Vacuum (quantum electrodynamic): Difference between revisions

Latest revision as of 09:33, 18 February 2012

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 \hbar \ ,

(with ΔE and Δt energy and time variations, and ℏ the Planck constant divided by 2π) 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.^[4]

This interpretation of the energy-time uncertainty relation is not universally accepted, however.^[5]^[6] One issue is the use of an uncertainty relation limiting measurement accuracy as though a time uncertainty Δt determines a "budget" for borrowing energy ΔE. 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) .^[7] 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.^[8]^[9] The very many approaches to the energy-time uncertainty principle are a long and continuing subject.^[9]

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.^[10]

An uncertainty principle applies to all quantum mechanical operators that do not commute.^[11] In particular, it applies also to the electromagnetic field. A digression follows to flesh out the role of commutators for the electromagnetic field.^[12]

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:^[12]

{\begin{aligned}\mathbf {B} &=\mathbf {\nabla \times A} ,\\\mathbf {E} &=-\mathbf {A-\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 ε₀ 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:^[13]

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

where r, r′ are spatial locations, ℏ is Planck's constant over 2π, δ_ij is the Kronecker delta and δ(r−r′) is the Dirac delta function.

Quantization can be achieved without introducing the vector potential, in terms of the underlying fields themselves:^[14]

\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 caret '^' denotes a Schrõdinger 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.^[15] 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.^[16]

Electromagnetic properties

As a result of quantization, the quantum electrodynamic vacuum can be considered as a dielectric medium, and is capable of vacuum polarization.^[17]^[18] In particular, the force law between charged particles is affected.^[19]^[20] The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple ε₀ of the classical vacuum. Likewise, its permeability can be calculated and differs slightly from μ₀. This medium is a dielectric with relative dielectric constant > 1, and is diamagnetic, with relative magnetic permeability < 1.^[21] Under some extreme circumstances (for example, in the very high fields found in the exterior regions of pulsars^[22]), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields.^[23] Many of electromagnetic effects of the vacuum are small, and only recently have experiments been designed to enable the observation of nonlinear effects.^[24] Calculations also indicate birefringence and dichroism at high fields.^[25]

Attainability

A perfect vacuum is itself only attainable in principle.^[26]^[27] It is an idealization, like absolute zero for temperature, that can be approached, but never actually realized:^[26]

“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

Because of virtual particles, a perfect vacuum cannot be realized. That situation leaves 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 and the vacuum of quantum gravity.^[28] It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.

References

↑ See Kurt Gottfried, Victor Frederick Weisskopf (1986). Concepts of particle physics, Volume 2. Oxford University Press, p. 266. ISBN 0195033930.
↑ Ramamurti Shankar (1994). Principles of quantum mechanics, 2nd ed.. Springer, p. 507. ISBN 0306447908.
↑ Joseph Silk (2005). On the shores of the unknown: a short history of the universe. Cambridge University Press, p. 62. ISBN 0521836271.
↑ For an example, see P. C. W. Davies (1982). The accidental universe. Cambridge University Press, p. 106. ISBN 0521286921.
↑ A vaguer description is provided by Jonathan Allday (2002). Quarks, leptons and the big bang, 2nd ed. CRC Press, pp. 224 ff. ISBN 0750308060. “The interaction will last for a certain duration Δt. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies ΔE.”
↑ 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, Moray B. King (2001). Quest for zero point energy: engineering principles for 'free energy' inventions. Adventures Unlimited Press, pp. 124 ff. ISBN 0932813941.
↑ 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 Kiyosi Itô (1993). “§ 351 (XX.23) C: Canonical commutation relations”, Encyclopedic dictionary of mathematics, 2nd ed. MIT Press, p. 1303. ISBN 0262590204.
↑ Paul Busch, Marian Grabowski, Pekka J. Lahti (1995). “§III.4: Energy and time”, Operational quantum physics. Springer, 77 ff. ISBN 3540593586.
↑ ^9.0 ^9.1 For a review, see Paul Busch (2008). “Chapter 3: The Time–Energy Uncertainty Relation”, J.G. Muga, R. Sala Mayato and Í.L. Egusquiza, editors: Time in Quantum Mechanics, 2nd ed. Springer, pp. 73 ff. ISBN 3540734724.
↑ Franz Schwabl (2007). “§ 3.1.3: The zero-point energy”, Quantum mechanics, 4rth ed.. Springer, p. 54. ISBN 3540719326.
↑ Peter Lambropoulos, David Petrosyan (2007). Fundamentals of quantum optics and quantum information. Springer, p. 30. ISBN 354034571X.
↑ ^12.0 ^12.1 Werner Vogel, Dirk-Gunnar Welsch (2006). “Chapter 2: Elements of quantum electrodynamics”, Quantum optics, 3rd ed.. Wiley-VCH, pp. 18 ff. ISBN 3527405070.
↑ The notation [a, b] denotes the commutator [ab–ba]. This commutation relation is oversimplified, and a correct version replaces the δ product on the right by the transverse δ-tensor:
$\delta _{\perp \ ij}(\mathbf {x-x'} )={\frac {1}{(2\pi )^{3}}}\int d^{3}\mathbf {k} \left(\delta _{ij}-{\hat {u}}_{i}{\hat {u}}_{j}\right)e^{i\mathbf {k\cdot (x-x')} }\ .$
where û is the unit vector of k, û = k/k. For a discussion see, G. Compagno, R. Passante, F. Persico (2005). “§2.1 Canonical quantization in the Coulomb gauge”, Atom-Field Interactions and Dressed Atoms; Vol. 17 of Cambridge Studies in Modern Optics. Cambridge University Press, 31 ff. ISBN 0521019729.
↑ Werner Vogel, Dirk-Gunnar Welsch (2006). “§2.2.1 Canonical quantization: Eq. (2.50)”, Quantum optics, 3rd. Wiley-VCH. ISBN 3527405070.
↑ Gilbert Grynberg, Alain Aspect, Claude Fabre (2010). Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light. Cambridge University Press, pp. 351 ff. ISBN 0521551129.
↑ Ian Parker (2003). Biophotonics, Volume 360, Part 1. Academic Press, p. 516. ISBN 012182263X.
↑ Kurt Gottfried, Victor Frederick Weisskopf (1986). Concepts of particle physics, Volume 2. Oxford University Press, 259 ff. ISBN 0195033930.
↑ Eberhard Zeidler (2011). “§19.1.9 Vacuum polarization in quantum electrodynamics”, Quantum Field Theory, Volume III: Gauge Theory: A Bridge Between Mathematicians and Physicists. Springer, 952. ISBN 3642224202.
↑ Michael Edward Peskin, Daniel V. Schroeder (1995). “§7.5 Renormalization of the electric charge”, An introduction to quantum field theory. Westview Press, pp. 244 ff. ISBN 0201503972.
↑ Silvan S Schweber (2003). “Elementary particles”, J. L. Heilbron, ed: The Oxford companion to the history of modern science. Oxford University Press, pp. 246-247. ISBN 0195112296. “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.”
↑ John F. Donoghue, Eugene Golowich, Barry R. Holstein (1994). Dynamics of the standard model, p. 47. ISBN 0521476526.
↑ Peter Mészáros (1992). “§2.6 Quantum electrodynamics in strong fields”, High-energy radiation from magnetized neutron stars. University of Chicago Press, pp. 56 ff. ISBN 0226520943.
↑ Frederic V. Hartemann (2002). High-field electrodynamics. CRC Press, p.428. ISBN 0849323789.
↑ José Tito Mendonça, Shalom Eliezer (2008). “Nuclear and particle physics with ultraintense lasers”, Shalom Eliezer, Kunioki Mima, eds: Applications of laser-plasma interactions. CRC Press, p. 145. ISBN 0849376041.
↑ Jeremy S. Heyl, Lars Hernquist (1997). "Birefringence and Dichroism of the QED Vacuum". J Phys A30: 6485-6492. DOI:10.1088/0305-4470/30/18/022. Research Blogging.
↑ ^26.0 ^26.1 Luciano Boi (2009). “Creating the physical world ex nihilo? On the quantum vacuum and its fluctuations”, Ernesto Carafoli, Gian Antonio Danieli, Giuseppe O. Longo, editors: The Two Cultures: Shared Problems. Springer, p. 55. ISBN 8847008689.
↑ PAM Dirac (2001). Jong-Ping Hsu, Yuanzhong Zhang, editors: Lorentz and Poincaré invariance: 100 years of relativity. World Scientific, p. 440. ISBN 9810247214.
↑ For example, see Rodolfo Gambini, Jorge Pullin (2010). “Chapter 1: Why quantize gravity?”, A First Course in Loop Quantum Gravity. Oxford University Press, pp. 1 ff. ISBN 0199590753. and Carlo Rovelli (2004). “§5.4.2 Much ado about nothing: the vacuum”, Quantum gravity. Cambridge University Press, pp. 202 ff. ISBN 0521837332. “We use three distinct notions of vacuum in quantum gravity”

[scatter-1] See Kurt Gottfried, Victor Frederick Weisskopf (1986). Concepts of particle physics, Volume 2. Oxford University Press, p. 266. ISBN 0195033930.

[Shankar-2] Ramamurti Shankar (1994). Principles of quantum mechanics, 2nd ed.. Springer, p. 507. ISBN 0306447908.

[Silk-3] Joseph Silk (2005). On the shores of the unknown: a short history of the universe. Cambridge University Press, p. 62. ISBN 0521836271.

[Davies-4] For an example, see P. C. W. Davies (1982). The accidental universe. Cambridge University Press, p. 106. ISBN 0521286921.

[Allday-5] A vaguer description is provided by Jonathan Allday (2002). Quarks, leptons and the big bang, 2nd ed. CRC Press, pp. 224 ff. ISBN 0750308060. “The interaction will last for a certain duration Δt. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies ΔE.”

[King-6] 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, Moray B. King (2001). Quest for zero point energy: engineering principles for 'free energy' inventions. Adventures Unlimited Press, pp. 124 ff. ISBN 0932813941.

[commutation-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 Kiyosi Itô (1993). “§ 351 (XX.23) C: Canonical commutation relations”, Encyclopedic dictionary of mathematics, 2nd ed. MIT Press, p. 1303. ISBN 0262590204.

[Busch0-8] Paul Busch, Marian Grabowski, Pekka J. Lahti (1995). “§III.4: Energy and time”, Operational quantum physics. Springer, 77 ff. ISBN 3540593586.

[Busch-9] 9.0 ^9.1 For a review, see Paul Busch (2008). “Chapter 3: The Time–Energy Uncertainty Relation”, J.G. Muga, R. Sala Mayato and Í.L. Egusquiza, editors: Time in Quantum Mechanics, 2nd ed. Springer, pp. 73 ff. ISBN 3540734724.

[Schwabl-10] Franz Schwabl (2007). “§ 3.1.3: The zero-point energy”, Quantum mechanics, 4rth ed.. Springer, p. 54. ISBN 3540719326.

[Petrosyan-11] Peter Lambropoulos, David Petrosyan (2007). Fundamentals of quantum optics and quantum information. Springer, p. 30. ISBN 354034571X.

[Vogel2-12] 12.0 ^12.1 Werner Vogel, Dirk-Gunnar Welsch (2006). “Chapter 2: Elements of quantum electrodynamics”, Quantum optics, 3rd ed.. Wiley-VCH, pp. 18 ff. ISBN 3527405070.

[Compagno-13] The notation [a, b] denotes the commutator [ab–ba]. This commutation relation is oversimplified, and a correct version replaces the δ product on the right by the transverse δ-tensor:
$\delta _{\perp \ ij}(\mathbf {x-x'} )={\frac {1}{(2\pi )^{3}}}\int d^{3}\mathbf {k} \left(\delta _{ij}-{\hat {u}}_{i}{\hat {u}}_{j}\right)e^{i\mathbf {k\cdot (x-x')} }\ .$
where û is the unit vector of k, û = k/k. For a discussion see, G. Compagno, R. Passante, F. Persico (2005). “§2.1 Canonical quantization in the Coulomb gauge”, Atom-Field Interactions and Dressed Atoms; Vol. 17 of Cambridge Studies in Modern Optics. Cambridge University Press, 31 ff. ISBN 0521019729.

[Vogel-14] Werner Vogel, Dirk-Gunnar Welsch (2006). “§2.2.1 Canonical quantization: Eq. (2.50)”, Quantum optics, 3rd. Wiley-VCH. ISBN 3527405070.

[Grynberg-15] Gilbert Grynberg, Alain Aspect, Claude Fabre (2010). Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light. Cambridge University Press, pp. 351 ff. ISBN 0521551129.

[Parker-16] Ian Parker (2003). Biophotonics, Volume 360, Part 1. Academic Press, p. 516. ISBN 012182263X.

[Weisskopf-17] Kurt Gottfried, Victor Frederick Weisskopf (1986). Concepts of particle physics, Volume 2. Oxford University Press, 259 ff. ISBN 0195033930.

[Zeidler-18] Eberhard Zeidler (2011). “§19.1.9 Vacuum polarization in quantum electrodynamics”, Quantum Field Theory, Volume III: Gauge Theory: A Bridge Between Mathematicians and Physicists. Springer, 952. ISBN 3642224202.

[Schroeder-19] Michael Edward Peskin, Daniel V. Schroeder (1995). “§7.5 Renormalization of the electric charge”, An introduction to quantum field theory. Westview Press, pp. 244 ff. ISBN 0201503972.

[Schweber-20] Silvan S Schweber (2003). “Elementary particles”, J. L. Heilbron, ed: The Oxford companion to the history of modern science. Oxford University Press, pp. 246-247. ISBN 0195112296. “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-21] John F. Donoghue, Eugene Golowich, Barry R. Holstein (1994). Dynamics of the standard model, p. 47. ISBN 0521476526.

[Meszaros-22] Peter Mészáros (1992). “§2.6 Quantum electrodynamics in strong fields”, High-energy radiation from magnetized neutron stars. University of Chicago Press, pp. 56 ff. ISBN 0226520943.

[Hartemann-23] Frederic V. Hartemann (2002). High-field electrodynamics. CRC Press, p.428. ISBN 0849323789.

[nonlinear-24] José Tito Mendonça, Shalom Eliezer (2008). “Nuclear and particle physics with ultraintense lasers”, Shalom Eliezer, Kunioki Mima, eds: Applications of laser-plasma interactions. CRC Press, p. 145. ISBN 0849376041.

[Heyl-25] Jeremy S. Heyl, Lars Hernquist (1997). "Birefringence and Dichroism of the QED Vacuum". J Phys A30: 6485-6492. DOI:10.1088/0305-4470/30/18/022. Research Blogging.

[Longo-26] 26.0 ^26.1 Luciano Boi (2009). “Creating the physical world ex nihilo? On the quantum vacuum and its fluctuations”, Ernesto Carafoli, Gian Antonio Danieli, Giuseppe O. Longo, editors: The Two Cultures: Shared Problems. Springer, p. 55. ISBN 8847008689.

[Dirac-27] PAM Dirac (2001). Jong-Ping Hsu, Yuanzhong Zhang, editors: Lorentz and Poincaré invariance: 100 years of relativity. World Scientific, p. 440. ISBN 9810247214.

[gravity-28] For example, see Rodolfo Gambini, Jorge Pullin (2010). “Chapter 1: Why quantize gravity?”, A First Course in Loop Quantum Gravity. Oxford University Press, pp. 1 ff. ISBN 0199590753. and Carlo Rovelli (2004). “§5.4.2 Much ado about nothing: the vacuum”, Quantum gravity. Cambridge University Press, pp. 202 ff. ISBN 0521837332. “We use three distinct notions of vacuum in quantum gravity”

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@@ Line 1: / Line 1: @@
 {{subpages}}
 <!-- Text is transcluded from the BASEPAGENAME/Definition subpage-->
-The term '''quantum electrodynamic vacuum''', or QED, refers to the ground state of the electromagnetic field, which is subject to fluctuations about a dormant zero average-field condition.
+{{dambigbox|Vacuum (quantum electrodynamic)|Vacuum}}
+<br/>
+{{Image|Photon-photon scattering.png|right|100px|Feynman diagram for photon-photon scattering in QED vacuum.<ref name=scatter/>}}
+{{TOC|right}}
+The term '''quantum electrodynamic vacuum''', or ''QED vacuum'', refers to the ground state of the electromagnetic field, which is subject to fluctuations about a dormant zero average-field condition:<ref name=Shankar/> Here is a description of the quantum vacuum:<ref name= Silk/>
+<blockquote><p> “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 form 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.”</p><p>
+&emsp;&emsp;&emsp;&emsp;-Joseph Silk ''On the shores of the unknown'', p. 62</p> </blockquote>
-===Quantization of the fields===
+===Virtual particles===
-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>
+It is sometimes attempted to provide an intuitive picture of virtual particles based upon the Heisenberg energy-time uncertainty principle:
+::<math>\Delta E \Delta t \ge \hbar \ , </math>
+(with ''ΔE'' and ''Δt'' [[energy]] and [[time]] variations, and ℏ the [[Planck constant]] divided by 2π) 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/>
-{{cite book |title=Quantum mechanics |author=Franz Schwabl |chapter=§ 3.1.3: The zero-point energy |pages=p. 54 |isbn=3540719326 |edition =4rth ed. |publisher=Springer |year=2007 |url=http://books.google.com/books?id=pTHb4NK2eZcC&pg=PA54}}
+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 ''Δt''  determines a "budget" for borrowing energy ''ΔE''.  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 {{nowrap|<nowiki>[</nowiki>''q'', ''p''<nowiki>] = iℏ</nowiki>) }}.<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 very many approaches to the energy-time uncertainty principle are a long and continuing subject.<ref name=Busch/>
-</ref>
-An uncertainty principle applies to all quantum mechanical operators that do not [[commutator|commute]].<ref name=Petrosyan>
+===Quantization of the fields===
+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/>
-{{cite book |title=Fundamentals of quantum optics and quantum information |author=Peter Lambropoulos, David Petrosyan |url=http://books.google.com/books?id=53bpU-41U8gC&pg=PA30 |pages=p. 30 |isbn=354034571X |year=2007 |publisher=Springer}}
-</ref> 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>
-{{cite book |title=Quantum optics |author=Werner Vogel, Dirk-Gunnar Welsch |pages=pp. 18 ''ff'' |chapter= Chapter 2: Elements of quantum electrodynamics |isbn=3527405070 |year=2006 |publisher=Wiley-VCH |edition =3rd ed. |url=http://books.google.com/books?id=eouwERvRrJEC&pg=PA18&lpg=PA18 }}
-</ref>
+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/>
-{|align=center style="width:70%;"
+{|align=center style="width:80%;"
 |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:<ref name=Vogel2/>
 <b> </b>
@@ Line 27: / Line 29: @@
 \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'' '''Π''', given by:
-::<math> \mathbf \Pi = \varepsilon_0 \frac{ \partial }{\partial t} \mathbf A \ . </math>
+::<math> \mathbf \Pi = \varepsilon_0 \frac{ \partial }{\partial t} \mathbf A \ , </math>
-Quantization is achieved by insisting that the momentum field and the vector potential do not commute.
+where 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>\left[\Pi_i(\mathbf{r},\ t),\ A_j(\mathbf{r'},\ t)\right]=-i\hbar \delta_{ij}\delta (\mathbf{r-r'})\ , </math>
+where '''r''', '''r′''' are spatial locations,  ℏ is [[Planck's constant]] over 2&pi;, &delta;<sub>ij</sub> is the [[Kronecker delta]] and &delta;('''r−r′''') is the [[Dirac delta function]].
+Quantization can be achieved without introducing the vector potential, in terms of the underlying fields themselves:<ref name=Vogel/>
+::<math>\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 caret '^' denotes a Schrõdinger time-independent field operator, and &epsilon;<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>
+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===
+As a result of quantization, the quantum electrodynamic vacuum can be considered as a dielectric medium, and 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 [[electric constant|ε<sub>0</sub>]] of the [[Vacuum (classical)|classical vacuum]]. Likewise, its permeability can be calculated and differs slightly from [[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/> Under some extreme circumstances (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/> 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/> Calculations also indicate birefringence and dichroism at high fields.<ref name=Heyl/>
+===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:<ref name=Longo/>
+{|align=center style="width:80%;"
+|“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
+|}
+Because of virtual particles, a ''perfect'' vacuum cannot be realized. That situation leaves 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 chromodynamics|quantum chromodynamic vacuum]] and the vacuum of [[quantum gravity]].<ref name=gravity/> It remains an open question whether further refinements in experimental technique ultimately will support another model for realizable vacuum.
+==References==
+{{reflist|refs=
+<ref name=Allday>
+A vaguer description is provided by {{cite book |title=Quarks, leptons and the big bang |author=Jonathan Allday |url=http://books.google.com/books?id=kgsBbv3-9xwC&pg=PA224 |pages=pp. 224 ''ff'' |quote=The interaction will last for a certain duration ''Δt''. This implies that the amplitude for the total energy involved in the interaction is spread over a range of energies ''ΔE''. |isbn=0750308060 |edition=2nd ed |publisher=CRC Press |year=2002}}
+</ref>
+<ref name=Busch>
+For a review, see {{cite book |title=Time in Quantum Mechanics |editor=J.G. Muga, R. Sala Mayato and Í.L. Egusquiza, editors |pages=pp. 73 ''ff'' |chapter=Chapter 3: The Time–Energy Uncertainty Relation |author=Paul Busch |url=http://arxiv.org/abs/quant-ph/0105049v3 |isbn=3540734724 |year=2008 |edition=2nd ed |publisher=Springer}}
+</ref>
+<ref name=Busch0>
+{{cite book |title=Operational quantum physics |author=Paul Busch, Marian Grabowski, Pekka J. Lahti |chapter=§III.4: Energy and time |pages=77 ''ff'' |url=http://www.amazon.com/Operational-Quantum-Physics-Lecture-Monographs/dp/3540593586/ref=sr_1_1?s=books&ie=UTF8&qid=1291503715&sr=1-1#reader_3540593586 |isbn=3540593586 |year=1995 |publisher=Springer}}
+</ref>
+<ref name=Compagno>
+The notation [a, b] denotes the [[commutator]] [ab–ba]. This commutation relation is oversimplified, and a correct version replaces the &delta; product on the right by the transverse ''&delta;-tensor'':
+:<math>\delta_{\perp\ ij}(\mathbf{x-x'}) = \frac{1}{(2\pi )^3}\int d^3 \mathbf{k} \left(\delta_{ij}-\hat{u}_i\hat{u}_j \right) e^{i\mathbf{k\cdot (x-x')}} \ . </math>
+where '''û''' is the unit vector of '''k''', '''û''' = '''k'''/''k''. For a discussion see, {{cite book |title=Atom-Field Interactions and Dressed Atoms;'' Vol. 17 of Cambridge Studies in Modern Optics'' |author=G. Compagno, R. Passante, F. Persico |url=http://books.google.com/books?id=R2GUQlBEeekC&pg=PA31 |chapter=§2.1 Canonical quantization in the Coulomb gauge |pages=31 ''ff'' |year=2005 |publisher=Cambridge University Press |isbn=0521019729 }}
+</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 |author=Kiyosi Itô |url=http://books.google.com/books?id=azS2ktxrz3EC&pg=PA1303 |pages=p. 1303 |chapter=§ 351 (XX.23) C: Canonical commutation relations |isbn=0262590204 |year=1993 |edition=2nd ed |publisher=MIT Press}}
+</ref>
+<ref name=Davies>
+For an example, see {{cite book |title=The accidental universe |author=P. C. W. Davies |url=http://books.google.com/books?id=s2s4AAAAIAAJ&pg=PA106 |pages=p. 106 |isbn=0521286921 |year=1982 |publisher=Cambridge University Press}}
+</ref>
+<ref name=Dirac>
+{{cite book |author=PAM Dirac |title=Lorentz and Poincaré invariance: 100 years of relativity |editor=Jong-Ping Hsu, Yuanzhong Zhang, editors |year=2001 |publisher=World Scientific |isbn=9810247214 |url=http://books.google.com/books?id=jryk42J8oQIC&pg=PA440 |pages=p. 440}}
+</ref>
+<ref name= Donoghue>
+{{cite book |title=Dynamics of the standard model |author=John F. Donoghue, Eugene Golowich, Barry R. Holstein |pages=p. 47 |url=http://books.google.com/books?id=hFasRlkBbpYC&pg=PA47 |isbn=0521476526 |Cambridge University Press |year=1994}}
+</ref>
+<ref name=gravity>
+For example, see {{cite book |title=A First Course in Loop Quantum Gravity |url=http://books.google.com/books?id=sSzPlPP8JEwC&pg=PA1 |pages=pp. 1 ''ff'' |chapter=Chapter 1: Why quantize gravity? |publisher=Oxford University Press |year=2010 |author=Rodolfo Gambini, Jorge Pullin |isbn=0199590753}} and {{cite book |title=Quantum gravity |author=Carlo Rovelli |url=http://books.google.com/books?id=HrAzTmXdssQC&pg=PA202 |pages=pp. 202 ''ff'' |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=0521837332}}
+</ref>
+<ref name=Grynberg>
 {{cite book |title=Introduction to Quantum Optics: From the Semi-classical Approach to Quantized Light |author=Gilbert Grynberg, Alain Aspect, Claude Fabre |url=http://books.google.com/books?id=l-l0L8YInA0C&pg=PA351 |pages=pp. 351 ''ff'' |§5.2.2 Vacuum fluctuations and their physical consequences |year=2010 |isbn=0521551129 |publisher=Cambridge University Press}}
+</ref>
-</ref> As a result, the quantum electrodynamic vacuum can be considered as a dielectric medium, and is capable of [[vacuum polarization]].<ref name=Weisskopf>
+<ref name=Hartemann>
+{{cite book |title=High-field electrodynamics |author=Frederic V. Hartemann |url=http://books.google.com/books?id=tIkflVrfkG0C&pg=PA428 |pages=p.428 |isbn=0849323789 |year=2002 |publisher=CRC Press}}
+</ref>
-{{cite book |title=Concepts of particle physics, Volume 2 |author=Kurt Gottfried, Victor Frederick Weisskopf |url=http://books.google.com/books?id=KXvoI-m9-9MC&pg=PA259 |pages=259 ''ff''   |isbn=	0195033930 |year=1986 |publisher=Oxford University Press}}
+<ref name=Heyl>
+{{cite journal |title=Birefringence and Dichroism of the QED Vacuum |author=Jeremy S. Heyl, Lars Hernquist |url=http://arxiv.org/abs/hep-ph/9705367v1 |journal=J Phys |volume=A30 |pages=6485-6492 |doi=10.1088/0305-4470/30/18/022 |year=1997}}
+</ref>
-</ref> In particular, the [[Coulomb's law|force law between charged particles]] is affected.<ref name=Schroeder>
+<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 |author=Moray B. King |url=http://books.google.com/books?id=0RmkmrFxHM0C&pg=PA124 |pages=pp. 124 ''ff'' |isbn=0932813941 |year=2001 |publisher=Adventures Unlimited Press}}
+</ref>
-{{cite book |title=An introduction to quantum field theory |author=Michael Edward Peskin, Daniel V. Schroeder |url=http://books.google.com/books?id=i35LALN0GosC&pg=PA244 |pages=pp. 244 ''ff'' |chapter=§7.5 Renormalization of the electric charge |isbn=0201503972 |publisher=Westview Press |year=1995}}
+<ref name=Longo>
+{{cite book |author=Luciano Boi |title=The Two Cultures: Shared Problems |chapter=Creating the physical world ''ex nihilo?'' On the quantum vacuum and its fluctuations |editor=Ernesto Carafoli, Gian Antonio Danieli, Giuseppe O. Longo, editors |url=http://books.google.com/books?id=Kz38u2qT36kC&pg=PA55 |pages=p. 55 |isbn=8847008689 |year=2009 |publisher=Springer}}
+</ref>
-</ref> The electrical permittivity of quantum electrodynamic vacuum can be calculated, and it differs slightly from the simple ε<sub>0</sub> of the classical vacuum. Likewise, its permeability can be calculated and differs slightly from 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=Meszaros>
+{{cite book |title=High-energy radiation from magnetized neutron stars |author=Peter Mészáros |url=http://books.google.com/books?id=L6DfR49OWd8C&pg=PA56#v=onepage&q&f=false |pages=pp. 56 ''ff'' |chapter=§2.6 Quantum electrodynamics in strong fields |isbn=0226520943 |year=1992  |publisher=University of Chicago Press}}
+</ref>
-{{cite book |title=Dynamics of the standard model |author=John F. Donoghue, Eugene Golowich, Barry R. Holstein |pages=p. 47 |url=http://books.google.com/books?id=hFasRlkBbpYC&pg=PA47 |isbn=0521476526 |Cambridge University Press |year=1994}}
+<ref name=nonlinear>
+{{cite book |title=Applications of laser-plasma interactions |chapter=Nuclear and particle physics with ultraintense lasers |author=José Tito Mendonça, Shalom Eliezer |url=http://books.google.com/books?id=inCD7lqR0K8C&pg=PA145 |pages=p. 145 |isbn=0849376041 |editor=Shalom Eliezer, Kunioki Mima, eds |year=2008 |publisher=CRC Press}}
+</ref>
-</ref> Under some extreme circumstances (for example, in the very high fields found in the exterior regions of [[pulsars]]<ref name=Mészáros>
+<ref name=Parker>
+{{cite book |title=Biophotonics, Volume 360, Part 1 |author=Ian Parker |pages=p. 516 |url=http://books.google.com/books?id=ACcfLIRrLLIC&pg=PA516 |isbn=012182263X |year=2003 |publisher=Academic Press}}
+</ref>
-{{cite book |title=High-energy radiation from magnetized neutron stars |author=Peter Mészáros |url=http://books.google.com/books?id=L6DfR49OWd8C&pg=PA530 |pages=pp. 59 ''ff'' |chapter=§2.6 (b) Wave propagation in the QED vacuum |isbn=0226520943 |year=1992  |publisher=University of Chicago Press}}
+<ref name=Petrosyan>
+{{cite book |title=Fundamentals of quantum optics and quantum information |author=Peter Lambropoulos, David Petrosyan |url=http://books.google.com/books?id=53bpU-41U8gC&pg=PA30 |pages=p. 30 |isbn=354034571X |year=2007 |publisher=Springer}}
+</ref>
-</ref> ), the quantum electrodynamic vacuum is thought to exhibit nonlinearity in the fields.<ref name=Hartemann>
+<ref name=scatter>
+See {{cite book |title=Concepts of particle physics, Volume 2 |author=Kurt Gottfried, Victor Frederick Weisskopf |pages=p. 266 |url=http://books.google.com/books?id=KXvoI-m9-9MC&pg=PA266#v=onepage&q&f=false |isbn=0195033930 |publisher=Oxford University Press |year=1986}}
+</ref>
-{{cite book |title=High-field electrodynamics |author=Frederic V. Hartemann |url=http://books.google.com/books?id=tIkflVrfkG0C&pg=PA428 |pages=p.428 |isbn=0849323789 |year=2002 |publisher=CRC Press}}
+<ref name=Schroeder>
+{{cite book |title=An introduction to quantum field theory |author=Michael Edward Peskin, Daniel V. Schroeder |url=http://books.google.com/books?id=i35LALN0GosC&pg=PA244 |pages=pp. 244 ''ff'' |chapter=§7.5 Renormalization of the electric charge |isbn=0201503972 |publisher=Westview Press |year=1995}}
+</ref>
-</ref> Calculations also indicate birefringence and dichroism at high fields.<ref name=Heyl>
+<ref name=Schwabl>
+{{cite book |title=Quantum mechanics |author=Franz Schwabl |chapter=§ 3.1.3: The zero-point energy |pages=p. 54 |isbn=3540719326 |edition =4rth ed. |publisher=Springer |year=2007 |url=http://books.google.com/books?id=pTHb4NK2eZcC&pg=PA54}}
+</ref>
-{{cite journal |title=Birefringence and Dichroism of the QED Vacuum |author=Jeremy S. Heyl, Lars Hernquist |url=http://arxiv.org/abs/hep-ph/9705367v1 |journal=J Phys |volume=A30 |pages=6485-6492 |doi=10.1088/0305-4470/30/18/022 |year=1997}}
+<ref name=Schweber>
+{{cite book |author=Silvan S Schweber |title=The Oxford companion to the history of modern science |editor=J. L. Heilbron, ed |url=http://books.google.com/books?id=abqjP-_KfzkC&pg=PA246 |pages=pp. 246-247 |quote=Thus in QED the presence of an electric charge ''e<sub>o</sub>'' polarizes the "vacuum" and the charge that is observed at a large distance differs from ''e<sub>o</sub>'' and is given by ''e=e<sub>o</sub>''/&epsilon; with &epsilon; the dielectric constant of the vacuum. |chapter=Elementary particles |isbn=0195112296 |year=2003 |publisher=Oxford University Press}}
+</ref>
+<ref name=Shankar>
+{{cite book|title=Principles of quantum mechanics |author=Ramamurti Shankar |url=http://books.google.com/books?id=2zypV5EbKuIC&pg=PA507 |pages=p. 507 |isbn=0306447908 |year=1994 |edition=2nd ed. |publisher=Springer}}
 </ref>
-===Attainability===
+<ref name= Silk>
-A perfect vacuum is itself only realizable in principle.<ref name=Longo>
+{{cite book |title=On the shores of the unknown: a short history of the universe |author=Joseph Silk |pages=p. 62 |url=http://books.google.com/books?id=JowbGgYiSDYC&pg=PA62 |isbn=0521836271 |publisher=Cambridge University Press |year=2005}}
+</ref>
-{{cite book |author=Luciano Boi |title=The Two Cultures: Shared Problems |chapter=Creating the physical world ''ex nihilo?'' On the quantum vacuum and its fluctuations |editor=Ernesto Carafoli, Gian Antonio Danieli, Giuseppe O. Longo, editors |url=http://books.google.com/books?id=Kz38u2qT36kC&pg=PA55 |pages=p. 55 |isbn=8847008689 |year=2009 |publisher=Springer}}
+<ref name=Vogel>
+{{cite book |author=Werner Vogel, Dirk-Gunnar Welsch |title=Quantum optics |chapter=§2.2.1 Canonical quantization: Eq. (2.50) |page=21 |url=http://books.google.com/books?id=AFdY_zNZciAC&pg=PA21 |isbn=3527405070 |publisher=Wiley-VCH |year=2006 |edition=3rd}}
+</ref>
-</ref><ref name=Dirac>
+<ref name=Vogel2>
+{{cite book |title=Quantum optics |author=Werner Vogel, Dirk-Gunnar Welsch |pages=pp. 18 ''ff'' |chapter= Chapter 2: Elements of quantum electrodynamics |isbn=3527405070 |year=2006 |publisher=Wiley-VCH |edition =3rd ed. |url=http://books.google.com/books?id=eouwERvRrJEC&pg=PA18&lpg=PA18 }}
+</ref>
-{{cite book |author=PAM Dirac |title=Lorentz and Poincaré invariance: 100 years of relativity |editor=Jong-Ping Hsu, Yuanzhong Zhang, editors |year=2001 |publisher=World Scientific |isbn=9810247214 |url=http://books.google.com/books?id=jryk42J8oQIC&pg=PA440 |pages=p. 440}}
+<ref name=Weisskopf>
+{{cite book |title=Concepts of particle physics, Volume 2 |author=Kurt Gottfried, Victor Frederick Weisskopf |url=http://books.google.com/books?id=KXvoI-m9-9MC&pg=PA259 |pages=259 ''ff''   |isbn=	0195033930 |year=1986 |publisher=Oxford University Press}}
+</ref>
-</ref> It is an idealization, like [[absolute zero]] for temperature, that can be approached, but never actually realized:<ref name=Longo/>
+<ref name=Zeidler>
-{|align=center style="width:70%;"
+{{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 |author=Eberhard Zeidler |url=http://books.google.com/books?id=miwuxaEXvOsC&pg=PA952#v=onepage&q&f=false |pages=952 |isbn=3642224202 |publisher=Springer |year=2011}}
-|“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...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''.”
+</ref>
-:::::''Luciano Boi'', "Creating the physical world ''ex nihilo?''"  p. 55
-|}
-==References==
+}}
-{{reflist}}

Vacuum (quantum electrodynamic): Difference between revisions

Latest revision as of 09:33, 18 February 2012

Contents

Virtual particles

Quantization of the fields

Electromagnetic properties

Attainability

References

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Vacuum (quantum electrodynamic): Difference between revisions

Latest revision as of 09:33, 18 February 2012

Virtual particles

Quantization of the fields

Electromagnetic properties

Attainability

References

Navigation menu

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