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Dirac operators and local invariants on perturbations of Minkowski space

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Pith's one-line read This paper proves that on small perturbations of Minkowski space, the squared Lorentzian Dirac operator has only real spectrum plus isolated resonances, and the residue of its spectral zeta density at $\alpha=n/2-1$ is a local invariant…

desk verdict Technically serious paper whose central residue formula appears to be off by a factor -2 and a sign from its own derivation. read the letter →

arxiv 2412.12714 v1 pith:XHGNEODK submitted 2024-12-17 math.AP math-phmath.MP

Nguyen Viet Dang , András Vasy , Micha{l} Wrochna This is my paper · ORCID
classification math.APmath-phmath.MP MSC 35L0558J4058J5053C27
keywords LorentzianDiracoperatorspectralzetafunctiondensityscatteringcalculusradialestimatesresonancesnon-trappingspacetimeslocalinvariantsBochner–Lichnerowiczformula
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper's goal is to give the Lorentzian Dirac operator the same spectral-zeta treatment that works for wave operators on asymptotically Minkowski spaces. For $P = -\slash{D}^2$ on a small perturbation of Minkowski space, it claims the spectrum consists of the real axis plus isolated resonances in a strip, with smooth resonant states. Its sharper claim is that the meromorphic residue of the spectral zeta density $\operatorname{tr}_E (P-i\varepsilon)^{-\alpha}(x,x)$ at $\alpha = n/2-1$, taken in the limit $\varepsilon\to 0^+$, is a local invariant built from the Lorentzian scalar curvature and the twisting curvature. If true, the spectral action principle of noncommutative geometry has a well-defined Lorentzian version, despite the Dirac operator's indefinite Hermitian form. The route is microlocal: radial estimates in a resolved scattering calculus control the resolvent uniformly along a complex contour, and a Hadamard parametrix converts the contour integral into curvature data.

What carries the argument

The central technical object is the resolved scattering phase space $\operatorname{sc,res}T^*M = [\operatorname{sc}T^*M; \operatorname{sc}S^*_{\partial M}M]$, obtained by blowing up the corner at fiber infinity and base infinity, together with the further blow-up at $h=0$ in fiber infinity that connects the semiclassical and classical pseudodifferential algebras. In the resolved calculus $\Psi^{m,k,\ell}_{sc,qsc}$ the relative order $k-s$ replaces the uncomfortable threshold $-\tfrac12$ in the radial estimates, so propagation and radial estimates give Fredholm estimates with adjustable trade-offs between regularity and decay. A second ingredient, the Bochner–Lichnerowicz formula $-\slash{D}^2 = \nabla^{E*}\nabla^E + F^E + \tfrac14 R_g\,\mathrm{Id}_E$, identifies the first transport coefficient in the Hadamard parametrix with the scalar and twisting curvature, turning the contour-integral residue into the displayed local invariant.

What would settle it

Take Minkowski space in dimension 4 and add a small compactly supported bump metric engineered to create a stable, trapped null geodesic (a photon sphere). Compute or numerically estimate whether the closure of $P=-\slash{D}^2$ still has spectrum only on $\mathbb{R}$ plus isolated resonances in a strip, and whether the residue at $\alpha=n/2-1$ still matches the scalar-curvature formula; a trapped geodesic producing resonances accumulating on $\mathbb{R}$ or a residue one full derivative different from the formula would falsify the small-perturbation claim as stated.

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Extended reading notes

Core claim

The paper establishes that, although $-\slash{D}^2$ is only formally self-adjoint for a non-positive Hermitian form, its spectral theory is controllable after endowing the spinor bundle with the auxiliary positive scalar product $\langle u,v\rangle = \langle u,\gamma(e)v\rangle_S$. The main theorem states that for every $\varepsilon>0$ the diagonal restriction $(P-i\varepsilon)^{-\alpha}(x,x)$ is meromorphic in $\alpha$ with poles at $n/2, n/2-1, \dots, 1$, and $$\lim_{\varepsilon\to0^+}\operatorname{res}_{\$\alpha$=n/2-1} \operatorname{tr}_E (P-i\varepsilon)^{-\$\alpha$}(x,x) = \frac{\operatorname{rk}(E)R_g(x)}{i6(4\pi)^{n/2}\Gamma(n/2-1)} + \frac{2\operatorname{tr}_E(F^E)(x)}{i(4\pi)^{n/2}\Gamma(n/2-1)},$$ where $R_g$ is the scalar curvature and $F^E$ the twisting curvature of the Clifford module $E$. In the same package, the resolvent is shown to be meromorphic with finite-multiplicity poles whose resonant states are smooth, and complex powers are defined up to finite-rank smoothing ambiguities that do not affect the residue. This is the Lorentzian analogue of the heat-kernel coefficient computation that underlies the spectral action.

Load-bearing premise

The load-bearing premise is that every sufficiently small perturbation of Minkowski space is non-trapping, meaning no lightlike geodesic is trapped: each one flows from a radial source at past infinity to a radial sink at future infinity, and no explicit bound or persistence proof is given for this property.

Editorial extensions

If this is right

  • The spectral action principle can be formulated for Lorentzian perturbations of Minkowski: $f(h(P+i\varepsilon))$ has a small-$h$ asymptotic expansion whose leading $h^{-n+2}$ coefficient contains $R_g/12 + \operatorname{tr}(F^E)$, the Lorentzian analogue of the Riemannian heat-kernel coefficient.
  • The resonance structure is qualitatively the same as for wave operators: off the real axis the spectrum is a discrete set of poles of finite multiplicity, and every generalized resonant state is smooth in the interior; this supports studying Dirac-type fields with the same Feynman-resolvent tools used for $\square_g$.
  • The definition of $(P-i\varepsilon)^{-\alpha}$ depends on the contour only by a finite-rank smoothing operator, so all diagonal residues and the associated local invariants are unambiguous.
  • Because the resolvent estimates replace the global hyperbolicity assumption used in earlier parametrix constructions with non-trapping dynamics, the same heat-kernel coefficients should be computable without assuming global hyperbolicity.

Reading between the lines

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

  • Beyond the paper, one testable extension is to compute the same residue for a metric whose null geodesic flow is non-trapping but has a normally hyperbolic trapped set; the framework suggests the local invariant would persist, but this is not claimed in the paper.
  • The formula pins the $h^{-n+2}$ coefficient of the Lorentzian spectral action to $R_g/12+\operatorname{tr}(F^E)$, which in a physical model would tie the gravitational and matter actions to the same constant; this numerical identification is an editorial extrapolation of the displayed residue.
  • A quantitative smallness bound for 'small perturbation' would let the result be verified by explicit examples, e.g. perturbing $\mathbb{R}^{1,3}$ by a bump metric and checking the resonance-strip width; the paper leaves the size implicit.
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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 studies P = -/D^2, the square of a Lorentzian Dirac operator on a small perturbation of Minkowski space, viewed as a non-self-adjoint scattering pseudodifferential operator with a small or decaying imaginary part relative to an auxiliary positive Hermitian form. The main results are: (i) Theorem 1.1 (cf. Theorems 3.3 and 3.9), asserting that the closure of P has spectrum consisting of the real line plus isolated resonances in a horizontal strip, with smooth resonant states; and (ii) Theorem 1.2 (cf. Theorem 5.7), asserting that the zeta-density trace tr_E((P - i epsilon)^{-alpha})(x,x) extends meromorphically with poles at n/2, n/2 - 1, ..., 1 and that the residue at alpha = n/2 - 1, in the limit epsilon -> 0+, is a local expression in the Lorentzian scalar curvature R_g and the twisting curvature F^E, intended as a Lorentzian spectral action coefficient. The proof introduces a resolved scattering calculus Psi^{m,k,ell}_{sc,qsc} and a further resolved classical-semiclassical calculus Psi^{m,k,ell,p,q,r}_{qsc,sc,hbar^2,hbar,cl} to prove radial, propagation, and large-parameter elliptic estimates, and then applies a vector-bundle Hadamard parametrix following the authors' earlier work [14].

Significance. The strengths are substantial if the central formula holds: the paper extends the Lorentzian spectral zeta program from the scalar wave operator to Dirac-type operators, overcoming the indefinite-form obstruction via the auxiliary positive scalar product (5.14); the two new calculi are original tools with likely wider applicability; the resolvent estimates are made uniform with O(<lambda>^{-1}) decay along the integration contour, allowing the zeta continuation without a global hyperbolicity assumption; and Lemma 5.6 quantifies the contour ambiguity as finite-rank smoothing, so the residues are unambiguous. The residue formula is a concrete, parameter-free, falsifiable prediction, and the vector-bundle Bochner-Lichnerowicz computation is explicit (u_0 and u_1 transport equations). These merits are real, but they are presently undercut by the internal inconsistency described in Major Comment 1: the displayed Theorem 5.7 does not follow from the paper's own Eq. (5.19) and u_1(0) computation, and differs from it by a factor of -2 on both geometric terms.

major comments (3)
  1. The displayed residue formula of Theorem 5.7 does not follow from the paper's own computation. Substituting k = 1 into Eq. (5.19) gives res_{alpha = n/2 - 1} (P - i epsilon)^{-alpha}(x,x) = i u_1(x,x)/(2^n pi^{n/2} Gamma(n/2 - 1)) = i u_1(x,x)/((4 pi)^{n/2} Gamma(n/2 - 1)). With u_1(0) = (R_g/12) 1_E + F^E as computed in Step 3, the traced residue equals i(rk(E) R_g/12 + tr_E(F^E))/((4 pi)^{n/2} Gamma(n/2 - 1)). The theorem instead states rk(E) R_g/(i 6 (4 pi)^{n/2} Gamma) + 2 tr_E(F^E)/(i (4 pi)^{n/2} Gamma) = -i rk(E) R_g/(6 (4 pi)^{n/2} Gamma) - 2i tr_E(F^E)/((4 pi)^{n/2} Gamma), which is exactly -2 times the derived value for each geometric term. Remark 5.8's small-h expansion lists rk(E) R_g/12 + tr(F^E) as the h^{-n+2} geometric coefficient, agreeing with the derivation and not with the displayed theorem. Thus the central advertised formula is internally inconsistent as written; the authors must determine which expression is correct, correct the other, and re-verify the consequences for the Lorentzian spectral action claim.
  2. The assertion that 'small perturbations of Minkowski space' satisfy the non-trapping hypothesis is not proved. The manuscripts states that version (2) of Definition 5.3 is verified for sufficiently small perturbations of the Minkowski metric in the sense of scattering metrics, but no smallness bound is given and no argument is supplied that the radial source/sink structure L_-/L_+ of Definition 2.9 persists under such perturbations, or that no trapped bicharacteristics appear (which would invalidate Propositions 2.14-2.15 and hence Theorems 3.3 and 5.7). Since the abstract and Theorems 1.1-1.2 are stated for 'small perturbations of Minkowski space', this is a load-bearing gap: the spectral and residue results are proved only for the abstract class of Definition 5.3, and the advertised application requires a quantitative stability lemma (e.g., explicit Psi^{2,0} seminorm bounds under which the radial points remain sources/sinks and all bicharacteristics escape). Lemma 5.4 (fast-decaying perturbations) proves only the decay of P - P*, not non-trapping, so it does not fill the gap.
  3. The large-|Im lambda| elliptic and propagation estimates are load-bearing for Theorem 3.3 and for the O(<lambda>^{-1}) resolvent decay used in Step 1 of the proof of Theorem 5.7, but they are presented as sketches. The proof of Proposition 2.12 explicitly defers the needed elliptic estimate to Section 4, and Theorem 4.2 is stated after a discussion in which microlocalizers and error terms are repeatedly suppressed and the passage from the resolved semiclassical algebra to classical spaces is described as 'obtained similarly as in [51]' with only a reference. For the framework to be verifiable, the composition rules, the normal operator and full ellipticity condition of Definition 4.1, and the derivation of (4.12) from the resolved calculus should be stated as lemmas with proofs, or at least with precise statements of the symbol classes and mapping properties involved.
minor comments (5)
  1. The phrase 'similarly as in as in [14]' contains a duplicated 'as in' and should be corrected.
  2. The sentence 'which is in dot C' uses an undefined symbol; presumably a fast-decay space is meant, but this should be spelled out.
  3. The endomorphism space is written End(S_x) although the setup of Theorem 5.7 uses a Clifford module E; the formula should read End(E_x), or the trace should be taken in E.
  4. The threshold is written 'k - s + beta_tilde_- < -1/2 on L_+' with a minus subscript on beta_tilde at the sink L_+; the subscript should be matched to L_+/L_- as in Eq. (2.16) to avoid confusion about which subprincipal correction is used.
  5. The caption spans nearly a full page and is very difficult to follow; moving the technical details of the alternative blow-up order into the text and shortening the caption would improve readability.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation is present: the spectral residue formula is computed from a Hadamard parametrix via transport equations, not fitted or defined by the claimed invariants, and the paper's self-citations are published technical support rather than load-bearing circularity.

full rationale

The paper's central claims are derived conditionally from microlocal hypotheses, not from their conclusions. Theorem 1.1/3.3 and Proposition 3.8 rest on propagation and radial estimates (Propositions 2.14, 2.15, 3.2, 4.2) under a non-trapping assumption that is stated as a domain hypothesis, not as a consequence of the theorem. Theorem 5.7's residue formula is obtained by a Hadamard parametrix in the spirit of [14], with the vector-bundle transport equation solved explicitly: the key step computes u1(0) from the Bochner–Lichnerowicz formula and then substitutes into Eq. (5.19). The residue is therefore computed from the operator's geometric data rather than fitted to the asserted local invariants, so the claimed Lorentzian spectral action coefficients are not presupposed. The paper does rely substantially on the authors' own prior work ([14], [48], [51]) for the microlocal and parametrix framework, but those are published results with independent content, and no uniqueness theorem or unverified ansatz is imported from same-author citations to force the conclusion. The absence of an explicit smallness bound for the non-trapping condition in Definition 5.3 is a gap in the verification of hypotheses, not a circularity. Separately, the displayed formula in Theorem 5.7 appears inconsistent with Eq. (5.19) together with u1(0) = R_g/12 + F^E: the displayed coefficient and sign differ by a factor of -2 from the derivation for each geometric term. This is a serious correctness concern, but it is an internal arithmetic inconsistency, not a reduction of the result to its inputs. Accordingly, no specific circular step is identified, and the score reflects only the notable same-author citation dependence of the technical machinery.

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

No free parameters are fitted; the only inputs are the spacetime geometry and the assumptions listed above. The paper introduces no new physical entities. The blow-up spaces in Sections 2 through 4 are analytical tools, not independent postulated objects.

assumptions (5)
  • domain assumption Non-trapping condition: each bicharacteristic of P in the characteristic set converges to a sink L+ in the past and a source L- in the future (Definition 3.1).
    Required for the global radial estimates (Propositions 2.14 and 2.15) and the Fredholm theory of Theorem 3.3.
  • domain assumption P - P* ∈ Ψ^{1,-1-δ} or P close to such P0 (Definition 5.3).
    This structural small-perturbation assumption ensures the positive commutator estimates have the stated thresholds and that the imaginary part is controlled.
  • domain assumption Existence of scattering spinor bundle S (or Clifford module E) with Clifford connection satisfying the Hermitian form properties a)-c) in Section 5.1.
    Defines /D and the positive scalar product (5.14) used for the Hilbert space formalism.
  • domain assumption Global hyperbolicity and existence of a future-directed time-like e ∈ V_sc(M) (Remark 5.1, Lemma 5.2).
    Needed to conclude P - P* ∈ Ψ^{1,-1} and to define the field-quantization scalar product.
  • standard math Validity of the Hadamard parametrix and transport hierarchy for P-iε on the bundle (from [14], Section 5.3).
    The residue computation assumes the transport equations have smooth solutions and that the u1(0) value equals Rg/12+F^E.

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Pith. "Pith review of Dirac operators and local invariants on perturbations of Minkowski space." pith.science (2026). https://pith.science/paper/XHGNEODK

@misc{pith2026241212714,
  author       = {Pith},
  title        = {Pith review of: Dirac operators and local invariants on perturbations of Minkowski space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHGNEODK}},
  note         = {Machine review of arXiv:2412.12714}
}
abstract

For small perturbations of Minkowski space, we show that the square of the Lorentzian Dirac operator $P= -D^2$ has real spectrum apart from possible poles in a horizontal strip. Furthermore, for $\varepsilon>0$ we relate the poles of the spectral zeta function density of $P-i\varepsilon$ to local invariants, in particular to the Lorentzian scalar curvature. The proof involves microlocal propagation and radial estimates in a resolved scattering calculus as well as high energy estimates in a further resolved classical-semiclassical calculus.

Figures

Figures reproduced from arXiv: 2412.12714 by the authors.

Figure 1
Figure 1. Blow-up sc,resT∗M = [scT∗M; scS ∗ ∂M M] of the corner scS ∗ ∂M M of scT∗M. The fi-face is the lift of fiber infinity scS ∗M = {ρ∞ = 0}, and the bi-face is the lift of base infinity scT∗ ∂M M = {ρ = 0}. henceforth denoted by ff. Fiber infinity and base infinity lift to two boundary hypersurfaces denoted respectively by fi and bi (see [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Resolutions of the fiber compactified semiclassical scattering cotangent bundle scT∗M × [0, 1]h. The initial space is on top left. Top right is the resolution of fiber infinity at x = 0, creating the semiclassical qsc face. Bottom right is the further resolution of h = 0 at fiber infinity, creating the h 2 -semiclassical qsc face. Finally bottom left adds the blow up of the zero section at h = 0 (which could have be… view at source ↗

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Cited by 1 Pith paper

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

  1. On the relationship between the semiclassical and standard pseudodifferential algebras

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Reference graph

Works this paper leans on

55 extracted references · 52 canonical work pages · cited by 1 Pith paper

  1. [14]

    N. V. Dang and M. Wrochna. Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces. J. Eur. Math. Soc. , 2023

    work page 2023

  2. [51]

    A. Vasy. On the relationship between the semiclassical and standard pseudodifferential algebras. 2024

    work page 2024

  3. [1]

    B¨ ar, P

    C. B¨ ar, P. Gauduchon, and A. Moroianu. Generalized cylinders in semi-Riemannian and spin geometry. Math. Zeitschrift , 249(3):545–580, 2005

    work page 2005

  4. [2]

    B¨ ar and A

    C. B¨ ar and A. Strohmaier. An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary. Am. J. Math. , 141(5):1421–1455, 2019

    work page 2019

  5. [3]

    B¨ ar and A

    C. B¨ ar and A. Strohmaier. Local index theory for Lorentzian manifolds. arXiv:2012.01364, 2020

    arXiv 2012

  6. [4]

    Berline, E

    N. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Operators . 2004

    work page 2004

  7. [5]

    A. H. Chamseddine and A. Connes. The spectral action principle. Commun. Math. Phys. , 1997

    work page 1997

  8. [6]

    A. H. Chamseddine, A. Connes, and M. Marcolli. Gravity and the standard model with neutrino mixing. Adv. Theor. Math. Phys. , 11(6):991–1089, 2007

    work page 2007

Show all 55 references
  1. [7]

    H. Chihara. Smoothing effects of dispersive pseudodifferential equations. Commun. Partial Differ. Equa- tions, 27(9-10):1953–2005, 2002

    1953

  2. [8]

    Colin de Verdi` ere and C

    Y. Colin de Verdi` ere and C. Le Bihan. On essential-selfadjointness of differential operators on closed manifolds. Ann. la Fac. des Sci. Toulouse Math´ ematiques, 31(5):1287–1302, 2022

    2022

  3. [9]

    A. Connes. Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math. Phys. , 182(1):155–176, 1996. DIRAC OPERATORS AND LOCAL INV ARIANTS ON PERTURBATIONS OF MINKOWSKI SPACE 39

    1996

  4. [10]

    Connes and M

    A. Connes and M. Marcolli. Noncommutative Geometry, Quantum Fields and Motives . American Math- ematical Society, Providence, RI, 2008

    2008

  5. [11]

    Damaschke

    O. Damaschke. Cauchy Problem for the Dirac operator on spatially non-compact spacetimes. arXiv:2409.17344, 2024

    2024 arXiv

  6. [12]

    D’Andrea, M

    F. D’Andrea, M. A. Kurkov, and F. Lizzi. Wick rotation and fermion doubling in noncommutative geometry. Phys. Rev. D , 94(2):025030, 2016

    2016

  7. [13]

    N. V. Dang and M. Wrochna. Dynamical residues of Lorentzian spectral zeta functions. J. l’ ´Ecole Polytech. — Math´ ematiques, 9:1245–1292, 2022

    2022

  8. [15]

    N. V. Dang and M. Wrochna. Lorentzian Spectral Zeta Functions on Asymptotically Minkowski Space- times. In J. K¨ ahler, U., Reissig, M., Sabadini, I., Vindas, editor, Anal. Appl. Comput. ISAAC 2021. , pages 501–514. Birkh¨ auser, 2023

    2021

  9. [16]

    Derezi´ nski and C

    J. Derezi´ nski and C. Gaß. Propagators in curved spacetimes from operator theory. arXiv:2409.03279, 2024

    2024

  10. [17]

    Derezi´ nski and D

    J. Derezi´ nski and D. Siemssen. Feynman propagators on static spacetimes. Rev. Math. Phys. , 2018

    2018

  11. [18]

    Devastato, S

    A. Devastato, S. Farnsworth, F. Lizzi, and P. Martinetti. Lorentz signature and twisted spectral triples. J. High Energy Phys. , 2018(3):89, 2018

    2018

  12. [19]

    Gell-Redman, N

    J. Gell-Redman, N. Haber, and A. Vasy. The Feynman propagator on perturbations of Minkowski space. Commun. Math. Phys. , 342(1):333–384, 2016

    2016

  13. [20]

    Gell-Redman, A

    J. Gell-Redman, A. Hassell, J. Shapiro, and J. Zhang. Existence and Asymptotics of Nonlinear Helmholtz Eigenfunctions. SIAM J. Math. Anal. , 52(6):6180–6221, 2020

    2020

  14. [21]

    G´ erard and M

    C. G´ erard and M. Wrochna. The massive Feynman propagator on asymptotically Minkowski spacetimes. Am. J. Math. , 141(6):1501–1546, 2019

    2019

  15. [22]

    G´ erard and M

    C. G´ erard and M. Wrochna. The massive Feynman propagator on asymptotically Minkowski spacetimes II. Int. Math. Res. Not. , 2019

    2019

  16. [23]

    Q. Jia, M. Molodyk, and E. Sussman. The essential self-adjointness of the wave operator on radiative spacetimes. arXiv:2412.03828, 2024

    2024

  17. [24]

    Kami´ nski

    W. Kami´ nski. Non-Self-Adjointness of the Klein–Gordon Operator on a Globally Hyperbolic and Geodesically Complete Manifold: An Example. Ann. Henri Poincar´ e, 23(12):4409–4427, 2022

    2022

  18. [25]

    C. Kottke. A Callias-type index theorem with degenerate potentials.Commun. Partial Differ. Equations, 40(2):219–264, 2015

    2015

  19. [26]

    H. B. Lawson and M.-L. Michelsohn. Spin Geometry, volume 38. Princeton University Press, Princeton, NJ, 2016

    2016

  20. [27]

    Martinetti and D

    P. Martinetti and D. Singh. Lorentzian fermionic action by twisting Euclidean spectral triples. J. Non- commutative Geom., 16(2):513–559, 2022

    2022

  21. [28]

    R. Melrose. The Atiyah–Patodi–Singer Index Theorem . CRC Press, Boca Raton, FL, 1993

    1993

  22. [29]

    R. Melrose. Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces. In Spectr. Scatt. Theory. Dekker, New York, 1994

    1994

  23. [30]

    Molodyk and A

    M. Molodyk and A. Vasy. An analogue of non-interacting quantum field theory in Riemannian signature. arXiv:2404.11821, 2024

    2024

  24. [31]

    V. Moretti. Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes. Rev. Math. Phys. , 15(10):1171–1217, 2003

    2003

  25. [32]

    Nakamura and K

    S. Nakamura and K. Taira. Essential self-adjointness of real principal type operators. Ann. Henri Lebesgue, 4:1035–1059, 2021

    2021

  26. [33]

    Nakamura and K

    S. Nakamura and K. Taira. A Remark on the Essential Self-adjointness for Klein–Gordon-Type Oper- ators. Ann. Henri Poincar´ e, 24(8):2587–2605, 2023

    2023

  27. [34]

    Nakamura and K

    S. Nakamura and K. Taira. Essential self-adjointness of Klein-Gordon type operators on asymptotically static, Cauchy-compact spacetimes. Commun. Math. Phys. , 398(3):1153–1169, 2023

    2023

  28. [35]

    Paschke and R

    M. Paschke and R. Verch. Local covariant quantum field theory over spectral geometries. Class. Quan- tum Gravity, 21(23):5299–5316, 2004

    2004

  29. [36]

    J. Roe. Elliptic Operators, Topology, and Asymptotic Methods . Chapman & Hall, 1999

    1999

  30. [37]

    Strohmaier

    A. Strohmaier. On noncommutative and pseudo-Riemannian geometry. J. Geom. Phys., 56(2):175–195, 2006

    2006

  31. [38]

    E. Sussman. Massive wave propagation near null infinity. arXiv:2305.01119, 2023. DIRAC OPERATORS AND LOCAL INV ARIANTS ON PERTURBATIONS OF MINKOWSKI SPACE 40

    2023 arXiv

  32. [39]

    Tadano and K

    Y. Tadano and K. Taira. Uniform bounds of discrete Birman–Schwinger operators. Trans. Am. Math. Soc., 372(7):5243–5262, 2019

    2019

  33. [40]

    K. Taira. Equivalence of classical and quantum completeness for real principal type operators on the circle. arXiv:2004.07547, 2020

    2004 arXiv

  34. [41]

    K. Taira. Limiting absorption principle and equivalence of Feynman propagators on asymptotically Minkowski spacetimes. Commun. Math. Phys. , 388(1):625–655, 2021

    2021

  35. [42]

    K. Taira. Remarks on the geodesically completeness and the smoothing effect on asymptotically Minkowski spacetimes. Lett. Math. Phys. , 112(2):22, 2022

    2022

  36. [43]

    Uhlmann and A

    G. Uhlmann and A. Vasy. The inverse problem for the local geodesic ray transform. Invent. Math. , 205(1):83–120, 2016

    2016

  37. [44]

    van den Dungen

    K. van den Dungen. Families of spectral triples and foliations of space(time). J. Math. Phys. , 59(6):063507, 2018

    2018

  38. [45]

    W. D. van Suijlekom. The noncommutative Lorentzian cylinder as an isospectral deformation. J. Math. Phys., 45(1):537–556, 2004

    2004

  39. [46]

    A. Vasy. A Minicourse on Microlocal Analysis for Wave Propagation. In Asymptot. Anal. Gen. Relativ. , pages 219–374. Cambridge University Press, 2017

    2017

  40. [47]

    A. Vasy. On the positivity of propagator differences. Ann. Henri Poincar´ e, 18(3):983–1007, 2017

    2017

  41. [48]

    A. Vasy. Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces. J. Spectr. Theory, 10(2):439–461, 2020

    2020

  42. [49]

    A. Vasy. Limiting absorption principle on Riemannian scattering (asymptotically conic) spaces, a La- grangian approach. Commun. Partial Differ. Equations , 46(5):780–822, 2021

    2021

  43. [50]

    A. Vasy. Resolvent near zero energy on Riemannian scattering (asymptotically conic) spaces. Pure Appl. Anal., 3(1):1–74, 2021

    2021

  44. [52]

    Vasy and M

    A. Vasy and M. Wrochna. Quantum fields from global propagators on asymptotically Minkowski and extended de Sitter spacetimes. Ann. Henri Poincar´ e, 2018

    2018

  45. [53]

    Vasy and M

    A. Vasy and M. Zworski. Semiclassical estimates in asymptotically Euclidean scattering. Commun. Math. Phys. , 212(1):205–217, 2000

    2000

  46. [54]

    Wrochna and R

    M. Wrochna and R. Zeitoun. The wave resolvent for compactly supported perturbations of Minkowski space. In Harmon. Anal. Partial Differ. Equations , pages 1–17. 2022

    2022

  47. [55]

    J. Wunsch. Propagation of singularities and growth for Schr¨ odinger operators. Duke Math. J. , 98(1), 1999. IRMA, Universit´e de Strasbourg, Strasbourg, France Institut Universitaire de France, Paris, France Email address : nvdang@unistra.fr Department of Mathematics, Stanfor...

    1999

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