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Second quantization can be shown in a more concise way without taking care of all kinds of components by using the the scalar field as an example. The general solution for the Klein-Gordon equation of Eq.(18) is just the expansion :

----------(54) |

- = i ---------- (55)

leading to the commutative relations after integrating over all space :

---------- (56) |

.

where the column matrix

If a number operator N

N

where n

c

c

Thus c

---------- (57) |

In particular, the vacuum state is:

---------- (58) |

Since c

by virtue of the commutative relation in Eq.(56). In case when k = k', Eq.(59) becomes :

c

which suggests that two or more scalar particles can occupy the same state. This turns out to be the property of all bosons having integral spin and they all obey the Bose-Einstein statistics.

For spin 1/2 particles, the Dirac spinor can be expanded as :

---------- (61) |

---------- (62) |

b

by virtue of the anticommutative relation in Eq.(62). In case when p = p', Eq.(63) becomes :

b

which can be true only for |2(p) = 0. Therefore, two spin 1/2 particles cannot occupy the same state - a characteristics of Fermi-Dirac statistics.

In terms of the b, b operators, the probability density can now be re-defined as charge density. The total charge operator is in the form :

---------- (65) |

Interaction between particles in quantum field theory is usually expressed by the interacting Hamiltonian. For example, the interaction between the pion (spin 0 particle) and nucleon (spin 1/2 particle) is H

theory in which the main concern is about energy levels, the quantum field theory works mostly with transition probability from an initial state i to the final state f, e.g., S = f
| H_{I} | i (known as the S matrix). Since the equation cannot be solved analytically with the addition of interaction, perturbation theory is employed to do the job approximately. Each term in the perturbation expansions can be represented by a
| |

## Figure 07e Nucleon Scattering [view large image] |
Feynman diagram, which has a mathematical correspondence to compute the S matrix. Figure 07e shows a simple Feynman diagram for nucleon-nucleon scattering via the exchange of virtual pion. |

It may have been noticed that the anti-particle in the Feynman diagrams within Figure 03 is represented by an arrow pointing backward in time. This is because the signs of energy E and time t are inter-changeable without altering the uncertainty rule in Eq.(2). Therefore, an anti-particle is represented by an arrow backward in time instead of labeling it by -|E| every time. It doesn't mean that anti-particle can be used to do time travel back to the past.

Another weird phenomena in Figure 03 is the entanglement associated with the particle pair created in many of the processes. By the law of conservation of angular momentum, no matter how far apart are the members of this pair, if the spin is flipped for one of the member, the spin for the other member will also be flipped to the opposite at precisely the same moment. This non-local influence (non-locality) occur instantaneously, as if some form of communication which Einstein called a "spooky action at a distance", operates not just faster than the speed of light, but infinitely fast. Nonetheless, it is now a routine occurrence in quantum computing (see also the refutation of Einstein's objection by Bell's theorem).

See "The Different Perspectives of the 1st and 2nd Quantizations".

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