# Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications

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## Abstract

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**PACS**03.65.Ge; 03.65.Pm

## 1. Introduction

**squeeze**transformations [1].

## 2. Lorentz or Squeeze Harmonics

## 3. The Physical Origin of Squeeze Transformations

- In 1927 [8], Dirac pointed out the time-energy uncertainty should be taken into consideration for efforts to combine quantum mechanics and special relativity.
- In 1945 [9], Dirac considered four-dimensional harmonic oscillator wave functions with$$exp\left\{-\frac{1}{2}\left({x}^{2}+{y}^{2}+{z}^{2}+{t}^{2}\right)\right\}$$
- In 1963 [10], Dirac constructed a representation of the (3 + 2) deSitter group using two harmonic oscillators. This deSitter group contains three (3 + 1) Lorentz groups as its subgroups.

## 4. Further Properties of the Lorentz Harmonics

#### 4.1. Lorentz-Invariant Orthogonality Relations

#### 4.2. Probability Interpretations

## 5. Two-Mode Squeezed States

## 6. Time-Separation Variable in Feynman’s Rest of the Universe

When we solve a quantum-mechanical problem, what we really do is divide the universe into two parts - the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the entire universe. To motivate the use of density matrices, let us see what happens when we include the part of the universe outside the system.

## 7. Lorentz-Covariant Quark Model

#### 7.1. Feynman’s Parton Picture and Feynman’s Decoherence

- The picture is valid only for hadrons moving with velocity close to that of light.
- The interaction time between the quarks becomes dilated, and partons behave as free independent particles.
- The momentum distribution of partons becomes widespread as the hadron moves fast.
- The number of partons seems to be infinite or much larger than that of quarks.

#### 7.2. Proton Form Factors and Lorentz Coherence

## 8. Conclusions

## References

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**Figure 1.**Space-time picture of quantum mechanics. In his 1927 paper, Dirac noted that there is a c-number time-energy uncertainty relation, in addition to Heisenberg’s position-momentum uncertainty relations, with quantum excitations. This idea is illustrated in the first figure (upper left). In his 1949 paper, Dirac produced his light-cone coordinate system as illustrated in the second figure (upper right). It is then not difficult to produce the third figure, for a Lorentz-covariant picture of quantum mechanics. This Lorentz-squeeze property is observed in high-energy laboratories through Feynman’s parton picture discussed in Section 7.

**Figure 2.**Orthogonality relations for the covariant harmonic oscillators. The orthogonality remains invariant. For the two wave functions in the orthogonality integral, the result is zero if they have different values of n. If both wave functions have the same value of n, the integral shows the Lorentz contraction property.

**Figure 3.**Localization property in the $zt$ plane. When the hadron is at rest, the Gaussian form is concentrated within a circular region specified by ${(z+t)}^{2}+{(z-t)}^{2}=1.$ As the hadron gains speed, the region becomes deformed to ${e}^{-2\eta}{(z+t)}^{2}+{e}^{2\eta}{(z-t)}^{2}=1.$ Since it is not possible to make measurements along the t direction, we have to deal with information that is less than complete.

**Figure 4.**The uncertainty from the hidden time-separation coordinate. The small circle indicates the minimal uncertainty when the hadron is at rest. More uncertainty is added when the hadron moves. This is illustrated by a larger circle. The radius of this circle increases by $\sqrt{cosh\left(2\eta \right)}$.

**Figure 5.**Lorentz-squeezed space-time and momentum-energy wave functions. As the hadron’s speed approaches that of light, both wave functions become concentrated along their respective positive light-cone axes. These light-cone concentrations lead to Feynman’s parton picture.

**Figure 6.**Coherence between the wavelength and the proton size. As the momentum transfer increases, the external signal sees Lorentz-contracting proton distribution. On the other hand, the wavelength of the signal also decreases. Thus, the cutoff is not as severe as the case where the proton distribution is not contracted.

**Table 1.**Cylindrical and hyperbolic equations. The cylindrical equation is invariant under rotation while the hyperbolic equation is invariant under squeeze transformation

Equation | Invariant under | Eigenvalue | ||

Cylindrical | Rotation | $\lambda ={n}_{x}+{n}_{y}+1$ | ||

Hyperbolic | Squeeze | $\lambda ={n}_{x}-{n}_{y}$ |

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Kim, Y.S.; Noz, M.E.
Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications. *Symmetry* **2011**, *3*, 16-36.
https://doi.org/10.3390/sym3010016

**AMA Style**

Kim YS, Noz ME.
Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications. *Symmetry*. 2011; 3(1):16-36.
https://doi.org/10.3390/sym3010016

**Chicago/Turabian Style**

Kim, Young S., and Marilyn E. Noz.
2011. "Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications" *Symmetry* 3, no. 1: 16-36.
https://doi.org/10.3390/sym3010016