# Partial Differential Equations and Quantum States in Curved Spacetimes

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hyperbolic Propagators

#### 2.1. Motivation

#### 2.2. The Wave Propagator on a Riemannian Manifold

**Definition**

**1**

**Lemma**

**1.**

- (i)
- ${\left(\right)}_{\phi}x={x}^{*}$,
- (ii)
- ${\left(\right)}_{{\partial}_{{x}^{\alpha}}}x={x}^{*}$,
- (iii)
- ${\left(\right)}_{det}x={x}^{*}$,
- (iv)
- $Im\phi \ge 0$.

- $\phi $ is the Levi–Civita phase function;
- $\chi $ is a cut-off function that serves the purpose of localizing the integration in a neighborhood of the orbit with initial condition $(y,\eta )$ and away from the zero section, see [13] (Section 5);
- the weight w is defined by$$w(t,x;y,\eta ):={\displaystyle \frac{1}{{\left[{\rho}_{g}\left(x\right){\rho}_{g}\left(y\right)\right]}^{1/2}}}{\left(\right)}^{{det}^{2}},$$$${\left(\right)}_{{\left(\right)}^{{det}^{2}}}1/4t=0$$

**Step one**. Set $\chi (t,x;y,\eta )=1$ and apply the wave operator

**Step two**. Construct a new oscillatory integral with x-independent amplitude $\mathfrak{b}=\mathfrak{b}(t;y,\eta )$, coinciding with (10) up to an integral operator with infinitely smooth integral kernel. Such a procedure is called reduction of the amplitude. This can be done by means of special operators, as described below.

**Step three**. Impose the condition that our oscillatory integral (10) satisfies the wave equation, modulo an integral operator with infinitely smooth kernel. This is achieved by solving transport equations obtained by equating to zero the homogeneous components of the reduced amplitude $\mathfrak{b}$,

**Remark**

**1.**

**Remark**

**2.**

## 3. The Notion of Wavefont Set and Propagation of Singularities

**Theorem**

**1.**

**Definition**

**2**

**Definition**

**3**

**Theorem**

**2.**

**Theorem**

**3**

## 4. Quantum Field Theory on Curved Spacetimes: Hadamard States

#### 4.1. Hadamard States

- (i)
- Some causal curves will intersect the initial curve more than once, in which case, for generic initial data, a single-valued solution will not exist.
- (ii)
- Some causal curves will not intersect the initial curve at all, in which case a portion of the plane will not be in the domain of dependence of the initial curve, and the values of the solution in that region will not depend on the initial conditions, resulting in non-uniqueness.

**Definition**

**4**

- (i)
- ${\mathcal{O}}_{\alpha}\subset \mathcal{M}$ is geodesically convex for every $\alpha $ and
- (ii)
- ${\mathcal{O}}_{\alpha}\cap {\mathcal{O}}_{{\alpha}^{\prime}}$ is either empty or geodesically convex.

**Definition**

**5.**

**Remark**

**3.**

- (a)
- The limit in the RHS of (18) has to be understood in the sense of distributions: First, one integrates against a test function, then one takes the limit for $\u03f5\to {0}^{+}$.
- (b)
- The smooth functions $u,{v}_{n}\in {C}^{\infty}(\mathcal{O}\times \mathcal{O})$ are known as Hadamard coefficients. They are obtained as unique solutions of a hierarchy of differential equations that arise by imposing that the RHS of (18) solves the Klein–Gordon equation in the variable x, interpreting y as a parameter and setting $w=0$. See, e.g., [38] (Appendix A) for further details.
- (c)
- Definition 5 immediately raises the question: does the series on the RHS of (18) converge? The answer, unfortunately, is negative. The convergence of the series is only guaranteed when $(\mathcal{M},\mathfrak{g})$ is analytic. In the general smooth case, the series appearing in (18) has to be understood as an asymptotic expansion ‘in smoothness’, namely, the identity (18) means -4.6cm0cm$${\omega}_{2}(x,y)-\underset{\u03f5\to {0}^{+}}{lim}\left(\right)open="("\; close=")">\frac{1}{4{\pi}^{2}}\frac{u(x,y)}{{\sigma}_{\u03f5}(x,y)}+\left(\right)open="("\; close=")">\sum _{n=0}^{N}{v}_{n}(x,y)\sigma {(x,y)}^{n}\in {C}^{N-1}(\mathcal{O}\times \mathcal{O})$$However, if one wants to work with a uniformly convergent series, the issue of non-convergence can be circumvented as follows. Choose a smooth cut-off $\chi :\mathbb{R}\to [0,1]$,$$\chi \left(\tau \right)=\left(\right)open="\{"\; close>\begin{array}{cc}1\hfill & \left|\tau \right|\le \frac{1}{2},\hfill \\ 0\hfill & \left|\tau \right|1.\hfill \end{array}$$Then, there exists a real sequence$$0<{c}_{1}<{c}_{2}<{c}_{3}<{c}_{n}<\dots \to +\infty $$$$v(x,y):=\sum _{n=0}^{+\infty}{v}_{n}(x,y){\sigma}^{n}(x,y)\chi \left({c}_{n}\sigma (x,y)\right)$$$$H(x,y):=\underset{\u03f5\to {0}^{+}}{lim}\left(\right)open="["\; close="]">\frac{1}{4{\pi}^{2}}\frac{u(x,y)}{{\sigma}_{\u03f5}(x,y)}+v(x,y)ln({\sigma}_{\u03f5}(x,y)/{\ell}^{2})$$$$\begin{array}{c}\underset{\u03f5\to {0}^{+}}{lim}{\int}_{\mathcal{O}\times \mathcal{O}}H(x,y)\left[(\square +{m}^{2}){f}_{1}\right]\left(x\right)\phantom{\rule{0.166667em}{0ex}}{f}_{2}\left(y\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\int}_{\mathcal{O}\times \mathcal{O}}k(x,y)\phantom{\rule{0.166667em}{0ex}}{f}_{1}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{f}_{2}\left(y\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy,\end{array}$$Different choices of the cut-off χ yield different smooth errors k.
- (d)
- Definition 5 completely prescribes the singular structure of the 2-point function, including the numerical prefactors. The definition of H only involves the geometry of our spacetime and the equation of motion, which enters in the Hadamard coefficients, but does not identify a particular state. The information about the ‘physics’ of the system—that is, about the individual state—is contained in the smooth term w.
- (e)
- Definition 5 prescribes the singular structure of ${\omega}_{2}$ locally but, prima facie, does not tell us anything about global properties of ω or ${\omega}_{2}$.

**Theorem**

**4.**

**Remark**

**4.**

**Theorem**

**5**

**Theorem**

**6**

**Theorem**

**7.**

#### 4.2. Construction of Hadamard States

**Definition**

**6**

- ${\tilde{X}}^{*}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}};s,y,\eta ):\varsigma \mapsto {\tilde{X}}^{*}(\varsigma ;s,y,\eta )$ is the unique null geodesic stemming from Y with initial cotangent vector ${\widehat{\eta}}_{+}$, parameterized by proper time;
- ${\tilde{\mathsf{\Xi}}}^{*}(\varsigma ;s,y,\eta )$ is the parallel transport along ${\tilde{X}}^{*}(\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}};s,y,\eta )$ of ${\eta}_{+}$, from Y to ${\tilde{X}}^{*}(\varsigma ;s,y,\eta )$.

**Definition**

**7**

**Theorem**

**8.**

**Remark**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bär, C.; Ginoux, N.; Pfäffle, F. Wave Equations on Lorenzian Manifolds and Quantization; ESI Lectures in Mathematics and Physics; EMS: Paris, France, 2007. [Google Scholar]
- Ikawa, M. Hyperbolic Partial Differential Equations and Wave Phenomena; Translations of Mathematical Monographs, Iwanami Series in Modern Mathematics; AMS: Providence, RI, USA, 2000. [Google Scholar]
- Capoferri, M.; Vassiliev, D. Spacetime diffeomorphisms as matter fields. J. Math. Phys.
**2020**, 61, 111508. [Google Scholar] [CrossRef] - Avetisyan, Z. A unified mode decomposition method for physical fields in homogeneous cosmology. Rev. Math. Phys.
**2014**, 26, 1430001-1–1430001-60. [Google Scholar] [CrossRef][Green Version] - Duistermaat, J.J.; Hörmander, L. Fourier integral operators. II. Acta Math.
**1972**, 128, 183–269. [Google Scholar] [CrossRef] - Hadamard, J. Lectures on Cauchy’s Problem in Linear Partial Differential Equations; Dover Publications: New York, NY, USA, 1953. [Google Scholar]
- Riesz, M. L’intégrale de Riemann–Liouville et le problème de Cauchy. Acta Math.
**1949**, 81, 1–223. [Google Scholar] [CrossRef] - Riesz, M. A geometric solution of the wave equation in space-time of even dimension. Comm. Pure Appl. Math.
**1960**, 13, 329–351. [Google Scholar] [CrossRef] - Hörmander, L. The Analysis of Linear Partial Differential Operators I; Classics in Mathematics; Springer: Berlin, Germany, 2003; Reprint of the second (1990) edition; III. Reprint of the 1994 edition, 2007; IV. Reprint of the 1994 edition, 2009. [Google Scholar]
- Shubin, M.A. Pseudodifferential Operators and Spectral Theory; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Trèves, F. Introduction to Pseudodifferential and Fourier Integral Operators; The University Series in Mathematics; Plenum Press: New York, NY, USA; London, UK, 1980; Volumes 1 & 2. [Google Scholar]
- Kumano-Go, H. Pseudo-Differential Operators; MIT Press: Cambridge, MA, USA, 1974. [Google Scholar]
- Capoferri, M.; Levitin, M.; Vassiliev, D. Geometric wave propagator on Riemannian manifolds. arXiv
**2019**, arXiv:1902.06982. [Google Scholar] - Chervova, O.; Downes, R.J.; Vassiliev, D. The spectral function of a first order elliptic system. J. Spectr. Theory
**2013**, 3, 317–360. [Google Scholar] [CrossRef][Green Version] - Laptev, A.; Safarov, Y.; Vassiliev, D. On global representation of Lagrangian distributions and solutions of hyperbolic equations. Comm. Pure Appl. Math.
**1994**, 47, 1411–1456. [Google Scholar] [CrossRef] - Safarov, Y.; Vassiliev, D. The Asymptotic Distribution of Eigenvalues of Partial Differential Operators; AMS: Providence, RI, USA, 1997. [Google Scholar]
- Avetisyan, Z.; Fang, Y.-L.; Vassiliev, D. Spectral asymptotics for first order systems. J. Spectr. Theory
**2016**, 6, 695–715. [Google Scholar] [CrossRef][Green Version] - Capoferri, M. Diagonalization of elliptic systems via pseudodifferential projections. arXiv
**2021**, arXiv:2106.07948. [Google Scholar] - Capoferri, M.; Dappiaggi, C.; Drago, N. Global wave parametrices on globally hyperbolic spacetimes. J. Math. Anal. Appl.
**2020**, 490, 124316. [Google Scholar] [CrossRef] - Capoferri, M.; Vassiliev, D. Global propagator for the massless Dirac operator and spectral asymptotics. arXiv
**2020**, arXiv:2004.06351. [Google Scholar] - Capoferri, M.; Vassiliev, D. Invariant subspaces of elliptic systems II: Spectral theory. arXiv
**2021**, arXiv:2103.14334; to appear in J. Spectr. Theory. [Google Scholar] - Capoferri, M.; Vassiliev, D. Invariant subspaces of elliptic systems I: Pseudodifferential projections. arXiv
**2021**, arXiv:2103.14325. [Google Scholar] - Levitan, B.M. On the asymptotic behaviour of the spectral function of a self-adjoint differential second order equation. Izv. Akad. Nauk SSSR Ser. Mat.
**1952**, 19, 325–352. [Google Scholar] - Avakumovic, V.G. Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten. Math. Z.
**1956**, 65, 327–344. [Google Scholar] [CrossRef] - Friedlander, F.G.; Joshi, M. Introduction to the Theory of Distributions, 2nd ed.; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Brunetti, R.; Dappiaggi, C.; Fredenhagen, K.; Yngvason, J. Advances in Algebraic Quantum Field Theory; Springer: Berlin/Heidelberg, Germany, 2015; 453p. [Google Scholar]
- Gérard, C. Microlocal Analysis of Quantum Fields on Curved Spacetimes; ESI Lectures in Mathematics and Physics; EMS: Paris, France, 2019. [Google Scholar]
- Khavkine, I.; Moretti, V. Algebraic QFT in Curved Spacetime and quasifree Hadamard states: An introduction. In Advances in Algebraic Quantum Field Theory; Brunetti, R., Dappiaggi, C., Fredenhagen, K., Yngvason, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Beem, J.; Ehrlich, P.; Easley, K. Global Lorentzian Geometry; Pure and Applied Mathematics; Chapman and Hall; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Bernal, A.; Sánchez, M. Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Commun. Math. Phys.
**2005**, 257, 43–50. [Google Scholar] [CrossRef][Green Version] - Avetisyan, Z. Global hyperbolicity and factorization in cosmological models. J. Math. Phys.
**2021**, 62, 033507. [Google Scholar] [CrossRef] - Brunetti, R.; Fredenhagen, K. Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Comm. Math. Phys.
**2000**, 208, 623–661. [Google Scholar] [CrossRef][Green Version] - Brunetti, R.; Fredenhagen, K.; Köhler, M. The microlocal spectrum conditionand and Wick polynomials of free fields on curved spacetimes. Comm. Math. Phys.
**1996**, 180, 633–652. [Google Scholar] [CrossRef] - Hollands, S.; Wald, R.M. Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Comm. Math. Phys.
**2001**, 223, 289–326. [Google Scholar] [CrossRef][Green Version] - Hollands, S.; Wald, R.M. Existence of local covariant time ordered products of quantum fields in curved spacetime. Comm. Math. Phys.
**2002**, 231, 309–345. [Google Scholar] [CrossRef][Green Version] - Hollands, S.; Wald, R.M. Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes. Rev. Math. Phys.
**2005**, 17, 227–311. [Google Scholar] [CrossRef][Green Version] - Kay, B.S.; Wald, R.M. Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate killing horizon. Phys. Rep.
**1991**, 207, 49–136. [Google Scholar] [CrossRef] - Moretti, V. Comments on the stress-energy tensor operator in curved spacetime. Comm. Math. Phys.
**2003**, 232, 189–221. [Google Scholar] [CrossRef][Green Version] - Wald, R.M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics; Chicago Lectures in Physics; University of Chicago Press: Chicago, IL, USA, 1994. [Google Scholar]
- Benini, M.; Capoferri, M.; Dappiaggi, C. Hadamard states for quantum Abelian duality. Ann. Henri Poincaré
**2017**, 18, 3325–3370. [Google Scholar] [CrossRef] - Benini, M.; Dappiaggi, C.; Murro, S. Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states. J. Math. Phys.
**2014**, 55, 082301. [Google Scholar] [CrossRef][Green Version] - Dappiaggi, C.; Siemssen, D. Hadamard states for the vector potential on asymptotically flat spacetimes. Rev. Math. Phys.
**2013**, 25, 1350002. [Google Scholar] [CrossRef][Green Version] - Dappiaggi, C.; Drago, N. Constructing Hadamard States via an Extended Møller Operator. Lett. Math. Phys.
**2016**, 106, 1587. [Google Scholar] [CrossRef][Green Version] - Drago, N.; Gérard, C. On the adiabatic limit of Hadamard states. Lett. Math. Phys.
**2017**, 107, 1409–1438. [Google Scholar] [CrossRef] - Fewster, C.J.; Pfenning, M.J. A quantum weak energy inequality for spin-one fields in curved space-time. J. Math. Phys.
**2003**, 44, 4480. [Google Scholar] [CrossRef][Green Version] - Finster, F.; Murro, S.; Röken, C. The Fermionic Projector in a Time-Dependent External Potential: Mass Oscillation Property and Hadamard States. J. Math. Phys.
**2016**, 57, 072303. [Google Scholar] [CrossRef][Green Version] - Gérard, C.; Wrochna, M. Hadamard states for the linearized Yang-Mills equation on curved spacetime. Comm. Math. Phys.
**2015**, 337, 253–320. [Google Scholar] [CrossRef][Green Version] - Gérard, C.; Wrochna, M. Hadamard states for the Klein–Gordon equation on Lorentzian manifolds of bounded geometry. Comm. Math. Phys.
**2017**, 352, 519–583. [Google Scholar] [CrossRef] - Drago, N.; Murro, S. A new class of Fermionic Projectors: Møller operators and mass oscillation properties. Lett. Math. Phys.
**2017**, 107, 2433–2451. [Google Scholar] [CrossRef] - Finster, F.; Murro, S.; Röken, C. The Fermionic Signature Operator and Quantum States in Rindler Space-Time. J. Math. Anal. Appl.
**2017**, 454, 385. [Google Scholar] [CrossRef][Green Version] - Gérard, C.; Wrochna, M. Analytic Hadamard states, Calderón projectors and Wick rotation near analytic Cauchy surfaces. Comm. Math. Phys.
**2019**, 366, 29–65. [Google Scholar] [CrossRef] - Dappiaggi, C.; Finster, F.; Murro, S.; Radici, E. The Fermionic Signature Operator in De Sitter Spacetime. J. Math. Anal. Appl.
**2020**, 485, 123808. [Google Scholar] [CrossRef][Green Version] - Murro, S.; Volpe, D. Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds. Ann. Glob. Anal. Geom.
**2021**, 59, 1–25. [Google Scholar] [CrossRef] - Hollands, S.; Wald, R.M. Quantum field theory in curved spacetime, the operator product expansion, and dark energy. Gen. Rel. Grav.
**2008**, 40, 2051–2059. [Google Scholar] [CrossRef][Green Version] - O’Neill, B. Semi-Riemannian Geometry with Applications to Relativity; Pure and Applied Mathematics 103; Academic Press: Cambridge, MA, USA, 1983. [Google Scholar]
- Radzikowski, M.J. Micro-local approach to the Hadamard condition in quantum field theory on curved space-time. Comm. Math. Phys.
**1996**, 179, 529–553. [Google Scholar] [CrossRef] - Sahlmann, H.; Verch, R. Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys.
**2001**, 13, 1203–1246. [Google Scholar] [CrossRef][Green Version] - Radzikowski, M.J. A local-to-global singularity theorem for quantum field theory on curved space-time. Comm. Math. Phys.
**1996**, 180, 1–22. [Google Scholar] [CrossRef] - Fulling, S.A.; Sweeney, M.; Wald, R.M. Singularity structure of the two-point function in quantum field theory in curved spacetime. Comm. Math. Phys.
**1978**, 63, 257–264. [Google Scholar] [CrossRef] - Fulling, S.A.; Narcowich, F.J.; Wald, R.M. Singularity Structure of the Two Point Function in Quantum Field Theory in Curved Space-time II. Ann. Phys.
**1981**, 136, 243–272. [Google Scholar] [CrossRef] - Verch, R. Local definiteness, primarity and quasiequivalence of quasifree Hadamard quantum states in curved spacetime. Comm. Math. Phys.
**1994**, 160, 507–536. [Google Scholar] [CrossRef] - Verch, R. Continuity of symplectically adjoint maps and the algebraic structure of Hadamard vacuum representations for quantum fields on curved spacetime. Rev. Math. Phys.
**1997**, 9, 635–674. [Google Scholar] [CrossRef][Green Version] - Fewster, C.J.; Verch, R. The necessity of the Hadamard condition. Class. Quantum Gravity
**2013**, 30, 235027. [Google Scholar] [CrossRef][Green Version] - Fewster, C.J. A general worldline quantum inequality. Class. Quantum Grav.
**2000**, 17, 1897. [Google Scholar] [CrossRef] - Rellich, F. Perturbation Theory of Eigenvalue Problems; Courant Institute of Mathematical Sciences, New York University: New York, NY, USA, 1954. [Google Scholar]
- Strohmaier, A.; Zelditch, S. A Gutzwiller trace formula for stationary space-times. Adv. Math.
**2021**, 376, 107434. [Google Scholar] [CrossRef] - Bär, C.; Strohmaler, A. An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary. Amer. J. Math.
**2019**, 141, 1421–1455. [Google Scholar] [CrossRef][Green Version] - Avetisyan, Z.; Fang, Y.-L.; Saveliev, N.; Vassiliev, D. Analytic definition of spin structure. J. Math. Phys.
**2017**, 58, 082301. [Google Scholar] [CrossRef][Green Version] - Capoferri, M.; Saveliev, N.; Vassiliev, D. Classification of first order sesquilinear forms. Rev. Math. Phys.
**2020**, 32, 2050027. [Google Scholar] [CrossRef][Green Version] - Moretti, V. Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence. Comm. Math. Phys.
**2006**, 268, 727–756. [Google Scholar] [CrossRef][Green Version] - Moretti, V. Quantum out-states holographically induced by asymptotic flatness: Invariance under space-time symmetries, energy positivity and Hadamard property. Comm. Math. Phys.
**2008**, 279, 31–75. [Google Scholar] [CrossRef][Green Version] - Gérard, C.; Wrochna, M. Construction of Hadamard states by pseudo-differential calculus. Comm. Math. Phys.
**2014**, 325, 713–755. [Google Scholar] [CrossRef][Green Version] - Gérard, C.; Wrochna, M. Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE
**2016**, 9, 111–149. [Google Scholar] [CrossRef][Green Version]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Avetisyan, Z.; Capoferri, M.
Partial Differential Equations and Quantum States in Curved Spacetimes. *Mathematics* **2021**, *9*, 1936.
https://doi.org/10.3390/math9161936

**AMA Style**

Avetisyan Z, Capoferri M.
Partial Differential Equations and Quantum States in Curved Spacetimes. *Mathematics*. 2021; 9(16):1936.
https://doi.org/10.3390/math9161936

**Chicago/Turabian Style**

Avetisyan, Zhirayr, and Matteo Capoferri.
2021. "Partial Differential Equations and Quantum States in Curved Spacetimes" *Mathematics* 9, no. 16: 1936.
https://doi.org/10.3390/math9161936