Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

On the Status of Frequency in de Broglie's Derivation of the Phase Wave

Philippe Orta

Published May 9, 2026• 0 citations
Publisher

Abstract

This article proposes a conceptual analysis of the reasoning developed by Louis de Broglie in his 1924 thesis Recherches sur la théorie des quanta, particularly the pages devoted to the derivation of the phase wave. By distinguishing the proper frequency associated with the particle at rest, the frequency observed from the stationary frame, and the frequency derived from the relativistic energy of the corpuscle, we show that these different frequencies do not appear to possess the same physical status in de Broglie’s original reasoning. We therefore examine the hypothesis according to which the introduction of the phase wave primarily responds to the necessity of maintaining relativistic coherence between two distinct interpretations of frequency. This study does not challenge the physical validity of the de Broglie relation linking momentum and wavelength, but rather proposes a critical re-examination of the conceptual status of the frequencies involved in the original 1924 construction. Finally, we discuss the possibility of future geometrical or topological reformulations of this relation without explicit recourse to a superluminal hase wave.

DOI
10.5281/zenodo.20099194

Related papers

De Broglie’s relativistic phase waves and wave groups

Harvey R. Brown, Roberto de Andrade Martins

Louis de Broglie’s original theory of matter waves has until very recently received relatively little treatment in the literature on history and foundations of physics. This paper attempts to clarify a number of aspects of de Broglie’s theory, as formulated primarily in his famous thesis, which still give rise to confusion or debate. To this end, a re-examination is made of the scope and validity of the relativistic physics employed in de Broglie’s arguments concerning the existence of phase waves and their physical implications. Recent doubts concerning the soundness of de Broglie’s introduction of a wave group associated with the quantum are also analyzed in detail, and it is argued that the wave group indeed has a problematic, if minor, role in the overall theory.

American Journal of Physics•Dec 1, 1984•10 citations

On the de Broglie waves

Janez Strnad, W Kuhn

On the occasion of the sixtieth anniversary of Louis de Broglie's thesis, the concept of phase waves, is re-examined. After a brief historical survey a modern conceptual presentation of these waves is described via phase invariance. In this respect de Broglie's hypothesis appears not to be self-consistent. Phase waves in galilean relativity are briefly discussed. As an interesting pedagogical device, inverse phase waves in two-dimensional special relativity are presented. Finally, some remarks are added concerning the development of concepts and theories.

European Journal of Physics•Jul 1, 1985•2 citations

De Broglie’s thesis: A critical retrospective

Edward M. MacKinnon

Louis de Broglie’s doctoral thesis developed a concept of waves associated with material particles that was soon incorporated into wave mechanics and later supported by experimental demonstrations. De Broglie’s original development, however, relied on an incorrect identification of two quite different relations: the relation between the velocity of a particle and the relation between the group velocity of a wave packet and the velocity of individual waves in the packet. A clarification of this difference and its historial significance is followed by a tentative account of the reasons why de Broglie’s well-known formula proved successful, though the theory supporting it rested on a conceptual confusion. Finally, this development and clarification are related to some of the difficulties involved in the attempts top fashion a coherent interpretation of quantum theory.

American Journal of Physics•Nov 1, 1976•38 citations

Revisiting de Broglie’s famous paper of 1924

Edward A. Davis

Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.

The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics•Apr 6, 2026•0 citations

Clock, Boost, Wave: A Kinematic Origin of the de Broglie Relation

Ping Zhang

The de~Broglie relation $\lambda = h/p$ is among the most well-confirmed results in modern physics, yet the kinematic origin of the associated matter wave remains conceptually obscure. This paper presents a kinematic analysis leading to the de~Broglie relation. The only assumption is the existence of a globally defined phase field with uniform rest-frame oscillation, a structure known to occur in superfluids and Bose--Einstein condensates. Under a Lorentz boost, the scalar transformation of the phase field converts temporal periodicity directly into spatial periodicity, yielding a wave number $k' = \gamma\omega_0 v/c_s^2$ and a phase velocity $v_{\mathrm{phase}} = c_s^2/v$. No wave equation, dynamical model, or quantization procedure is required. The result follows solely from the relativity of simultaneity applied to a scalar phase field. When the characteristic frequency and invariant speed are identified as $\omega_0 = mc^2/\hbar$ and $c_s = c$, the resulting wavelength and phase velocity reproduce the standard de~Broglie relations. The logical architecture inverts that of de~Broglie's original construction: rather than postulating a separate clock and wave linked by phase harmony, the wave emerges as the Lorentz-transformed image of a single coherent phase clock. The phase harmony that de~Broglie introduced as an independent postulate becomes a consequence of Lorentz covariance. The analysis establishes a precise kinematic correspondence without asserting that the physical vacuum is a coherent medium. Whether this correspondence reflects a deeper physical principle or a mathematical analogy remains an open question.

•Jun 1, 2026•0 citations

The Covariant Relativistic Derivation of De Broglie Relation

Samuel Bueno Soltau

This paper provides an examination of the de Broglie relation, tracing its historical development from the quantum hypotheses proposed by Planck and Einstein to its covariant relativistic derivation. The discussion begins by situating de Broglie's seminal insight within the early framework of quantum theory. We then reconstruct his original heuristic derivation. The primary focus of this work, however, is the derivation of the de Broglie relation directly from the principles of special relativity, employing the four-momentum formalism. A comparative analysis between the heuristic and relativistic approaches underscores the universality and conceptual coherence of the latter. The paper concludes by highlighting the significance of relativistic mechanics in establishing a consistent foundation for wave-particle duality, further reinforcing this through a quantum field theoretical perspective.

arXiv:2508.03908•Sitientibus Série Ciências Físicas•Nov 1, 2025•0 citations
This article proposes a conceptual analysis of the reasoning developed by Louis de Broglie in his 1924 thesis Recherches sur la théorie des quanta, particularly the pages devoted to the derivation of the phase wave. By distinguishing the proper frequency associated with the particle at rest, the frequency observed from the stationary frame, and the frequency derived from the relativistic energy of the corpuscle, we show that these different frequencies do not appear to possess the same physical status in de Broglie’s original reasoning. We therefore examine the hypothesis according to which the introduction of the phase wave primarily responds to the necessity of maintaining relativistic coherence between two distinct interpretations of frequency. This study does not challenge the physical validity of the de Broglie relation linking momentum and wavelength, but rather proposes a critical re-examination of the conceptual status of the frequencies involved in the original 1924 construction. Finally, we discuss the possibility of future geometrical or topological reformulations of this relation without explicit recourse to a superluminal hase wave.
Louis de Broglie’s original theory of matter waves has until very recently received relatively little treatment in the literature on history and foundations of physics. This paper attempts to clarify a number of aspects of de Broglie’s theory, as formulated primarily in his famous thesis, which still give rise to confusion or debate. To this end, a re-examination is made of the scope and validity of the relativistic physics employed in de Broglie’s arguments concerning the existence of phase waves and their physical implications. Recent doubts concerning the soundness of de Broglie’s introduction of a wave group associated with the quantum are also analyzed in detail, and it is argued that the wave group indeed has a problematic, if minor, role in the overall theory.
On the occasion of the sixtieth anniversary of Louis de Broglie's thesis, the concept of phase waves, is re-examined. After a brief historical survey a modern conceptual presentation of these waves is described via phase invariance. In this respect de Broglie's hypothesis appears not to be self-consistent. Phase waves in galilean relativity are briefly discussed. As an interesting pedagogical device, inverse phase waves in two-dimensional special relativity are presented. Finally, some remarks are added concerning the development of concepts and theories.
Louis de Broglie’s doctoral thesis developed a concept of waves associated with material particles that was soon incorporated into wave mechanics and later supported by experimental demonstrations. De Broglie’s original development, however, relied on an incorrect identification of two quite different relations: the relation between the velocity of a particle and the relation between the group velocity of a wave packet and the velocity of individual waves in the packet. A clarification of this difference and its historial significance is followed by a tentative account of the reasons why de Broglie’s well-known formula proved successful, though the theory supporting it rested on a conceptual confusion. Finally, this development and clarification are related to some of the difficulties involved in the attempts top fashion a coherent interpretation of quantum theory.
Louis-Victor de Broglie secured his place in the history of quantum physics by proposing the existence of matter waves. Following the concept accepted early in the twentieth century that light waves could display properties characteristic of particles (photons), de Broglie suggested the complementary principle that particles could display properties characteristic of waves. Within a few years, this far-reaching proposal was experimentally verified. The well-known formula for the wavelength λ of the wave associated with a particle of momentum p, namely λ=h/p, does not appear in de Broglie’s original paper, which was published in this journal in 1924. What is derived from the starting assumption that the frequency of the wave is the rest energy of the particle divided by Planck’s constant h, is the wave's velocity, which turns out to be c2/v, where v is the particle’s velocity. This velocity, being superluminal, means that the wave cannot transport energy. De Broglie called it a phase wave and postulated a group of such waves having virtually, but not exactly, the same velocity and wavelength. The group velocity of such a large number of almost identical phase waves is then shown, very satisfactorily, to be equal to the velocity of the particle. The proposal by de Broglie inspired Schrödinger to develop the mathematical foundations of wave mechanics. Whereas in conventional quantum theory, phase waves are considered fictional, de Broglie considered them to have a certain physical reality, renaming them pilot or guiding waves – a concept developed later by Bohm in 1952.
The de~Broglie relation λ=h/p\lambda = h/pλ=h/p is among the most well-confirmed results in modern physics, yet the kinematic origin of the associated matter wave remains conceptually obscure. This paper presents a kinematic analysis leading to the de~Broglie relation. The only assumption is the existence of a globally defined phase field with uniform rest-frame oscillation, a structure known to occur in superfluids and Bose--Einstein condensates. Under a Lorentz boost, the scalar transformation of the phase field converts temporal periodicity directly into spatial periodicity, yielding a wave number k′=γω0v/cs2k' = \gamma\omega_0 v/c_s^2k′=γω0​v/cs2​ and a phase velocity vphase=cs2/vv_{\mathrm{phase}} = c_s^2/vvphase​=cs2​/v. No wave equation, dynamical model, or quantization procedure is required. The result follows solely from the relativity of simultaneity applied to a scalar phase field. When the characteristic frequency and invariant speed are identified as ω0=mc2/ℏ\omega_0 = mc^2/\hbarω0​=mc2/ℏ and cs=cc_s = ccs​=c, the resulting wavelength and phase velocity reproduce the standard de~Broglie relations. The logical architecture inverts that of de~Broglie's original construction: rather than postulating a separate clock and wave linked by phase harmony, the wave emerges as the Lorentz-transformed image of a single coherent phase clock. The phase harmony that de~Broglie introduced as an independent postulate becomes a consequence of Lorentz covariance. The analysis establishes a precise kinematic correspondence without asserting that the physical vacuum is a coherent medium. Whether this correspondence reflects a deeper physical principle or a mathematical analogy remains an open question.
This paper provides an examination of the de Broglie relation, tracing its historical development from the quantum hypotheses proposed by Planck and Einstein to its covariant relativistic derivation. The discussion begins by situating de Broglie's seminal insight within the early framework of quantum theory. We then reconstruct his original heuristic derivation. The primary focus of this work, however, is the derivation of the de Broglie relation directly from the principles of special relativity, employing the four-momentum formalism. A comparative analysis between the heuristic and relativistic approaches underscores the universality and conceptual coherence of the latter. The paper concludes by highlighting the significance of relativistic mechanics in establishing a consistent foundation for wave-particle duality, further reinforcing this through a quantum field theoretical perspective.