# Partial Differential Equations and Quantum States in Curved Spacetimes

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## Abstract

**:**

## 1. Introduction

## 2. Hyperbolic Propagators

#### 2.1. Motivation

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

**Definition**

**1**

**Lemma**

**1.**

- (i)
- ${\left.\phi \right|}_{x={x}^{*}}=0$,
- (ii)
- ${\left.{\partial}_{{x}^{\alpha}}\phi \right|}_{x={x}^{*}}={\xi}_{\alpha}$,
- (iii)
- ${\left.det{\partial}_{{x}^{\alpha}{\eta}_{\beta}}^{2}\phi \right|}_{x={x}^{*}}\ne 0$,
- (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$$w(t,x;y,\eta ):={\displaystyle \frac{1}{{\left[{\rho}_{g}\left(x\right){\rho}_{g}\left(y\right)\right]}^{1/2}}}{\left[{det}^{2}\left({\partial}_{{x}^{\alpha}{\eta}_{\beta}}^{2}\phi (t,x;y,\eta )\right)\right]}^{1/4},$$$${\left.{\left[{det}^{2}\left({\partial}_{{x}^{\alpha}{\eta}_{\beta}}^{2}\phi (t,x;y,\eta )\right)\right]}^{1/4}\right|}_{t=0}=1\phantom{\rule{0.166667em}{0ex}}.$$

**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$${\omega}_{2}(x,y)-\underset{\u03f5\to {0}^{+}}{lim}\left(\frac{1}{4{\pi}^{2}}\frac{u(x,y)}{{\sigma}_{\u03f5}(x,y)}+\left(\sum _{n=0}^{N}{v}_{n}(x,y)\sigma {(x,y)}^{n}\right)ln({\sigma}_{\u03f5}(x,y)/{\ell}^{2})\right)\in {C}^{N-1}(\mathcal{O}\times \mathcal{O})$$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 $\chi :\mathbb{R}\to [0,1]$,$$\chi \left(\tau \right)=\left\{\begin{array}{cc}1\hfill & \left|\tau \right|\le \frac{1}{2},\hfill \\ 0\hfill & \left|\tau \right|>1.\hfill \end{array}\right.$$Then, there exists a real sequence$$0<{c}_{1}<{c}_{2}<{c}_{3}<{c}_{n}<\dots \to +\infty $$$$v(x,y):=\sum _{n=0}^{+\infty}{v}_{n}(x,y){\sigma}^{n}(x,y)\chi \left({c}_{n}\sigma (x,y)\right)$$$$H(x,y):=\underset{\u03f5\to {0}^{+}}{lim}\left[\frac{1}{4{\pi}^{2}}\frac{u(x,y)}{{\sigma}_{\u03f5}(x,y)}+v(x,y)ln({\sigma}_{\u03f5}(x,y)/{\ell}^{2})\right]$$$$\begin{array}{c}\underset{\u03f5\to {0}^{+}}{lim}{\int}_{\mathcal{O}\times \mathcal{O}}H(x,y)\left[(\square +{m}^{2}){f}_{1}\right]\left(x\right)\phantom{\rule{0.166667em}{0ex}}{f}_{2}\left(y\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\int}_{\mathcal{O}\times \mathcal{O}}k(x,y)\phantom{\rule{0.166667em}{0ex}}{f}_{1}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{f}_{2}\left(y\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(x\right)\phantom{\rule{0.166667em}{0ex}}{\rho}_{\mathfrak{g}}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}dy,\end{array}$$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

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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