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where is the Lagrangian density,

and the integration is over a four-volume R. Similar to the case of particle Lagrangan in Eq.(1a)

Symmetry | Group | Conservation Law |
---|---|---|

Space-time translation | (Translational) Poincare (x^{} x^{} + a^{}) |
Energy-momentum |

Space-time rotation | (Spatial) Lorentz _{} |
Angular momentum |

Interal rotation of complex field | U(1) in scalar field | Particle number |

Internal rotation of lepton or quark field | SU(2) in weak interaction | 3rd component of weak isospin |

Internal rotation of quark field | SU(3) in strong interaction | Number of color charges |

The "Action Principle" for field theories states that if we perform an arbitrary variation of the field, + , subject to the boundary condition

---------- (1k) |

which yields the Klein-Gordon Equation according to Eq.(1k):

---------- (1l) |

are the d'Alembertian operator in 4-dimensional space-time and the Laplacian operator in 3-dimensional space respectively. The repeated dummy index in the equations is understood to be summed over the 4 space-time coordinates.

The field can be expressed in a series expansion in terms of the harmonic functions and the coefficients c

---------- (2) |

---------- (3) |

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