Pith. sign in

REVIEW 3 major objections 5 minor 38 references

The EPR Paradox of Quantum Mechanics in the Light of Four Unpublished Letters between A. Einstein and the Mathematician J. L. B. Cooper

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper reproduces what it argues is the complete Einstein–Cooper correspondence of 1949, showing Einstein's contention that quantum theory is either incomplete or implies instantaneous action at a distance, and Cooper's mathematical…

desk verdict Four genuinely new letters with a useful commentary, but the editorial apparatus needs archival backing before the completeness claim can stand. read the letter →

arxiv 2507.11736 v1 pith:UONVZOZM submitted 2025-07-01 physics.hist-ph math-phmath.MPquant-ph

P. L. Butzer (1) , D. E. Edmunds (2) , G. Roepstorff , G. Schmeisser (3) , R. L. Stens (1) ((1) Lehrstuhl A für Mathematik , RWTH Aachen , Aachen , Germany
show 8 more authors
(2) Department of Mathematics University of Sussex Brighton U. K. (3) Department of Mathematics University of Erlangen-Nuremberg Erlangen Germany)
This is my paper · ORCID
classification physics.hist-phmath-phmath.MPquant-ph PACS 03.65.Ta
keywords EPRparadoxEinsteinlettersCooperquantumincompletenessaction-at-a-distanceself-adjointoperatorshistoryofmechanics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper presents four letters exchanged between Albert Einstein and the mathematician J. L. B. Cooper between October and December 1949, which it identifies as the complete correspondence on the EPR paradox. Its central claim is that the exchange documents two irreconcilable standards: Einstein contends that statistical quantum theory cannot simultaneously uphold the independent existence of spatially separated parts and the wave function as the complete description of an individual system, so one must choose incompleteness or immediate action-at-a-distance; Cooper contends that the paradox evaporates once the systems are separated by an infinite potential barrier, because then the momentum operator on a half-line is not self-adjoint and the spectral-representation theorem central to EPR breaks down. The paper analyzes the standoff from a physicist's and a mathematician's viewpoint, arguing that the disagreement reflects different demands on mathematical models. A sympathetic reader would care because these letters give a direct, non-formal statement of Einstein's later EPR position and an early mathematician's critique.

What carries the argument

The argument rides on two mechanisms. First, Cooper's operator-theoretic observation: on a half-line, with an isolating potential wall, the momentum operator is Hermitian and maximal but not self-adjoint, so no spectral resolution and no Dirac representation theorem exist; this is the step that would block the EPR reasoning. Second, Cooper's Fourier-analytic point: a wave function vanishing outside a finite interval has an analytic momentum-space transform with at most isolated zeros, implying a finite probability of interaction at any later time, no matter the separation. Einstein's answer works by a limiting-case principle: the wave function never vanishes exactly, but it can vanish to any approximation, and that is enough for the EPR argument. The paper isolates these two standards—exact self-adjointness and approximate realization—as the crux of the Einstein–Cooper disagreement.

What would settle it

A search of Einstein's and Cooper's archived papers that turns up a fifth letter between 11 October and 18 December 1949, or a scan showing a date or wording that differs from the retyped texts, would settle that the claimed completeness and fidelity are wrong.

Watch

Extended reading notes

Core claim

The paper claims to have assembled, from archived originals, the full Einstein–Cooper correspondence, and it reproduces the letters verbatim, with an English translation of Einstein's German reply. Its discovery is that Einstein's second letter states the EPR situation as an explicit alternative: either the quantum-mechanical description of an individual system is incomplete, or one must accept an immediate action-at-a-distance, or both. Cooper's letters, in turn, put forward the mathematical thesis that a true separation of two systems requires an infinite potential barrier, and that on a half-space the momentum operator is merely Hermitian and not self-adjoint, so the expansion theorem used by EPR and Schrödinger is invalid. The paper's analysis shows that Einstein did not engage Cooper's mathematical proposal because his concern was not a logical inconsistency in quantum theory but the compatibility of the theory with the separability principle.

Load-bearing premise

The claim that these four letters are the complete Einstein–Cooper correspondence on EPR, accurately retyped, rests on archive matching without scans or call numbers; if one letter is lost or mistranscribed, the paper's central evidence is compromised.

Editorial extensions

If this is right

  • The four letters become a primary source for Einstein's 1949 formulation of the EPR incompleteness argument, distinct from the 1935 paper's formalism.
  • Einstein's second letter, now translated, provides a crisp statement of the completeness-or-action-at-a-distance alternative.
  • Cooper's mathematical objection—that an impassable barrier makes momentum non-self-adjoint—is documented as the origin of his 1950-published analysis of separated systems.
  • The exchange shows that the disagreement was not a technical error but a clash between an engineer's tolerance for approximate vanishing and a mathematician's demand for exactness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not draw a connection, but Einstein's 'independent existence of spatially separated parts' is a separability axiom that later experimental tests of Bell inequalities probe directly.
  • An unstated consequence of Einstein's dichotomy is that its two horns are not exhaustive; later no-go theorems show that any hidden-variable completion must be nonlocal or contextual.
  • A testable extension would be to compare Cooper's letters to his 1950 published paper to see which arguments survived the editorial process, and to search digitized archival catalogs for any further Einstein correspondence on this topic.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper presents and comments on four letters exchanged between Albert Einstein and the mathematician J. L. B. Cooper between October and December 1949, concerning the EPR paradox. Cooper's two letters argue that a genuine separation of two quantum systems requires an infinite potential barrier, which makes the momentum operator on a half-line non-self-adjoint and invalidates the eigenfunction expansions used in EPR; Einstein's replies reject the barrier as irrelevant to his real concern, which is the incompatibility of locality ('independent existence of spatially separated parts') with the completeness of the ψ-function. The paper retypes the letters, provides an English translation of Einstein's German letter, and analyzes the disagreement from a physicist's and a mathematician's point of view, drawing an analogy with time-limited versus band-limited signals. The authors claim that the four letters constitute the complete Einstein–Cooper correspondence on the subject.

Significance. If the documentary claims are accepted, the letters have historical value: they show Einstein formulating in 1949 the completeness-versus-locality alternative in his own words, and they document a mathematically literate correspondent pressing a technical objection (non-self-adjointness of momentum on a half-line) that Einstein did not engage. The mathematical commentary is sound where checkable: the half-line momentum operator is indeed Hermitian but not self-adjoint, and the Fourier transform of a compactly supported function is entire, hence cannot have non-isolated zeros. The sampling-theory analogy is instructive. The paper's main contribution, however, is archival, and that contribution is currently under-evidenced: no facsimiles, call numbers, or independent collations are supplied, so the accuracy and completeness of the transcription cannot be assessed from the submission. The paper also contains a concrete dating contradiction in §4.2. Because the historical claim is the heart of the paper, these issues must be resolved before publication.

major comments (3)
  1. [Section 1 (pp. 2–3)] The negative existential claim that 'There was no further exchange of views between Einstein and Cooper' and the characterization of the four letters as 'the complete correspondence thus obtained' are load-bearing for the paper's historical value, but the evidence described in Section 1—two letters in Cooper's estate and two in the Einstein Archive—does not establish completeness. Absence from these two repositories does not rule out letters held elsewhere or later discovered. No facsimiles, archival call numbers, finding aids, or independent collations are provided, and the retyped texts contain unmarked irregularities (e.g., 'A nad B', 'The the point') that could be authorial or editorial. The authors should either supply archival documentation or replace the completeness claim with a claim limited to the correspondence located in the two named collections.
  2. [Section 4.2 (p. 12)] The text dates Einstein's reply to Cooper's second letter as 'November 11', but the letter itself in Section 3 is dated December 18, 1949, and the Introduction states the correspondence ran 'between October and December 1949'. This internal contradiction affects the reconstruction of the exchange and must be corrected; all dates in the paper should be checked against the letters.
  3. [Section 4.1 (p. 10)] The physicist's commentary attributes to Cooper the assumption 'that related observables like momentum, position and energy are then restricted to a halfspace' and calls this 'definitely wrong'. Cooper's letters do not make that assumption; they argue that an infinite barrier confines the wave function to a half-space and that the momentum operator on the half-line is Hermitian but not self-adjoint, so the spectral expansion fails. The commentary's own next sentence—'Defined on a halfspace they are merely Hermitian operators without any physical interpretation whatsoever'—restates Cooper's point. This section should be rewritten to represent Cooper's argument accurately.
minor comments (5)
  1. [Introduction (p. 2)] Reference [9] is cited for a 'detailed biography' of Cooper, but the reference is a 2016 paper on Sobolev spaces; the biographical references appear to be [6] and [12]. Please correct the citation.
  2. [References, [19]] The entry for Fine's Stanford Encyclopedia article ends with '2916', which should presumably be '2016' or a proper access date.
  3. [Section 2 (p. 2)] There is a typo: 'transated into English' should be 'translated into English'.
  4. [Section 3 (pp. 6–9)] The retyped letters contain numerous irregularities such as 'you letter', 'A nad B', 'nor' for 'not', and 'The the point'. If these appear in the originals, they should be marked with '[sic]' or discussed in an editorial note; if they are retyping errors, they should be corrected or recorded. The current presentation leaves the transcription policy unclear.
  5. [Section 3 (p. 8) and translation (p. 8)] Einstein's German letter refers to 'Dielektrica' while the English translation and the rest of the paper use 'Dialectica'. If this is Einstein's own misspelling, a note would help; otherwise it should be corrected for consistency.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the four letters are external primary sources and the interpretive analysis rests on quotations and independent scholarship; the only self-references are minor, non-load-bearing citations to the authors' own sampling-theory papers.

full rationale

The paper's central content is the four retyped letters, which are external primary sources; there is no mathematical derivation, fitted parameter, or predicted quantity that could reduce to the paper's own inputs. The analysis of the Einstein-Cooper disagreement is grounded in quotations from the letters and in independent prior scholarship (Jammer [25], Bohr [4], Schrodinger [34-36], Furry [20,21], Fine [19], Howard [24]), so the interpretive conclusion is not equivalent by construction to its evidence. The only self-referential material is illustrative: Section 4.2's band-limited-signal analogy cites the authors' own sampling-theory papers ([7], [8], [9]), and the biography of Cooper cites two works by the present authors ([6], [12]); these are independently published, standard, peer-reviewed mathematics and do not support the historical completeness claim, so under the review rule for independent support they do not raise the circularity score. Two weaknesses are noted for correctness rather than circularity: (i) the negative-existential claim that 'There was no further exchange of views between Einstein and Cooper' rests on searches of only the Einstein Archive and Cooper's estate, with no call numbers, scans, or finding aids supplied; and (ii) Section 4.2 misdates Einstein's reply to Cooper's second letter as 'November 11' even though the letter itself is dated December 18, 1949, and the retyped texts contain unmarked irregularities such as 'you letter', 'A nad B', and 'The the point'. These are evidentiary and editorial concerns, not instances of the derivation reducing to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on historical documents and standard mathematical facts. No free parameters or invented entities are introduced. The main unexamined inputs are archival authenticity and completeness of the letter set.

assumptions (5)
  • standard math A compactly supported wave function has an analytic Fourier transform whose zeros are isolated.
    Used in Cooper's second letter and Section 4.2 to argue finite probability of particles coming together.
  • standard math The momentum operator on a half-line is Hermitian but not self-adjoint, so no spectral resolution into eigenstates exists.
    Central to Cooper's first letter and to the physicist commentary in Section 4.1.
  • domain assumption Physical observables must correspond to self-adjoint operators.
    Section 4.1 invokes this to reject Cooper's half-space momentum as unphysical; it is the standard quantum mechanics postulate, not a theorem.
  • domain assumption The four letters are authentic and accurately transcribed.
    Section 3 reproduces them; no scans are provided for verification.
  • ad hoc to paper The correspondence is complete after December 1949.
    Section 1 asserts no further exchange based on absence of traces; this completeness is part of the paper's framing.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The EPR Paradox of Quantum Mechanics in the Light of Four Unpublished Letters between A. Einstein and the Mathematician J. L. B. Cooper." pith.science (2026). https://pith.science/paper/UONVZOZM

@misc{pith2026250711736,
  author       = {Pith},
  title        = {Pith review of: The EPR Paradox of Quantum Mechanics in the Light of Four Unpublished Letters between A. Einstein and the Mathematician J. L. B. Cooper},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UONVZOZM}},
  note         = {Machine review of arXiv:2507.11736}
}
read the original abstract

This paper presents correspondence between Albert Einstein and the mathematical analyst J. L. B. Cooper on the Einstein-Podolsky-Rosen (EPR) paradox of quantum theory published in 1935. Two letters written by Cooper, and the replies from Einstein, all written between October and December 1949, are retyped from the original ones. Furthermore, Einstein's second letter, which he wrote in German, is translated into English. The lack of agreement, arising from very different points of view, is analysed, taking into account the complex underlying mathematical and physical factors that arise naturally in connection with the EPR paradox.

Figures

Figures reproduced from arXiv: 2507.11736 by the authors.

Figure 1
Figure 1. Sixth Solvay Conference on Physics, Brussels, 1930. Standing, from left to right [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 35 canonical work pages

  1. [1]

    Audretsch

    J. Audretsch. Entangled Systems: New Directions in Quantum Physics. Wiley- VCH, Weinheim, 2007

    work page 2007

  2. [2]

    M. G. Beaty, M. M. Dodson, S. P. Eveson, and J. R. Higgins. On the approx- imate form of Kluvánek’s theorem. J. Approx. Theory, 160(1):281–303, 2009. doi:10.1016/j.jat.2009.02.013

    work page doi:10.1016/j.jat.2009.02.013 2009

  3. [3]

    Blaylock

    G. Blaylock. The EPR paradox, Bell’s inequality, and the question of locality.Amer. J. Phys., 78(1):111–120, 2010. doi:10.1119/1.3243279

    work page doi:10.1119/1.3243279 2010

  4. [4]

    N. Bohr. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 48:696–702, Oct 1935. doi:10.1103/PhysRev.48.696

    doi:10.1103/physrev.48.696 1935

  5. [5]

    Brown, Jr

    J. Brown, Jr. On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem.J. Math. Anal. Appl., 18(1):75–84; Er- ratum. Ibid. 21 (1968), no. 3, 699, 1967. doi:10.1016/0022-247X(67)90183-7. and doi:10.1016/0022-247X(68)90271-0

    work page doi:10.1016/0022-247x(67)90183-7 1968

  6. [6]

    P. L. Butzer. Lionel Bakst Cooper—in memoriam. In P. L. Butzer, B. Sz.- Nagy, and E. Görlich, editors,Functional Analysis and Approximation, pages 19–23. Birkhäuser, Basel, 1981

    work page 1981

  7. [7]

    P. L. Butzer and R. L. Stens. Sampling theory for not necessarily band-limited func- tions: A historical overview.SIAM Review, 34(1):40–53, 1992. doi:10.1137/1034002

    work page doi:10.1137/1034002 1992

  8. [8]

    P. L. Butzer, W. Splettstößer, and R. L. Stens. The sampling theorem and linear prediction in signal analysis.Jahresber. Deutsch. Math.-Verein., 90(1), 1988. 14 P.L. Butzer, D.E. Edmunds, G. Roepstorff, G. Schmeisser, R.L. Stens

    work page 1988

Show all 38 references
  1. [9]

    P. L. Butzer, G. Schmeisser, and R. L. Stens. Sobolev spaces of fractional order, Lipschitzspaces, readaptedmodulationspacesandtheirinterrelations; applications. J. Approx. Theory, 212:1–40, 2016. doi:10.1016/j.jat.2016.08.001

    2016 doi

  2. [10]

    J. L. B. Cooper. The paradox of separated systems in quantum theory. Proc. Cambridge Philos. Soc., 46:620–625, 1950

    1950

  3. [11]

    J. L. B. Cooper. The foundations of thermodynamics. J. Math. Anal. Appl., 17: 172–193, 1967. doi:10.1016/0022-247X(67)90174-6

    1967 doi

  4. [12]

    D. E. Edmunds. Jacob Lionel Bakst Cooper. Bull. London Math. Soc., 13(5): 429–450, 1981. doi:10.1112/blms/13.5.429

    1981 doi

  5. [13]

    Einstein

    A. Einstein. Physik und Realität. Journal of the Franklin Institute, 221(3):313 – 347, 1936. doi:10.1016/S0016-0032(36)91045-1

    1936 doi

  6. [14]

    Einstein

    A. Einstein. Physics and reality.Journal of the Franklin Institute, 221(3):349 – 382,

  7. [15]

    Einstein

    A. Einstein. Quanten-Mechanik und Wirklichkeit.Dialectica, 2(3-4):320–324, 1948. doi:10.1111/j.1746-8361.1948.tb00704.x. [English translation published after letter 88 in Born, M., editor, The Born-Einstein letters, 1916–1955. Macmillan, New York, 2005]

    1948

  8. [16]

    Einstein

    A. Einstein. Autobiographical notes. In P. A. Schilpp, editor,Einstein: Philosopher- Scientist, volume 7 ofThe Library of Living Philosophers, pages 2–94. Northwestern Univ. Press, Evanston, Ill., 1949

    1949

  9. [17]

    Einstein

    A. Einstein. Reply to criticisms. In P. A. Schilpp, editor,Einstein: Philosopher- Scientist, volume 7 ofThe Library of Living Philosophers, page 648. Northwestern Univ. Press, Evanston, Ill., 1949

    1949

  10. [18]

    Einstein, B

    A. Einstein, B. Podolsky, and N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Phys. Rev., 47:777–780, May 1935. doi:10.1103/PhysRev.47.777

    1935 doi

  11. [19]

    A. Fine. The Einstein-Podolsky-Rosen argument in quantum theory. In E. N. Zalta, editor, The Stanford Encyclopedia of Philosophy (Fall 2016 Edition). 2916

    2016

  12. [20]

    W. H. Furry. Note on the quantum-mechanical theory of measurement.Phys. Rev., 49:393–399, Mar 1936. doi:10.1103/PhysRev.49.393

    1936 doi

  13. [21]

    W. H. Furry. Remarks on measurements in quantum theory.Phys. Rev., 49:476–476, Mar 1936. doi:10.1103/PhysRev.49.476

    1936 doi

  14. [22]

    J. B. Götte, S. Franke-Arnold, and M. S. Barnett. Angular EPR paradox.J. Modern Opt., 53(5–6):627–645, 2016. doi:10.1080/009500340500227075

    2016 doi

  15. [23]

    Handsteiner, A

    J. Handsteiner, A. S. Friedman, D. Rauch, J. Gallicchio, B. Liu, H. Hosp, J. Kofler, D. Bricher, M. Fink, C. Leung, A. Mark, H. T. Nguyen, I. Sanders, F. Steinlechner, R. Ursin, S. Wengerowsky, A. H. Guth, D. I. Kaiser, T. Scheidl, and A. Zeilinger. Cosmic bell test: Measureme...

    2017 doi

  16. [24]

    Nicht sein kann was nicht sein darf,

    D. Howard. “Nicht sein kann was nicht sein darf,” or the Prehistory of EPR, 1909– 1935: Einstein’s early worries about the quantum mechanics of composite systems. In A. I. Miller, editor,Sixty-Two Years of Uncertainty – Historical, Philosophical, and Physical Inquiries into th...

    1909 doi

  17. [25]

    M. Jammer. The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective. Wiley-Interscience, New York, 1974

    1974

  18. [26]

    E. C. Kemble. The correlation of wave functions with the states of physical systems. Phys. Rev., 47:973–974, Jun 1935. doi:10.1103/PhysRev.47.973.2

    1935 doi

  19. [27]

    E. C. Kemble.The Fundamental Principles of Quantum Mechanics. McGraw-Hill, New York, 1937

    1937

  20. [28]

    Kupczynski

    M. Kupczynski. EPR paradox, quantum nonlocality and physical reality.J. Phys.: Conf. Ser., 701:012021, 2016. doi:10.1088/1742-6596/701/1/012021

    2016 doi

  21. [29]

    F. Levi. EPR paradox: Nonlocal legacy. Nature Phys., 11(5):384, 2015. doi:10.1038/nphys3328

    2015 doi

  22. [30]

    Peise, I

    J. Peise, I. Kruse, K. Lange, B. Lücke, L. Pezzè, J. Arlt, W. Ertmer, K. Hammerer, L. Santos, A. Smerzi, and C. Klempt. Satisfying the Einstein- Podolsky-Rosen criterion with massive particles.Nature Commun., 6:8984, 2015. doi:10.1038/ncomms9984

    2015 doi

  23. [31]

    K. Popper. The Logic of Scientific Discovery. Hutchinson, London, 1959

    1959

  24. [32]

    Reichenbach.Philosophic Foundations of Quantum Mechanics

    H. Reichenbach.Philosophic Foundations of Quantum Mechanics. Univ. of Califor- nia Press, Berkeley, 1944

    1944

  25. [33]

    T. Sauer. An Einstein manuscript on the EPR paradox for spin observables.Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38(4):879–887, 2007. doi:10.1016/j.shpsb.2007.03.002

    2007 doi

  26. [34]

    Discussionofprobabilityrelationsbetweenseparatedsystems

    E.Schrödinger. Discussionofprobabilityrelationsbetweenseparatedsystems. Proc. Cambridge Philos. Soc., 31:555–563, 1935

    1935

  27. [35]

    Schrödinger

    E. Schrödinger. Die gegenwärtige Situation in der Quantenmechanik.Die Natur- wissenschaften, 23:807–812; 823–828, 1935

    1935

  28. [36]

    Schrödinger

    E. Schrödinger. Probability relations between separated systems.Proc. Cambridge Philos. Soc., 32:446–452, 1936

    1936

  29. [37]

    P. Weiss. An estimate of the error arising from misapplication of the sampling theorem. Notices Amer. Math. Soc., 10:351, 1963

    1963

  30. [1936]

    doi:10.1016/S0016-0032(36)91047-5

    doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.