# Foundations of the Quaternion Quantum Mechanics

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## Abstract

**:**

## 1. Introduction

“Either the wavefunction, as given by the Schrödinger equation, is not everything, or it is not right.”

“My own first paper [4] on this subject starts with a summary of the EPR argument from locality to deterministic hidden variables. But the commentators have almost universally reported that it begins with deterministic hidden variables.”

and Murray Gell-Mann statement in his lecture at the 1976 Nobel Conference that“It is safe to say that no one understands quantum mechanics”[8]

are both actual. The deterministic interpretation that was considered as basically equivalent to the Copenhagen orthodox understanding is not in use. It was reasoned that since this theory is empirically indistinguishable from the standard theory, it should be considered an example of “bad science”. S Weinberg in a letter to S Goldstein [10,11] explicitly expressed such a way of thinking:“Niels Bohr brainwashed the whole generation of theorists into thinking that the job (of finding an interpretation of quantum mechanics) was done 50 years ago”[9]

“At the regular weekly luncheon meeting today of our Theory Group, I asked my colleagues what they think of Bohm’s version of quantum mechanics. The answers were pretty uniform and much what I would have said myself. First, as we understand it, Bohm’s quantum mechanics uses the same formalism as ordinary quantum mechanics, including a wavefunction that satisfies the Schrödinger equation, but adds an extra element, the particle trajectory. The predictions of the theory are the same as for ordinary quantum mechanics, so, there seems little point in the extra complication, except to satisfy some a priori ideas about what a physical theory should be like… In any case… we are all too busy with our own work to spend time on something that doesn’t seem likely to help us make progress with our real problems.”

- $\mathbb{R}$, the real numbers,
- $\mathbb{C}$, the complex numbers,
- $\mathbb{Q}$, the quaternions.

“Quaternions came from Hamilton…and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including Clerk Maxwell.”

_{P}, that obey the laws of mass, momentum and energy conservation. Each particle exerts a short-range force at the Planck length. The Kleinert concept linked with the Cauchy model of the elastic continuum has been later analyzed with the arbitrary assumption of the complex potential field [35]. Recently, the Cauchy theory was rigorously combined with the Helmholtz decomposition of the vector field of deformations together with quaternion algebra [36] and such representation of the Cauchy equation of motion produced the Klein–Gordon wave equation [37].

## 2. Methods

#### 2.1. Essentials of the Quaternion Algebra

“The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus are fitted to be of the greatest in all parts of science.”

**Remark**

**1.**

**Remark**

**2.**

#### 2.2. Cauchy Classical Theory of Elasticity

- mathematical analysis of the various phenomena;
- the Cauchy and to the same degree the majority of physical problems cannot be reduced to vectorial models (the vector product does not permit the formulation of algebra with unity, for example, the division operation is not defined [48]);
- the Hamilton algebra of quaternions, $\mathbb{Q}$, and Hamilton concept of the four-dimensional space allow combining Cauchy theory with the Helmholtz decomposition theorem.

- The continuum is treated as a closed system occupying the constant volume $\mathsf{\Omega}\subset {\mathbb{R}}^{3}$.
- The continuum density, ${\rho}_{P}$, is high and we consider the small deformation limit only, ${l}_{P}=const.$, thus the density changes are negligible and ${\rho}_{P}=4{m}_{P}/{l}_{P}^{3}=const.$
- The small deformation limit implies invariant transverse wave velocity: $c=\sqrt{0.4Y/{\rho}_{P}}=const.$, where Y is the Young modulus [46], Equation (16).
- In agreement with the Helmholtz decomposition theorem [46], every lattice deformation
**u**can be expressed as a sum of compression and twist, $u={u}_{0}+{u}_{\varphi}$.

**u**belongs to the ${C}^{3}$ class of functions then $u={u}_{0}+{u}_{\varphi}$, where $\mathrm{rot}{u}_{0}=0$ and $\mathrm{div}{u}_{\varphi}=0$. Upon acting on Equation (14) by the divergence and rotation operators, we decompose it and get well known transverse and the longitudinal wave equations in the usual form ${u}_{tt}=k\Delta u$:

## 3. Results

#### 3.1. The Cauchy Deformation Field in the Quaternion Representation

**Remark**

**3.**

#### 3.2. Quaternion Quantum Mechanics

“On our theory, it (energy) resides in the electromagnetic field, in the space surrounding the electrified and magnetic bodies, as well as in those bodies themselves, may be described according to a very probable hypothesis, as the motion and the strain of one and the same medium (elastic ether).”

“…assumption, therefore, that gravitation arises from the action of the surrounding medium leads to the conclusion that every part of this medium possesses, when undisturbed, an enormous intrinsic energy As I am unable to understand in what way a medium can possess such properties, I cannot go any further in this direction in searching for the cause of gravitation.”

**Remark**

**4.**

#### 3.3. Relativistic Stationary Waves in the Cauchy Continuum: the Quaternion Klein–Gordon Equation

**Remark**

**5.**

#### 3.4. Non-Relativistic Waves in the Cauchy Continuum: the Quaternion Schrödinger Equation

- the overall energy, $E={E}^{0}+Q$, where ${E}^{0}$ and Q are the ground and excess energies, and the overall energy density, ${\rho}_{E}$;
- the equivalent mass interrelated to the wave overall energy;
- the wave mass center and its translation velocity $\upsilon $ [19].

- the existence assumption of the quasi-stationary wave implies an equal duration of the periodic cycles in the whole volume occupied by the wave, $\Delta t=const.$ Consequently implies, that the s- and k-actions are equal$${\int}_{t}^{t+\Delta t}s\left(\tau ,x\right)\mathrm{d}}\tau ={\displaystyle {\int}_{t}^{t+\Delta t}k\left(\tau ,x\right)\mathrm{d}}\tau =\gamma \left(x\right)\Delta t;$$
- the sum of the overall strain, S, and the kinetic energy, K, in relation (31) equals the overall wave energy $E={E}^{0}+Q$, and is time-invariant;
- spans of the strain and the kinetic energy terms are equal,$$\left[0,\mathrm{max}\left\{k\left(t,x\right)\right\}\right]=\left[0,\mathrm{max}\left\{s\left(t,x\right)\right\}\right]=\left[0,{\rho}_{E}\left(t,x\right)/\left({c}^{2}{\rho}_{P}\right)\right]=\left[0,{\rho}_{E}\left(t,x\right)/\left(0.4Y\right)\right].$$

**Remark**

**6.**

- (1)
- the excess, Q, and ground, ${E}^{0}$, energies are entangled in (38) and can’t be separated;
- (2)
- the overall wave energy can be increased by translation term, accordingly also all displacements and velocities are affected;
- (3)
- the wave periodicity implies that by solving the relation (39), one should expect only the discrete values if excess energy Q;
- (4)
- when the wave overall energy equals its ground energy, Q = 0, then relation (38) results in$$0={\displaystyle {\int}_{\mathsf{\Omega}}\left({\rho}_{P}{c}^{2}{\tilde{\sigma}}^{0}\cdot {\left({\tilde{\sigma}}^{0}\right)}^{*}-{E}^{0}\psi \cdot {\psi}^{*}\right)\mathrm{d}x},$$

#### 3.5. Waves in the Time-Invariant Potential Field

#### 3.6. Quaternion Time Dependent Schrödinger Equation

**Remark**

**7.**

## 4. Summary

- The quaternion form of the time-dependent Schrödinger Equation (53) was obtained and the special case when it solves complex Schrödinger equation was demonstrated. Thus, the origin of complex numbers in QM was explained.
- The Klein–Gordon and Poisson equations were derived from assumptions that are independent of the postulates of quantum mechanics and prove the origin of the wave function. The problem of the indefinite probability of the density, present in the classical Klein–Gordon equation, is ruled out in its quaternion form.
- All presented family of the quaternion wave functions is ontic, directly represent a state of elastic continuum showing properties of the Planck–Kleinert crystal.

- Experimental verification to reveal superdeterminism, e.g., using the method proposed by Hossenfelder [59].
- The experimental verification of the particle mass center, Equation (52).
- The avoiding of the assumption of the small deformation limit implies the varying transverse wave velocity. The dependence of the energy density on deformation will result in:
- the extended quaternion form of relation (29) and
- the geometrical theory of gravitation when an invariant velocity of the transverse wave is avoided.

- A reexamination of Schrödinger’s charge density hypothesis. Namely, the Noether theorem can be used to decrease the order of the quaternion form of the Schrödinger equation [60].
- The time-dependent Schrödinger requires the rigorous proof of the quaternion form of the Hamilton–Jacobi equation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

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Label Used in This Work | Planck Constants | Symbol for Unit | Value | SI Unit | Reference |
---|---|---|---|---|---|

Transverse wave velocity | Light velocity in vacuum | c | $2.99792458\times {10}^{8}$ | m∙s^{−1} | [52] |

Lattice parameter | Planck length | ${l}_{P}$ | 1.616229(38) × 10^{−35} | m | [52] |

Poisson ratio | $\nu $ | 0.25 | - | [52] | |

Mass of the Planck particle | Planck mass | ${m}_{P}$ | 2.176470(51) × 10^{−8} | kg | [52] |

Planck–Kleinert crystal density | ${\rho}_{P}$ | 2.062072 × 10^{97} | kg∙m^{−3} | [52] | |

Duration of the internal process | Planck time | ${t}_{P}$ | 5.39116(13) × 10^{−44} | s^{−1} | [52] |

Young modulus | Intrinsic energy density | Y | 4.6332447 × 10^{114} | kg∙m^{−1}s^{−2} | $c=\sqrt{0.4Y/{\rho}_{P}}$ |

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Danielewski, M.; Sapa, L.
Foundations of the Quaternion Quantum Mechanics. *Entropy* **2020**, *22*, 1424.
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