# Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations

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## Abstract

**:**

## 1. Introduction

## 2. Two Quasi-Hermitian Formulations of Quantum Theory

#### 2.1. Non-Hermitian Schrödinger Picture (NSP)

#### 2.2. Non-Hermitian Interaction Picture (NIP)

**Lemma 1.**

**Proof.**

## 3. Samples of Application

#### 3.1. Wheeler-DeWitt Equation

#### 3.2. Closed versus Open Quantum Systems

#### 3.3. Pure States in Dyadic Representation

## 4. Quantum Gravity in a Toy Model

#### 4.1. Classical Singularities

- The classical space-time geometry of the Universe has to remain “next to trivial”. We will employ the not exceedingly revolutionary kinematics working with the non-covariant concept of absolute time. The quantum-theory-controlled evolution of the Universe will then be assumed to be unitary, i.e., unitary in the language of the more or less conventional quantum mechanics of the so-called closed systems.
- In both the classical and quantum settings, the naively physical non-relativistic parameter of time t will be assumed positive and set equal to zero at Big Bang. On the classical non-relativistic level also the 3D spatial coordinates will be assumed time-dependent, therefore, $x=x\left(t\right)$, $y=y\left(t\right)$ and $z=z\left(t\right)$.
- All this would lead to a still nontrivial version of the background independence because the observable values of the spatial nodes $x=x\left(t\right)$, $y=y\left(t\right)$, and $z=z\left(t\right)$ (i.e., say, point-particle positions [47]) have to be defined (i.e., prepared and/or measured) as eigenvalues of operators, in principle at least.
- The last three operators have to be self-adjoint in a physical Hilbert space ${\mathcal{H}}_{phys}$ in which the inner product has the property of being time-dependent and degenerating at $t=0$. In other words, a “non-Hermitian” NIP version of QM will have to be used.

#### 4.2. The Radius of the Universe in a Solvable Toy Model

## 5. The Consistent Model-Building Process

#### 5.1. The First Step: The Construction of the Metric

**Theorem 1.**

#### 5.2. Coriolis Force and the Evolution Equations

**Theorem 2.**

## 6. Discussion

#### 6.1. Conventional Time-Asymmetric QM Concept of the Evolution

#### 6.2. More Realistic Frameworks like Loop Quantum Gravity

#### 6.3. A Broader Physical Context

## 7. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The “multiverse” eigenvalues of the toy-model operator ${R}^{\left(4\right)}\left(t\right)$ of Equation (19) representing the eligible instantaneous size of the quantized Universe expanding after Big Bang.

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**MDPI and ACS Style**

Znojil, M.
Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations. *Axioms* **2023**, *12*, 644.
https://doi.org/10.3390/axioms12070644

**AMA Style**

Znojil M.
Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations. *Axioms*. 2023; 12(7):644.
https://doi.org/10.3390/axioms12070644

**Chicago/Turabian Style**

Znojil, Miloslav.
2023. "Quasi-Hermitian Formulation of Quantum Mechanics Using Two Conjugate Schrödinger Equations" *Axioms* 12, no. 7: 644.
https://doi.org/10.3390/axioms12070644