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:''Not to be confused with the [[Born rule]], which relates to the probability of the outcome of a measurement on a quantum system.''

In physics, '''Born reciprocity''', also called '''reciprocal relativity''' or '''Born–Green reciprocity''', is a principle set up by theoretical physicist [[Max Born]] that calls for a [[duality (mathematics)|duality]]-[[symmetry]] among [[space]] and [[momentum]]. Born and his co-workers expanded his principle to a framework that is also known as '''reciprocity theory'''.<ref name="born-1938">M. Born, A suggestion for unifying quantum theory and relativity, Proceedings of the Royal Society London A (1938), vol.&nbsp;165, pp.&nbsp;291–303, {{DOI|10.1098/rspa.1938.0060}} [http://rspa.royalsocietypublishing.org/content/165/921/291.full.pdf full text]</ref><ref name="born-1949">M. Born (1949), Reciprocity Theory of Elementary Particles, Review of Modern Physics vol.&nbsp;21, no.&nbsp;3, pp.&nbsp;463–473 {{DOI|10.1103/RevModPhys.21.463}}</ref>

Born noticed a symmetry among [[configuration space]] and [[momentum space]] representations of a [[free particle]], in that its wave function description is [[Invariant (physics)|invariant]] to a change of variables ''x''&nbsp;→&nbsp;''p'' and ''p''&nbsp;→&nbsp;&minus;''x''. (It can also be worded such as to include scale factors, e.g. invariance to ''x''&nbsp;→&nbsp;''ap'' and ''p''&nbsp;→&nbsp;&minus;''bx'' where ''a'', ''b'' are constants.) Born hypothesized that such symmetry should apply to the [[four-vector]]s of [[special relativity]], that is, to the four-vector space coordinates
:<math> \mathbf{X} = X^{\mu} := \left(X^0, X^1, X^2, X^3 \right) = \left(ct, x, y, z \right) </math>
and the four-vector momentum ([[four-momentum]]) coordinates
:<math> \mathbf{P} = P_{\nu} := \left(P_0, P_1, P_2, P_3 \right) = \left(E, p_x, p_y, p_z \right) </math>
Both in classical and in quantum mechanics, the Born reciprocity conjecture postulates that the transformation ''x''&nbsp;→&nbsp;''p'' and ''p''&nbsp;→&nbsp;&minus;''x'' leaves invariant the [[Hamilton equations]]:
:<math>\dot{x}_i = \partial H / \partial p_i</math> and <math>\dot{p}_i = - \partial H / \partial x_i</math>

From his reciprocity approach, Max Born conjectured the invariance of a space-time-momentum-energy [[line element]].<ref>M. Born, Reciprocity theory of elementary particles, Reviews of Modern Physics, vol.&nbsp;21, no.&nbsp;3 (1949), pp.&nbsp;463–473 ([http://rmp.aps.org/abstract/RMP/v21/i3/p463_1 abstract], [http://rmp.aps.org/pdf/RMP/v21/i3/p463_1 full text])</ref> Born and H.S. Green similarly introduced the notion an invariant (quantum) metric operator <math>x_k x^k + p_k p^k</math> as extension of the [[Minkowski metric]] of special relativity to an invariant metric on [[phase space]] coordinates.<ref>See for example the introductory sections of: Jan Govaerts et al: World-line Quantisation of a Reciprocally Invariant System, [http://arxiv.org/abs/0706.3736v1 arXiv:0706.3736v1] (submitted 26 June 2007)</ref> The metric is invariant under the group of [[quaplectic transformation]]s.<ref>Stuart Morgan: [http://eprints.utas.edu.au/10689/ A Modern Approach to Born Reciprocity], PhD Thesis, University of Tasmania, 2011</ref><ref>Jan Govaerts, [[Peter D. Jarvis]], Stuart O. Morgan, Stephen G. Low, World-line quantization of a reciprocally invariant system, Journal of Physics A: Mathematical and Theoretical, vol.&nbsp;40 (2007), pp.&nbsp;12095–12111, {{DOI|10.1088/1751-8113/40/40/006}} ([http://eprints.utas.edu.au/4450/1/4450.pdf PDF])</ref>

Such a reciprocity as called for by Born can be observed in much, but not all, of the formalism of classical and quantum physics. Born's reciprocity theory was not developed much further for reason of difficulties in the mathematical foundations of the theory.

However Born's idea of a quantum metric operator was later taken up by [[Hideki Yukawa]] when developing his nonlocal quantum theory in the 1950s.<ref>[[Eduard Prugovečki]]: Stochastic Quantum Mechanics and Quantum Spacetime, Kluwer Academic Publishers, 1984, ISBN 978-9027716170, ''Section 4.5 Reciprocity Theory and Born's Quantum Metric Operator'', pp.&nbsp;199 ff.</ref><ref>Y. S. Kim, Marilyn E. Noz, Physical basis for minimal time-energy uncertainty relation, Foundations of Physics, vol.&nbsp;9, no.&nbsp;5-6 (1979), pp.&nbsp;375-387, {{DOI|10.1007/BF00708529}}</ref> In 1981, [[Eduardo R. Caianiello]] proposed a "maximal acceleration", similarly as there is a [[Planck Length|minimal length at Planck scale]], and this concept of maximal acceleration has been expanded upon by others.<ref>Is there a maximal acceleration? Lettere al Nuovo Cimento, vol.&nbsp;32, no.&nbsp;3 (1981), pp.&nbsp;65–70, {{DOI|10.1007/BF02745135}}</ref><ref>Carlos Castro: Maximal-acceleration phase space relativity from Clifford algebras, [http://arxiv.org/abs/hep-th/0208138v2 arXiv:hep-th/0208138v2] (submitted 20 August 2002, version of 8 September 2002)</ref> It has also been suggested that Born reciprocity may be the underlying physical reason for the [[T-duality]] symmetry in string theory,<ref>Carlos Castro (2008) [http://www.rxiv.org/pdf/0908.0094v1.pdf On Born's deformed reciprocal complex gravitational theory and noncommutative geometry]</ref> and that Born reciprocity may be of relevance to developing a [[quantum geometry]].<ref>[[Eduard Prugovečki]]: Principles of Quantum General Relativity, World Scientific Pub. Co., 1995, ISBN 978-9810221386, ''Section 3.8 Fundamental Special-Relativistic Quantum Lorentz Frames'', pp.&nbsp;106–111</ref><ref>Giovanni Amelino-Camelia, Laurent Friedel, Jerzy Kowalski-Glikman, [[Lee Smolin]]: Relative locality: A deepening of the relativity principle [http://arxiv.org/pdf/1106.0313.pdf arXiv 1106.0313], 1 June 2011</ref>

Born chose the term "reciprocity" for the reason that in a [[Crystal structure|crystal]] [[Lattice model (physics)|lattice]], the motion of a particle can be described in ''p''-space by means of the [[reciprocal lattice]].<ref name="born-1938"/>

== References ==
{{reflist}}

==Further reading==
* P. D. Jarvis und S. O. Morgan, Born Reciprocity and the Granularity of Spacetime, Foundations of Physics Letters, vol.&nbsp;19, no.&nbsp;6 (2006), pp.&nbsp;501-517, {{DOI|10.1007/s10702-006-1006-5}}
* Stephen G. Low: Reciprocal Relativity of Noninertial Frames and the Quaplectic Group, Foundations of Physics, vol.&nbsp;36, no.&nbsp;7 (2006), pp.&nbsp;1036-1069, {{DOI|10.1007/s10701-006-9051-2}}
* R. Delbourgo, D. Lashmar, Born Reciprocity and the 1/r Potential, Foundations of Physics, vol.&nbsp;38, no.&nbsp;11 (2008), pp.&nbsp;995-1010, {{DOI|10.1007/s10701-008-9247-8}}

[[Category:Concepts in physics]]
[[Category:Duality theories]]