by [[Paul Dirac]]<ref>{{harvnb|Dirac|1934}}</ref> and [[Werner Heisenberg]]<ref>{{harvnb|Heisenberg|1934}}</ref> in 1934. Effects of vacuum polarization were calculated to first order in the coupling constant by [[Robert Serber]]<ref>{{harvnb|Serber|1935}}</ref> and [[Edwin Albrecht Uehling]]<ref>{{harvnb|Uehling|1935}}</ref> in 1935.<ref>{{harvnb|Gell-Mann|Low|1954}}</ref>

== Explanation ==
According to [[quantum field theory]], the vacuum between interacting particles is not simply empty space. Rather, it contains short-lived [[virtual particle|virtual particle–antiparticle pairs]] ([[leptons]] or [[quarks]] and [[gluons]]). These short-lived pairs are called [[Feynman diagram#Vacuum bubbles|vacuum bubbles]]. It can be shown that they have no measurable impact on any process.<ref>{{harvnb|Greiner|Reinhardt|1996|loc=Chapter 8.}}</ref><ref group=nb>They yield a phase factor to the vacuum to vacuum transition amplitude.</ref>

Virtual particle–antiparticle pairs can also occur as a photon propagates.<ref>{{harvnb|Weinberg|2002|loc=Chapters 10&ndash;11}}</ref> In this case, the '''effect on other processes is measurable'''. The one-loop contribution of a fermion&ndash;antifermion pair to the vacuum polarization is represented by the following diagram:

:[[Image:Vacuum polarization.svg]]

These particle–antiparticle pairs carry various kinds of charges, such as [[color charge]] if they are subject to [[quantum chromodynamics]] such as [[quarks]] or [[gluons]], or the more familiar electromagnetic charge if they are electrically charged [[leptons]] or [[quarks]], the most familiar charged [[lepton]] being the [[electron]] and since it is the lightest in [[mass]], the most numerous due to the energy–time [[uncertainty principle]] as mentioned above; e.g., virtual electron–positron pairs. Such charged pairs act as an [[electric dipole]].  In the presence of an electric field, e.g., the [[electromagnetic field]] around an electron, these particle–antiparticle pairs reposition themselves, thus partially counteracting the field (a partial screening effect, a [[dielectric]] effect). The field therefore will be weaker than would be expected if the vacuum were completely empty. This reorientation of the short-lived particle–antiparticle pairs is referred to as ''vacuum polarization''.

== Electric and magnetic fields ==

Extremely strong electric and magnetic fields cause an excitation of electron–positron pairs. [[Maxwell's equations]] are the classical limit of the quantum electrodynamics which cannot be described by any classical theory. A point charge must be modified at extremely small distances less than the reduced [[Compton wavelength]] <math>\bar\lambda_\text{c}</math> (<math display="inline">\,  = \frac{\hbar}{m c} = 3.86 \times 10^{-13} \text{ m}</math>). To lowest order in the [[fine-structure constant]], <math>\alpha</math>, the QED result for the electrostatic potential of a point charge is:<ref>{{harvnb|Berestetskii|Lifshitz|Pitaevskii|1980|loc=Section 114.}}</ref>
<math display="block">\phi(r) = \frac{q}{4 \pi \epsilon_0 r} \times \begin{cases}
  1 - \frac{2 \alpha}{3 \pi} \ln\left( \frac{r}{\bar\lambda_\text{c}}\right)& r \ll \bar\lambda_\text{c} \\[2pt]
  1 + \frac{\alpha}{4 \sqrt{\pi}} \left( \frac{r}{\bar\lambda_\text{c}}\right)^{-3/2} e^{-2r/\bar\lambda_\text{c}} & r \gg \bar\lambda_\text{c}
\end{cases} </math>

This can be understood as a screening of a point charge by a medium with a dielectric permittivity, which is why the term vacuum polarization is used. When observed from distances much greater than <math>\bar\lambda_\text{c}</math>, the charge is renormalized to the finite value <math>q</math>. See also the [[Uehling potential]].

The effects of vacuum polarization become significant when the external field approaches the [[Schwinger limit]], which is:
<math display="block">E_\text{c} = \frac{ m c^2}{e \bar\lambda_\text{c}} = 1.32 \times 10^{18} \text{ V/m}</math>
<math display="block">B_\text{c} = \frac{ m c}{e \bar\lambda_\text{c}} = 4.41 \times 10^9 \text{ T} \,.</math>

These effects break the linearity of Maxwell's equations and therefore break the [[superposition principle]]. The QED result for slowly varying fields can be written in non-linear relations for the vacuum. To lowest order <math>\alpha</math>, virtual pair production generates a vacuum polarization and magnetization given by:
<math display="block">\mathbf P = \frac{2 \epsilon_0 \alpha}{E_\text{c}^2} \left ( 2 \left( E^2 - c^2 B^2\right) \mathbf E + 7 c^2 \left(\mathbf E \cdot \mathbf B\right) \mathbf B\right )</math>
<math display="block">\mathbf M = - \frac{2  \alpha}{\mu_0 E_\text{c}^2} \left ( 2 \left( E^2 - c^2 B^2\right) \mathbf E + 7 c^2 \left(\mathbf E \cdot \mathbf B\right) \mathbf B \right ).</math>

{{As of|2019|post=,}} this polarization and magnetization has not been directly measured.

== Vacuum polarization tensor ==

The vacuum polarization is quantified by the vacuum polarization tensor {{math|Π<sup>''μν''</sup>(''p'')}} which describes the dielectric effect as a function of the four-momentum ''p'' carried by the photon.  Thus the vacuum polarization depends on the momentum transfer, or in other words, the [[electric constant]] is scale dependent.  In particular, for electromagnetism we can write the [[fine-structure constant]] as an effective momentum-transfer-dependent quantity; to first order in the corrections, we have
<math display="block"> \alpha_\text{eff}(p^2) = \frac{\alpha}{1 - [\Pi_2(p^2) - \Pi_2(0)]} </math>
where {{math|1=Π<sup>''μν''</sup>(''p'') = (''p''<sup>2</sup> ''g''<sup>''μν''</sup> − ''p''<sup>''μ''</sup>''p''<sup>''ν''</sup>) Π(''p''<sup>2</sup>)}} and the subscript 2 denotes the leading order-''e''{{i sup|2}} correction.  The tensor structure of {{math|Π<sup>''μν''</sup>(''p'')}} is fixed by the [[Ward identity]].

==Note==
Vacuum polarization affecting spin interactions has also been reported based on experimental data and also treated theoretically in [[quantum chromodynamics]], as for example in considering the [[hadron]] [[Spin (physics)|spin]] structure.

== See also ==
{{div col|colwidth=20em}}
* [[Polarization density]]
* [[Infraparticle]]
* [[Renormalization]]
* [[Virtual particles]]
* [[QED vacuum]]
* [[QCD vacuum]]
* [[Schwinger limit]]
* [[Schwinger effect]]
* [[Uehling potential]]
* [[Vacuum birefringence]]
* [[Nonoblique correction]]
{{div col end}}

== Remarks ==
{{reflist|group=nb}}

== Notes ==
{{reflist|4}}

== References ==
*{{Cite book|last1=Berestetskii|first1=V. B.|last2=Lifshitz|first2=E. M.|authorlink2=Evgeny Lifshitz|last3= Pitaevskii|first3=L.| authorlink3=Lev Pitaevskii|title=Quantum Electrodynamics|edition=2nd|publisher=Butterworth-Heinemann|year=1980|chapter=Section 114 | series=[[Course of Theoretical Physics]]|volume=4|isbn=978-0750633710}}
*{{cite journal|last=Bethe|first=H. A.|authorlink=Hans Bethe|year=1947|title=The Electromagnetic Shift of Energy Levels| journal=[[Physical Review]]|volume=72|issue=4|pages=339–341|doi=10.1103/PhysRev.72.339|bibcode=1947PhRv...72..339B}}
*{{cite journal|last1=Brown|first1=Douglas H.|last2=Worstell|first2=William A|title=The Lowest Order Hadronic Contribution to the Muon g − 2 Value with Systematic Error Correlations|journal=Physical Review D|year=1996|volume=54| pages=3237–3249| doi=10.1103/PhysRevD.54.3237|issue=5|pmid=10020994|arxiv=hep-ph/9607319|bibcode=1996PhRvD..54.3237B| s2cid=37689024}}
*{{cite journal|last=Dirac|first=P. A. M.|authorlink=Paul Dirac|title=Discussion of the infinite distribution of electrons in the theory of the positron|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|volume=30|issue=2| date=1934|pages=150–163| doi=10.1017/S030500410001656X|url=https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/issue/E8607B18CBE63969305630B2583E5844|bibcode=1934PCPS...30..150D|zbl=0009.13704|jfm=60.0790.02}}
*{{cite journal|last1=Gell-Mann|first1=M.|authorlink1=Murray Gell-Mann|last2=Low|first2=F. E.|authorlink2=Francis E. Low|year=1954|title=Quantum Electrodynamics at Small Distances|journal=[[Physical Review]] |volume=95|issue=5|pages=1300&ndash;1312|doi=10.1103/PhysRev.95.1300|bibcode=1954PhRv...95.1300G|url=https://journals.aps.org/pr/issues/95/5}}
*{{cite book|last1=Greiner|first1=W.|authorlink1=Walter Greiner|last2=Reinhardt|first2=J.|year=1996|title=Field Quantization| publisher=[[Springer Publishing]]|isbn=978-3-540-59179-5|url-access=registration|url=https://archive.org/details/fieldquantizatio0000grei}}
*{{cite journal|last=Heisenberg|first=W.|authorlink=Werner Heisenberg|title=Bemerkungen zur Diracschen Theorie des Positrons| language=German|journal=Zeitschrift für Physik|volume=90|issue=3–4|pages=209–231|year=1934|issn=0044-3328| doi=10.1007/BF01333516 | bibcode=1934ZPhy...90..209H|s2cid=186232913}}
*{{cite journal|last=Levine|first=I.|collaboration=TOPAZ Collaboration|title=Measurement of the Electromagnetic Coupling at Large Momentum Transfer|journal=Physical Review Letters|year=1997|volume=78|pages=424–427|doi=10.1103/PhysRevLett.78.424 | bibcode=1997PhRvL..78..424L|issue=3}}
*{{cite journal|last=Serber|first=R.|authorlink=Robert Serber|year=1935|title=Linear Modifications in the Maxwell Field Equations | journal=[[Physical Review]] |volume=48|issue=1|pages=49–54|doi=10.1103/PhysRev.48.49|bibcode=1935PhRv...48...49S|jfm=61.1250.03}}
*{{cite journal|last=Uehling|first=E. A.|year=1935|title=Polarization Effects in the Positron Theory|journal=[[Physical Review]] | volume=48 | issue=1|pages=55–63|doi=10.1103/PhysRev.48.55|bibcode=1935PhRv...48...55U}}
*{{cite book|last=Weinberg|first=S.|authorlink=Steven Weinberg|year=2002|title=Foundations|series=The Quantum Theory of Fields| volume=I|isbn=978-0-521-55001-7|publisher=[[Cambridge University Press]]|url-access=registration|url=https://archive.org/details/quantumtheoryoff00stev}}

== Further reading ==
* For a derivation of the vacuum polarization in QED, see section 7.5 of M.E. Peskin and D.V. Schroeder, ''An Introduction to Quantum Field Theory'', Addison-Wesley, 1995.

{{QED}}

[[Category:Gauge theories]]
[[Category:Quantum electrodynamics]]