Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

The Cosmological Constant from a Quantum Gravitational $\theta$-Vacua and the Gravitational Hall Effect

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The cosmological constant in general relativity is claimed to be topologically protected, tied to a theta parameter by θ = 12π²/(Λℓ_Pl²) mod 2π.

desk verdict A clean derivation of the Euclidean CSK theta–Lambda relation, but the topological-protection claim for Lorentzian GR goes beyond what is shown. read the letter →

arxiv 2506.14886 v1 pith:REYJZZUV submitted 2025-06-17 gr-qc astro-ph.COcond-mat.mes-hallhep-thquant-ph

Stephon Alexander , Heliudson Bernardo , Aaron Hui This is my paper
classification gr-qcastro-ph.COcond-mat.mes-hallhep-thquant-ph
keywords cosmologicalconstantthetavacuumChern-Simons-KodamastatequantumHalleffectWheeler-DeWittequationAshtekarvariablestopologicalprotection
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

This paper proposes that the cosmological constant $\Lambda$ is not a free parameter but is fixed by a topological angle $\theta$ through $\theta = 12\pi^2/(\Lambda \ell_{\mathrm{Pl}}^2) \mod 2\pi$. The claim comes from demanding that the Chern-Simons-Kodama state, an exact non-perturbative solution of the Wheeler-DeWitt equation, transform consistently under large gauge transformations. In this picture $\Lambda$ is protected against perturbative graviton loop corrections, just as the quantized Hall conductance is immune to disorder. The authors also show that the Euclidean CSK state's probability current looks exactly like a Hall current, with $\Lambda$ acting as a gravitational Hall resistivity. If right, this would recast the cosmological constant problem as a topological selection problem rather than a radiative stability problem.

What carries the argument

The Chern-Simons-Kodama (CSK) state, $\Psi[A] = N \exp\!\bigl(3i\,\mathrm{CS}[A]/(2\Lambda \ell_{\mathrm{Pl}}^2)\bigr)$ for the Euclidean ($\beta=-1$) self-dual connection, is the load-bearing object. It is an exact solution to all GR constraints in the Ashtekar connection variables. Under a large gauge transformation of winding number $n$, the Chern-Simons functional $\mathrm{CS}[A]$ shifts by $8\pi^2 n$, so the state transforms by a phase $e^{i 12\pi^2 n/(\Lambda \ell_{\mathrm{Pl}}^2)}$; matching this with the $\theta$-sector phase $e^{i\theta n}$ yields the quantization relation $\theta = 12\pi^2/(\Lambda \ell_{\mathrm{Pl}}^2) \mod 2\pi$. The same state, through its probability current, realizes a gravitational Hall current with conductance $3/(2\Lambda \ell_{\mathrm{Pl}}^2)$.

What would settle it

Compute the one-loop graviton correction to the cosmological constant in the CSK background: any non-vanishing shift that moves $\Lambda$ off the values satisfying $6\pi/(\Lambda \ell_{\mathrm{Pl}}^2)\in\mathbb{Z}$ would falsify the topological-protection claim. Alternatively, test whether a physical Lorentzian state can be assigned a phase $e^{i\theta w(g)}$ under large gauge transformations consistent with the constraint.

Watch

Extended reading notes

Core claim

The central discovery is the constraint $\theta = 12\pi^2/(\Lambda \ell_{\mathrm{Pl}}^2) \mod 2\pi$, derived from the transformation of the CSK state under large $SU(2)$ gauge transformations. Because the Chern-Simons functional shifts by $8\pi^2 n$ under winding-$n$ gauge transformations, the state acquires a phase $e^{i 12\pi^2 n/(\Lambda \ell_{\mathrm{Pl}}^2)}$; consistency with the $\theta$-sector rule $\Psi^g = e^{i\theta w(g)}\Psi$ then fixes $\theta$ in terms of $\Lambda$. Consequently the superselection of $\theta$ quantizes $1/\Lambda$, and fixing a CP-preserving sector ($\theta=\pi$) gives discrete values of $\Lambda$. The paper further shows that in the Euclidean case the conserved probability current of the CSK state is a Hall-type current with conductance $3/(2\Lambda \ell_{\mathrm{Pl}}^2)$, so the cosmological constant plays the role of a quantum gravitational Hall resistivity. The authors argue that this topological protection makes $\Lambda$ immune to perturbative graviton loop corrections, paralleling the non-renormalization of $\theta$ in QCD.

Load-bearing premise

The argument assumes the Euclidean Chern-Simons-Kodama state ($\beta=-1$) is the physically relevant quantum state of gravity, and that a discrete $\theta$ sector is selected by external input; the bridge to Lorentzian spacetime is not established.

Editorial extensions

If this is right

  • The value of $\theta$ and $\Lambda$ are locked by $\theta = 12\pi^2/(\Lambda \ell_{\mathrm{Pl}}^2) \mod 2\pi$, so measuring one determines the other.
  • Perturbative graviton loop corrections to $\Lambda$ would be renormalization-free, removing the perturbative UV part of the cosmological constant problem.
  • Fixing the CP-preserving sector $\theta=\pi$ yields discrete allowed values $\Lambda = 12\pi/(\ell_{\mathrm{Pl}}^2(1+2n))$.
  • The probability current of the Euclidean CSK state is a gravitational Hall current with quantized conductance $3/(2\Lambda \ell_{\mathrm{Pl}}^2)$ when $\theta$ is fixed.
  • The large-gauge-invariance issue of the CSK state raised in earlier work is resolved by assigning the state to a definite $\theta$-sector.

Reading between the lines

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

  • If the relation survives, the cosmological constant becomes a discrete superselection label; measuring $\Lambda$ would reveal which topological vacuum we inhabit, and parity could be observably violated if $\theta\neq 0,\pi$.
  • The Hall analogy suggests a gravitational analogue of topological insulators: regions with different $\theta$ values would be separated by boundary currents, a direction the paper mentions but does not develop.
  • Promoting $\theta$ to a dynamical axion-like field would turn the relation into a potential for $\Lambda$, potentially connecting this picture to relaxation or axion models of the cosmological constant.
  • Because the derivation uses the Euclidean signature, extending the quantization condition to the Lorentzian theory requires resolving the reality conditions on complex Ashtekar connections, which the paper leaves open.
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. The paper derives a relation between the cosmological constant and a gravitational theta parameter, θ = 12π²/(Λℓ_Pl²) mod 2π, by demanding that the Chern-Simons-Kodama (CSK) state transform covariantly under large SU(2) gauge transformations. It then argues that this relation implies Λ is topologically protected against perturbative graviton loop corrections, in analogy with the quantized Hall conductance. The authors also propose a gravitational analogue of the Hall effect in which the Hamiltonian constraint generates a current proportional to 3/(2Λℓ_Pl²) times the curvature, and they use this to suggest a quantization of the gravitational Hall conductance. The derivation is carried out in the canonical Ashtekar-variable formalism and relies on the Euclidean (β = −1) CSK state.

Significance. If the central claim could be established for Lorentzian-signature general relativity, the relation between Λ and θ would be a striking non-perturbative result with potential implications for the cosmological constant problem. The manuscript is commendably explicit: the algebraic steps from the CSK state to Eq. (14) are clear, the relation to earlier concerns about the CSK state (Refs. [25,35]) is acknowledged, and the authors are transparent about some limitations, such as the vanishing probability current and complex Hall current in the Lorentzian case. Nevertheless, the main physical conclusion is currently supported only in the Euclidean sector, which substantially limits its significance as stated.

major comments (3)
  1. [§II.B, Eqs. (12)–(14)] The derivation of the theta–Lambda relation uses the Euclidean CSK state (β = −1, Eq. (A20)). For the Lorentzian CSK state (β = i, Eq. (A18)), a large gauge transformation changes the state by the real factor exp[−12π²n/(Λℓ_Pl²)], not by a phase e^{iθ n}. Since the theta-sector transformation (10) is defined by a pure phase, Eq. (14) cannot be obtained for β = i. The manuscript does not supply a real-section argument or a physical inner product showing that the Euclidean state defines the Lorentzian vacuum; it even notes, in Section III, that the Lorentzian CSK probability current vanishes and the Hall current becomes complex. Therefore the central claim that Λ in general relativity is topologically protected is not established as stated.
  2. [§II.B, last paragraph; §IV] The statement that consistency of the CSK state with perturbative quantization implies that Λ is robust to graviton loop corrections is an inference, not a derivation. The paper does not show that perturbative corrections preserve the θ-sector, nor that the exact CSK state is the vacuum selected by the full quantum theory (including a physical inner product). Without such a demonstration, the sentence 'there is no point in computing perturbative corrections' overreaches; at most the paper establishes a property of an exact solution in the Euclidean sector, not a general non-renormalization theorem for Λ.
  3. [§III, Eqs. (17)–(24)] The gravitational Hall effect analogy is explicitly Euclidean-only: for β = i, the current (17) is complex and the probability current (23) of the CSK state vanishes. Consequently, the 'quantized gravitational Hall conductance' invoked in Section IV is not connected to Lorentzian-signature physics. The relation σ_H = 3/(2Λℓ_Pl²) and its quantization via θ require a Lorentzian counterpart before they can support the paper's physical conclusions.
minor comments (5)
  1. [Eq. (21)] The second term inside the brackets is identical to the first term; it should contain Ψ[A]∇_A Ψ*[A] (or the analogue of the second term in Eq. (20)), otherwise the current is identically zero by antisymmetry.
  2. [§III, first paragraph and later text] The sentence 'The results in this section do not depend on the relation between Λ and the θ-vacua (see Eq. (2))' is inconsistent with the later statement that fixing θ quantizes the gravitational Hall conductance via Eq. (2); the scope of independence should be clarified.
  3. [§I and Appendix A] The reduced Planck length ℓ_Pl appears in Eq. (2) before it is defined; the definition ℓ_Pl² = 8πGℏ in Appendix A should be introduced at first use.
  4. [Eq. (25)] For θ = π, the expression Λ = 12π/(ℓ_Pl²(1+2n)) gives negative values for n ≤ −1; the allowed range of n should be restricted (or the negative-Λ case discussed) if the cosmological constant is intended to be positive.
  5. [Acknowledgments] The name 'Friedel' appears to be a typo for 'Freidel', which is the spelling used in Ref. [19].

Circularity Check

2 steps flagged · score 6.0 of 10

The θ–Λ relation is a read-off of the CSK state's own exponent, and the claimed quantization of Λ reduces to the hand-imposed input θ=π; the central topological-protection conclusion is partly definitional.

  1. self definitional [Section II.B, Eqs. (8), (13)–(14)]
    "Since the CSK state (8) depends exponentially on CS[A], its transformation under a large gauge transformation is given by ΨCSK[A^g] = e^{i 12π²/(Λℓ²_Pl) n} ΨCSK[A]. (13) Comparing this with our θ-sector constraint in Eq. (10), this implies θ= 12π²/(Λℓ²_Pl) mod 2π. (14)"

    The phase in (13) is obtained by substituting the shift (12) into the CSK state (8), whose exponent already contains Λ as a free parameter. Eq. (10) defines θ as precisely the phase of a state under large gauge transformations, so (14) is a read-off of the input state's exponent: θ is defined through Λ, not constrained independently. Consistency alone restricts nothing, since e^{iθn} is a valid large-gauge representation for every real θ; the integrality claim that follows Eq. (14) silently restricts θ to the trivial sector, making the 'quantization' a restatement of the mod-2π period. The alleged constraint is therefore equivalent, by construction, to the Λ already present in the state.

  2. self definitional [Section IV, Eq. (25); Section III–IV discussion of quantized Hall conductance]
    "θ=π =⇒ Λ = 12π/(ℓ²_Pl(1 + 2n)). (25) for n∈Z. This is in addition to the trivial θ= 0 case [35]. ... Furthermore, for fixed θ, Eq. (2) implies that the gravitational Hall conductance is quantized."

    The discrete spectrum of Λ is obtained by inserting the extra input θ=π, justified only by CP conservation, into (14) and inverting the relation. Since (14) defines θ from the Λ already in the CSK state, the 'quantization' of Λ is the same relation run backwards: the chosen superselection value is the premise and the allowed Λ values are its algebraic image. Nothing in the WdW formalism or the θ-sector construction selects θ=π; the topological protection of Λ and the quantized gravitational Hall conductance therefore reduce by construction to the imposed input. The paper's own Section III disclaimer ('The results in this section do not depend on the relation between Λ and the θ-vacua') applies only to the Hall-current analogy, not to the quantization claim, which reuses Eq. (2).

full rationale

The derivation chain is internally consistent: Appendix A constructs the CSK state (A18) as an exact Wheeler-DeWitt solution, and Section II.B computes its large-gauge phase, identifying θ through Eq. (10). That calculation, however, is not an independent constraint on Λ. The phase coefficient 12π²/(Λℓ_Pl²) comes directly from the Λ sitting in the exponent of the input state (8), so Eq. (14) defines θ as that coefficient; consistency imposes nothing because e^{iθn} is a valid large-gauge representation for every real θ. The integrality statement 'the superselection choice of θ constrains the quantity 6π/(Λℓ_Pl²)∈Z' silently restricts θ to the trivial sector, making the 'quantization' a restatement of the mod-2π period. The discrete spectrum of Eq. (25) is then obtained only by adding the unforced input θ=π (CP conservation) and inverting the relation, so the paper's headline conclusion — that Λ is topologically protected — reduces by construction to that imposed choice rather than emerging from the formalism. A separate scope caveat is flagged here: the derivation uses the explicitly Euclidean state (8) (β=-1 in (A20)); for Lorentzian β=i the general state (A18) transforms by a real exponential, not a phase, so Eqs. (10)-(14) do not apply to Lorentzian GR. The paper acknowledges the failure only for the probability current ('Note that (23) vanishes for the Lorentzian signature CSK state, while JH in (18) is complex valued in that case'), not for Eq. (14), leaving the abstract's claim about the GR cosmological constant overreaching. On self-citation: [17]-[24] include two papers co-authored by a present author ([22], [24]), but they are background; the exact-solution status is demonstrated in Appendix A by direct computation, and the Hartle-Hawking/Vilenkin connection relies on [23] (Magueijo). No load-bearing self-citation chain is present. The Euclidean gravitational Hall-effect correspondence is a genuine structural observation and is explicitly independent of Eq. (2), but the quantized-Hall-conductance claim reuses Eq. (2) and inherits the same definitional circularity. Overall: partial circularity (score 6), because the central physical predictions reduce by construction to the Λ-dependence of the input state and the hand-imposed θ=π choice.

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

The central claim rests on the Euclidean CSK state, the equivalence of gravitational and YM large gauge transformations, and an external CP argument to discretize theta. No new particles or forces are introduced. The free parameters are the theta angle and the Euclidean signature choice, both of which are put in by hand.

free parameters (2)
  • theta (θ) superselection angle = unconstrained; chosen as θ=π for CP invariance in Sec. IV
    The theta angle is a superselection choice. The quantization of Lambda only appears when theta is assumed discrete, e.g., θ=π, which is an ad hoc input rather than a derived value.
  • beta (β) signature choice = -1 (Euclidean)
    The paper uses the Euclidean signature to obtain a pure phase transformation under large gauge transformations. The Lorentzian case (β=i) would give an exponential factor and is not discussed, so the Euclidean choice is load-bearing for the derivation.
assumptions (5)
  • standard math The Chern-Simons functional shifts by 8π²n under large gauge transformations with winding number n.
    This is a standard property of the Chern-Simons integral, used in Eq. (12) to compute the transformation of the CSK state.
  • domain assumption Large gauge transformations of the Ashtekar connection are classified by π_3(SU(2)) = Z, as in Yang-Mills theory.
    The paper cites Ashtekar, Balachandran, and Jo [29] for this equivalence. It is load-bearing for constructing gravitational theta-sectors.
  • domain assumption The CSK state is an exact solution to all GR constraints for Λ≠0.
    Taken from Kodama [17] and references therein. The paper does not re-derive this, and the physical validity of the state (including normalizability) is not addressed.
  • ad hoc to paper The Euclidean (β=-1) CSK state is the relevant state for physical predictions.
    The paper selects the Euclidean case to get a pure phase under large gauge transformations. The Lorentzian case would not produce the same structure, and the physical relevance of the Euclidean state to the real universe is not justified.
  • ad hoc to paper CP invariance requires θ=0 or θ=π.
    Used in Sec. IV to produce discrete Lambda values. This is an external principle not derived from the formalism and is the actual source of the claimed quantization.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Cosmological Constant from a Quantum Gravitational $\theta$-Vacua and the Gravitational Hall Effect." pith.science (2026). https://pith.science/paper/REYJZZUV

@misc{pith2026250614886,
  author       = {Pith},
  title        = {Pith review of: The Cosmological Constant from a Quantum Gravitational $\theta$-Vacua and the Gravitational Hall Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/REYJZZUV}},
  note         = {Machine review of arXiv:2506.14886}
}
abstract

We provide a new perspective on the cosmological constant by exploring the background-independent Wheeler-DeWitt quantization of general relativity. The Chern-Simons-Kodama state of quantum gravity, a generalization of the Hartle-Hawking and Vilenkin states, has a striking structural similarity to the topological field theory of the quantum Hall effect. As a result, we study the gravitational topological $\theta$-sectors in analogy to Yang-Mills theory. We find that the cosmological constant $\Lambda$ is intimately linked to the $\theta$-parameter by $\theta=12\pi^2/(\Lambda \ell^2_{\rm Pl}) \mod 2\pi$ due to the fact that Chern-Simons-Kodama state must live in a particular $\theta$-sector. This result is shown in the canonical, non-perturbative formalism. Furthermore, we explain how the physics of the Hamiltonian constraint is analogous to the quantum Hall effect, with the cosmological constant playing the role of a quantum gravitational Hall resistivity. These relations suggest that $\Lambda$ is topologically protected against perturbative graviton loop corrections, analogous to the robustness of quantized Hall conductance against disorder in a metal.

Discussion (0). Sign in to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Thermodynamics of black and white holes in ensemble of Planckons

    gr-qc 2025-06 conditional novelty 6.0 of 10

    A toy model counting pairs of Planckons gives integer black hole entropy, negative white hole entropy, charge-independent Reissner-Nordstrom entropy, and a quantized cosmological constant.

Reference graph

Works this paper leans on

55 extracted references · 33 canonical work pages · cited by 1 Pith paper

  1. [1]

    E. H. Hall, On a new action of the magnet on electric cur- rents, American Journal of Mathematics2, 287 (1879)

    work page

  2. [2]

    von Klitzing, G

    K. von Klitzing, G. Dorda, and M. Pepper, New method for high accuracy determination of the fine structure con- stant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980)

    work page 1980

  3. [3]

    R. B. Laughlin, Quantized Hall conductivity in two- dimensions, Phys. Rev. B23, 5632 (1981)

    work page 1981

  4. [4]

    D. C. Tsui, H. L. Stormer, and A. C. Gossard, Two- dimensional magnetotransport in the extreme quantum limit, Phys. Rev. Lett.48, 1559 (1982)

    1982

  5. [5]

    R. B. Laughlin, Anomalous quantum Hall effect: An In- compressible quantum fluid with fractionallycharged ex- citations, Phys. Rev. Lett.50, 1395 (1983)

    work page 1983

  6. [6]

    H. L. Stormer, D. C. Tsui, and A. C. Gossard, The frac- tional quantum Hall effect, Rev. Mod. Phys.71, S298 (1999)

    1999

  7. [7]

    T. J. B. M. Janssen, J. M. Williams, N. E. Fletcher, R. Goebel, A. Tzalenchuk, R. Yakimova, S. Lara-Avila, S. Kubatkin, and V. I. Fal’ko, Precision comparison of the quantum hall effect in graphene and gallium arsenide, Metrologia49, 294–306 (2012)

    work page 2012

  8. [8]

    D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B14, 2239 (1976)

    1976

Show all 55 references
  1. [9]

    D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall Conductance in a Two- Dimensional Periodic Potential, Phys. Rev. Lett.49, 405 (1982)

    1982

  2. [10]

    Bellissard, A

    J. Bellissard, A. van Elst, and H. Schulz-Baldes, The non- commutative geometry of the quantum Hall effect, Jour- nal of Mathematical Physics35, 5373–5451 (1994)

    1994

  3. [11]

    Witten, Quantum Field Theory and the Jones Poly- nomial, Commun

    E. Witten, Quantum Field Theory and the Jones Poly- nomial, Commun. Math. Phys.121, 351 (1989)

    1989

  4. [12]

    S. C. Zhang, T. H. Hansson, and S. Kivelson, Effective- field-theory model for the fractional quantum hall effect, Phys. Rev. Lett.62, 82 (1989)

    1989

  5. [13]

    Blok and X

    B. Blok and X. G. Wen, Effective theories of the frac- tional quantum hall effect: Hierarchy construction, Phys. Rev. B42, 8145 (1990)

    1990

  6. [14]

    A. P. Balachandran and A. M. Srivastava, Chern-Simons dynamics and the quantum Hall effect (1991), arXiv:hep- th/9111006

    1991

  7. [15]

    Fr¨ ohlich, A

    J. Fr¨ ohlich, A. H. Chamseddine, F. Gabbiani, T. Kerler, C. Kling, P. A. Marchetti, U. M. Studer, and E. Thi- ran, The Fractional Quantum Hall Effect, Chern-Simons Theory, and Integral Lattices, inInternational Congress of Mathematicians(1995)

    1995

  8. [16]

    Susskind, The Quantum Hall fluid and noncommuta- tive Chern-Simons theory (2001), arXiv:hep-th/0101029

    L. Susskind, The Quantum Hall fluid and noncommuta- tive Chern-Simons theory (2001), arXiv:hep-th/0101029

    2001 arXiv

  9. [17]

    Kodama, Holomorphic wave function of the universe, Phys

    H. Kodama, Holomorphic wave function of the universe, Phys. Rev. D42, 2548 (1990)

    1990

  10. [18]

    Smolin, Quantum gravity with a positive cosmological constant (2002), arXiv:hep-th/0209079

    L. Smolin, Quantum gravity with a positive cosmological constant (2002), arXiv:hep-th/0209079

    2002 arXiv

  11. [19]

    Freidel and L

    L. Freidel and L. Smolin, The Linearization of the Kodama state, Class. Quant. Grav.21, 3831 (2004), arXiv:hep-th/0310224

    2004 arXiv

  12. [20]

    Randono, Generalizing the Kodama state

    A. Randono, Generalizing the Kodama state. I. Con- struction (2006), arXiv:gr-qc/0611073

    2006 arXiv

  13. [21]

    Randono, Generalizing the Kodama state

    A. Randono, Generalizing the Kodama state. II. Prop- erties and physical interpretation (2006), arXiv:gr- qc/0611074

    2006

  14. [22]

    S. H. S. Alexander and G. Calcagni, Quantum grav- ity as a Fermi liquid, Found. Phys.38, 1148 (2008), arXiv:0807.0225 [hep-th]

    2008 arXiv

  15. [23]

    Magueijo, Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave-functions, Phys

    J. Magueijo, Equivalence of the Chern-Simons state and the Hartle-Hawking and Vilenkin wave-functions, Phys. Rev. D102, 044034 (2020), arXiv:2005.03381 [gr-qc]

    2020 arXiv

  16. [24]

    Alexander, G

    S. Alexander, G. Herczeg, and L. Freidel, An inner product for 4D quantum gravity and the Chern–Simons–Kodama state, Class. Quant. Grav.40, 145010 (2023), arXiv:2212.07446 [hep-th]

    2023 arXiv

  17. [25]

    Witten, A Note on the Chern-Simons and Kodama wave functions (2003), arXiv:gr-qc/0306083

    E. Witten, A Note on the Chern-Simons and Kodama wave functions (2003), arXiv:gr-qc/0306083

    2003 arXiv

  18. [26]

    B. S. DeWitt, Quantum theory of gravity. i. the canonical theory, Phys. Rev.160, 1113 (1967)

    1967

  19. [27]

    J. A. Wheeler, Superspace and the nature of quantum geometrodynamics, Adv. Ser. Astrophys. Cosmol.3, 27 (1987)

    1987

  20. [28]

    Jackiw, Topological investigations of quantized gauge theories, Conf

    R. Jackiw, Topological investigations of quantized gauge theories, Conf. Proc. C8306271, 221 (1983)

    1983

  21. [29]

    Ashtekar, A

    A. Ashtekar, A. P. Balachandran, and S. Jo, The CP Problem in Quantum Gravity, Int. J. Mod. Phys. A4, 1493 (1989)

    1989

  22. [30]

    Jackiw and C

    R. Jackiw and C. Rebbi, Vacuum Periodicity in a Yang- Mills Quantum Theory, Phys. Rev. Lett.37, 172 (1976)

    1976

  23. [31]

    C. G. Callan, Jr., R. F. Dashen, and D. J. Gross, Toward a Theory of the Strong Interactions, Phys. Rev. D17, 2717 (1978)

    1978

  24. [32]

    Klimek-Chudy and W

    S. Klimek-Chudy and W. Kondracki, On the theta theo- ries and the multivalued wave functions, J. Geom. Phys. 1N3, 1 (1984)

    1984

  25. [33]

    Ashtekar, New variables for classical and quantum gravity, Phys

    A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett.57, 2244 (1986)

    1986

  26. [34]

    Ashtekar, New hamiltonian formulation of general rel- ativity, Phys

    A. Ashtekar, New hamiltonian formulation of general rel- ativity, Phys. Rev. D36, 1587 (1987)

    1987

  27. [35]

    Wieland, Complex Ashtekar variables, the Kodama state and spinfoam gravity (2011), arXiv:1105.2330 [gr- qc]

    W. Wieland, Complex Ashtekar variables, the Kodama state and spinfoam gravity (2011), arXiv:1105.2330 [gr- qc]

    2011 arXiv

  28. [36]

    Marino,Instantons and large N: an introduction to non-perturbative methods in quantum field theory(Cam- bridge University Press, 2015)

    M. Marino,Instantons and large N: an introduction to non-perturbative methods in quantum field theory(Cam- bridge University Press, 2015)

    2015

  29. [37]

    Witten, Instatons, the quark model, and the 1/n ex- pansion, Nuclear Physics B149, 285 (1979)

    E. Witten, Instatons, the quark model, and the 1/n ex- pansion, Nuclear Physics B149, 285 (1979)

    1979

  30. [38]

    Vilenkin, Quantum Cosmology and the Initial State of the Universe, Phys

    A. Vilenkin, Quantum Cosmology and the Initial State of the Universe, Phys. Rev. D37, 888 (1988)

    1988

  31. [39]

    Sekine and K

    A. Sekine and K. Nomura, Axion electrodynamics in topological materials, Journal of Applied Physics129, 141101 (2021)

    2021

  32. [40]

    X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B78, 195424 (2008)

    2008

  33. [41]

    A. A. Burkov, Chiral anomaly and transport in weyl met- als, Journal of Physics: Condensed Matter27, 113201 (2015)

    2015

  34. [42]

    Wen, Topological orders and edge excitations in fractional quantum hall states, Advances in Physics44, 405 (1995), https://doi.org/10.1080/00018739500101566

    X.-G. Wen, Topological orders and edge excitations in fractional quantum hall states, Advances in Physics44, 405 (1995), https://doi.org/10.1080/00018739500101566

    1995 doi

  35. [43]

    Zee, Quantum hall fluids, inField Theory, Topology and Condensed Matter Physics, edited by H

    A. Zee, Quantum hall fluids, inField Theory, Topology and Condensed Matter Physics, edited by H. B. Geyer (Springer Berlin Heidelberg, Berlin, Heidelberg, 1995) pp. 99–153. 6

    1995

  36. [44]

    Soo, Selfdual variables, positive semidefinite action, and discrete transformations in 4-d quantum gravity, Phys

    C. Soo, Selfdual variables, positive semidefinite action, and discrete transformations in 4-d quantum gravity, Phys. Rev. D52, 3484 (1995), arXiv:gr-qc/9504042

    1995 arXiv

  37. [45]

    J. B. Hartle and S. W. Hawking, Wave Function of the Universe, Phys. Rev. D28, 2960 (1983)

    1983

  38. [46]

    Vilenkin, Approaches to quantum cosmology, Phys

    A. Vilenkin, Approaches to quantum cosmology, Phys. Rev. D50, 2581 (1994), arXiv:gr-qc/9403010

    1994 arXiv

  39. [47]

    Arnowitt, S

    R. Arnowitt, S. Deser, and C. W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev.116, 1322 (1959)

    1959

  40. [48]

    Sen, Gravity as a spin system, Phys

    A. Sen, Gravity as a spin system, Phys. Lett. B119, 89 (1982)

    1982

  41. [49]

    Ashtekar,Lectures on nonperturbative canonical grav- ity, Advanced Series In Astrophysics And Cosmology, Vol

    A. Ashtekar,Lectures on nonperturbative canonical grav- ity, Advanced Series In Astrophysics And Cosmology, Vol. 6 (World Scientific Publishing Company, 1991)

    1991

  42. [50]

    Kiefer,Quantum gravity, Vol

    C. Kiefer,Quantum gravity, Vol. 124 (Clarendon, Oxford, 2004)

    2004

  43. [51]

    Rovelli,Quantum gravity, Cambridge Monographs on Mathematical Physics (Univ

    C. Rovelli,Quantum gravity, Cambridge Monographs on Mathematical Physics (Univ. Pr., Cambridge, UK, 2004)

    2004

  44. [52]

    Thiemann,Modern Canonical Quantum General Rela- tivity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    T. Thiemann,Modern Canonical Quantum General Rela- tivity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    2007

  45. [53]

    Immirzi, Real and complex connections for canonical gravity, Class

    G. Immirzi, Real and complex connections for canonical gravity, Class. Quant. Grav.14, L177 (1997), arXiv:gr- qc/9612030

    1997

  46. [54]

    J. F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D51, 5507 (1995), arXiv:gr-qc/9410014

    1995 arXiv

  47. [55]

    dynamics

    G. Immirzi, Quantum gravity and Regge calculus, Nucl. Phys. B Proc. Suppl.57, 65 (1997), arXiv:gr-qc/9701052. Appendix A: The Gravitational Hamiltonian with a Cosmological Constant To make the discussion in this letter self-contained, we review the equivalence between the Arno...

    1997 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.