pith. machine review for the scientific record. sign in

arxiv: 2601.07932 · v1 · submitted 2026-01-12 · 🪐 quant-ph · physics.hist-ph

Bohmian mechanics: A legitimate hydrodynamic picture for quantum mechanics, and beyond

A. S. Sanz This is my paper

Pith reviewed 2026-05-16 14:40 UTC · model grok-4.3

classification 🪐 quant-ph physics.hist-ph
keywords Bohmian mechanicshydrodynamic pictureSchrödinger equationquantum representationspractical applicationshidden variablesnumerical methods
0
0 comments X

The pith

Bohmian mechanics provides a legitimate hydrodynamic picture of quantum mechanics through its practical analytical uses.

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

Bohmian mechanics began as a hidden-variable approach meant to sidestep von Neumann's theorem but met strong initial resistance. Applications to concrete problems in physics since the late 1990s shifted its role toward a pragmatic tool for analysis and computation based on the Schrödinger equation. This change supports treating it as a valid quantum representation that belongs in elementary courses. The paper supports the view by re-examining the Schrödinger equation and numerical examples under a less restrictive lens than usual.

Core claim

Bohmian mechanics, once framed as a hidden-variable model, now functions as a hydrodynamic representation for quantum mechanics, where the wave function guides fluid-like trajectories; this pragmatic shift, shown through re-analysis of the Schrödinger equation and specific examples, establishes it as a legitimate quantum framework suitable for teaching and extension to other fields.

What carries the argument

The hydrodynamic picture of the Schrödinger equation, in which quantum evolution is recast as a fluid flow with trajectories determined by the wave function.

If this is right

  • Bohmian mechanics serves as an effective analytical and computational tool for problems across branches of physics.
  • It can be taught in elementary quantum mechanics courses as a standard representation.
  • The hydrodynamic approach allows transfer of ideas to benefit other fields.
  • Moderate acceptance replaces early controversy as operational value becomes clear.

Where Pith is reading between the lines

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

  • Trajectory-based methods may offer computational advantages in simulating certain many-particle quantum systems.
  • Similar pragmatic acceptance could apply to other alternative quantum pictures without full resolution of foundational issues.
  • Extensions to relativistic domains or open quantum systems could follow the same operational path.

Load-bearing premise

Community attitude shifts and practical examples are enough to establish legitimacy without directly resolving foundational objections such as von Neumann's theorem.

What would settle it

A specific quantum problem where Bohmian trajectory calculations produce results that systematically disagree with both standard quantum predictions and experimental measurements.

Figures

Figures reproduced from arXiv: 2601.07932 by A. S. Sanz.

Figure 1
Figure 1. Figure 1: Probability density (a) and velocity field (b) for a linear superposition of two Gaussian [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Transverse cuts of the velocity field at various times for a two-Gaussian wave-packet super [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: This does not mean that we the phase coherence has been lost; it has only been lost for a single [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bohmian trajectories associated with a Bell-type bipartite continuous variable state (a)-(b), [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Propagation of an ideal Airby beam (a) and a finite-energy Airy beam (b). The contour [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: In the central part, Bohmian trajectories associated with a superposition of two counter [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

Since its inception, Bohmian mechanics has been surrounded by a halo of controversy. Originally proposed to bypass the limitations imposed by von Neumann's theorem on the impossibility of hidden-variable models in quantum mechanics, it faced strong opposition from the outset. Over time, however, its use in tackling specific problems across various branches of physics has led to a gradual shift in attitude, turning the early resistance into a more moderate acceptance. A plausible explanation for this change may be that, since the late 1990s and early 2000s, Bohmian mechanics has been taking on a more operational and practical role. The original hidden-variable idea has gradually faded from its framework, giving way to a more pragmatic approach that treats it as a suitable analytical and computational tool. This discussion explores how and why such a shift in perspective has occurred and, therefore, answers questions such as whether Bohmian mechanics should be considered once and for all a legitimate quantum representation (i.e., worth being taught in elementary quantum mechanics courses) or, by extension, whether these ideas can be transferred to and benefit other fields. Here, the Schr\"odinger equation and several specific numerical examples are re-examined in the light of a less restrictive view than the standard one usually adopted in quantum mechanics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that Bohmian mechanics, originally introduced to circumvent von Neumann's theorem on the impossibility of hidden-variable models, has undergone a pragmatic shift since the late 1990s toward operational and computational use as a hydrodynamic picture. This evolution, away from strict hidden-variable interpretations, is presented as establishing it as a legitimate quantum representation suitable for elementary teaching, with the Schrödinger equation and selected numerical examples re-examined under a less restrictive viewpoint to illustrate broader applicability across physics.

Significance. If the historical and pragmatic narrative holds, the paper could support greater inclusion of Bohmian trajectories in quantum mechanics education and computational practice, potentially aiding analytical tools in related fields. However, the absence of new derivations, quantitative benchmarks, or direct engagement with foundational no-go theorems limits its technical impact to interpretive discussion rather than advancing core methods or resolving longstanding objections.

major comments (3)
  1. [Abstract] Abstract and introduction: The assertion that observed community acceptance and practical applications since the 1990s suffice to declare Bohmian mechanics a 'legitimate quantum representation' is not supported by any derivation showing how the hydrodynamic formulation evades or modifies the non-contextuality assumptions underlying von Neumann's theorem, which the paper itself identifies as the original motivation.
  2. [Schrödinger equation discussion] Section re-examining the Schrödinger equation: The claim of a 'less restrictive view' is advanced without explicit formal comparison (e.g., to the standard derivation of the continuity equation or probability current) or quantitative metrics demonstrating improved insight or computational advantage over conventional treatments.
  3. [Numerical examples] Numerical examples section: The manuscript references specific examples but provides no error-controlled benchmarks, convergence data, or direct comparisons to alternative methods (such as standard wave-function propagation), undermining the claim of suitability as a practical analytical tool.
minor comments (2)
  1. The abstract is lengthy and could be condensed to focus more sharply on the central pragmatic argument.
  2. Additional citations to post-1990s applications of Bohmian mechanics would strengthen the historical narrative of the shift in attitude.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive report. Our manuscript is an interpretive discussion of the observed pragmatic shift in Bohmian mechanics toward hydrodynamic use, not a source of new derivations or quantitative benchmarks. We address each major comment below, indicating revisions where they strengthen clarity without altering the paper's scope.

read point-by-point responses

  1. Referee: [Abstract] Abstract and introduction: The assertion that observed community acceptance and practical applications since the 1990s suffice to declare Bohmian mechanics a 'legitimate quantum representation' is not supported by any derivation showing how the hydrodynamic formulation evades or modifies the non-contextuality assumptions underlying von Neumann's theorem, which the paper itself identifies as the original motivation.

    Authors: We agree the manuscript provides no new derivation addressing von Neumann's theorem or non-contextuality. The claim of legitimacy is grounded in documented community practice and applications since the 1990s, where the hydrodynamic picture functions operationally regardless of hidden-variable ontology. The original motivation is presented only as historical background. We will revise the abstract and introduction to explicitly separate historical context from the pragmatic argument. revision: yes

  2. Referee: [Schrödinger equation discussion] Section re-examining the Schrödinger equation: The claim of a 'less restrictive view' is advanced without explicit formal comparison (e.g., to the standard derivation of the continuity equation or probability current) or quantitative metrics demonstrating improved insight or computational advantage over conventional treatments.

    Authors: The less restrictive view treats the hydrodynamic equations as a valid analytical representation without requiring ontological commitment to trajectories as hidden variables. We will add an explicit side-by-side comparison to the standard derivation of the continuity equation and probability current from the Schrödinger equation to clarify equivalence and the trajectory-based perspective. Quantitative metrics are outside the manuscript's interpretive scope and will not be added. revision: partial

  3. Referee: [Numerical examples] Numerical examples section: The manuscript references specific examples but provides no error-controlled benchmarks, convergence data, or direct comparisons to alternative methods (such as standard wave-function propagation), undermining the claim of suitability as a practical analytical tool.

    Authors: The examples are drawn from existing literature to illustrate pragmatic adoption, not new computations performed here. We will revise the section to state this explicitly and add citations to published works containing error-controlled benchmarks and comparisons to standard propagation methods, thereby supporting the practical-utility claim through reference rather than new data. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the historical and pragmatic argument

full rationale

The paper advances a historical and interpretive discussion of shifting attitudes toward Bohmian mechanics, citing its practical use as an analytical tool and re-examining the Schrödinger equation plus numerical examples. No load-bearing derivation, equation, or parameter fit is presented that reduces by construction to its own inputs, self-citations, or ansatzes. The central claim equates observed community acceptance with legitimacy for teaching without invoking uniqueness theorems, fitted predictions, or self-referential definitions; the argument remains self-contained as opinion grounded in external usage trends rather than internal loops.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard quantum mechanics via the Schrödinger equation and historical observation of community attitudes; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • standard math The Schrödinger equation remains the valid dynamical law for quantum systems.
    The paper states it re-examines the Schrödinger equation under a less restrictive view.

pith-pipeline@v0.9.0 · 5523 in / 1223 out tokens · 48425 ms · 2026-05-16T14:40:58.857879+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages

  1. [1]

    Bohm D 1989Quantum Theory(New York: Dover); first printed by Prentice Hall in 1951

    work page 1951

  2. [2]

    de Broglie L 1924Recherches sur la th´ eorie des QuantaTheses Migration - universit´ e en cours d’affectationhttps://theses.hal.science/tel-00006807

    work page

  3. [3]

    de Broglie L 1927J. Phys. Radium8225–241 https://doi.org/10.1051/jphysrad:0192700805022500

    work page doi:10.1051/jphysrad:0192700805022500

  4. [4]

    A Causal Interpretation(Amsterdam: Elsevier)

    de Broglie L 1960Non-Linear Wave Mechanics. A Causal Interpretation(Amsterdam: Elsevier)

    work page

  5. [5]

    Instituts Solvay 1928 ´Electrons et photons: Rapports et discussions du cinqui` eme Conseil de physique tenu ` a Bruxelles du 24 au 29 octobre 1927 sous les auspices de l’Institut international de physique Solvay(Paris: Gauthier-Villars et cie.) https://catalog.hathitrust.org/Record/102101097

    work page arXiv 1928

  6. [6]

    Philippidis C, Dewdney C and Hiley B J 1979Nuovo Cimento52B15–28 https://link.springer.com/article/10.1007/BF02743566

    work page doi:10.1007/bf02743566

  7. [7]

    B7157–88 https://doi.org/10.1007/BF02721695

    Philippidis C, Bohm D and Kaye R D 1982Nuovo Cim. B7157–88 https://doi.org/10.1007/BF02721695

    work page doi:10.1007/bf02721695

  8. [8]

    A suggested interpretation of the quan- tum theory in terms of ”hidden” variables

    Bohm D 1952Phys. Rev.85166–179https://link.aps.org/doi/10.1103/PhysRev.85.166

    work page doi:10.1103/physrev.85.166

  9. [9]

    A suggested interpretation of the quantum theory in terms of ”hidden” variables

    Bohm D 1952Phys. Rev.85180–193https://link.aps.org/doi/10.1103/PhysRev.85.180

    work page doi:10.1103/physrev.85.180

  10. [10]

    Von Neumann J and Beyer R T (transl) 1955Mathematical Foundations of Quantum Mechanics (Princeton: Princeton University Press)

    work page

  11. [11]

    Phys.40322–326 https://link.springer.com/article/10.1007/BF01400372

    Madelung E 1926Z. Phys.40322–326 https://link.springer.com/article/10.1007/BF01400372

    work page doi:10.1007/bf01400372

  12. [12]

    de Broglie L 1926Comptes Rendus183272–274 https://gallica.bnf.fr/ark:/12148/bpt6k3136h/f272.item.r=broglie

    work page

  13. [13]

    de Broglie L 1926Comptes Rendus183447–448 https://gallica.bnf.fr/ark:/12148/bpt6k3136h/f447.item.r=broglie

    work page

  14. [14]

    Phys.1411301 https://link.springer.com/article/10.1007/s11467-018-0853-4

    Sanz A S 2019Front. Phys.1411301 https://link.springer.com/article/10.1007/s11467-018-0853-4

    work page doi:10.1007/s11467-018-0853-4

  15. [15]

    Today51(3) 42–46https://doi.org/10.1063/1.882184

    Goldstein S 1998Phys. Today51(3) 42–46https://doi.org/10.1063/1.882184

    work page doi:10.1063/1.882184

  16. [16]

    Today51(4) 38–42https://doi.org/10.1063/1.882241

    Goldstein S 1998Phys. Today51(4) 38–42https://doi.org/10.1063/1.882241

    work page doi:10.1063/1.882241

  17. [17]

    Bohm D 1980Wholeness and the Implicate Order(New York: Routledge)

    work page

  18. [18]

    Bohm D and Hiley B J 1993The Undivided Universe(New York: Routledge)

    work page

  19. [19]

    Holland P R 2002The Quantum Theory of Motion: An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics(Cambridge University Press) https://archive.org/details/quantumtheoryofm0000holl

    work page

  20. [20]

    Cushing J T 1994Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony (Chicago: University of Chicago Press)

    work page

  21. [21]

    Cushing J T, Fine A and Goldstein S (eds) 1996Bohmian Mechanics: An Appraisal(Boston: Kluwer)

    work page

  22. [22]

    Cushing J T 1999From Physics to Philosophy(Cambridge: Cambridge University Press)

    work page

  23. [23]

    Wyatt R E 2005Quantum Dynamics with Trajectories(New York: Springer)

    work page

  24. [24]

    The Physics and Mathematics of Quantum Theory (Berlin: Springer)

    D¨ urr D and Teufel S 2009Bohmian Mechanics. The Physics and Mathematics of Quantum Theory (Berlin: Springer)

    work page

  25. [25]

    D¨ urr D, Goldstein S and Zangh` ı N 2013Quantum Physics without Quantum Philosophy(Berlin: Springer)

    work page

  26. [26]

    Sanz A S and Miret-Art´ es S 2012A Trajectory Description of Quantum Processes. I. Fundamentals (Lecture Notes in Physicsvol 850) (Berlin: Springer)

    work page

  27. [27]

    Sanz A S and Miret-Art´ es S 2014A Trajectory Description of Quantum Processes. II. Applications (Lecture Notes in Physicsvol 831) (Berlin: Springer)

    work page

  28. [28]

    Oriols X and Mompart J (eds) 2012Applied Bohmian Mechanics: From Nanoscale Systems to Cosmology(Singapore: Pan Standford Publishing)

    work page

  29. [29]

    Bricmont J 2016Making Sense of Quantum Mechanics(Berlin: Springer)

    work page

  30. [30]

    Norsen T 2017Foundations of Quantum MechanicsUndergraduate Lecture Notes in Physics (Cham, Switzerland: Springer)

    work page

  31. [31]

    McCullough E A and Wyatt R E 1969J. Chem. Phys.511253–1254 https://doi.org/10.1063/1.1672133

    work page doi:10.1063/1.1672133

  32. [32]

    McCullough E A and Wyatt R E 1971J. Chem. Phys.543578–3591 https://doi.org/10.1063/1.1675384

    work page doi:10.1063/1.1675384

  33. [33]

    McCullough E A and Wyatt R E 1971J. Chem. Phys.543592–3600 https://doi.org/10.1063/1.1675385

    work page doi:10.1063/1.1675385

  34. [34]

    Hirschfelder J O, Christoph A C and Palke W E 1974J. Chem. Phys.615435–5455 https://doi.org/10.1063/1.1681899

    work page doi:10.1063/1.1681899

  35. [35]

    Hirschfelder J O, Goebel C J and Bruch L W 1974J. Chem. Phys.615456–5459 https://doi.org/10.1063/1.1681900

    work page doi:10.1063/1.1681900

  36. [36]

    Hirschfelder J O and Tang K T 1976J. Chem. Phys.64760–785 https://doi.org/10.1063/1.432223

    work page doi:10.1063/1.432223

  37. [37]

    Hirschfelder J O and Tang K T 1976J. Chem. Phys.65470–486 https://doi.org/10.1063/1.432790

    work page doi:10.1063/1.432790

  38. [38]

    Rev.96208–216https://doi.org/10.1103/PhysRev.96.208

    Bohm D and Vigier J P 1954Phys. Rev.96208–216https://doi.org/10.1103/PhysRev.96.208

    work page doi:10.1103/physrev.96.208

  39. [39]

    Rep.17293–122 https://doi.org/10.1016/0370-1573(89)90160-9

    Bohm D and Hiley B J 1989Phys. Rep.17293–122 https://doi.org/10.1016/0370-1573(89)90160-9

    work page doi:10.1016/0370-1573(89)90160-9

  40. [40]

    Rev.1501079–1085https://doi.org/10.1103/PhysRev.150.1079

    Nelson E 1966Phys. Rev.1501079–1085https://doi.org/10.1103/PhysRev.150.1079

    work page doi:10.1103/physrev.150.1079

  41. [41]

    Sanz A S and Miret-Art´ es S 2012Am. J. Phys.80525–533https://doi.org/10.1119/1.3698324

    work page doi:10.1119/1.3698324

  42. [42]

    Phys.: Conf

    Sanz A S 2012J. Phys.: Conf. Ser.361012016 https://doi.org/10.1088/1742-6596/361/1/012016

    work page doi:10.1088/1742-6596/361/1/012016

  43. [43]

    Schiff L I 1968Quantum Mechanics3rd ed (Singapore: McGraw-Hill)

    work page

  44. [44]

    Bohm D and Hiley B J 1985Phys. Rev. Lett.552511–2514 https://doi.org/10.1103/PhysRevLett.55.2511

    work page doi:10.1103/physrevlett.55.2511

  45. [45]

    Sanz A S 2021Ann. Fond. Louis de Broglie4619–63 https://fondationlouisdebroglie.org/AFLB-461/aflb461m955.htm

    work page

  46. [46]

    Phys.33911–21 https://doi.org/10.1016/j.aop.2013.08.004

    Sanz A S and Miret-Art´ es S 2013Ann. Phys.33911–21 https://doi.org/10.1016/j.aop.2013.08.004

    work page doi:10.1016/j.aop.2013.08.004 2013

  47. [47]

    Ehrenberg W and Siday R E 1949Proc. Phys. Soc. B628–21 https://doi.org/10.1088/0370-1301/62/1/303

    work page doi:10.1088/0370-1301/62/1/303

  48. [48]

    Significance of electromagnetic potentials in the quantum theory,

    Aharonov Y and Bohm D 1959Phys. Rev.115485–491 https://doi.org/10.1103/PhysRev.115.485

    work page doi:10.1103/physrev.115.485

  49. [49]

    Rev.1231511–1524 https://doi.org/10.1103/PhysRev.123.1511

    Aharonov Y and Bohm D 1961Phys. Rev.1231511–1524 https://doi.org/10.1103/PhysRev.123.1511

    work page doi:10.1103/physrev.123.1511

  50. [50]

    Tonomura A, Osakabe N, Matsuda T, Kawasaki T, Endo J, Yano S and Yamada H 1986Phys. Rev. Lett.56(8) 792–795https://link.aps.org/doi/10.1103/PhysRevLett.56.792

    work page doi:10.1103/physrevlett.56.792

  51. [51]

    Tonomura A and Nori F 2008Nature452298–299https://doi.org/10.1038/452298a

    work page doi:10.1038/452298a

  52. [52]

    Today6238–43https://doi.org/10.1063/1.3226854

    Batelaan H and Tonomura A 2009Phys. Today6238–43https://doi.org/10.1063/1.3226854

    work page doi:10.1063/1.3226854

  53. [53]

    Aharonov Y, Albert D Z and Vaidman L 1988Phys. Rev. Lett.601351–1354 https://doi.org/10.1103/PhysRevLett.60.1351

    work page doi:10.1103/physrevlett.60.1351

  54. [54]

    Aharonov Y and Vaidman L 1990Phys. Rev. A4111–20 https://doi.org/10.1103/PhysRevA.41.11

    work page doi:10.1103/physreva.41.11

  55. [55]

    Duck I M, Stevenson P M and Sudarshan E C G 1989Phys. Rev. D402112–2117 https://doi.org/10.1103/PhysRevD.40.2112

    work page doi:10.1103/physrevd.40.2112

  56. [56]

    Phys.35469–491https://doi.org/10.1007/s10701-004-1984-8

    Leavens C R 2005Found. Phys.35469–491https://doi.org/10.1007/s10701-004-1984-8

    work page doi:10.1007/s10701-004-1984-8 1984

  57. [57]

    Phys.9165https://doi.org/10.1088/1367-2630/9/6/165

    Wiseman H M 2007New J. Phys.9165https://doi.org/10.1088/1367-2630/9/6/165

    work page doi:10.1088/1367-2630/9/6/165

  58. [58]

    Phys.: Conf

    Hiley B J 2012J. Phys.: Conf. Ser.361012014 http://doi.org/10.1088/1742-6596/361/1/012014

    work page doi:10.1088/1742-6596/361/1/012014

  59. [59]

    Hirschfelder J O 1978J. Chem. Phys.685151–5162https://doi.org/10.1063/1.435635

    work page doi:10.1063/1.435635

  60. [60]

    Feynman R P, Leighton R B and Sands M 1965The Feynman Lectures on Physicsvol 3 (Reading, MA: Addison-Wesley)

    work page

  61. [61]

    Feynman R P and Hibbs A R 1965Quantum Mechanics and Path Integrals(New York: McGraw-Hill)

    work page

  62. [62]

    Sanz A S and Miret-Art´ es S 2008J. Phys. A: Math. Theor.41435303 https://doi.org/10.1088/1751-8113/41/43/435303

    work page doi:10.1088/1751-8113/41/43/435303

  63. [63]

    Kocsis S, Braverman B, Ravets S, Stevens M J, Mirin R P, Shalm L K and Steinberg A M 2011 Science3321170–1173https://doi.org/10.1126/science.1202218

    work page doi:10.1126/science.1202218 2011

  64. [64]

    Sanz A S 2023Entropy251077https://www.mdpi.com/1099-4300/25/7/1077

    work page

  65. [65]

    Feynman R P, Leighton R B and Sands M 1965The Feynman Lectures on Physicsvol 2 (Reading, MA: Addison-Wesley)

    work page

  66. [66]

    Prosser R D 1976Int. J. Theor. Phys.15169–180https://doi.org/10.1007/BF01807089

    work page doi:10.1007/bf01807089

  67. [67]

    Prosser R D 1976Int. J. Theor. Phys.15181–193https://doi.org/10.1007/BF01807090

    work page doi:10.1007/bf01807090

  68. [68]

    Braunbek W and Laukien G 1952Optik9174–179

    work page

  69. [69]

    Phys.325763–784 https://doi.org/10.1016/j.aop.2009.12.005

    Sanz A S, Davidovi´ c M, Boˇ zi´ c M and Miret-Art´ es S 2010Ann. Phys.325763–784 https://doi.org/10.1016/j.aop.2009.12.005

    work page doi:10.1016/j.aop.2009.12.005 2009

  70. [70]

    Boˇ zi´ c M, Davidovi´ c M, Dimitrova T L, Miret-Art´ es S, Sanz A S and Weis A 2010J. Russ. Laser Res.31117–128https://doi.org/10.1007/s10946-010-9131-9

    work page doi:10.1007/s10946-010-9131-9

  71. [71]

    Sanz A S, Campos-Mart´ ınez J and Miret-Art´ es S 2012J. Opt. Am. Soc. A29695–701 https://doi.org/10.1364/JOSAA.29.000695

    work page doi:10.1364/josaa.29.000695

  72. [72]

    Sci.101808 https://doi.org/10.3390/app10051808

    Sanz A S, Davidovi´ c M and Boˇ zi´ c M 2020Appl. Sci.101808 https://doi.org/10.3390/app10051808

    work page doi:10.3390/app10051808

  73. [73]

    Phys.: Conf

    Sanz A S 2024J. Phys.: Conf. Ser.2883012011 https://doi.org/10.1088/1742-6596/2883/1/012011

    work page doi:10.1088/1742-6596/2883/1/012011

  74. [74]

    Phys.353205–221 https://doi.org/10.1016/j.aop.2014.11.012

    Sanz A S, Davidovi´ c M and Boˇ zi´ c M 2015Ann. Phys.353205–221 https://doi.org/10.1016/j.aop.2014.11.012

    work page doi:10.1016/j.aop.2014.11.012 2014

  75. [75]

    Berry M V and Balazs N L 1979Am. J. Phys.47264–267https://10.1119/1.11855

    work page doi:10.1119/1.11855

  76. [76]

    Siviloglou G A, Broky J, Dogariu A and Christodoulides D N 2007Phys. Rev. Lett.99213901 https://link.aps.org/doi/10.1103/PhysRevLett.99.213901

    work page doi:10.1103/physrevlett.99.213901

  77. [77]

    Lett.32979–981 http://opg.optica.org/ol/abstract.cfm?URI=ol-32-8-979

    Siviloglou G A and Christodoulides D N 2007Opt. Lett.32979–981 http://opg.optica.org/ol/abstract.cfm?URI=ol-32-8-979

    work page

  78. [78]

    Sanz A S 2022J. Opt. Soc. Am. A39C79–C85 https://opg.optica.org/josaa/abstract.cfm?URI=josaa-39-12-C79

    work page

  79. [79]

    Mart´ ınez-Herrero R and Sanz A S 2022Phys. Rev. A106053512 https://link.aps.org/doi/10.1103/PhysRevA.106.053512

    work page doi:10.1103/physreva.106.053512

  80. [80]

    Takabayasi T 1952Prog. Theor. Phys.8143–182 https://academic.oup.com/ptp/article/8/2/143/1883374

    work page

Showing first 80 references.