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Particle

Newtonian Mechanics

Relativistic Theory

Quantum Mechanics

Quantum Field Theory

The change of heat energy dQ can be further specified for isothermal process (where the temperature T = constant, i.e., in equilibrium) as : dQ = TdS + dN, where dS is the change of entropy (randomness), is the chemical potential associated with the change of state such as phase change etc, and dN the change of particle number in the system. | |

## Figure 01 Energy of a System [view large image] |
While dW = pdV signifies the change of volume V in response to pressure p. |

- Galaxy - The galaxies in the universe become points in the pressure-less fluid. Cosmology is a macroscopic domain in which those points are lumped into the density = Nm/V, where m is the averaged mass of the galaxies, N the total number, and V the volume of the universe. The size to system scale ratio R ~ 10
^{-6}. The dynamics is governed by the "Friedmann Equation". - Planet - Its behavior is summarized in the Kepler's Law derived from Newtonian mechanics, and characterized by its mass m with the ratio R ~ 10
^{-7}. - Electron - The electrons are considered as points in the hydrogen atom and other atomic nuclei. It is parameterized by its mass, charge, and spin. It is governed by the Schrodinger Equation, and can be solved analytically in hydrogen atom. The ratio R ~ 10
^{-7}similar to the planetary motion since the interaction is long range for both cases with 1/r^{2}dependence. - Quark - The point-like quarks are mediated by gluons to form nucleon. It is supposed to be

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## Figure 02 Particle, Definition |
characterized by its electric charge, mass, spin, and color charge. The ratio R ~ 10^{-3}, which is much larger because the strong interaction in this case is a short range force. Actually, not much is known about this system. |

- Classical Mechanics : E = E
_{p}+ E_{k}for bound or free state. - Relativity Theory : E = (m
^{2}c^{4}+ p^{2}c^{2})^{1/2}for free particle. - Non-relativistic Quantum Mechanics : E = E
_{p}+ E_{k}for bound or free state. - Quantum Field Theory : E = (m
^{2}c^{4}+ p^{2}c^{2})^{1/2}for free particle.

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## Figure 03 The 4 Domains of Theory [view large image] |
Obviously, the expressions are different between non-relativistic and relativistic theories. |

## Figure 04 Escape Velocity |

The mainly difference is the existence of two different kinds of charge, i.e., the positive (+) and negative (-) varieties. As shown in Figure 05, like charges are repulsive while opposite charges are attractive to each other. Thus, the letter case behaves the same as gravity with GMm replaced by Qq. In case Q = - q = e (the charge of the electron), E_{p} = - e^{2}/r. For the case of like charge E_{p} = + e^{2}/r, there is no bound state (Figure 06).
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## Figure 05 Electro-static Forces [view large image] |
## Figure 06 Electro-static PE |

. Then E = kx ^{2}/2 + mv^{2}/2 which would never become negative. Instead, the degree of localization of the particle is determined by the boundary a = (2E/k)^{1/2}, which is obtained by setting v = 0 in E (Figure 07(b)). It also shows the values of E_{p} and E_{k} inter-changed between x = 0 and a. The same kind of energies swapping also occurs in planetary motion Between perihelion and aphelion (Figure 07(a)).
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## Figure 07 Potential Curves [view large image] |

## Figure 08 4-Momemtum [view large image] |
In addition, the velocity of the particle v > c for p^{2} > |p_{0}|^{2} with m^{2} < 0 (Figure 08). It implies that the particle becomes a "tachyon", which does not exist. |

## Figure 09 Non-relativistic Quantum Mechanics [view large image] |
Analytic solutions have been obtained for many potential forms including the hydrogen atom and harmonic oscillator. The novel features include discrete energy level and there is a certain probability for penetrating the boundary (Figure 09). |

Quantum Field Theory (QFT) is devised for studying high energy particle collision. As such, the energy has to be relativistic. The field equation is the starting point for its formulation. The Maxwell's Equations from the era of 1800's are the most ancient one. It is now assigned to cover the photon. The Klein-Gordon Equation for the spin-0 particle (such as the Higg's) was derived in 1926 by quantizing the relativistic energy equation E^{2} = m^{2}E^{4} + p^{2}c^{2} according to the rules mentioned earlier (also see "Klein-Gordon Equation of the Scalar Field"). This formulation avoids the issue of negative energy by using the squared version. In 1928 P.A.M. Dirac tried to resolve the problem with negative energy by splitting the energy equation into two linear forms (see "Derivation of the Dirac Equation and the Weyl Spinor"). The end products are two coupled linear equations, one of which turns out to associate with negative energy in the wave function (see "Interpretation of the Dirac Equation"). He then invented a sea of un-seen particles filling out all the negative energy levels in an attempt to solve the problem. The accepted interpretation now is to identify the negative energy particle with positive energy "anti-particle" moving backward in time (see the colorful illustration below). It is something like treating the -|E| traveling wave exp[(px+|E|t)/_{}] = exp{[(px-(-|E|)t]/_{}]} = exp{[px-|E|(-t)]/_{}}. This is known as "Feynman–Stueckelberg interpretation". A lot of questions arise since then about "moving backward in time" like going back to the Jurassic Era (Figure 10) via such mechanism (aka Time Machine).
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## Figure 10 Time Machine |
_{} Feynman diagram for electron-positron annihilation. |

A particle with charge q in a constant magnetic field B moves in circular path according to the formula for the angular velocity d/dt, i.e.,

## Figure 11 Negative Energy=Anti-Particle, and Backward Time [view large image] |
In this example, the circular orbit is equivalent to the path of the traveling wave, the anti-fermion with charge "-q" and "-t" behaves similarly to a normal fermion. The negative time has no practical significance and can be banished as just a mathematical trick (Figure 11). |