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.
Green similarly introduced the notion an invariant (quantum) metric operator
[citation needed] The metric is invariant under the group of quaplectic transformations.
[3][4] Such a reciprocity as called for by Born can be observed in much, but not all, of the formalism of classical and quantum physics.
[7][8] It has also been suggested that Born reciprocity may be the underlying physical reason for the T-duality symmetry in string theory,[citation needed] and that Born reciprocity may be of relevance to developing a quantum geometry.