Born reciprocity

{{Short description|Principle in theoretical physics}}

{{distinguish|text=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-symmetry among space and momentum. Born and his co-workers expanded his principle to a framework that is also known as reciprocity theory.{{Cite journal|doi=10.1098/rspa.1938.0060|title=A suggestion for unifying quantum theory and relativity|year=1938|last1=Born|first1=Max|last2=Whittaker|first2=Edmund Taylor|journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=165|issue=921|pages=291–303|bibcode=1938RSPSA.165..291B|s2cid=121816621|doi-access=|citeseerx=10.1.1.205.4432}}{{Cite journal|doi=10.1103/RevModPhys.21.463|title=Reciprocity Theory of Elementary Particles|year=1949|last1=Born|first1=Max|journal=Reviews of Modern Physics|volume=21|issue=3|pages=463–473|bibcode=1949RvMP...21..463B|doi-access=free|url=http://cds.cern.ch/record/1062652/files/RevModPhys.21.463.pdf}}

Born noticed a symmetry among configuration space and momentum space representations of a free particle, in that its wave function description is invariant to a change of variables x → p and p → −x. (It can also be worded such as to include scale factors, e.g. invariance to x → ap and p → −bx where a, b are constants.) Born hypothesized that such symmetry should apply to the four-vectors of special relativity, that is, to the four-vector space coordinates

:$\backslash mathbf\{X\}\; =\; X^\{\backslash mu\}\; :=\; \backslash left(X^0,\; X^1,\; X^2,\; X^3\; \backslash right)\; =\; \backslash left(ct,\; x,\; y,\; z\; \backslash right)$

and the four-vector momentum (four-momentum) coordinates

:$\backslash mathbf\{P\}\; =\; P^\{\backslash nu\}\; :=\; \backslash left(P^0,\; P^1,\; P^2,\; P^3\; \backslash right)\; =\; \backslash left(\backslash frac\{E\}\{c\},\; p\_x,\; p\_y,\; p\_z\; \backslash right)$

Both in classical and in quantum mechanics, the Born reciprocity conjecture postulates that the transformation x → p and p → −x leaves invariant the Hamilton equations:

:$\backslash dot\{x\}\_i\; =\; \backslash partial\; H\; /\; \backslash partial\; p\_i$ and $\backslash dot\{p\}\_i\; =\; -\; \backslash partial\; H\; /\; \backslash partial\; x\_i$

From his reciprocity approach, Max Born conjectured the invariance of a space-time-momentum-energy line element. Born and H.S. Green similarly introduced the notion an invariant (quantum) metric operator $x\_k\; x^k\; +\; p\_k\; p^k$ as extension of the Minkowski metric of special relativity to an invariant metric on phase space coordinates.{{cn|date=March 2021}} The metric is invariant under the group of quaplectic transformations.Stuart Morgan: [http://eprints.utas.edu.au/10689/ A Modern Approach to Born Reciprocity], PhD Thesis, University of Tasmania, 2011{{Cite journal|doi = 10.1088/1751-8113/40/40/006|title = World-line quantization of a reciprocally invariant system|year = 2007|last1 = Govaerts|first1 = Jan|last2 = Jarvis|first2 = Peter D.|last3 = Morgan|first3 = Stuart O.|last4 = Low|first4 = Stephen G.|journal = Journal of Physics A: Mathematical and Theoretical|volume = 40|issue = 40|pages = 12095–12111|arxiv = 0706.3736|bibcode = 2007JPhA...4012095G|s2cid = 16995610}}

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.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. 199 ff.{{Cite journal|doi = 10.1007/BF00708529|title = Physical basis for minimal time-energy uncertainty relation|year = 1979|last1 = Kim|first1 = Y. S.|last2 = Noz|first2 = Marilyn E.|journal = Foundations of Physics|volume = 9|issue = 5–6|pages = 375–387|bibcode = 1979FoPh....9..375K|s2cid = 121121484}} In 1981, Eduardo R. Caianiello proposed a "maximal acceleration", similarly as there is a minimal length at Planck scale, and this concept of maximal acceleration has been expanded upon by others.{{Cite journal|doi = 10.1007/BF02745135|title = Is there a maximal acceleration?|year = 1981|last1 = Caianiello|first1 = E. R.|journal = Lettere al Nuovo Cimento |series=Series 2|volume = 32|issue = 3|pages = 65–70|s2cid = 122974218}}{{Cite arXiv|eprint = hep-th/0208138v2|last1 = Castro|first1 = Carlos|title = Maximal-acceleration phase space relativity from Clifford algebras|year = 2002}} It has also been suggested that Born reciprocity may be the underlying physical reason for the T-duality symmetry in string theory,{{cn|date=March 2021}} and that Born reciprocity may be of relevance to developing a quantum geometry.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. 106–111{{Cite journal|arxiv = 1106.0313|doi = 10.1007/s10714-011-1212-8|title = Relative locality: A deepening of the relativity principle|year = 2011|last1 = Amelino-Camelia|first1 = Giovanni|last2 = Freidel|first2 = Laurent|last3 = Kowalski-Glikman|first3 = Jerzy|last4 = Smolin|first4 = Lee|journal = General Relativity and Gravitation|volume = 43|issue = 10|pages = 2547–2553|bibcode = 2011GReGr..43.2547A|s2cid = 118607387}}

Born chose the term "reciprocity" for the reason that in a crystal lattice, the motion of a particle can be described in p-space by means of the reciprocal lattice.