# Entangled Harmonic Oscillators and Space-Time Entanglement

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## Abstract

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## 1. Introduction

## 2. Two-Dimensional Harmonic Oscillators

#### 2.1. Squeezed Gaussian Function

#### 2.2. Sheared Gaussian Function

## 3. Dirac’s Entangled Oscillators

## 4. Entangled Oscillators in the Phase-Space Picture

## 5. Entangled Excited States

## 6. E(2)-Sheared States

## 7. Feynman’s Rest of the Universe

## 8. Space-Time Entanglement

- The parton picture is valid only for protons moving with velocity close to that of light.
- The interaction time between the quarks becomes dilated, and partons are like free particles.
- The momentum distribution becomes wide-spread as the proton moves faster. Its width is proportional to the proton momentum.
- The number of partons is not conserved, while the proton starts with a finite number of quarks.

## 9. Concluding Remarks

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**One mathematics for two branches of physics. Let us look at Equations (1) and (2) applicable to quantum optics and special relativity, respectively. They are the same formula from the Lorentz group with different variables as in the case of the Inductor-Capacitor-Resistor (LCR) circuit and the mechanical oscillator sharing the same second-order differential equation.

**Figure 2.**Transformations in the two-dimensional space. The object can be rotated, squeezed or sheared. In all three cases, the area remains invariant.

**Figure 3.**Squeeze along the 45 ${}^{\circ}$C direction, discussed most frequently in the literature.

**Figure 4.**Transformations generated by ${Q}_{3}$ and ${K}_{3}$. As the parameter η becomes larger, both the space and momentum distribution becomes larger.

**Figure 5.**Sear transformation of the Gaussian form given in Equation (11).

**Figure 6.**Feynman’s rest of the universe. As the Gaussian function is squeezed, the x and y variables become entangled. If the y variable is not measured, it affects the quantum mechanics of the x variable.

**Figure 7.**Dirac’s form of Lorentz-covariant quantum mechanics. In addition to Heisenberg’s uncertainty relation, which allows excitations along the spatial direction, there is the “c-number” time-energy uncertainty without excitations. This form of quantum mechanics can be combined with Dirac’s light-cone picture of Lorentz boost, resulting in the Lorentz-covariant picture of quantum mechanics. The elliptic squeeze shown in this figure can be called the space-time entanglement.

**Figure 8.**Orthogonality relations for two covariant oscillator wave functions. The orthogonality relation is preserved for different frames. However, they show the Lorentz contraction effect for two different frames.

**Figure 9.**Feynman’s rest of the universe. This figure is the same as Figure 6. Here, the space variable z and the time variable t are entangled.

**Figure 10.**The transition from the quark to the parton model through space-time entanglement. When $\eta =0$, the system is called the quark model where the space separation and the time separation are dis-entangled. Their entanglement becomes maximum when $\eta =\infty .$ The quark model is transformed continuously to the parton model as the η parameter increases from zero to ∞. The mathematics of this transformation is given in terms of circles and ellipses.

**Figure 11.**Entropy and temperature as functions of ${[tanh\left(\eta \right)]}^{2}={\beta}^{2}$. They are both zero when the hadron is at rest, but they become infinitely large when the hadronic speed becomes close to that of light. The curvature for the temperature plot changes suddenly around ${[tanh\left(\eta \right)]}^{2}=0.8$, indicating a phase transition.

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Başkal, S.; Kim, Y.S.; Noz, M.E.
Entangled Harmonic Oscillators and Space-Time Entanglement. *Symmetry* **2016**, *8*, 55.
https://doi.org/10.3390/sym8070055

**AMA Style**

Başkal S, Kim YS, Noz ME.
Entangled Harmonic Oscillators and Space-Time Entanglement. *Symmetry*. 2016; 8(7):55.
https://doi.org/10.3390/sym8070055

**Chicago/Turabian Style**

Başkal, Sibel, Young S. Kim, and Marilyn E. Noz.
2016. "Entangled Harmonic Oscillators and Space-Time Entanglement" *Symmetry* 8, no. 7: 55.
https://doi.org/10.3390/sym8070055