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Spin-1 Lagrangian
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> free spin 1指proca lagrangian，不知道是怎么在数学上推出来的。就好像 > dirac lagrangian为什么描述的是spin 1/2的自由粒子呢。 These are in the Standard field theory books. here is short summary For spin-1, first we decide which field to use. Since it is spin-1, we know its wave-function must include l=1 spherical harmonics. Therefore, by looking at those l=1spherical harmonics, we know it must be space vectors. however, just space vector cannot be lorentz invariant. therefore, it must be a four vector, A. let's worry about massless case first. In this case, we know that p^2=0. Therefore, we could just write down something like L=[(\partia_mul) A^mu]^2. This way, the equation of motion is \partial^2 A=0 whose Fourier transformation gives p^2 A=0 However, there could be other forms which also works, such as (\partial_mu A_nu)(\partial^nu A^mu), etc. Great simplification occurs when we demand that there is a gauge symmetry A->A+\partial \phi This tell us the the Lagrangian should be expressed in terms of F_{mu nu}=\partial_mu A_nu - \partial_nu A_mu Therefore, we have the Lagrangian for a massless spin-1 particle with gauge symmetry as L=1/4 F^2. What happened is we have a massive spin-1. the most naive thing one can do is just add a mass term, we have then 1/4 F^2 -1/2 m^2 A^2. Fourier transformation shows that p^2-m^2=0 which is consistent with a massive spin-1. This is the Proca Lagrangian. However, this is a very bad one. It breaks the gauge symmetry explicitly. Therefore, there is no reason why there is no additional terms such as (\partial A)^2 (this term also explicitly breaks gauge symmetry). This lagrangian is also not renormalizable. There is a similar story with spin-1/2. Basically, lorentz invariance, mass-shell condition and spin-1/2 fixs everything. |