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REVIEW 1 major objections 5 minor 34 references

The symbol i can be removed from the quantum wave equation, but the symplectic and complex structure it encoded cannot—the coherent real form requires restoring the missing component as a canonical momentum.

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2026-08-01 10:46 UTC pith:ANT55XIB

load-bearing objection Careful diagnostic paper: the one-field real Schrödinger equation is dynamically sufficient but not locally complete; the lifted phase-space part is known, but Proposition 2 and the Chen correction are genuinely new and solid. the 1 major comments →

arxiv 2607.20104 v1 pith:ANT55XIB submitted 2026-07-22 quant-ph physics.hist-ph

Schr\"odinger's real-valued equation revisited

Oliver Passon , Bernd Rosenow This is my paper
classification quant-ph physics.hist-ph
keywords real-valued wave equationcomplex structuresymplectic structureBorn rulenon-localityHamiltonian liftSO(2) gauge symmetrytime-reversal
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper revisits the old idea that the complex quantum wave equation can be rewritten as a second-order equation for a single real field, and diagnoses exactly where that reduction fails. It shows the reduced field is dynamically sufficient but not locally complete: the Born density, probability current, momentum, and the standard derivation of the uncertainty relation all become non-local in the Cauchy data (u, u̇), because the eliminated imaginary part must be reconstructed through the inverse Hamiltonian H⁻¹. The constructive claim is that these failures are symptoms, not refutations: the reduced equation is the configuration-space projection of a real Hamiltonian phase-space system, whose canonical partner π = ℏ²H⁻¹u̇ completes the picture and makes ψ = u + iπ/ℏ satisfy the original complex equation. Thus, a fully real formulation of quantum mechanics exists, but it must carry the symplectic/complex structure that the symbol i was compactly encoding. A sympathetic reader cares because this clarifies what is essential about complex numbers in quantum mechanics: the notation is removable, the structure is not.

Core claim

The paper's central discovery is a precise diagnosis-and-lift theorem for the real-valued wave equation. Starting from the reduced second-order equation −ℏ²ü = H²u (with the natural generalization when the potential is time-dependent), every attempt to express the Born density, current, or momentum in terms of u and its time derivative alone introduces the non-local operator H⁻¹: ρ = u² + (ℏH⁻¹u̇)², and the current has a comparable form. Proposition 2 sharpens this: two Cauchy data sets can coincide, with all spatial derivatives, in a neighbourhood of a point yet assign different Born densities there, so the density is not a local function of the reduced data. The constructive step is the H

What carries the argument

The Hamiltonian lift of the reduced real equation: the canonical pair (u,π) with π = ℏ²H⁻¹u̇ (equivalently the dimensionless second quadrature v = π/ℏ), governed by u̇ = ℏ⁻²Hπ, π̇ = −Hu, and the identification ψ = u + iπ/ℏ. This object does the argument's work by replacing the non-local inversion H⁻¹ with an explicit phase-space coordinate, restoring the local symplectic structure and turning the phase symmetry into an internal SO(2) rotation; the inverse Hamiltonian survives only as the dictionary between the reduced Cauchy data and the canonical partner.

Load-bearing premise

The equivalence with the standard complex equation holds only because the physical sign branch of the lift (K = H rather than K = |H|) is selected by importing the very first-order dynamics the reconstruction is meant to reproduce; if that branch choice were not available, the real phase-space system could conserve the Born norm while producing empirically different dynamics.

What would settle it

Take a Hamiltonian whose spectrum contains both signs, prepare a superposition of a bound state with E₁ < 0 and a scattering state with E₂ > 0, and compare the time dependence of the local density under the two lifts that share the same reduced second-order equation: the standard-sign branch gives beats at (|E₁|+E₂)/ℏ, the |H| branch at ||E₁|−E₂|/ℏ. Observing which beat frequency occurs (numerically or experimentally) determines whether the physical branch follows from the reduced data or from the imported sign convention.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any attempt to interpret the single real field (plus its time derivative) as the complete state of a quantum system must confront either non-local probability rules or a Hamiltonian-dependent Born rule; a local and autonomous Born rule requires the canonical pair.
  • The uncertainty relation is not a casualty of real formulations: once the canonical partner is restored, the standard Cauchy–Schwarz proof goes through unchanged, so the bound is a reflection of local symplectic geometry rather than of the complex symbol.
  • Magnetic fields can be included in real variables: minimal coupling emerges as the gauging of the internal SO(2) symmetry of the (u,π/ℏ) plane, with gyroscopic terms in the reduced equation and a covariant current containing the standard diamagnetic correction.
  • For composite systems, the reduced reconstruction map π_AB = ℏ²H_AB⁻¹u̇_AB does not factorize across subsystems even when the Hamiltonian has no interaction term; hence the single-field variables do not carry the local tensor-product structure assumed in operational no-go arguments, while the lifted pair reproduces standard complex predictions exactly.
  • Time-reversal for half-integer spin is represented by a real orthogonal map squaring to −1, giving a constructive proof of the half-integer-spin degeneracy and distinguishing it from the spinless reflection.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The branch ambiguity in the lift (every K = ε(H)H with ε(H)²=1 conserves the Born norm) suggests a sharp empirical discriminator: for a superposition of a negative-energy bound state and a positive-energy scattering state, the standard-sign branch and the |H| branch predict different beat frequencies in the density from identical reduced trajectories; a measurement would settle whether the physica
  • The paper's dichotomy — local but dynamics-dependent, or autonomous but non-local — invites a general no-go theorem over all admissible single-field variables; the paper explicitly does not attempt one, so characterizing the allowed reconstruction maps and proving (or disproving) the dichotomy is a natural open problem.
  • The SO(2) gauge reading of minimal coupling suggests a geometric continuation: if the internal phase-plane rotation is interpreted as a connection, the formalism may extend to curved-space or gravitational settings, connecting this real phase-space form to geometric quantum mechanics without the complex symbol.
  • For composite systems, the non-factorization of H_AB⁻¹ parallels the known result that simulating complex quantum theory with real Hilbert spaces demands a global resource; one inference is that any real-variable simulation of quantum circuits must carry this global reference frame, with resource cost scaling with the number of subsystems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper revisits the Schrödinger–Chen reduction of the complex Schrödinger equation to a second-order equation for a single real field u. It argues that this reduced single-field formulation is dynamically sufficient but not locally complete: the Born density, probability current, momentum, and the standard uncertainty derivation become non-local in the operator-theoretic sense, because the missing imaginary part is recovered only through H^{-1} \dot u. The paper then constructs a real Hamiltonian phase-space lift with canonical momentum \pi = \hbar^2 H^{-1} \dot u, proves in Proposition 1 an exact equivalence with the complex Schrödinger equation, and shows that the local structures are restored: local density and current, momentum as symplectic generator, an internal SO(2) gauge structure under magnetic coupling, real time-reversal and Kramers degeneracy, and non-factorization for composite systems. Proposition 2 gives a state-level proof that the Born density is non-local in the reduced Cauchy data even when the data coincide locally at a point.

Significance. If correct, the paper gives a precise and well-delimited answer to an old question: the symbol i can be removed from the equations, but the real symplectic/complex structure it encodes cannot. The paper’s strengths are the explicit real-linear isomorphism in Proposition 1; the careful admissibility argument in Proposition 2; the exact Gaussian and coherent-packet solutions; the magnetic generalization with its gyroscopic term; the real construction of Kramers degeneracy; and the sharp analysis of the network no-go theorems, including the non-factorizing reconstruction map for composites. The authors are honest about scope: the ‘local-or-autonomous’ dichotomy is stated only for the two natural single-field parametrizations, not as a general no-go theorem. The construction is parameter-free, and the H versus |H| branch distinction gives a concrete, in-principle observable difference built on identical reduced trajectory data.

major comments (1)
  1. [Sec. 5.2, Eq. (60)] The statement that for sign-definite H the lift is unique up to π → −π is too strong when H has degenerate eigenvalues. Any self-adjoint K with K² = H² is a lift; such K need not be a function of H. On a degenerate subspace H = E I₂, the operator K = E σ_z (σ_z the Pauli matrix) satisfies K² = H², is self-adjoint, conserves the norm (57), and commutes with the complex structure, yet it is not of the form ε(H)H. Thus the norm criterion does not fix K even for sign-definite H unless the spectrum is simple. This does not undermine the main diagnostic conclusion—it actually strengthens the non-uniqueness point—but the uniqueness sentence and the phrase ‘only up to these subspace-wise signs’ should be corrected.
minor comments (5)
  1. [Sec. 8.3, Eq. (152)] The factor-1/2 convention introduced in Sec. 5.7 is not repeated here. Without a reminder, the magnetic linear and diamagnetic coefficients in Eq. (152) appear smaller by a factor of 2 relative to the standard minimal-coupling Hamiltonian. Please add a one-sentence note that the same convention is in force.
  2. [Sec. 5.2, Eqs. (60)–(61)] The branch choice K = H is made by matching the first-order dynamics (3), i.e., by importing the target Schrödinger evolution. This is acknowledged in the text and is consistent with the paper’s diagnostic framing, but the abstract and Introduction should state this explicitly so the reconstruction is not mistaken for an autonomous derivation from the reduced equation.
  3. [Typos and formatting] ‘Acknoledgement’ in the acknowledgment heading; ‘veriables’ in Appendix C; ‘short-time cheque’ in Sec. 6.3 should be ‘short-time check’; Ref. [17] has a duplicated arXiv identifier.
  4. [Sec. 6.3, Prop. 2] The proof says the ball B is closed and does not contain r₀, which guarantees a positive distance from r₀ and hence a neighborhood on which the data coincide. A one-line clarification would make this explicit and block a possible objection.
  5. [Appendix A, item (vi)] The statement that the propositions are rigorous for finite-dimensional spaces and are ‘to be read on natural domains’ for continuous spectra could be sharpened for Proposition 2: the authors should explicitly note that the dipole decay ∼ |r|⁻² is square-integrable in three dimensions, since the volume element supplies the needed r² factor.

Circularity Check

0 steps flagged

No significant circularity; central isomorphism is self-contained, with one explicitly acknowledged branch-selection caveat.

full rationale

After walking the derivation chain, I find no significant circularity. The central constructive result is Proposition 1, an explicit real-linear isomorphism T(u,π)=u+iπ/ℏ between the lifted real Hamiltonian system and the complex Schrödinger equation; this is a mathematical equivalence rather than a fit or a renamed input, and the proof proceeds by direct substitution with no free parameters. The reduced single-field equation is obtained by eliminating v via v=ℏ H^-1 u̇, which is algebraic elimination from the same complex equation, and the subsequent non-locality analysis (Secs. 3–4, Prop. 2) is derived from that definition, not assumed. The only place where the target dynamics is imported is the branch selection after Eq. (60): K^2=H^2 fixes the lift only up to subspace-wise signs, and the paper explicitly selects K=H by matching the first-order dynamics (3). This is a genuine reliance on the input equation for a constructive uniqueness claim, but it is fully acknowledged, and the paper's stated diagnostic aim is exactly to show that the reduced equation alone does not encode the sign of H and hence the complex structure. Because this caveat is explicit and the central equivalence is independently stated as a theorem, it does not amount to hidden or load-bearing circularity. There are no fitted parameters, no load-bearing self-citations, no ansatz imported through prior work of the present authors, and the worked examples (free packet, oscillator, Landau levels, spin) are internal consistency checks rather than circular predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No parameters are fitted to data. The central reconstruction uses standard quantum mechanics as input: a real symmetric Hamiltonian, inverse-Hamiltonian reconstruction on admissible data, and an explicit branch choice for the lift. No new entities (particles, forces, dimensions) are postulated.

axioms (5)
  • domain assumption The Hamiltonian Ĥ is self-adjoint and real symmetric in the position representation; in the magnetic case Ĥ_R is real self-adjoint and Ĥ_I is real skew-adjoint.
    Needed for the split ψ=u+iv and for the reconstruction v=ℏĤ⁻¹u̇; stated in Appendix A(i) and used throughout.
  • domain assumption Ĥ⁻¹ is defined on the subspace orthogonal to zero modes (or via Moore-Penrose pseudoinverse), and admissible Cauchy data satisfy u̇∈Ran Ĥ.
    The reconstruction map and Proposition 2 depend on this; Appendix A(ii)–(iii).
  • domain assumption The physical lift branch is selected by requiring the lifted first-order system to reproduce the original spectral sign of Ĥ, not merely Ĥ²; K=|Ĥ| is rejected as empirically inequivalent.
    Sec. 5.2 after Eq. (60) and Appendix A(iv): the norm criterion leaves subspace-wise sign freedom, and equivalence to Schrödinger dynamics is used to fix it.
  • domain assumption For real time-reversal-invariant Ĥ, gyroscopic diagonal blocks in the lift ansatz are excluded by requiring covariance under (u,π)→(u,−π); magnetic coupling reintroduces them.
    Footnote in Sec. 5.2; restricts the generality of the Hamiltonian lift ansatz in Eq. (54).
  • domain assumption The conserved norm of the lift is taken to be the rotationally invariant quadratic form ∫(u²+π²/ℏ²)d³r, i.e., the real form of the Born norm.
    Sec. 5.2 Step 2: this is demanded so that the lifted theory carries the correct Born probability interpretation.

pith-pipeline@v1.3.0-alltime-deepseek · 50195 in / 21905 out tokens · 207149 ms · 2026-08-01T10:46:05.675829+00:00 · methodology

0 comments
read the original abstract

The Schr\"odinger equation can be rewritten as a second-order equation for a single real-valued scalar field. Taken at face value, however, this one-field reduction obscures several local structures of quantum mechanics: the Born density and current, the role of momentum as a generator, and the usual derivation of the uncertainty relation. We argue that these difficulties do not show that real variables fail. They show that the reduced equation is not, by itself, a complete local representation of the theory. Its coherent real form is obtained by restoring the underlying Hamiltonian phase-space structure, where the missing component reappears as a canonical partner. In this real phase-space formulation, the standard structures reappear locally, and minimal magnetic coupling reveals an internal \(SO(2)\) gauge structure. Thus, the symbol \(i\) can be removed, but the symplectic and complex structure it encodes cannot.

Figures

Figures reproduced from arXiv: 2607.20104 by Bernd Rosenow, Oliver Passon.

Figure 1
Figure 1. Figure 1: The local real phase plane (𝑢, 𝜋∕ℏ) at a fixed point. (a) Free evolution rotates every energy mode clockwise at its frequency 𝜔𝑘 = 𝐸𝑘∕ℏ, drawn for 𝐸𝑘 > 0 (Sec. 5.3); the Born density is the squared radius 𝑢 2 + 𝜋 2∕ℏ 2 , the conserved 𝐴 2 𝑘 of each mode. (b) Spinless time reversal 𝔗0 ∶ (𝑢, 𝜋) ↦ (𝑢,−𝜋) is the reflection in the 𝑢-axis – the real form of complex conjugation (Sec. 10). (c) A gauge transformati… view at source ↗
Figure 2
Figure 2. Figure 2: Real form of the orbital Zeeman effect in an (𝑛, 𝓁, 𝑚) plane with 𝑚 > 0, drawn for an electron (𝜔𝐿 = 𝑒𝐵∕2𝑀 > 0). Left: the two counter-rotating normal modes in the coefficient plane (𝑎, 𝑏) of (cos 𝑚𝜙,sin 𝑚𝜙), Eq. (238). With 𝑢 = Re 𝜓, counterclockwise motion corresponds to 𝑚𝓁 = +𝑚 and rotates at Ω+ = 𝜔𝑛𝓁 + 𝜔𝐿𝑚, clockwise motion to 𝑚𝓁 = −𝑚 at Ω− = 𝜔𝑛𝓁 − 𝜔𝐿𝑚, Eq. (292). Right: the field resolves the degenera… view at source ↗

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Reference graph

Works this paper leans on

34 extracted references · 7 canonical work pages

  1. [1]

    Geometrical formulation of quantum mechanics, in: Harvey, A

    Ashtekar, A., Schilling, T.A., 1999. Geometrical formulation of quantum mechanics, in: Harvey, A. (Ed.), On Einstein’s Path: Essays in Honor of Engelbert Schücking. Springer, New York, pp. 23–65. arXiv:gr-qc/9706069

    Pith/arXiv arXiv 1999

  2. [2]

    A direct road to majorana fields

    Aste, A., 2010. A direct road to majorana fields. Symmetry 2, 1776–1809. URL:https://doi.org/10.3390/sym2041776,doi:10.3390/ sym2041776,arXiv:0806.1690

    Pith/arXiv arXiv 2010

  3. [3]

    Generalized Inverses: Theory and Applications

    Ben-Israel, A., Greville, T.N.E., 2003. Generalized Inverses: Theory and Applications. 2 ed., Springer, New York. doi:10.1007/b97366

    doi:10.1007/b97366 2003

  4. [4]

    Quantum Theory

    Bohm, D., 1951. Quantum Theory. Prentice-Hall, Englewood Cliffs, N.J

    1951

  5. [5]

    Asuggestedinterpretationofthequantumtheoryin terms of “hidden” variables (part 1)

    Bohm,D.,1952. Asuggestedinterpretationofthequantumtheoryin terms of “hidden” variables (part 1). Physical Review 85, 166–193. doi:10.1103/PhysRev.85.166

    doi:10.1103/physrev.85.166 1952

  6. [6]

    Quantum mechanics: Keeping it real? Brit

    Callender, C., 2023. Quantum mechanics: Keeping it real? Brit. J. Phil. Sci. 74, 837–851

    2023

  7. [7]

    Derivationoftherealformofschrödinger’sequation for a nonconservative system and the unique relation between re(𝜓) and im(𝜓)

    Chen,R.,1989. Derivationoftherealformofschrödinger’sequation for a nonconservative system and the unique relation between re(𝜓) and im(𝜓). J. Math. Phys. 30, 83–86

    1989

  8. [8]

    Concerning the gauge invariance and the apparent nonlocality of the real form of schrödinger’s equation

    Chen, R., 1991. Concerning the gauge invariance and the apparent nonlocality of the real form of schrödinger’s equation. J. Math. Phys. 32, 464–465

    1991

  9. [9]

    Journal of Mathemat- ical Physics 3, 1199–1215

    Dyson,F.J.,1962.Thethreefoldway.Algebraicstructureofsymmetry groups and ensembles in quantum mechanics. Journal of Mathemat- ical Physics 3, 1199–1215

    1962

  10. [10]

    Einigediequantenmechanikbetreffendeerkundi- gungsfragen

    Ehrenfest,P.,1932. Einigediequantenmechanikbetreffendeerkundi- gungsfragen. Z. Phys. 78, 555–559

    1932

  11. [11]

    The symplectic camel and the uncertainty principle

    de Gosson, M.A., 2009. The symplectic camel and the uncertainty principle. Foundations of Physics 39, 194–214. doi:10.1007/ s10701-009-9272-2

    2009

  12. [12]

    Schrödinger equation from an exact uncertainty principle

    Hall, M.J.W., Reginatto, M., 2002. Schrödinger equation from an exact uncertainty principle. Journal of Physics A: Mathematical and General 35, 3289–3303. doi:10.1088/0305-4470/35/14/310

    work page doi:10.1088/0305-4470/35/14/310 2002

  13. [13]

    Quantum mechanics as a classical theory

    Heslot, A., 1985. Quantum mechanics as a classical theory. Physical Review D 31, 1341–1348. doi:10.1103/PhysRevD.31.1341

    work page doi:10.1103/physrevd.31.1341 1985

  14. [14]

    Real spinor fields

    Hestenes, D., 1967. Real spinor fields. Journal of Mathematical Physics 8, 798–808. doi:10.1063/1.1705279

    work page doi:10.1063/1.1705279 1967

  15. [15]

    Quantum mechanics based on real numbers: A consistent description

    Hita, P.B., Trushechkin, A., Kampermann, H., Epping, M., Bruß, D., 2025. Quantum mechanics based on real numbers: A consistent description. ArXiv:2503.17307

    Pith/arXiv arXiv 2025

  16. [16]

    Quantum theory does not need complex numbers

    Hoffreumon, T., Woods, M.P., 2025. Quantum theory does not need complex numbers. ArXiv:2504.02808. O. Passon and B. Rosenow:Preprint submitted to ElsevierPage 34 of 35 Schrödinger’s real-valued equation revisited

    arXiv 2025

  17. [17]

    Quantum theory based on real numbers cannot be experimentally falsified.arXiv:2603.19208

    Hoffreumon, T., Woods, M.P., 2026. Quantum theory based on real numbers cannot be experimentally falsified.arXiv:2603.19208. arXiv:2603.19208v1

    arXiv 2026

  18. [18]

    Schrödinger’s original struggles with a complex wave function

    Karam, R., 2020. Schrödinger’s original struggles with a complex wave function. Am. J. Phys 88, 433–438

    2020

  19. [19]

    Geometrization of quantum mechanics

    Kibble, T.W.B., 1979. Geometrization of quantum mechanics. Com- munications in Mathematical Physics 65, 189–201. doi:10.1007/ BF01225149

    1979

  20. [20]

    Quantentheorie in hydrodynamischer form

    Madelung, E., 1927. Quantentheorie in hydrodynamischer form. ZeitschriftfürPhysik40,322–326. URL:https://link.springer.com, doi:10.1007/BF01400372

    doi:10.1007/bf01400372 1927

  21. [21]

    Quantum me- chanics over real numbers fully reproduces standard quantum theory

    Maioli, A.C., Curado, E.M.F., Gazeau, J.P., 2026. Quantum me- chanics over real numbers fully reproduces standard quantum theory. arXiv:2604.19482. arXiv:2604.19482v2

    Pith/arXiv arXiv 2026

  22. [22]

    Teoria simmetrica dell’elettrone e del positrone

    Majorana, E., 1937. Teoria simmetrica dell’elettrone e del positrone. Il Nuovo Cimento 14, 171–184. URL:https://doi.org/10.1007/ BF02961314, doi:10.1007/BF02961314

    doi:10.1007/bf02961314 1937

  23. [23]

    A real-valued description of quan- tum mechanics with schrödinger’s 4th-order matter-wave equation

    Makris, N., Dargush, G.F., 2025. A real-valued description of quan- tum mechanics with schrödinger’s 4th-order matter-wave equation. Physics Open 23, 100262

    2025

  24. [24]

    Simulating quan- tum systems using real hilbert spaces

    McKague, M., Mosca, M., Gisin, N., 2009. Simulating quan- tum systems using real hilbert spaces. Phys. Rev. Lett. 102, 020505. URL:https://link.aps.org/doi/10.1103/PhysRevLett.102. 020505, doi:10.1103/PhysRevLett.102.020505

    doi:10.1103/physrevlett.102 2009

  25. [25]

    Dirac, majorana, and weyl fermions

    Pal, P.B., 2011. Dirac, majorana, and weyl fermions. American Journal of Physics 79, 485–498. URL:https://doi.org/10.1119/1. 3549729, doi:10.1119/1.3549729,arXiv:1006.1718

    Pith/arXiv arXiv 2011

  26. [26]

    Dieallgemeinenprinzipienderwellenmechanik,in: Geiger, H., Scheel, K

    Pauli,W.,1933a. Dieallgemeinenprinzipienderwellenmechanik,in: Geiger, H., Scheel, K. (Eds.), Handbuch der Physik. 2 ed.. Springer, Berlin, pp. 83–272

  27. [27]

    Einige die quantenmechanik betreffenden erkundi- gungsfragen

    Pauli, W., 1933b. Einige die quantenmechanik betreffenden erkundi- gungsfragen. Z. Phys. 80, 573–586

  28. [28]

    Quantum theory based on real numbers can be experimentally falsified

    Renou, M.O., Trillo, D., Weilenmann, M., Le, T.P., Tavakoli, A., Gisin, N., Acín, A., Navascués, M., 2021. Quantum theory based on real numbers can be experimentally falsified. Nature 600, 625–629

    2021

  29. [29]

    Kramers degeneracy without eigenvectors

    Roberts, B.W., 2012. Kramers degeneracy without eigenvectors. Physical Review A 86, 034103. doi:10.1103/PhysRevA.86.034103

    work page doi:10.1103/physreva.86.034103 2012

  30. [30]

    Quantisierung als eigenwertproblem (vierte mitteilung)

    Schrödinger, E., 1926. Quantisierung als eigenwertproblem (vierte mitteilung). Ann. Phys. 81, 109–139

    1926

  31. [31]

    Time reversal and reciprocity

    Sigwarth, O., Miniatura, C., 2022. Time reversal and reciprocity. AAPPS Bulletin 32, 23. doi:10.1007/s43673-022-00053-4

    work page doi:10.1007/s43673-022-00053-4 2022

  32. [32]

    Complex coordinates and quantum mechanics

    Strocchi, F., 1966. Complex coordinates and quantum mechanics. ReviewsofModernPhysics38,36–40.doi:10.1103/RevModPhys.38.36

    work page doi:10.1103/revmodphys.38.36 1966

  33. [33]

    Quantum theory in real hilbert space

    Stueckelberg, E.C.G., 1960. Quantum theory in real hilbert space. Helvetica Physica Acta 33, 727–752

    1960

  34. [34]

    Partial independence suffices to rule out real quantum theory experimentally

    Weilenmann, M., Gisin, N., Sekatski, P., 2025. Partial independence suffices to rule out real quantum theory experimentally. Physical Review Letters 135, 180201. doi:10.1103/PhysRevLett.135.180201. 𝑎 𝑏 𝑚𝓁=+𝑚:Ω + 𝑚𝓁=−𝑚:Ω − 𝐸𝑛𝓁 𝐸𝑛𝓁+ℏ𝜔𝐿𝑚 𝐸𝑛𝓁−ℏ𝜔𝐿𝑚 𝐵=0 𝐵 >0 Figure 2:Real form of the orbital Zeeman effect in an(𝑛,𝓁,𝑚) plane with𝑚 >0, drawn for an electron (𝜔𝐿 =...

    work page doi:10.1103/physrevlett.135.180201 2025