The Hermitian inner product selects the time axis, the Born rule measures it
Abstract
The correspondence between Hermitian matrices and Minkowski -vectors recovers Lorentzian symmetries from the internal degrees of freedom of a qubit, with no reference to an external spacetime. Recent work characterises the resulting Lorentz invariants and leaves the \emph{mechanism} of emergence -- what singles out a time direction -- as an explicit open question. We give an elementary answer and, in doing so, correct a natural misattribution. The bare spin space is -symmetric and singles out no axis; so is the null cone it generates. What selects a future-timelike axis is the choice of a Hermitian inner product, equivalently a positive reference form : this choice -- made in passing from a normed space to a Hilbert space, \emph{before} any probability is assigned -- reduces to its maximal compact , the stabiliser of . The Born rule enters one level later: is the projection of the state's null vector onto , i.e. its energy in that frame, and under a boost it rescales as a Doppler shift. Thus the Hilbert structure selects the axis; the Born rule is where that axis becomes a measurable energy and where the frame-dependence of becomes empirical. The ingredients are classical; what we add is their identification as the mechanism the recent programme leaves open, with the symmetry-breaking step located precisely. This is a kinematic identification of that step, not a dynamical account of why a particular axis is selected. We close by handing back the many-qubit case, where the datum is a tuple of such choices.
Cite
@article{arxiv.2607.05447,
title = {The Hermitian inner product selects the time axis, the Born rule measures it},
author = {Sebastian Zając},
journal= {arXiv preprint arXiv:2607.05447},
year = {2026}
}