Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

A single invariant equation in quantum phase space unifies Planck length and de Sitter radius

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-02 17:08 UTC pith:4QIIB6K7

load-bearing objection The paper's headline equation is an ansatz wearing a derivation's clothes; the one solid result is the LCT invariance of Gamma. the 3 major comments →

arxiv 2603.28836 v3 pith:4QIIB6K7 submitted 2026-03-30 quant-ph

Geometric structure of the relativistic quantum phase space

Philippe Manjakasoa Randriantsoa , Ravo Tokiniaina Ranaivoson , Raoelina Andriambololona , Roland Raboanary , Wilfrid Chrysante Solofoarisina , Anjary Feno Hasina Rasamimanana This is my paper
classification quant-ph
keywords relativistic quantum phase spacelinear canonical transformationsscalar invariantvariance-covariance matrixde Sitter spacetimecurved momentum spaceBorn reciprocityuncertainty relations
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.

This paper extends the relativistic quantum phase space formalism, in which a quantum state is described both by mean values and by a variance-covariance matrix, to a spacetime with signature (1,4). It constructs a scalar from the mean values and the inverse variance-covariance matrix, proves that this scalar is invariant under linear canonical transformations, and argues that for states saturating the uncertainty relations the invariant equals L²/ℓ², where L is the de Sitter radius and ℓ the Planck length. From this invariant it derives a single geometric equation that binds mean values and quantum fluctuations together. In the limit ℓ→0 the equation reduces to the de Sitter spacetime equation, while in the limit L→∞ it becomes a de Sitter-like curved momentum space, with the combined Minkowski limit recovering familiar relativistic relations.

Core claim

The paper's central claim is that the quantum phase space for signature (1,4) possesses a scalar invariant, Γ = (⟨p⟩ ⟨x⟩) (P Q; Qᵀ X)⁻¹ (⟨p⟩ ⟨x⟩)ᵀ, which is unchanged under linear canonical transformations. For states that saturate the uncertainty relations, the paper sets the variance-covariance matrix to a diagonal form with variances ℏ²/(4ℓ²) in momentum and L² in position, and a specific position-momentum correlation term; substituting these into Γ yields Γ = L²/ℓ². The unified geometric equation, (⟨p⟩ ⟨x⟩) (P Q; Qᵀ X)⁻¹ (⟨p⟩ ⟨x⟩)ᵀ = L²/ℓ², then follows. In the ℓ→0 limit it forces the mean coordinates to satisfy the de Sitter spacetime equation; in the L→∞ limit it forces the mean moment

What carries the argument

The key object is the invariant Γ, a scalar formed from the ten mean values (⟨pμ⟩, ⟨xμ⟩) and the inverse of the 10×10 variance-covariance matrix. Under a linear canonical transformation, the mean-value vector and the variance-covariance matrix transform in mutually inverse ways, leaving Γ invariant. The subsequent derivation depends on choosing a reference frame in which the saturated state has a block-diagonal variance-covariance matrix, Pμν = δμν ℏ²/(4ℓ²), Xμν = δμν L², Qμν = δμν (ℏ/(2ℓ))√(L²−ℓ²), and on setting Γ = L²/ℓ²; this choice converts the invariant identity into the geometric equation that drives both limits.

Load-bearing premise

The derivation rests on the assumed diagonal form of the variance-covariance matrix in Eq. (18) and on the subsequent choice Γ = L²/ℓ²; the paper does not derive this matrix from the saturation condition, and the limiting results depend on that specific form.

What would settle it

Compute the variance-covariance matrix for the actual Gaussian saturated states defined in Eq. (12) and verify whether it equals the diagonal block form of Eq. (18) in some reference frame; if the correlation terms or the product PμμXμμ − (Qμμ)² fail to match the assumed values, the invariant Γ = L²/ℓ² and the derived limits collapse.

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

If this is right

  • If the central equation is correct, the Planck length and the de Sitter radius cease to be independent inputs; they are tied together through the symplectic structure of the quantum phase space.
  • The ℓ→0 limit recovers classical de Sitter spacetime, suggesting that a positive cosmological constant could be a geometric remnant of quantum phase space rather than a separate parameter.
  • The L→∞ limit yields a curved momentum space with curvature scale ℏ/(2ℓ), connecting the formalism to doubly special relativity and to the idea of an observer-independent maximum momentum uncertainty.
  • The Minkowski limit reproduces standard relativistic relations for rest mass and proper time, so known physics emerges as the joint limit of the two quantum scales.
  • The unified equation implies that quantum fluctuations are part of the geometry itself, not a small correction to a classical phase space.

Where Pith is reading between the lines

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

  • Beyond the paper: if Γ = L²/ℓ² were derived rather than chosen, the ratio L/ℓ would be fixed by the state's saturation condition, giving a direct numerical link between the cosmological constant and the Planck scale.
  • Beyond the paper: a direct derivation of the diagonal variance-covariance matrix from the Gaussian saturated state of Eq. (12) would test whether the assumed form actually holds; a mismatch would change the limiting equations.
  • Beyond the paper: the same invariant construction could be applied to other spacetime signatures, and checking signature (3,1) would reveal whether the de Sitter limit survives when the extra spacelike dimension is absent.
  • Beyond the paper: promoting the variance-covariance matrix to a dynamical variable would turn the geometric equation into an equation of motion, potentially connecting the formalism to quantum field theory.

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

3 major / 4 minor

Summary. The paper studies the relativistic quantum phase space (QPS) formalism for signature (1,4), whose symmetry group is Sp(2,8). It defines a scalar Γ = (⟨p⟩ ⟨x⟩) C^{-1} (⟨p⟩ ⟨x⟩)^T from the mean values and inverse variance-covariance matrix and correctly proves its invariance under linear canonical transformations. For states saturating the uncertainty relation, the paper chooses a particular reference frame in which the covariance matrix is assumed diagonal and isotropic, Eq. (18), and then sets Γ = L^2/ℓ^2, Eq. (25). This leads to the 'geometric equation of the quantum phase space', Eq. (29). Taking ℓ→0 and L→∞ yields the advertised de Sitter spacetime equation and curved momentum space equation. The paper concludes that the de Sitter radius and Planck length emerge from the QPS symplectic structure and saturation condition.

Significance. If the derivation were sound, the paper would offer a unified geometric constraint linking quantum fluctuations to the de Sitter radius and the Planck length, with concrete limiting regimes and a realization of Born reciprocity. The invariance lemma in Sec. 2 is correct, and the algebraic steps leading from the assumed covariance matrix to the limiting equations are straightforward and checkable. However, the central result is not derived from the saturation condition; it is imposed by two unproven assumptions: the specific variance-covariance matrix (18) and the value Γ = L^2/ℓ^2 (25). Since these assumptions are load-bearing, the claimed geometric equation and its physical implications do not follow from the QPS formalism as stated.

major comments (3)
  1. [§3, Eq. (25)] The central equation (29) is not derived from the saturation condition (11). In the privileged frame F0, Eq. (21) gives Γ = 4L²κ²/ℏ² − (4/ℓℏ)√(L²−ℓ²)κλ + λ²/ℓ², which depends on the mean values κ and λ. Setting Γ = L²/ℓ² in Eq. (25) imposes a quadratic relation among κ, λ, L, and ℓ; it is not a consequence of saturation, which constrains only the variance-covariance matrix. Thus Eq. (29) holds at most for a subset of saturating states satisfying that extra constraint. The conclusion in Sec. 4 that the value Γ = L²/ℓ² is 'determined by the states themselves' is therefore unsupported; it is fixed by assumption. The limiting equations (27) and (28) are valid only conditional on that choice.
  2. [§3, Eq. (18)] The variance-covariance matrix (18) is a pure ansatz. The text says 'We may also suppose' and checks only that it satisfies the saturation condition (11). No derivation is given from the Gaussian state (12), from LCT covariance, or from a minimization principle. The specific entries Pμν = δμν ℏ²/(4ℓ²), Xμν = δμν L², and Qμν = δμν (ℏ/(2ℓ))√(L²−ℓ²) are load-bearing: the inverse matrix (20), the quadratic form (21), and therefore Eq. (29) all depend on them. Since infinitely many covariance matrices satisfy Eq. (11), this step requires an independent justification; without it, the geometric equation is an assumption about the state, not a derived property.
  3. [§3, Eqs. (27)–(28); abstract] The two advertised limits are not autonomous results; they are consequences of the particular choice Γ = L²/ℓ² in Eq. (25). With a different Γ, the prefactors in Eq. (26) would change and the limits would fail or take different forms. Additionally, the arXiv abstract claims a Minkowski limit with both ℓ→0 and L→∞, but the body treats only the separate limits. In Eq. (26), taking both limits simultaneously makes all three terms tend to zero (unless a ratio is held fixed), which would reduce the equation to 0=1. This claim should be removed or replaced by a precise limiting procedure.
minor comments (4)
  1. [§3, Eq. (12)] The Gaussian wavefunction is written with unclear notation (⟨xμ|z⟩) and the matrix B is not defined. More importantly, this state is never connected to the assumed covariance matrix (18), so it plays no role in the derivation. Either make the connection explicit or remove the state as an unnecessary element.
  2. [Abstract and §4] The arXiv abstract contains a Minkowski-limit sentence that is absent from the full-text abstract and, as noted above, is not substantiated by the body. The two abstracts should be harmonized.
  3. [§3, text near Eq. (26)] Typographical error: 'explicitely' should be 'explicitly'. The end of §4 contains 'his perspective' which should be 'This perspective'.
  4. [Throughout] The paper uses 'momenta space' and 'momentum space' interchangeably; choose one terminology for consistency.

Circularity Check

2 steps flagged

The central geometric equation is fixed by fiat: Eq. (25) chooses Γ = L²/ℓ², so Eqs. (27)-(29) are imposed by construction rather than derived.

specific steps
  1. self definitional [Sec. 3, Eq. (25), leading to Eqs. (26)-(29); see also abstract]
    "The Eqs. 13 and 14 can be deduced respectively from Eqs. 24 if we choose: Γ = L2/ℓ2 (25) In fact, with this choice Eqs. 24 becomes: ..."

    Γ is not obtained from the saturation condition (11) or from the state; it is set to L²/ℓ² by fiat. Equation (29), the paper's central geometric equation, is just Eq. (7) with this chosen value substituted. The advertised de Sitter limit (27) and curved momentum limit (28) are then read off after that choice. Since changing Γ would change or destroy the limits, the predictions are equivalent to the input normalization. The later claim that 'This value is determined by the states themselves' is contradicted by the explicit 'if we choose'.

  2. other [Sec. 3, Eq. (18)]
    "We may also suppose that the variance-covariance values associated with this state in this particular frame and which satisfy the uncertainties saturation condition in Eqs.(11) are: Pµν = δµν ℏ2/(4ℓ2), Xµν = δµν L2, Qµν = δµν (ℏ/(2ℓ))√(L2−ℓ2)."

    The variance-covariance matrix is posited, not derived from the saturation condition (11) or the Gaussian state (12). The inverse (20), and hence the coefficients in Eq. (21), are fixed by this assumed form. Together with Eq. (25), this choice makes the cross term subleading and lets the ℓ→0 and L→∞ limits produce exactly the de Sitter and momentum-space equations. A different matrix satisfying (11) would yield a different invariant and different limits, so the 'predicted' spacetime equations are built into the assumed covariances.

full rationale

The formal invariance of Γ under LCTs (Eqs. 7-10) is self-contained, but the advertised physical content is not. Eq. (21) leaves Γ dependent on the arbitrary mean values λ, κ; saturation (11) does not fix them. The paper then explicitly chooses Γ = L²/ℓ² (Eq. 25) and further assumes the covariance matrix (18). Both moves are inputs, not consequences. Consequently Eq. (29) reduces by construction to the chosen normalization, and the limits (27)-(28) are imposed rather than predicted. This is a central circularity: the claimed derivation of de Sitter spacetime and curved momentum space from quantum phase space reduces to the choice of Γ and of the covariance matrix. The paper is not self-citation-circular; the issue is self-definitional/fitted input. Score 8: the headline result is forced by the chosen input, while the ancillary LCT invariance proof remains valid.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central equation depends on two free length scales (L, ℓ) and a chosen value of the invariant Γ. The variance matrix (18) and the value Γ = L²/ℓ² are assumed without derivation, making them ad hoc inputs rather than consequences of the stated principles.

free parameters (3)
  • L = identified with de Sitter radius ~1.3×10^26 m
    Large length scale setting maximal coordinate uncertainty; identified with de Sitter radius by assumption (Section 2).
  • ℓ = identified with Planck length ~1.6×10^-35 m
    Small length scale setting minimal coordinate uncertainty; identified with Planck length by assumption (Section 2).
  • Γ = L²/ℓ²
    Scalar invariant; the paper 'chooses' this value in Eq. 25 so that the desired limits are reproduced, rather than deriving it from the state conditions.
axioms (5)
  • domain assumption Canonical commutation relations [p_μ, x_ν] = iℏ η_μν (Eq. 1) hold in (1,4) signature.
    Starting point of the QPS formalism, taken from prior work [1,2].
  • domain assumption The physical states defining the QPS saturate the Robertson-Schrödinger uncertainty relation P_μμ X_μμ − (Q_μμ)² = ℏ²/4 (Eq. 11).
    Definition of QPS states, following prior work [2,3,6,14].
  • ad hoc to paper In some reference frame F0, a saturating state has the diagonal variance-covariance matrix (18).
    The paper 'supposes' this form; it is not derived from the saturation condition and is a load-bearing choice.
  • ad hoc to paper The scalar invariant Γ is set to L²/ℓ² (Eq. 25).
    This choice is needed for the limits to reproduce de Sitter spacetime and curved momentum space; no independent justification is given.
  • domain assumption L and ℓ are identified with the de Sitter radius and Planck length respectively.
    Physical interpretation of the two scales; plausible but not derived.

pith-pipeline@v1.3.0-alltime-deepseek · 9342 in / 25007 out tokens · 220106 ms · 2026-08-02T17:08:03.347127+00:00 · methodology

0 comments
read the original abstract

The quest to reconcile quantum mechanics with gravitational theory motivates the exploration of frameworks that treat quantum uncertainty and spacetime geometry under a unified approach. A promising candidate that emerges from this pursuit is the relativistic quantum phase space (QPS) formalism, which extends classical phase space by incorporating both mean values and variance-covariance matrices of quantum states, providing a unified setting where the uncertainty principle and relativistic covariance coexist. For the signature $(1,4)$, we construct a scalar from the mean values and the inverse variance-covariance matrix and prove its invariance under linear canonical transformations (LCTs). Motivated by the form of the variance-covariance matrix in a particular reference frame, we identify this invariant as $\Gamma = L^2/\ell^2$ for states that saturate the uncertainty relations, where $L$ and $\ell$ are two fundamental length scales that can be identified with the de Sitter radius and the Planck length, respectively. From this invariant, we obtain a geometric equation that unifies mean values and quantum fluctuations. In the limit $\ell \to 0$, the equation reduces to the de Sitter spacetime equation; in the limit $L \to \infty$, it yields a curved momentum space reminiscent of Born reciprocity. In the Minkowski limit (both $\ell \to 0$ and $L \to \infty$), the familiar relativistic relations for rest mass and proper time emerge. These limiting cases show how the Planck length and the cosmological constant can be unified within a single geometric constraint, establishing the QPS geometry as a promising framework for exploring the interplay between quantum mechanics and gravity.

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Contractions of the relativistic quantum LCT group and the emergence of spacetime symmetries

    physics.class-ph 2026-03 conditional novelty 5.0

    Contractions of the LCT Lie algebra for signature (1,4) yield the de Sitter algebra so(1,4) and the Poincaré algebra iso(1,3) in the respective limits of minimum length ℓ and maximum length L.

Reference graph

Works this paper leans on

27 extracted references · 14 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Ranaivoson, Raoelina Andriambololona, H

    R.T. Ranaivoson, Raoelina Andriambololona, H. Rakotoson, R. Raboanary,Linear Canonical Transformations in Relativistic Quantum Physics, Phys. Scr. 96, 065204 (2021), arXiv:1804.10053 [quant-ph]

    Pith/arXiv arXiv 2021

  2. [2]

    Ranaivoson, Raoelina Andriambololona, H

    R.T. Ranaivoson, Raoelina Andriambololona, H. Rakotoson, R.H.M. Ravelonjato, Invariant quadratic operators associated with Linear Canonical Transformations and their eigenstates, J. Phys. Commun. 6, 095010 (2022), arXiv:2008.10602 [quant-ph]

    Pith/arXiv arXiv 2022

  3. [3]

    Ravelonjato, R.T

    R.H.M. Ravelonjato, R.T. Ranaivoson, Raoelina Andriambololona, R. Raboanary, H. Rakotoson, N. Rabesiranana,Quantum and Relativistic Corrections to Maxwell- Boltzmann Ideal Gas Model from a Quantum Phase Space Approach, Found. Phys. 53, 88 (2023)

    2023

  4. [4]

    Raoelina Andriambololona, R. T. Ranaivoson, H. D. Randriamisy , H. Rakotoson , Dispersion Operators Algebra and Linear Canonical Transformations, Int. J. Theor. Phys. 56, 1258–1273 (2017)

    2017

  5. [5]

    Ranaivoson, H

    Raoelina Andriambololona, R.T. Ranaivoson, H. Rakotoson, R. Raboanary,Sterile neutrino existence suggested from LCT covariance, J. Phys. Commun. 5, 091001 (2021), arXiv:2109.03807 [hep-ph]

    Pith/arXiv arXiv 2021

  6. [6]

    Ranaivoson, Raoelina Andriambololona, H

    R.T. Ranaivoson, Raoelina Andriambololona, H. Rakotoson, R. Raboanary, J. Rajao- belison, P.M. Randriantsoa,Quantum Phase Space Symmetry and Sterile Neutrinos, J. Subat. Part. Cosmol. 3, 100039 (2025)

    2025

  7. [7]

    Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

    Mike D. Schneider,Empty space and the (positive) cosmological con- stant,arXiv:2303.14974 [physics.hist-ph], Studies in History and Philosophy of Science, 100, 12-21,ISSN 0039-3681 (2023)

    Pith/arXiv arXiv 2023

  8. [8]

    Heavens,The cosmological model: an overview and an outlook, J

    A. Heavens,The cosmological model: an overview and an outlook, J. Phys.: Conf. Ser. (2008)

    2008

  9. [9]

    Martin Lopez-Corredoira,Tests and problems of the standard model in Cosmology, arXiv:1701.08720 [astro-ph.CO], https://arxiv.org/abs/1701.08720, Found Phys 47, 711–768 (2017)

    Pith/arXiv arXiv 2017

  10. [10]

    Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M

    Ricardo T. Génova-Santos,The establishment of the Standard Cosmological Model through observations, arXiv:2001.08297, [astro-ph.CO],In: Kabáth, P., Jones, D., Skarka, M. (eds) Reviews in Frontiers of Modern Astrophysics. Springer, Cham.(2020)

    Pith/arXiv arXiv 2001

  11. [11]

    Naumov,Sterile Neutrino

    D.V. Naumov,Sterile Neutrino. A short introduction, EPJ Web Conf. 207, 04004 (2019), arXiv:1901.00151 [hep-ph]

    Pith/arXiv arXiv 2019

  12. [12]

    Boser, C

    S. Boser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft, M. Wurm,Status of light sterile neutrino searches, Prog. Part. Nucl. Phys. 111, 103736 (2020), arXiv:1906.01739 [hep-ex]

    Pith/arXiv arXiv 2020

  13. [13]

    Drewes,The phenomenology of right handed neutrinos, Int

    M. Drewes,The phenomenology of right handed neutrinos, Int. J. Mod. Phys. E 22, 1330019 (2013), arXiv:1303.6912 [hep-ph]. 11

    Pith/arXiv arXiv 2013

  14. [14]

    Randriantsoa, R.T

    P.M. Randriantsoa, R.T. Ranaivoson, Raoelina Andriambololona, R. Raboanary, W.C. Solofoarisina, A.F.H. Rasamimanana,Casimir operators for the relativistic quantum phase space symmetry group,arXiv:2512.18262 [quant-ph], (2025)

    Pith/arXiv arXiv 2025

  15. [15]

    Raoelina Andriambololona,Algèbre linéaire et multilinéaire, Collection LIRA, (1985)

    1985

  16. [16]

    Meschini ,Planck-scale physics: facts and beliefsarXiv:gr-qc/0601097, Found

    D. Meschini ,Planck-scale physics: facts and beliefsarXiv:gr-qc/0601097, Found. Sci. 12: 277, Springer Netherlands (2007)

    Pith/arXiv arXiv 2007

  17. [17]

    Einstein,Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsber

    A. Einstein,Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie, Sitzungsber. Preuss. Akad. Wiss. 1, 142-152 (1917)

    1917

  18. [18]

    de Sitter,On the curvature of space, Proc

    W. de Sitter,On the curvature of space, Proc. Kon. Ned. Acad. Wet. 20, 229-243 (1917)

    1917

  19. [19]

    Amelino-Camelia,Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale, Int

    G. Amelino-Camelia,Relativity in spacetimes with short-distance structure governed by an observer-independent (Planckian) length scale, Int. J. Mod. Phys. D 11, 35 (2002), arXiv:gr-qc/0012051

    Pith/arXiv arXiv 2002

  20. [20]

    Magueijo, L

    J. Magueijo, L. Smolin,Lorentz invariance with an invariant energy scale, Phys. Rev. Lett. 88, 190403 (2002), arXiv:hep-th/0112090

    Pith/arXiv arXiv 2002

  21. [21]

    Born,Reciprocity theory of elementary particles, Rev

    M. Born,Reciprocity theory of elementary particles, Rev. Mod. Phys. 21, 463 (1949)

    1949

  22. [22]

    Castro Perelman,Born Reciprocal (Non-inertial) Relativity, Phase Space Trajecto- ries and Strings with variable Tension,Mod

    C. Castro Perelman,Born Reciprocal (Non-inertial) Relativity, Phase Space Trajecto- ries and Strings with variable Tension,Mod. Phys. Lett. A 2650078, arXiv:2508.17333 (2025)

    Pith/arXiv arXiv 2025

  23. [23]

    Lévy-Leblond,Une nouvelle limite non-relativiste du groupe de Poincaré, Ann

    J.-M. Lévy-Leblond,Une nouvelle limite non-relativiste du groupe de Poincaré, Ann. Inst. Henri Poincaré 3, 1 (1965)

    1965

  24. [24]

    Lévy-Leblond,On the unexpected fate of scientific ideas

    J.-M. Lévy-Leblond,On the unexpected fate of scientific ideas. An archeology of the Carroll group, in: 34th International Colloquium on Group Theoretical Methods in Physics, Strasbourg, 18–22 July 2022

    2022

  25. [25]

    Lévy-Leblond,Quand Galilée et Carroll engendrent Lorentz, Ann

    J.-M. Lévy-Leblond,Quand Galilée et Carroll engendrent Lorentz, Ann. Henri Poincaré 24, 3209 (2023)

    2023

  26. [26]

    Feynman,Space–Time Approach to Non-Relativistic Quantum Mechanics, Rev

    R.P. Feynman,Space–Time Approach to Non-Relativistic Quantum Mechanics, Rev. Mod. Phys. 20, 367 (1948)

    1948

  27. [27]

    Hawking,The quantum state of the universe, Nucl

    S.W. Hawking,The quantum state of the universe, Nucl. Phys. B 239, 257 (1984). 12

    1984