English

The Hermitian inner product selects the time axis, the Born rule measures it

Quantum Physics 2026-07-04 v1

Abstract

The correspondence between 2×22\times 2 Hermitian matrices and Minkowski 44-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 (C2,ε)(\mathbb{C}^2,\varepsilon) is SL(2,C)SL(2,\mathbb{C})-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 σ0\sigma^0: this choice -- made in passing from a normed space to a Hilbert space, \emph{before} any probability is assigned -- reduces SL(2,C)SL(2,\mathbb{C}) to its maximal compact SU(2)SU(2), the stabiliser of σ0\sigma^0. The Born rule enters one level later: ξξ=tr(σ0ξξ)\langle \xi \vert \xi \rangle = \text{tr}(\sigma^0 \, \xi\xi^\dagger) is the projection of the state's null vector onto σ0\sigma^0, 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 ψ2\lvert\psi\rvert^2 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.

Keywords

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}
}