Born reciprocity

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. 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. 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 and the four-vector momentum (four-momentum) coordinates 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: 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 {displaystyle 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. The metric is invariant under the group of quaplectic transformations.