A 2022 letter by M. Bauer showed that a dynamical time operator T=α⋅r/c+βτ0T=\alpha\cdot r/c+\beta\tau_0, derived together with the Dirac Hamiltonian from a canonical quantization of special relativity, is not invariant under the combined charge-conjugation–parity (CP) transformation: its expectation value shifts by −2⟨β⟩h/(m0c2)-2\langle\beta\rangle h/(m_0c^2), a CP-odd effect inversely proportional to the fermion mass, and of opposite sign for particle-like and antiparticle-like wave packets. Because neutrino masses are six to nine orders of magnitude below the lightest-quark masses, this operator-induced CP violation would be correspondingly amplified in the neutrino sector, and its sign structure was argued qualitatively to agree with the indication from the T2K long-baseline neutrino-oscillation experiment (Tokai-to-Kamioka, Japan) that muon neutrinos oscillate into electron neutrinos more readily than antineutrinos do. The letter, however, derived no CP-violating transition rate, performed no baryogenesis computation, and produced no quantitative oscillation predictions. The open problem is to determine quantitatively whether this mechanism can account for the observed baryon asymmetry of the universe ηB≈6×10−10\eta_B\approx 6\times10^{-10} when embedded in a concrete baryogenesis scenario satisfying the Sakharov conditions, and whether it yields definite, testable neutrino–antineutrino oscillation asymmetries — or, alternatively, to demonstrate that the operator non-invariance induces no observable CP asymmetry at all.

Background

The universe contains baryonic matter but essentially no antibaryons; the baryon asymmetry, conventionally measured by the baryon-to-photon ratio ηB≈6×10−10\eta_B\approx 6\times10^{-10}, is one of the central unexplained facts of cosmology. Sakharov's conditions state that a dynamical origin of this asymmetry requires (i) violation of baryon number, (ii) violation of C (charge conjugation) and CP (the combined charge-conjugation–parity transformation, which interchanges particles with antiparticles), and (iii) departure from thermal equilibrium. In the Standard Model (SM), the CP violation entering quark mixing (described by the Cabibbo–Kobayashi–Maskawa matrix) is far too small to produce the observed asymmetry, which is why extensions are studied; the mainstream route is leptogenesis, in which CP-violating decays of heavy neutral leptons create a lepton asymmetry that weak-interaction sphaleron processes (thermal transitions between vacua with different baryon and lepton number) convert into a baryon asymmetry.

The candidate mechanism comes from a different direction. Starting from a canonical quantization of the invariant scalars of special relativity, pμpμp_\mu p^\mu, xμxμx_\mu x^\mu, and {xμpμ−pμxμ}\{x_\mu p^\mu-p_\mu x^\mu\}, supplemented by Born's reciprocity principle (the proposal that position and momentum should enter the foundations on an equal footing), Bauer and co-workers derived, alongside the Dirac Hamiltonian HD=c α⋅p+βm0c2H_D=c\,\alpha\cdot p+\beta m_0c^2, a dynamical ("internal") time operator

T=α⋅r/c+β τ0,τ0=h/(m0c2),T=\alpha\cdot r/c+\beta\,\tau_0,\qquad \tau_0=h/(m_0c^2),

where α\alpha and β\beta are the Dirac matrices, rr the position operator, and τ0\tau_0 the de Broglie period (the characteristic internal clock period of a particle of rest mass m0m_0). This operator is not conjugate to HDH_D, so it does not literally contradict Pauli's theorem (the standard result that no self-adjoint time operator can be conjugate to a Hamiltonian whose spectrum is bounded below); its spectrum is continuous from −∞-\infty to +∞+\infty with a gap 2τ02\tau_0 at the origin, and it evolves with a Zitterbewegung-type oscillation (the rapid interference trembling of Dirac wave packets) superposed on linear evolution.

Under CP, the expectation value of TT is not invariant:

⟨T⟩CP−⟨T⟩=−2 ⟨β⟩ τ0=−2 ⟨β⟩ h/(m0c2),\langle T\rangle_{CP}-\langle T\rangle=-2\,\langle\beta\rangle\,\tau_0=-2\,\langle\beta\rangle\,h/(m_0c^2),

which the letter interprets as a CP violation (CPV) inversely proportional to the fermion mass. Since neutrino masses, though not yet fully determined, are expected to lie around 10−310^{-3}–10−110^{-1} eV — six to nine orders of magnitude below the up- and down-quark masses — this CPV would be amplified by the same factor in the neutrino sector, and the normal mass ordering (the arrangement in which the lightest of the three neutrino mass eigenstates is the one with the largest electron-neutrino component) maximizes it for νe\nu_e and νμ\nu_\mu. Because ⟨β⟩≃+1\langle\beta\rangle\simeq +1 for particle-like and ≃−1\simeq -1 for antiparticle-like wave packets, the CPV has opposite signs for neutrinos and antineutrinos; the letter argues qualitatively that this matches the observation of the T2K experiment (a long-baseline accelerator-neutrino experiment in Japan) that νμ→νe\nu_\mu\to\nu_e oscillations occur more readily than the corresponding antineutrino oscillations. Crucially, what the letter does not do: it derives no CP-violating transition amplitude or rate asymmetry, constructs no baryon-number-violating out-of-equilibrium dynamics, computes no baryon asymmetry, and gives no energy- or baseline-dependent oscillation predictions. It states only that the effect "may be sufficient to account for the matter antimatter asymmetry observed". Whether that sufficiency claim survives quantitative analysis — and whether the operator-level non-invariance can be connected to observable particle–antiparticle rate asymmetries at all — is the open question.

Problem Statement

Determine, quantitatively, whether the internal-time-operator CP violation — the CP non-invariance ⟨T⟩CP−⟨T⟩=−2⟨β⟩h/(m0c2)\langle T\rangle_{CP}-\langle T\rangle=-2\langle\beta\rangle h/(m_0c^2) of the dynamical time operator T=α⋅r/c+βτ0T=\alpha\cdot r/c+\beta\tau_0 (with α\alpha and β\beta the Dirac matrices, rr the position operator, m0m_0 the fermion rest mass, and τ0=h/(m0c2)\tau_0=h/(m_0c^2) the de Broglie internal-clock period) — can account for the observed matter–antimatter asymmetry, by completing the program the source letter left open. Concretely:

  1. Establish the bridge from the operator-level non-invariance to observable physics: derive whether, and through what mechanism, it induces CP-asymmetric transition rates or cross-sections for particle versus antiparticle processes (this bridge is a prerequisite for everything else, since the letter derives no CP-violating amplitude at all).

  2. Baryogenesis sufficiency: insert the induced CPV into a concrete baryogenesis scenario satisfying all three Sakharov conditions (baryon-number violation, C and CP violation, and departure from thermal equilibrium) — a specified baryon- or lepton-number-violating process with out-of-equilibrium dynamics — and compute the resulting baryon asymmetry, including its dependence on the neutrino masses and their (still experimentally undetermined) ordering, in order to decide whether the observed baryon asymmetry, ηB≈6×10−10\eta_B\approx 6\times10^{-10} (the baryon-to-photon number-density ratio), can be produced.

  3. Oscillation phenomenology: derive the neutrino–antineutrino oscillation rate asymmetry implied by the mechanism, with definite energy and baseline dependence, so that the prediction is testable against current long-baseline accelerator-neutrino observations such as those of T2K.

Items 1–3 are jointly the stated objective: they are the quantitative elaboration of the single mechanism the letter proposes as the origin of the matter–antimatter asymmetry, with item 3 serving as its experimental test. Items 2 and 3 apply insofar as item 1 yields an affirmative bridge to CP-asymmetric rates; if the derivation of item 1 shows instead that the operator non-invariance cannot affect physical rates, items 2 and 3 have no content. Whether the mechanism is sufficient or insufficient for the observed asymmetries on quantitative grounds — or, by explicit derivation, cannot affect physical rates at all — is what remains to be determined. The problem is to be answered at the generality of the source; no restriction to particular neutrino masses, orderings, or model classes applies.

References

  1. M. Bauer, "Matter-antimatter Asymmetry, CP Violation and the Time Operator in Relativistic Quantum Mechanics", arXiv:2208.13864 (2022). DOI: 10.48550/arXiv.2208.13864, https://arxiv.org/abs/2208.13864
  2. C. A. Aguillón, M. Bauer, G. E. García, "Time and energy operators in the canonical quantization of special relativity", Eur. J. Phys. 41, 035601 (2020). DOI: 10.1088/1361-6404/ab67d6, https://doi.org/10.1088/1361-6404/ab67d6
  3. Mariano Bauer, "A dynamical time operator in Dirac's relativistic quantum mechanics", Int. J. Mod. Phys. A 29, 1450036 (2014). arXiv:0908.2789, DOI: 10.1142/S0217751X14500365, https://doi.org/10.1142/S0217751X14500365
  4. T2K Collaboration (K. Abe et al.), "Constraint on the matter–antimatter symmetry-violating phase in neutrino oscillations", Nature 580, 339–344 (2020). DOI: 10.1038/s41586-020-2177-0, https://doi.org/10.1038/s41586-020-2177-0
  5. A. D. Sakharov, "Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe", Sov. Phys. Usp. 34, 392–393 (1991) [JETP Lett. 5, 24 (1967)]. DOI: 10.1070/pu1991v034n05abeh002497, https://doi.org/10.1070/pu1991v034n05abeh002497
  6. W. Buchmüller, P. Di Bari, M. Plümacher, "Leptogenesis for Pedestrians", Ann. Phys. 315, 305–351 (2005). arXiv:hep-ph/0401240, DOI: 10.1016/j.aop.2004.02.003, https://arxiv.org/abs/hep-ph/0401240
  7. T2K Collaboration (K. Abe et al.), "Measurements of neutrino oscillation parameters from the T2K experiment using 3.6×10^21 protons on target", Eur. Phys. J. C 83, 782 (2023). DOI: 10.1140/epjc/s10052-023-11819-x, https://doi.org/10.1140/epjc/s10052-023-11819-x