Relativistic implications of entropy and purity

Joseph Balsells; Martin Bojowald

REVIEW 3 major objections 5 minor 26 references

Relativistic implications of entropy and purity

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Quantum position correlations in at least two directions change the effective spacetime geometry from Riemannian to Finsler, with entropy and purity entering the time-dilation law.

desk verdict Underneath a technically clean quasiclassical derivation, the paper's central prediction rests on an unproven postulate for quantum proper time; worth refereeing, but not acceptable as stated. read the letter →

arxiv 2506.14705 v2 pith:ZGJ3ZLM7 submitted 2025-06-17 quant-ph gr-qc

Joseph Balsells , Martin Bojowald This is my paper · ORCID

classification quant-phgr-qc

keywords quantumpropertimeFinslergeometryentropyandpuritydilationquasiclassicalmechanicsnon-GaussianstatesgeodesicmotionHawkingradiation

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

Quantum uncertainty makes a particle spatially extended, so in a gravitational field different parts of its wave function experience different relativistic effects. The paper argues that when the state has position correlations in at least two directions, these effects cannot be captured by a Riemannian metric: proper time depends on a non-quadratic square root $P=\sqrt{C_2^4-C_1^4+(C_1^2-4p_\alpha^2)^2}$, making the effective geometry Finsler. The two conserved parameters $C_1,C_2$ encode the purity and entropy of the state, so those quantum-information quantities enter time dilation, the effective mass, and the Newtonian potential. If true, this gives a universal state-dependent time-dilation law that could be tested with free-falling clocks and would affect predictions for photon dispersion and Hawking radiation.

What carries the argument

The load-bearing mechanism is the canonical representation of second-order moments: the ten moments of a two-degree-of-freedom state are rewritten, to first order in $\hbar$, as six canonical pairs ($s_1,p_{s1}$), ($s_2,p_{s2}$), ($\beta,p_\beta$), ($\alpha,p_\alpha$) plus two conserved quantities $C_1,C_2$. The conserved quantities are tied to the symplectic eigenvalues $\nu_\pm\geq\hbar/2$ of the covariance matrix by $C_1^2=\nu_+^2+\nu_-^2$ and $C_2^2=\nu_+^2-\nu_-^2$, so they determine entropy via Eq. (7) and purity via Eq. (8). The identity that carries the argument is the square root in Eq. (9), $P=\sqrt{C_2^4-C_1^4+(C_1^2-4p_\alpha^2)^2}$, which appears inside the time-dilation formula; its non-polynomial dependence on $p_\alpha$ is what forces the Finsler interpretation. Finsler geometry is the generalization of Riemannian geometry in which the length of a curve is not necessarily quadratic in its velocities or momenta.

What would settle it

A free-fall comparison of two states with identical mass and velocity but different purity (one Gaussian, one non-Gaussian, or two states with different squeezing) would test Eq. (6): the $P$-dependent term predicts a relative proper-time difference between the two clocks at the level set by their momentum variances, and a null result at that sensitivity would falsify the central claim.

Watch

Extended reading notes

Core claim

The central discovery is a proper-time law for quantum geodesics that is non-quadratic in momenta. Starting from a quasiclassical Hamiltonian in which $\langle \hat H\rangle$ is a function of classical variables plus canonical variables for second-order moments, the paper defines quantum proper time by $\Delta\tau=\int\sqrt{1-2\langle \hat H\rangle/(mc^2)}\,d\tau$. When the moment relations are substituted, the radicand contains $P=\sqrt{C_2^4-C_1^4+(C_1^2-4p_\alpha^2)^2}$, which is polynomial in the momenta only for a Gaussian state; for non-Gaussian states it is non-polynomial in the correlation momentum $p_\alpha$. Since a Riemannian metric always gives proper time quadratic in momenta, the paper concludes that quantum geodesic motion does not experience Riemannian spacetime, and the extended configuration space is instead Finslerian, with $\alpha$ as the non-Riemannian direction. The same structure produces an effective mass $m_{\mathrm{eff}}=\sqrt{m^2+g_{ab}\Delta(p^ap^b)/c^2}$, making entropy and purity contribute to the gravitational weight of even classically massless objects, and reveals a Minkowski metric on the $(p_\alpha,p_\beta)$ subspace that governs logarithmic negativity.

Load-bearing premise

The whole chain rests on the ansatz $\Delta\tau=\int\sqrt{1-2\langle \hat H\rangle/(mc^2)}\,d\tau$ for quantum proper time, which the paper postulates rather than derives from Schrödinger evolution or from a physical clock model; if the true relation between quantum evolution and proper time differs, the Finsler geometry and the purity-dependent time dilation do not follow.

Editorial extensions

If this is right

Free-falling clocks show purity-dependent gravitational redshift: two otherwise identical clock states with different entropy or purity run at slightly different rates, with the correction universal across internal clock mechanisms.
Classically massless objects acquire a state-dependent effective mass from momentum fluctuations; in non-radial motion, entropy and purity modify the Newtonian potential, with a leading $1/r$ correction proportional to $-\frac{1}{2}P\cos\alpha\sin\beta/(mc^2)$ in Schwarzschild spacetime.
Photon dispersion becomes nonlinear in vacuum: the group velocity $\partial\langle\hat E\rangle/\partial p_\alpha$ depends on $p_\alpha$, producing nonlinear-optics signatures that cold-atom dispersion experiments can constrain.
The standard Hawking spectrum is preserved for radial photons because radial motion produces no effective mass, while non-radial Hawking emission and photon-ring orbits are predicted to show purity-dependent corrections.
Logarithmic negativity is governed by the Minkowski distance $p_\beta^2-p_\alpha^2$ on an entanglement subspace, giving a dynamical geometric measure of entanglement at the level of second moments.

Reading between the lines

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

The cleanest extension is an experiment built around Eq. (6): two free-falling clocks of identical mass and velocity but different purity should show a relative gravitational-redshift shift proportional to $P$, isolating the Finsler term from classical contributions.
The reality condition on $P$ forbids the range $\frac{1}{2}(\nu_+-\nu_-)\leq |p_\alpha|\leq \frac{1}{2}(\nu_++\nu_-)$, which suggests a forbidden band of correlation momenta; a next step is to test whether that band is fundamental or an artifact of the first-order truncation.
Because the time-dilation correction is claimed to be independent of the clock's internal mechanism, the same prediction could be searched with composite or macroscopic clocks whose external degrees of freedom are controlled, without modeling their internal workings.
Deriving Eq. (6) from a fully quantum clock model is the natural sequel; any modification of the radicand would change which geometry the state sees, so the Finsler conclusion would move with it.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the paper is technically inventive but rests on an unproven postulate, so its headline claims are not established. Eq. (6) is the fulcrum. Everything after it—the non-quadratic radicand, Finsler geometry, state-dependent time dilation, effective mass—follows from replacing the classical Hamiltonian by its expectation value in the proper-time integrand. That replacement is never derived from the Schrödinger equation or from a physical clock model. The paper itself just says it 'implies' the quasiclassical geodesic and that all variables affect time dilation 'through quantum proper time (6).' For a result whose selling point is a new universal time-dilation law, this is a big gap.

What is genuinely new: applying the authors' canonical-moment formalism from [2,3] to the relativistic geodesic Hamiltonian, and isolating the correlation parameter α as the non-Riemannian direction. Eq. (11) connecting α to specific uncertainty products is a real piece of work, and the expression for purity/entropy in the effective mass (Eqs. 8, 13) is concrete and checkable. The paper is honest about using the companion paper [4] for the Finsler-from-entropy concept, but that means the incremental novelty is partly a repackaging of [4].

The soft spots are concentrated exactly where the stress-test put them. Universal independence of clock mechanism is asserted without any clock degrees of freedom in the model. The Finsler claim is interpretive—they show the radicand is non-quadratic and cite Finsler literature, but no Finsler metric is constructed. And there are no magnitude estimates, so the claimed experimental relevance is open. The Hawking radiation discussion is qualitative and shouldn't be mistaken for a calculation.

I also checked the algebraic structure: the canonical relationships in the appendix are consistent with [2,3], and the positivity conditions on P are handled correctly. The paper is not internally contradictory; it's just built on a postulate.

Bottom line: this deserves a serious referee, not a desk reject, because the mathematical scaffolding is substantial and the question—does quantum proper time depend on entropy and purity—is important. But I would not accept it in this form. The referee should insist on either a derivation of Eq. (6) from a clock model or an explicit downgrade of the claims to 'under this ansatz.' I wouldn't cite it in my own work yet.

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a quasiclassical extension of relativistic geodesic motion in which quantum fluctuations and correlations enter the proper-time integral and effective mass. Using the canonical moment representation developed in Refs. [2,3], the authors express second-order moments in terms of canonical variables and identify the square-root expression P (Eq. 9) as the source of a non-quadratic dependence of the proper-time integrand on the momentum p_alpha. They interpret this as evidence that correlated quantum states require Finsler geometry rather than Riemannian geometry, and they derive consequences for time dilation, entanglement, effective mass, and Hawking radiation.

Significance. If the central proper-time assumption were justified, the paper would be a valuable bridge between quantum information measures and relativistic observables, with concrete predictions for free-fall clocks, photon dispersion in vacuum, and black-hole physics. The algebraic moment relations are presented in detail and are checkable, and the identification of the covariance-matrix invariants C1 and C2 with purity is explicit and useful. However, the physical conclusions are conditional on the proper-time ansatz in Eq. (6) and on the Finsler interpretation in Section 4, so the significance is not yet established.

major comments (3)

[Section 2, Eq. (6)] The quantum proper time formula Delta tau = integral sqrt(1 - 2<H-hat>/(m c^2)) d tau is introduced as a postulate, not derived from the Schroedinger equation or from a physical clock model. It is the load-bearing input for the paper's main claims: the non-quadratic radicand in Eq. (9), the Finsler conclusion in Section 4, and the universal time-dilation law stated in the abstract all follow from this formula. The authors should either derive Eq. (6) from an action principle or a clock model, or clearly frame it as an assumption and temper the universality claim. In addition, if the geodesic Hamiltonian (2) is imposed as a constraint with physical states satisfying H-hat|psi>=0, then <H-hat>=0 and Eq. (6) is trivial; the paper must specify how the expectation value is computed in the constrained setting.
[Section 4] The conclusion that quantum geodesic motion 'does not experience Riemannian space-time' is an interpretation rather than a demonstrated property. Eq. (9) is non-polynomial in p_alpha, which is a momentum conjugate to an internal (quantum) degree of freedom alpha in the extended phase space. To establish Finsler geometry of spacetime, one must Legendre transform the extended Hamiltonian, reduce to the spacetime tangent bundle, and show that the resulting Finsler function is not quadratic in the spacetime velocities. The paper does not perform this reduction, so the central geometric claim is not yet supported.
[Section 2, Eqs. (3)-(6)] The canonical moment variables of Refs. [2,3] were derived for unitary evolution with respect to a coordinate time t via Eq. (3). Geodesic motion generated by Hamiltonian (2) is parameterized by an affine parameter, with H=0 on shell. The paper does not justify that the moment relations (15)-(24) and the conserved quantities C1 and C2 remain valid along the proper-time flow. This is important because the state evolves along the geodesic and C1 and C2 (hence purity) are used as conserved inputs in the subsequent analysis.

minor comments (5)

[Eq. (6)] The same symbol d tau appears on both sides of Eq. (6); please clarify whether the differential on the right-hand side is coordinate time, an affine parameter, or the unperturbed proper time.
[Section 2, after Eq. (3)] The text says 'two pairs of canonical coordinates, (s1, ps2) and (s2, ps2)', which should read (s1, ps1) and (s2, ps2).
[Eq. (12)] The notation Delta^{B1B2}_{A1A2} is used before it is defined; please define it explicitly in the main text rather than only in the displayed equation.
[Abstract and Section 1] There are typographical artifacts such as 'uncertaint y' and 'eects'; please proofread the manuscript.
[Section 5, Hawking radiation discussion] The argument that radial Hawking photons acquire no effective mass relies on the energy being linear in the single nonzero momentum; this should be stated more carefully, since the metric component g_rr is position-dependent and position fluctuations may enter.

Circularity Check

1 steps flagged · score 6.0 of 10

Quantum time dilation and Finsler geometry are unpacked from the postulated Eq. (6); the entropy/purity dependence is placed into the premise via ⟨Ĥ⟩, so the 'first-principles demonstration' restates the ansatz.

self definitional [Section 2, around Eq. (6); used in Sections 4-6 and Conclusions.]
""The Hamiltonian formulation makes it possible to extend this dual role to a quantum treatment without requiring further assumptions about how gravity might couple to a wave function or density matrix: The classical Hamiltonian can directly be replaced with the expectation value ⟨ ˆH⟩ in a suitable state." ... "Moreover, all variables affect time dilation through quantum proper time ∆τ = ∫ sqrt(1 − 2/(mc²) ⟨ ˆH⟩) dτ (6) integrated along the geodesic.""

Eq. (6) is introduced as the definition of quantum proper time, with the accompanying claim that replacing H by ⟨Ĥ⟩ requires no further assumptions. No derivation from the unitary evolution (3) or from a physical clock model is given. All later physical conclusions—non-quadratic proper time via P (9), entropy/purity corrections to time dilation, the universal clock-independent law, and Finsler geometry—are obtained by inserting the quasiclassical expansion of ⟨Ĥ⟩ (Eqs. 4, 5, 22-23) into this postulated integrand. The state dependence of time dilation is therefore an input of the definition, not an independent result. Calling the outcome a 'first-principles demonstration' in the Conclusions restates the ansatz as a derivation.

full rationale

The quasiclassical canonical-variable construction cited to [2,3] and the companion paper [4] is legitimate, parameter-free mathematics: it parametrizes second-order moments by canonical coordinates and conserved quantities via Hamilton equations and does not assume the target time-dilation result. The relations of C1 and C2 to symplectic eigenvalues, purity, and entropy are standard and are derived in the paper. Thus the self-citations are not the circularity. The circularity is localized in Eq. (6), the physical bridge that defines quantum proper time through ⟨Ĥ⟩; because this definition is the only channel through which entropy and purity enter the time-dilation law, the central claim reduces by construction to the input. No fitted parameters or data are involved, so the score is 6 (partial circularity) rather than 8-10.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper contains no fitted free parameters; the conserved quantities C1 and C2 are state-dependent inputs, not tunable constants. The central load-bearing assumptions are the quasiclassical equivalence from prior work [2,3] and the postulated quantum proper-time formula (Eq. 6). The Finsler interpretation adds a domain assumption about non-quadratic Hamiltonians.

assumptions (5)

domain assumption Unitary evolution of a semiclassical state is equivalent, to first order in hbar, to Hamiltonian evolution on an extended phase space with canonical coordinates including second-order moments (from [2,3]).
Invoked in Section 2; the present paper takes this equivalence as given and does not re-derive it.
ad hoc to paper Quantum proper time is given by Delta tau = integral sqrt(1 - 2<H-hat>/(mc^2)) d tau.
Eq. (6) is a postulate; it is the foundation of all time-dilation and Finsler claims.
domain assumption Truncation to second-order moments; higher-order moments are neglected for semiclassical states.
Stated in Section 2; higher moments would contribute higher-order corrections in hbar.
standard math The relation between C1, C2 and symplectic eigenvalues nu_plus, nu_minus, and the purity formula mu = hbar^2/4/(nu_plus*nu_minus), are taken from the continuous-variable literature [5].
Used in Section 3 to connect conserved quantities to entropy and purity.
domain assumption A non-quadratic proper-time Hamiltonian corresponds to Finsler geometry.
Section 4; the paper cites [6-9] for this correspondence, but does not construct a Finsler metric explicitly.

invented entities (2)

correlation parameter alpha
purpose: Canonical variable representing a specific combination of second-order moments; claimed to be the non-Riemannian (Finsler) direction in state space.
alpha is introduced via the canonical formalism and its physical meaning is only partially characterized (Eq. 11); no direct measurement protocol is given, making it difficult to falsify independently.
Minkowski structure on a subspace of entanglement degrees of freedom independent evidence
purpose: The quantity p_beta^2 - p_alpha^2 appears as a dynamical measure of entanglement in the logarithmic negativity formula (Eq. 10).
Logarithmic negativity is an experimentally measurable entanglement criterion, so the combination is indirectly falsifiable.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Relativistic implications of entropy and purity." pith.science (2026). https://pith.science/paper/ZGJ3ZLM7

@misc{pith2026250614705,
  author       = {Pith},
  title        = {Pith review of: Relativistic implications of entropy and purity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZGJ3ZLM7}},
  note         = {Machine review of arXiv:2506.14705}
}

read the original abstract

A quantum object is extended by virtue of uncertainty. When subjected to gravity, different parts of its wave function experience distinct local relativistic effects, leading to tidal and interference phenomena absent in the classical limit. These effects can be incorporated into a geometric extension of classical spacetime. For states quantum correlated in at least two directions, a complete description of motion requires a non-Riemannian geometry whose form is controlled by the state's entropy and purity and affects a broad range of phenomena from lab measurements to Hawking radiation. A specific implication of this framework is the appearance of quantum parameters in the time-dilation law in addition to the usual dependence on velocity and gravitational potential. The quantum-corrected time-dilation law is universal: the corrections depend solely on the external degrees of freedom and are independent of internal details of the clock mechanism.

Figures

Figures reproduced from arXiv: 2506.14705 by the authors.

**Figure 1.** Logarithmic negativity (β = π/2, C1 = 2.7~, C2 = 2~)) is a function of the Minkowski distance of (pα, pβ), as shown by hyperbolic structures. Gray regions are excluded by positivity conditions on (9) and have boundaries represented by Riemannian geometry on extended space-time, while it is Finslerian in the exteriors. The red dashed hyperbolae where p 2 β − p 2 α = 1 4 (ν 2 − − ~ 2/4)(ν 2 + − ~ 2/4) separate entang… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references · 8 canonical work pages

[4]

Balsells and M

J. Balsells and M. Bojowald, Quantum proper time: A Finsler space f rom entropy and purity, [arXiv:2503.06667]

arXiv
[1]

Peres and D

A. Peres and D. Terno, Quantum Information and Relativity Theo ry, Rev. Mod. Phys. 76 (2004) 93–123, [quant-ph/0212023]

arXiv 2004
[2]

Bayta¸ s, M

B. Bayta¸ s, M. Bojowald, and S. Crowe, Faithful realizations of semiclassical trunca- tions, Ann. Phys. 420 (2020) 168247, [arXiv:1810.12127]

arXiv 2020
[3]

Bayta¸ s, M

B. Bayta¸ s, M. Bojowald, and S. Crowe, Eﬀective potentials from canonical realizations of semiclassical truncations, Phys. Rev. A 99 (2019) 042114, [arXiv:1811.00505]

arXiv 2019
[5]

Seraﬁni, Quantum continuous variables , CRC Press, London, England, 2021

A. Seraﬁni, Quantum continuous variables , CRC Press, London, England, 2021

work page 2021
[6]

Girelli, S

F. Girelli, S. Liberati, and L. Sindoni, Planck-scale modiﬁed dispersio n relations and Finsler geometry, Phys. Rev. D 75 (2007) 064015, [gr-qc/0611024]

arXiv 2007
[7]

Raetzel, S

D. Raetzel, S. Rivera, and F. P. Schuller, Geometry of physical d ispersion relations, Phys. Rev. D 83 (2011) 044047, [arXiv:1010.1369]

arXiv 2011
[8]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys. Rev. D 85 (2012) 064009, [arXiv:1112.5641]

arXiv 2012

Show all 26 references

[9]

Hohmann, C

M. Hohmann, C. Pfeifer, and N. Voicu, Finsler gravity action from variational com- pletion, Phys. Rev. D 100 (2019) 064035, [arXiv:1812.11161]

2019 arXiv
[10]

Amelino-Camelia, C

G. Amelino-Camelia, C. L¨ ammerzahl, F. Mercati, and G. M. Tino, C onstraining the Energy-Momentum Dispersion Relation with Planck-Scale Sensitiv ity Using Cold Atoms, Phys. Rev. Lett. 103 (2009) 171302, [arXiv:0911.1020] 9

2009 arXiv
[11]

M. Zych, F. Costa, I. Pikovski, and C. Brukner, Quantum inter ferometric visibil- ity as a witness of general relativistic proper time, Nat. Commun. 2 (2011) 505, [arXiv:1105.4531]

2011 arXiv
[12]

M. Zych, F. Costa, I. Pikovski, T. C. Ralph, and C. Brukner, Ge neral relativistic eﬀects in quantum interference of photons, Class. Quantum Grav. 29 (2012) 224010, [arXiv:1206.0965]

2012 arXiv
[13]

A. R. H. Smith and M. Ahmadi, Quantizing time: Interacting clocks and systems, Quantum 3 (2019) 160, [arXiv:1712.00081]

2019 arXiv
[14]

A. R. H. Smith and M. Ahmadi, Quantum clocks observe classical a nd quantum time dilation, Nat. Commun. 11 (2020) 5360, [arXiv:1904.12390]

2020 arXiv
[15]

J. F. Donoghue, Leading Quantum Correction to the Newtonian Potential, Phys. Rev. Lett. 72 (1994) 2996–2999, [gr-qc/9310024]

1994 arXiv
[16]

J. F. Donoghue, General relativity as an eﬀective ﬁeld theory: The leading quantum corrections, Phys. Rev. D 50 (1994) 3874–3888, [gr-qc/9405057]

1994 arXiv
[17]

C. P. Burgess, Quantum Gravity in Everyday Life: General Rela tivity as an Eﬀective Field Theory, Living Rev. Relativity 7 (2004) 5, [gr-qc/0311082], http://www.livingreviews.org/lrr-2004-5

2004 arXiv
[18]

I. J. Muzinich and S. Vokos, Long Range Forces in Quantum Grav ity, Phys. Rev. D 52 (1995) 3472–3483, [hep-th/9501083]

1995 arXiv
[19]

H. W. Hamber and S. Liu, On the Quantum Corrections to the New tonian Potential, Phys. Lett. B 357 (1995) 51–56, [hep-th/9505182]

1995 arXiv
[20]

Akhundov, S

A. Akhundov, S. Bellucci, and A. Shiekh, Gravitational interact ion to one loop in eﬀective quantum gravity, Phys. Lett. B 395 (1997) 16–23, [gr-qc/9611018]

1997 arXiv
[21]

N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Qu antum Gravitational Corrections to the Nonrelativistic Scattering Potential of Two Mas ses, Phys. Rev. D 67 (2003) 084033, [hep-th/0211072]

2003 arXiv
[22]

N. E. J. Bjerrum-Bohr, J. F. Donoghue, and B. R. Holstein, Qu antum Corrections to the Schwarzschild and Kerr Metrics, Phys. Rev. D 68 (2003) 084005, [hep-th/0211071]

2003 arXiv
[23]

Faller, Eﬀective Field Theory of Gravity: Leading Quantum Gra vitational Corrections to Newtons and Coulombs Law, Phys

S. Faller, Eﬀective Field Theory of Gravity: Leading Quantum Gra vitational Corrections to Newtons and Coulombs Law, Phys. Rev. D 77 (2008) 124039, [arXiv:0708.1701] 10 A Canonical moments Complete set of second-order moments in terms of canonical varia bles [2, 3]: ∆(x2

2008 arXiv
[24]

= s2 2 (15) ∆(x1p1) = s1ps1 , ∆(x2p2) = s2ps2 (16) ∆(x1x2) = s1s2 cos β (17) ∆(x1p2) = s1ps2 cos(β) − sin(β) s1 s2 (pα + pβ) (18) ∆(x2p1) = s2ps1 cos(β) + sin(β) s2 s1 (pα − pβ) (19) ∆(p2
[25]

= p2 s1 + U1 s2 1 , ∆(p2
[26]

= p2 s2 + U2 s2 2 (20) (21) where U1 = (pα − pβ)2 + 1 2 sin2 (β) ( C 2 1 − 4p2 α − √ C 4 2 − C 4 1 + (C 2 1 − 4p2 α)2 sin (α + β) ) (22) and U2 = (pα + pβ)2 + 1 2 sin2(β) ( C 2 1 − 4p2 α − √ C 4 2 − C 4 1 + (C 2 1 − 4p2 α)2 sin(α − β) ) . (23) Finally, ∆(p1p2) = ps1ps2 cos(β) ...

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [4]

Balsells and M

J. Balsells and M. Bojowald, Quantum proper time: A Finsler space f rom entropy and purity, [arXiv:2503.06667]

arXiv

[2] [1]

Peres and D

A. Peres and D. Terno, Quantum Information and Relativity Theo ry, Rev. Mod. Phys. 76 (2004) 93–123, [quant-ph/0212023]

arXiv 2004

[3] [2]

Bayta¸ s, M

B. Bayta¸ s, M. Bojowald, and S. Crowe, Faithful realizations of semiclassical trunca- tions, Ann. Phys. 420 (2020) 168247, [arXiv:1810.12127]

arXiv 2020

[4] [3]

Bayta¸ s, M

B. Bayta¸ s, M. Bojowald, and S. Crowe, Eﬀective potentials from canonical realizations of semiclassical truncations, Phys. Rev. A 99 (2019) 042114, [arXiv:1811.00505]

arXiv 2019

[5] [5]

Seraﬁni, Quantum continuous variables , CRC Press, London, England, 2021

A. Seraﬁni, Quantum continuous variables , CRC Press, London, England, 2021

work page 2021

[6] [6]

Girelli, S

F. Girelli, S. Liberati, and L. Sindoni, Planck-scale modiﬁed dispersio n relations and Finsler geometry, Phys. Rev. D 75 (2007) 064015, [gr-qc/0611024]

arXiv 2007

[7] [7]

Raetzel, S

D. Raetzel, S. Rivera, and F. P. Schuller, Geometry of physical d ispersion relations, Phys. Rev. D 83 (2011) 044047, [arXiv:1010.1369]

arXiv 2011

[8] [8]

Pfeifer and M

C. Pfeifer and M. N. R. Wohlfarth, Finsler geometric extension of Einstein gravity, Phys. Rev. D 85 (2012) 064009, [arXiv:1112.5641]

arXiv 2012