A Yang-Mills-Dirac Quantum Field Theory Emerging From a Dirac Operator on a Configuration Space

Jesper M. Grimstrup; Johannes Aastrup

REVIEW 4 major objections 4 minor 23 references

A Yang-Mills-Dirac Quantum Field Theory Emerging From a Dirac Operator on a Configuration Space

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read A twisted inner fluctuation of a Dirac operator on the configuration space of SU(2) gauge connections yields, when squared, the Hamiltonian of Yang-Mills quantum field theory together with a fermionic Dirac Hamiltonian.

desk verdict A plausible but formally incomplete extension of the spectral-geometry program: the twisted fluctuation is new, but the uncomputed remainder term makes the central identity an assertion rather than a verified result. read the letter →

arxiv 2501.00005 v1 pith:XO33LQZE submitted 2024-11-22 hep-th math-phmath.MP

Johannes Aastrup , Jesper M. Grimstrup This is my paper · ORCID

classification hep-thmath-phmath.MP MSC 81T1381T7558B34

keywords Yang-MillsquantumfieldtheoryDiracoperatorconfigurationspaceinnerfluctuationsChern-SimonstermnoncommutativegeometryfermionicHamiltonianSU(2)gaugeconnections

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 tries to show that a single geometric object—a Dirac operator on the space of SU(2) gauge connections—can, through a twisted inner fluctuation, encode both bosonic and fermionic quantum field theory. The authors exhibit a unitary element built from the Chern-Simons term and conjugate the Dirac operator with it in a twisted way; the square of the resulting operator equals the Yang-Mills Hamiltonian plus a fermionic Hamiltonian. In a first form the fermions are one-forms on the three-dimensional manifold. If a metric and triad exist, a change of basis rewrites the fermionic Hamiltonian as a Dirac Hamiltonian for Lie-algebra-valued fermions that are no longer one-forms. The point is to carry the noncommutative-geometry unification mechanism from the classical to the quantized level.

What carries the argument

The carrying object is the Dirac operator $D = \mathrm{diag}(D_1,D_2)$ on the Hilbert space $L^2(F) \oplus L^2(F)$ tensored with the exterior algebra of $\Omega^1(M,S\oplus S)$, where $D_1 = \sum_i \bar c(\psi_i)\nabla_{\xi_i}$ and $D_2 = \sum_i \bar c(i\psi_i)\nabla_{\xi_i}$. The twisted fluctuation $\tilde D = D + \gamma u[D,u^{-1}]\gamma^{-1}$ with the Chern-Simons phase $u$ and the twist through $\gamma$ interchanges Clifford elements via $c(i\psi) = i\bar c(\psi)$, which turns the second functional derivative $\partial^2 \mathrm{CS}/\partial x_i \partial x_j$ into the fermionic Hamiltonian. The change of basis uses the triad to map the one-form basis $\{\xi_i\}$ to $\{\tilde\varphi_m M_{mi}\}$, producing the spatial Dirac operator $D_A$.

What would settle it

Compute $\tilde D^2$ on a configuration-space metric that is not locally flat, keeping the nonvanishing commutator $[\partial/\partial\xi_i, \bar c(\psi_j)]$; if the residual term $\Xi$ does not assemble into $H_{\mathrm{YM}}+H_{\mathrm{fermionic}}$, the identification fails. Alternatively, check whether the kernel of $\tilde D$ can be represented in $L^2(F)$; the paper notes the kernel is a real Chern-Simons phase, so a representation exists only if the metric counterbalances it.

Watch

Extended reading notes

Core claim

The central claim is the identity $\tilde D^2 = \mathrm{diag}(H_{\mathrm{YM}} + H_{\mathrm{fermionic}}, H_{\mathrm{YM}} + H_{\mathrm{fermionic}})$, with $\tilde D = D + \gamma u[D,u^{-1}]\gamma^{-1}$, $u = \mathrm{diag}(e^{i\mathrm{CS}(A)}, e^{-i\mathrm{CS}(A)})$, and $H_{\mathrm{YM}}$ the Yang-Mills Hamiltonian obtained from squares of covariant derivatives and commutators with the Chern-Simons term. The fermionic piece $H_{\mathrm{fermionic}}$ comes from the second functional derivative of the Chern-Simons term contracted with Clifford elements; in the original basis it describes one-form fermions. Given a metric $g_{\mu\nu}=e^a_\mu e^a_\nu$ and triad $e$, a change of basis $\xi_i = \sum_m \tilde\varphi_m M_{mi}$ transforms $H_{\mathrm{fermionic}}$ into $(1/3!)\int d\mathrm{Vol}\, \mathrm{Tr}_{g_1\otimes g_2}(\Psi D_A \Psi^\dagger - \Psi^\dagger D_A \Psi) + \Xi$, with $D_A = -i\sigma^a e_a^\mu(\nabla^A_\mu + \omega_\mu)$ a spatial Dirac operator and $\Psi$ Lie-algebra-valued fermionic fields obeying canonical anti-commutation relations in the local limit. The authors state this as: quantized Yang-Mills-Dirac theory emerges from the square of a fluctuated Dirac operator on a configuration space.

Load-bearing premise

The computation presupposes that the gauge-fixed configuration space F carries a well-defined metric and Dirac operator on the BRST Hilbert space $L^{2}$(F), and the paper explicitly leaves the rigorous Hilbert-space representation of the fluctuated operator $\tilde D$ open.

Editorial extensions

If this is right

If the identity $\tilde D^2 = H_{\mathrm{YM}} + H_{\mathrm{fermionic}}$ holds as an operator statement, then Yang-Mills and Dirac Hamiltonians are not separate inputs but both derive from one Dirac operator on configuration space.
The fermionic fields obtained after the change of basis take values in the Lie algebra of SU(2) and obey canonical anti-commutation relations in the local limit, so a quantized fermionic field on a curved background emerges without being inserted by hand.
The construction makes the second functional derivative of the Chern-Simons term the source of the fermionic Hamiltonian, linking fermions to the geometry of the configuration space.
Because the metric on configuration space encodes information about the underlying three-manifold, the same mechanism also carries information about gravity.

Reading between the lines

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

If the Hilbert-space representation issue is resolved, the same twisted-fluctuation pattern could be probed for gauge groups beyond SU(2); the Clifford embedding used here is SU(2)-specific, so a concrete test would be to see whether an analogous embedding exists for SU(3).
The distinction between the complex-i twist and the real structure suggests there may be two inequivalent ways for spinor structure to enter the configuration-space Dirac operator, and comparing the two could decide which one is physically realized.
The need for a metric and triad to obtain the Dirac form suggests the fermionic Hamiltonian is background-dependent; one could test whether varying the triad changes the fermionic spectrum covariantly, as in a gravitational background.

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.

Desk Editor's Note

read the letter

You should know two things about this paper. First, the twisted inner fluctuation (6) with gamma u [D,u^{-1}] gamma^{-1} is genuinely new relative to the authors' earlier work, and it produces a diagonal square with a fermionic term rather than only self-dual/anti-self-dual sectors. Second, the central identity D-tilde^2 = H_YM + H_fermionic is not actually established, because the correction term Xi is never evaluated.

The paper does some things well. The algebraic core is assessable: the definitions of the Dirac operator, the unitary with the Chern-Simons phase, and the twist are explicit, and the computation that leads to the claimed form of D-tilde^2 is plausible. The change of basis in Section 5 is also original and gives a Dirac Hamiltonian for fermions that are no longer one-forms. The authors are honest about what they have not done: they explicitly say they ignore gauge-fixing issues and have not checked whether the fluctuated Dirac operator admits a rigorous Hilbert space representation. That candor is worth respecting.

The soft spot is load-bearing. Section 4.1 defines Xi as 'an additional term due to (3)' and never computes it. The paper needs Xi to vanish or be harmless to get exactly H_YM + H_fermionic, but the only situation where it is clearly absent is the trivial-geometry assumption nabla_xi = partial/partial xi. That same assumption is used to recognize H_YM as the Yang-Mills Hamiltonian. However, the Dirac operator (4) was constructed with a nontrivial, A-dependent metric, and Eq. (3) says the Clifford basis is A-dependent. So the paper faces a dilemma: take the nontrivial metric seriously and Xi is expected nonzero, potentially modifying the fermionic term; take the trivial geometry and the construction of the Dirac operator loses its metric input. No consistent set of assumptions is supplied under which the central identity holds. The stress-test note is correct on this point.

Secondary concerns are real but less damaging. The fermionic anti-commutation relations are non-local and reduce to canonical ones only in a flat local limit. The Hilbert space representation of the fluctuated Dirac operator remains unaddressed. The citation pattern is heavily self-referential, but that is reasonable for a continuation of a long program; the earlier papers are the actual scaffolding.

Who gets value from this? Researchers working on noncommutative geometry approaches to quantum field theory and quantum gravity, and anyone interested in whether Yang-Mills-Dirac systems can emerge from configuration-space Dirac operators. It is not a closed result; it is a progress report with a new computation and an honest list of open items.

I would send it to a serious referee. The new step deserves referee time, and a good referee could either compute Xi or push the authors to weaken the claim to a formal identity under trivial geometry. I would not cite it in my own work until the remainder term is understood. For a reading group, maybe; it would spark a good discussion about how much formal computation counts as evidence.

Referee Report

4 major / 4 minor

Summary. The paper proposes a spectral-geometric construction in which a twisted inner fluctuation of a Dirac operator on the configuration space of SU(2) connections has a square equal to diag(H_YM + H_fermionic, H_YM + H_fermionic), where H_YM is identified with the Yang-Mills Hamiltonian and H_fermionic with a fermionic Hamiltonian. It then argues that a triad-based change of basis of the configuration space turns the fermionic sector, initially composed of one-form fermions, into Lie-algebra-valued fermions governed by a Dirac-type Hamiltonian. The computation is entirely formal: it is carried out on a gauge-fixed configuration space F with a BRST Hilbert space from the authors' earlier work, using an A-dependent metric and Clifford basis, and it leaves an unspecified correction term Xi in the central operator identity.

Significance. If the central identity were fully established, this would be a striking constructive route from a Dirac operator on a configuration space to a quantized Yang-Mills-Dirac system, with no fitted parameters and with the fermionic sector arising from the second functional derivative of the Chern-Simons term. The algebraic mechanism involving the twist by diag(exp(iCS), exp(-iCS)) and the Clifford relation c(iψ)=i\bar c(ψ) is interesting and potentially valuable. The paper is also commendably explicit about several limitations: the gauge-fixing issues are deferred to [7], the Hilbert-space representation of the fluctuated Dirac operator is not checked, and the correction term Xi is not evaluated. These gaps are not merely cosmetic, however; they affect the verification of the paper's main claim, so the present version is not yet conclusive.

major comments (4)

[Section 4.1] The correction term Xi, introduced in the definition of H_fermionic as "an additional term due to (3)", is never computed, bounded, or shown to vanish. Since equation (3) explicitly states that the Clifford basis elements have nonvanishing commutators with the derivatives ∂/∂ξ_i because the inner product on Ω¹(M,S⊕S) depends on A, the displayed identity D̃² = diag(H_YM + H_fermionic, H_YM + H_fermionic) is not an established operator identity. It is an assertion with an unknown remainder. The paper must either compute Xi under the stated assumptions or prove that it cancels before the central claim can be accepted.
[Section 4.1] The identification of H_YM as the Yang-Mills Hamiltonian is made under the assumption of trivial geometry on F, i.e. ∇_{ξ_i} = ∂/∂ξ_i, but the Dirac operator (4) is defined using an A-dependent inner product, and equation (3) records the nonzero commutators that this dependence produces. The paper does not present a consistent regime in which both the Dirac operator is genuinely nontrivial and Xi is absent or harmless. If the geometry is taken to be trivial, the connection dependence of the Clifford basis and the configuration-space metric that motivated the construction are lost; if it is taken to be nontrivial, Xi is expected to be nonzero and may modify H_fermionic. This tension bears directly on the central identity.
[Section 4.1, equations (7)-(8)] The rewriting of H_fermionic as 2∫ Tr(Φ∇_AΦ† - Φ†∇_AΦ) + Xi and the claim that the fields (Φ,Φ†) satisfy canonical anticommutation relations rely on the statement that the integral kernel ∑_i ξ_i(x)ξ_i(y) is a Dirac delta function in the local and flat limits. This is only an approximation, and the paper does not quantify the corrections or state the precise limiting procedure. Consequently the advertised interpretation as a quantum field theory of fermions on M is not yet a rigorous statement, and the nonlocal corrections to the CAR could affect the physics of the model.
[Section 2 and final paragraph of Section 6] The whole computation is performed on a gauge-fixed configuration space F using a BRST-constructed Hilbert space from [7], and the paper explicitly states that it ignores all issues emerging from the gauge fixing and that it has not checked whether the fluctuated Dirac operator (6) admits a rigorous Hilbert-space representation. The final paragraph notes that the kernel of (6) is a real phase involving the Chern-Simons term and that consistency must be restored by the metric on the configuration space. Because the square-of-Dirac-operator result is only a formal algebraic identity unless D̃ is a well-defined self-adjoint operator on a Hilbert space, this acknowledged gap concerns a load-bearing premise of the paper.

minor comments (4)

[Section 5, equation (9)] The derivation of equation (9) is described only as "A simple computation"; since this equation is the bridge from one-form fermions to the Dirac Hamiltonian for Lie-algebra-valued fermions, the computation should be included or a detailed reference provided.
[Section 2] The gauge-fixing issue is mentioned but deferred to [7]; given that the Hilbert space L²(F) and the Dirac operator depend on this choice, a brief statement of which properties of the construction are gauge-fixing independent would help the reader assess the robustness of the result.
[Throughout] There are several typos and infelicities, for example "similar to the to the Kodama ground state" in Section 6, and "with a complex ’i’" should be "with the complex number i". A careful proofreading pass is needed.
[Section 5, footnote 6] The role of the two copies g1 and g2 of su(2) is clear locally, but a more explicit statement about why g2 can in principle be a different Lie group while g1 must be su(2) would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central operator identity is a direct computation from explicit definitions, with the uncomputed term and formal Hilbert-space issue being correctness caveats rather than circular reductions.

full rationale

The paper's central claim, D̃² = diag(H_YM + H_fermionic, H_YM + H_fermionic), is presented as a 'straightforward computation' from explicit definitions: the Dirac operator (4), the unitary u built from the Chern-Simons term, and the twist γ. No data are fitted, no parameter is tuned to an output, and the physical identification of H_YM and H_fermionic is made through standard identities such as ∂CS/∂ξ_i = 2∫ Tr(ξ_i ∧ F(A)), not by defining the input in terms of the output. The paper does rely on the authors' prior work [7] for the configuration-space metric, Hilbert space, and gauge-fixing construction, and on [15,16] for the Chern-Simons unitary ansatz; this is a structural premise and a self-citation chain, but the result is not imported from those papers — it is recomputed here. The main caveats are explicitly disclosed in the paper: footnote 3 says the gauge-fixing issues are ignored, Section 4.1 leaves the term Ξ unevaluated, and Section 6 admits that a rigorous Hilbert-space representation of the fluctuated Dirac operator has not been checked. These are formal or rigor gaps that make the derivation conditional, but they do not make the derivation circular, because no step reduces the claimed conclusion to its own assumptions by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numbers are fitted to data; the construction has no free parameters in the usual sense. The freedom enters through structural choices: gauge group SU(2), the twisted fluctuation with the complex i, the Chern-Simons phase u, and the assumption of a trivial geometry on the configuration space. These choices are listed as axioms rather than fitted parameters. No new physical particles, forces, or conserved quantities are proposed; the fermionic operator-valued fields Phi and Psi are constructed objects rather than independently postulated entities.

assumptions (7)

domain assumption A fibered metric on the configuration space A exists, A maps to <.,.>_A, and admits a Dirac operator and Hilbert space as constructed in [7].
Section 2 invokes [7] for the metric and L^2(A); the existence of this structure is not proved in this paper.
domain assumption The BRST gauge-fixed Hilbert space L^2(F) correctly represents the configuration space and all BRST issues can be ignored in the computation.
Section 2 states 'we shall work with F instead of A and ignore all issues', and footnote 3 in Section 4 defers to [7]; the paper does not re-derive the BRST quantization.
domain assumption For G=SU(2), the embedding chi from Omega^1(M,g) to Omega^1(M,S tensor S) of [15] exists and is independent of the chosen spinors under the stated conditions.
Section 2 invokes this embedding to replace Lie-algebra-valued one-forms by spinor-valued one-forms; it is necessary for half-integer spin.
ad hoc to paper The geometry of the configuration space is trivial, gradient_{xi_i} = partial_{xi_i}, when identifying H_YM as the Yang-Mills Hamiltonian.
Section 4.1 states 'If we assume that we have a trivial geometry on F, i.e. gradient_{xi_i}=partial_{xi_i}' and then recognizes H_YM; this simplifying assumption is needed for the QFT interpretation.
domain assumption A metric g and triad field e exist on the underlying three-manifold, and the transformed spin connection (omega_mu)^b_a = e^nu_a partial_mu e^b_nu, properly symmetrized, produces a spatial Dirac operator.
Section 5 introduces g, e, and omega to change basis and obtain equation (10); the paper notes that omega must be symmetrized to be a spin connection.
ad hoc to paper The operator product kernels sum_i xi_i(x) xi_i(y) and sum_m phi_m(x) phi_m(y) are proportional to delta functions in the local flat and L^2-norm limits.
These limits are used to claim canonical anti-commutation relations (8) and (11); the paper states they are non-local relations that become delta functions only in a limit.
domain assumption The operator D_A = -i sigma^a e_a^mu (gradient_A,mu + omega_mu) is the spatial Dirac operator, with the understanding that the paper works on a dense domain and ignores the full domain of the operator.
Footnote 7 in Section 5 restricts the statement to expectation values and notes that the full domain would require expanding the g1 part to two-by-two complex matrices.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Yang-Mills-Dirac Quantum Field Theory Emerging From a Dirac Operator on a Configuration Space." pith.science (2026). https://pith.science/paper/XO33LQZE

@misc{pith2026250100005,
  author       = {Pith},
  title        = {Pith review of: A Yang-Mills-Dirac Quantum Field Theory Emerging From a Dirac Operator on a Configuration Space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XO33LQZE}},
  note         = {Machine review of arXiv:2501.00005}
}

abstract

Starting with a Dirac operator on a configuration space of $SU(2)$ gauge connections we consider its fluctuations with inner automorphisms. We show that a certain type of twisted inner fluctuations leads to a Dirac operator whose square gives the Hamiltonian of Yang-Mills quantum field theory coupled to a fermionic sector that consist of one-form fermions. We then show that if a metric exists on the underlying three-dimensional manifold then there exists a change of basis of the configuration space for which the transformed fermionic sector consists of fermions that are no-longer one-forms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[7]

Constructing spectral t riples over holonomy-diﬀeomorphisms and the problem of reconciling gen eral rel- ativity with quantum ﬁeld theory,

J. Aastrup and J. M. Grimstrup, “Constructing spectral t riples over holonomy-diﬀeomorphisms and the problem of reconciling gen eral rel- ativity with quantum ﬁeld theory,” (2023) [arXiv:2309.063 74]. 12

work page 2023
[1]

Gravity coupled with matter and the foundati on of non- commutative geometry,

A. Connes, “Gravity coupled with matter and the foundati on of non- commutative geometry,” Commun. Math. Phys. 182 (1996) 155

work page 1996
[2]

The Spectral action pri nciple,

A. H. Chamseddine and A. Connes, “The Spectral action pri nciple,” Commun. Math. Phys. 186 (1997) 731

work page 1997
[3]

A universal action form ula,

A. H. Chamseddine and A. Connes, “A universal action form ula,” [arXiv:9606056]

work page
[4]

Gravity an d the standard model with neutrino mixing,

A. H. Chamseddine, A. Connes and M. Marcolli, “Gravity an d the standard model with neutrino mixing,” [arXiv:0610241]

work page
[5]

Noncommutative Geometry,

A. Connes, “Noncommutative Geometry,” Academic Press, 1994

work page 1994
[6]

Noncommutative Geometry, Qu antum Fields and Motives,

A. Connes and M. Marcolli, “Noncommutative Geometry, Qu antum Fields and Motives,” www.alainconnes.org/docs/bookwebﬁ nal.pdf, (2008)

work page 2008
[8]

Spectral triples of holo nomy loops,

J. Aastrup and J. M. Grimstrup, “Spectral triples of holo nomy loops,” Commun. Math. Phys. 264 (2006), 657-681

work page 2006

Show all 23 references

[9]

C*-algebras of Holonomy - Diﬀeomorphisms and Quantum Gravity I,

J. Aastrup and J. M. Grimstrup, “C*-algebras of Holonomy - Diﬀeomorphisms and Quantum Gravity I,” Class. Quant. Grav. 30 (2013) 085016

2013
[10]

C*-algebras of Holonom y- Diﬀeomorphisms and Quantum Gravity II

J. Aastrup and J. M. Grimstrup, “C*-algebras of Holonom y- Diﬀeomorphisms and Quantum Gravity II”, J. Geom. Phys. 99 (2016) 10

2016
[11]

Quantum Holonomy Theor y,

J. Aastrup and J. M. Grimstrup, “Quantum Holonomy Theor y,” Fortsch. Phys. 64 (2016) no.10, 783-818

2016
[12]

The quantum holonomy- diﬀeomorphism algebra and quantum gravity,

J. Aastrup and J. M. Grimstrup, “The quantum holonomy- diﬀeomorphism algebra and quantum gravity,” Int. J. Mod. Phy s. A 31 (2016) no.10, 1650048

2016
[13]

The metric nature of mat ter,

J. Aastrup and J. M. Grimstrup, “The metric nature of mat ter,” J. Geom. Phys. 171 (2022), 104408

2022
[14]

Non-perturbative Quan tum Field Theory and the Geometry of Functional Spaces,

J. Aastrup and J. M. Grimstrup, “Non-perturbative Quan tum Field Theory and the Geometry of Functional Spaces,” Fortsch. Phy s. 69 (2021) no.10, 2100106

2021
[15]

Dirac Operators on Conﬁ guration Spaces: Fermions with Half-integer Spin, Real Structure, a nd Yang- Mills Quantum Field Theory,

J. Aastrup and J. M. Grimstrup, “Dirac Operators on Conﬁ guration Spaces: Fermions with Half-integer Spin, Real Structure, a nd Yang- Mills Quantum Field Theory,” [arXiv:2410.07290]

arXiv
[16]

Dirac Operators on Conﬁ guration Spaces and Yang-Mills Quantum Field Theory,

J. Aastrup and J. M. Grimstrup, “Dirac Operators on Conﬁ guration Spaces and Yang-Mills Quantum Field Theory,” [arXiv:2410. 03699]
[17]

The Qualitative Behavior of Yang-Mills Theory in (2+1)-Dimensions,

R. P. Feynman, “The Qualitative Behavior of Yang-Mills Theory in (2+1)-Dimensions,” Nucl. Phys. B 188 (1981) 479

1981
[18]

The Geometry of the Orbit Space for Nonabe lian Gauge Theories. (Talk),

I. M. Singer, “The Geometry of the Orbit Space for Nonabe lian Gauge Theories. (Talk),” Phys. Scripta 24 (1981) 817

1981
[19]

The Metric on the space of Yang-Mills conﬁgu rations,

P. Orland, “The Metric on the space of Yang-Mills conﬁgu rations,” [arXiv:9607134]
[20]

Representations of the Quantum Holonomy-Diﬀeomorphism Algebra,

J. Aastrup and J. M. Grimstrup, “Representations of the Quantum Holonomy-Diﬀeomorphism Algebra,” [arXiv:1709.02943]. 13

arXiv
[21]

Quantization of Nonabelian Gauge Theori es,

V. N. Gribov, “Quantization of Nonabelian Gauge Theori es,” Nucl. Phys. B 139 (1978), 1

1978
[22]

Specialization of Ashtekar’s Formalism to Bianchi Cos- mology,

H. Kodama, “Specialization of Ashtekar’s Formalism to Bianchi Cos- mology,” Prog. Theor. Phys. 80 (1988) 1024

1988
[23]

Quantum gravity with a positive cosmologic al constant,

L. Smolin, “Quantum gravity with a positive cosmologic al constant,” [arXiv:0209079]. 14

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [7]

Constructing spectral t riples over holonomy-diﬀeomorphisms and the problem of reconciling gen eral rel- ativity with quantum ﬁeld theory,

J. Aastrup and J. M. Grimstrup, “Constructing spectral t riples over holonomy-diﬀeomorphisms and the problem of reconciling gen eral rel- ativity with quantum ﬁeld theory,” (2023) [arXiv:2309.063 74]. 12

work page 2023

[2] [1]

Gravity coupled with matter and the foundati on of non- commutative geometry,

A. Connes, “Gravity coupled with matter and the foundati on of non- commutative geometry,” Commun. Math. Phys. 182 (1996) 155

work page 1996

[3] [2]

The Spectral action pri nciple,

A. H. Chamseddine and A. Connes, “The Spectral action pri nciple,” Commun. Math. Phys. 186 (1997) 731

work page 1997

[4] [3]

A universal action form ula,

A. H. Chamseddine and A. Connes, “A universal action form ula,” [arXiv:9606056]

work page

[5] [4]

Gravity an d the standard model with neutrino mixing,

A. H. Chamseddine, A. Connes and M. Marcolli, “Gravity an d the standard model with neutrino mixing,” [arXiv:0610241]

work page

[6] [5]

Noncommutative Geometry,

A. Connes, “Noncommutative Geometry,” Academic Press, 1994

work page 1994

[7] [6]

Noncommutative Geometry, Qu antum Fields and Motives,

A. Connes and M. Marcolli, “Noncommutative Geometry, Qu antum Fields and Motives,” www.alainconnes.org/docs/bookwebﬁ nal.pdf, (2008)

work page 2008

[8] [8]

Spectral triples of holo nomy loops,

J. Aastrup and J. M. Grimstrup, “Spectral triples of holo nomy loops,” Commun. Math. Phys. 264 (2006), 657-681

work page 2006