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#REDIRECT [[Vacuum (classical)]]
{{dambigbox|Free space (electromagnetism)|Free space}}
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'''Free space''' usually refers to a perfect [[vacuum]], devoid of all particles. The term is most often used in classical [[electromagnetism]] where it refers to a reference state,<ref name=Weiglhofer>{{cite book|title=Introduction to complex mediums for optics and electromagnetics |author=Werner S. Weiglhofer and Akhlesh Lakhtakia |year=2003 |url=http://books.google.com/books?id=QtIP_Lr3gngC&pg=PA34&hl=en#v=onepage&q&f=false|publisher=SPIE Press |isbn=0819449474 |chapter=§4.1: The classical vacuum as reference medium }}</ref> and in quantum physics where it refers to the ground state of the [[Electromagnetic wave|electromagnetic field]], which is subject to fluctuations about a dormant zero average-field condition.<ref name=Shankar>{{cite book|title=Principles of quantum mechanics |author=Ramamurti Shankar |url=http://books.google.com/books?id=2zypV5EbKuIC&pg=PA507#v=onepage&q=free%20space&f=false |pages=p. 507 |isbn=0306447908 |year=1994 |edition=2nd ed. |publisher=Springer}}</ref> The classical case of vanishing fields implies all fields are source-attributed, while in the quantum case field moments can arise without sources by virtual [[photon]] creation and destruction.<ref name=Vogel>{{cite book |title=Quantum optics |author=Werner Vogel, Dirk-Gunnar Welsch |url=http://books.google.com/books?id=qRtnP1dPGmQC&pg=PA337&hl=en#v=onepage&q&f=false |pages=p. 337 |publisher=Wiley-VCH |year=2006 |edition=3rd ed.  |isbn=3527405070}}</ref> The description of free space varies somewhat among authors, with some authors requiring only the absence of substances with electrical properties,<ref name=Pathria>{{cite book|title=The Theory of Relativity |author= RK Pathria |url=http://books.google.com/books?id=Ma4ZFefVKIYC&pg=PA119&hl=en#v=onepage&q&f=false |pages=p. 119 | |year=2003 |isbn=0486428192 |publisher=Courier Dover Publications |edition=Reprint of Hindustan 1974 2nd ed.}}</ref> or of charged matter ([[ion]]s and [[electron]]s, for example).<ref name=Morris>{{cite book |title=Academic Press dictionary of science and technology |editor=Christopher G. Morris, editor |publisher=Academic |url=http://books.google.com/books?id=nauWlPTBcjIC&pg=PA880&hl=en#v=onepage&q&f=false|pages=p. 880 |year=1992 |isbn=0122004000}}</ref>
 
===Classical case===
In classical physics, free space is a concept of electromagnetic theory, corresponding to a theoretically perfect vacuum and sometimes referred to as the ''vacuum of free space'', or as ''classical vacuum'', and is appropriately viewed as a reference medium.<ref name=Weiglhofer/> In the classical case, free space is characterized by the electrical permittivity ε<sub>0</sub> and the magnetic permeability μ<sub>0</sub>.<ref name=Messier>
 
{{cite book |title=Sculptured thin films: nanoengineered morphology and optics |author=Akhlesh Lakhtakia, R. Messier |chapter=§6.2: Constitutive relations |url=http://books.google.com/books?id=yCzDND-vIhMC&pg=PA105#v=onepage&q&f=false |pages=p. 105 |publisher=SPIE Press |year=2005 |isbn=0819456063}}
 
</ref> The exact value of ε<sub>0</sub> is provided by [[NIST]] as the [[electric constant]] <ref name="NIST1">{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?ep0 |title=Electric constant |accessdate=2010-11-28 |author=[[CODATA]] |work=2006 CODATA recommended values |publisher=[[NIST]] }}</ref> and the defined value of μ<sub>0</sub> as the [[magnetic constant]]:<ref name="NIST2">{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?mu0 |title=Magnetic constant |accessdate=2010-11-28 |author=[[CODATA]] |work=2006 CODATA recommended values |publisher=[[NIST]] }}</ref>
::ε<sub>0</sub> ≈ 8.854 187 817... × 10<sup>−12</sup> F m<sup>−1</sup>
 
::μ<sub>0</sub> = 4π × 10<sup>−7</sup> ≈ 12.566 370 614... x 10<sup>−7</sup> N A<sup>−2</sup>
 
where the approximation is not a physical uncertainty (such as a measurement error) but a result of the inability to express these [[irrational numbers]] with a finite number of digits. The [[SI units]] farad, metre, newton and ampere are denoted by 'F', 'm', 'N', and 'A'.
 
One consequence of these electromagnetic properties coupled with [[Maxwell's equations]] is that the [[speed of light]] in free space is related to ε<sub>0</sub> and μ<sub>0</sub> via the relation:<ref name=Baschek>
 
{{cite book |title= The new cosmos: an introduction to astronomy and astrophysics |author=Albrecht Unsöld, B. Baschek |url=http://books.google.com/books?id=nNnmR8ljctoC&pg=PA101 |pages=p. 101 |chapter=§4.1: Electromagnetic radiation, Equation 4.3 |isbn=3540678778 |year=2001 |publisher=Springer |edition=5th ed.}}
 
</ref>
 
::<math>c_0 = 1/\sqrt{\mu_0 \varepsilon_0}\ . </math>
 
Using the defined valued for the speed of light provided by NIST as:<ref name="NIST3">{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?c |accessdate=2010-11-28 |author=[[CODATA]] |work=2006 CODATA recommended values |publisher=[[NIST]]|title=Speed of light in vacuum  }} A ''defined value'' for the speed of light is a consequence of adoption of ''time of transit'' as the measure of length, so lengths are measured in seconds. See [[Metre (unit)|metre]].</ref>
 
::c<sub>0</sub> = 299 792 458 m s <sup>−1</sup>,
 
and the already mentioned defined value for μ<sub>0</sub>, this relationship leads to the exact value given above for ε<sub>0</sub>.
 
Another consequence of these electromagnetic properties is that the ratio of electric to magnetic field strengths in an [[electromagnetic wave]] propagating in free space is an exact value provided by NIST as the [[characteristic impedance of vacuum]]:<ref name="NIST4">{{cite web |url=http://physics.nist.gov/cgi-bin/cuu/Value?z0 |accessdate=2010-11-28 |author=[[CODATA]] |work=2006 CODATA recommended values |publisher=[[NIST]] |title=Characteristic impedance of vacuum Z<sub>0</sub>  }}</ref>
 
:: <math>Z_0 =  \sqrt{\mu_0 /\varepsilon_0} \  </math>
:::= 376.730 313 461... ohms.
 
It also can be noted that the electrical permittivity ε<sub>0</sub> and the magnetic permeability μ<sub>0</sub> do not depend upon direction, field strength, polarization, or frequency. Consequently, free space is isotropic, linear, non-dichroic, and dispersion free. Linearity, in particular, implies that the fields and/or potentials due to an assembly of charges is simply the addition of the fields/potentials due to each charge separately (that is, the  principle of superposition applies).<ref name=Pramanik>
{{cite book |title=Electro-Magnetism: Theory and Applications |author=A. Pramanik |url=http://books.google.com/books?id=gnEEwy12S5cC&pg=PT23 |pages=pp. 37-38 |chapter=§1.3 The principle of superposition |isbn=8120319575 |year=2004 |publisher=PHI Learning Pvt. Ltd}}</ref>
 
===Quantum case===
The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle in one dimension does not allow a state of rest 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 [[zero-point energy]] and a range of momentum and spread in location attributable to quantum fluctuations.
 
An uncertainty principle applies to all quantum mechanical operators that do not [[commutator|commute]].<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> 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>
 
: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/>
 
:::<math>\mathbf B =\mathbf {\nabla  \times A} \ </math>
:::<math>\mathbf E = -\mathbf{ A - \nabla }V \ . </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>
 
:Quantization is achieved by insisting that the momentum field and the vector potential do not commute.
 
Because of the non-commutation of field variables, the variances of the fields cannot be zero, although their averages are zero.<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> As a result, the vacuum can be considered as a dielectric medium, and is capable of [[vacuum polarization]].<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> In particular, the [[Coulomb's law|force law between charged particles]] is affected.<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> The electrical permittivity of 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>.
 
===Attainability===
A perfect vacuum is itself only realizable in principle.<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 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> It is an idealization, like [[absolute zero]] for temperature, that can be approached, but never actually realized:<ref name=Longo/>
<blockquote>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 <u>vacuum fluctuation</u>.
::::::Luciano Boi, <u>Creating the physical world ''ex nihilo?''</u>  p. 55</blockquote>
 
 
 
And classical vacuum is one step further removed from attainability because its permittivity ε<sub>0</sub> and permeability μ<sub>0</sub> do not allow for quantum fluctuations. Nonetheless, outer space and good terrestrial vacuums are modeled adequately by classical vacuum for many purposes.
 
==References==
{{reflist}}

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