# Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Quantum Scalar Field on a Curved Space-Time: The Canonical and Precanonical Descriptions

**Ψ**, ${x}^{\mu}=(t,\mathbf{x})=(t,{x}^{i})$ are space-time coordinates, ${g}_{\mu \nu}$ is the space-time metric tensor whose components depend on ${x}^{\mu}$, $g=|det({g}_{\mu \nu}\left)\right|$. In this equation, one uses the space-time coordinates adapted to the space-like foliation such as the induced metric on the space-like leaves of the foliation is ${g}_{ij}$, the lapse $N=\sqrt{{g}_{00}}$ and the shift functions ${N}_{i}={g}_{0i}=0$.

## 3. Relating the Precanonical Wave Function and the Schrödinger Wave Functional

#### 3.1. The Restriction of Precanonical Schrödinger Equation to $\Sigma $

#### 3.1.1. The Total Covariant Derivative

#### 3.2. The Time Evolution of the Schrödinger Wave Functional from pSE

#### 3.3. The Functional Derivatives of $\mathbf{\Psi}$

#### 3.4. The Correspondence between Terms I–V in Equation (14) and the Canonical Hamiltonian Operator in (1)

#### 3.4.1. The Potential Term V

#### 3.4.2. The Second Functional Derivative Term

#### 3.4.3. The Non-Ultralocality Term and the Wave Functional $\mathbf{\Psi}$ in Terms of Precanonical ${\Psi}_{\Sigma}$

#### 3.4.4. The Vanishing Contribution from the Terms I and $IIIa$

- (i)
- the first boundary term is the result of the covariant Stokes theorem and it vanishes if ${\Psi}_{\Sigma}$ vanishes on the boundary $\partial \Sigma $, i.e., the spatial infinity;
- (ii)
- the following three terms follow from the Leibniz rule for the total covariant derivative ${\nabla}_{i}^{tot}$ with respect to the Clifford products of tensor Clifford-algebra-valued functions;
- (iii)
- in the second term, ${\nabla}_{i}^{tot}\left({\gamma}_{0}{\gamma}^{i}\right)=0$ due to the covariant constancy of Dirac matrices (12);
- (iv)
- in the third term, the metric compatibility yields ${\nabla}_{i}\sqrt{h}=0$;
- (v)
- in the fourth term, the explicit formula for $\mathbf{\Phi}\left(\mathbf{x}\right)$ in (31) yields$${\nabla}_{i}^{tot}\mathbf{\Phi}\left(\mathbf{x}\right)=\frac{-\mathrm{i}}{\varkappa}\mathbf{\Phi}\left(\mathbf{x}\right)\left({\partial}_{i}\varphi {\gamma}^{l}{\partial}_{l}\varphi +\varphi {\gamma}^{l}{\partial}_{il}\varphi +\varphi \left({\nabla}_{i}^{tot}{\gamma}^{l}\right){\partial}_{l}\varphi \right).$$$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \phantom{\rule{-10.0pt}{0ex}}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\mathrm{Tr}\left\{\mathbf{\Phi}\left(\mathbf{x}\right)\frac{1}{\varkappa}{\gamma}_{0}{\Psi}_{\Sigma}\left(\mathbf{x}\right)\left({g}^{il}{\partial}_{i}\varphi {\partial}_{l}\varphi +\varphi {g}^{il}{\partial}_{il}\varphi \right)\right\}& =\mathbf{\Psi}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\sqrt{g}\left({g}^{il}{\partial}_{i}\varphi {\partial}_{l}\varphi +\varphi {g}^{il}{\partial}_{il}\varphi \right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =-\mathbf{\Psi}\int \phantom{\rule{-0.166667em}{0ex}}\mathrm{d}\mathbf{x}\phantom{\rule{4pt}{0ex}}\sqrt{h}{\nabla}_{i}^{}\left(\sqrt{{g}_{00}}{g}^{il}\right)\frac{1}{2}{\partial}_{l}{\varphi}^{2}=0,\hfill \end{array}\end{array}$$

## 4. Static Space-Times with ${\mathbf{\omega}}_{\mathbf{0}}=\mathbf{0}$

`prodint`) is used

## 5. Non-Static Space-Times with ${\mathbf{\omega}}_{\mathbf{0}}\ne \mathbf{0}$

## 6. Conclusions

- (i)
- (ii)
- (iii)
- The required cancellation of one of those additional terms with a similar term $II$ in (14), which also has no obvious counterpart in (1), leads to the expression of the Schrödinger wave functional as the trace of the continuous product of restricted precanonical wave functions (40) over all spatial points, which we later interpret as a multidimensional product integral, Equation (41);
- (iv)
- The expression of the wave functional in terms of precanonical wave functions substituted into the second additional term mentioned in (ii) reproduces the second term in the right-hand side of (1), which is responsible for non-ultralocality;
- (v)
- The expression of the wave functional in terms of precanonical wave functions obtained in (iii) also implies that under the boundary conditions of vanishing fields $\varphi \left(\mathbf{x}\right)$ and ${\Psi}_{\Sigma}(\varphi \left(\mathbf{x}\right),\mathbf{x},t)$ at the spatial infinity the terms I and $IIIa$ in (14) do not contribute to the canonical Schrödinger equation (1);
- (vi)
- As a consequence of (i)–(v), in static space-times with ${\omega}_{0}=0$, the functional Schrödinger equation (1) is thus derived from the precanonical Schrödinger equation (2) and the Schrödinger wave functional is expressed as the trace of the product integral of precanonical wave functions;
- (vii)
- In non-static space-times with ${\omega}_{0}\ne 0$, the transformation (55) absorbs the contribution of the term $IIIb$ in (14) thus allowing us again to obtain the functional Schrödinger equation (1) from the precanonical Schrödinger equation (2) and to express the Schrödinger wave functional in terms of transformed precanonical wave functions;
- (viii)

## Funding

## Acknowledgments

## Conflicts of Interest

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Kanatchikov, I.V. Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time. *Symmetry* **2019**, *11*, 1413.
https://doi.org/10.3390/sym11111413

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Kanatchikov IV. Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time. *Symmetry*. 2019; 11(11):1413.
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Kanatchikov, Igor V. 2019. "Precanonical Structure of the Schrödinger Wave Functional of a Quantum Scalar Field in Curved Space-Time" *Symmetry* 11, no. 11: 1413.
https://doi.org/10.3390/sym11111413