kanex
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丢失了不少帖子，重新问QFT问题，望昌海兄也看看
(1) spin 1, 3/2, 2....粒子的Lagrangian是怎么写出来的呢？0, 1/2的方法挺简单，再上去的话书上就没说了。
(2) local gauge invariance很令人奇怪，现在有没有更深层的解释呢。我们知道，微观与宏观不一定一样.......例如metric和topology。
江畔何人初见月`江月何年初照人`
| 发表时间：2005-03-22, 08:09:15 |
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轩轩
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
we get the lagrangian from the field equation.
the field equation is from the group thoery and the lorentz invariance.
http://zhangxuanzhong.blogone.net
我的主页
(2004-06-01 13:58:27) 轩轩
http://dzh.mop.com/topic/readSub.jsp?sid=5229875#
| 发表时间：2005-03-22, 08:19:01 |
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kanex
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
上面的方法我也大致可以确定，但没见过有哪里有具体推导。
江畔何人初见月`江月何年初照人`
| 发表时间：2005-03-22, 08:26:12 |
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sage
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
(1) spin 1, 3/2, 2....粒子的Lagrangian是怎么写出来的呢？0, 1/2的方法挺简单，再上去的话书上就没说了。
what do you mean? spin-1 is gauge theory. very very well explained in all quantum field theory books.
Spin 2 is called general relativity. it is again everywhere.
spin 3/2 is called gravotino. you could find it in Wess and Bagger 'supersymmetry and supergravity''
we do not know how to write down a local, lorentz invairant, theory with higher spins.
(2) local gauge invariance很令人奇怪，现在有没有更深层的解释呢。我们知道，微观与宏观不一定一样.......例如metric和topology。
In simple words, if we just try to write naive theories with spin-1 particles, the theory will give rise to non-sensical result due to extra unphysical degrees of freedom. The only way we know to deal with it is to use some symmetry to get rid of those unphysical degrees of freedom. If you want to get rid of a field, the symmetry better be local since a field is local.
| 发表时间：2005-03-25, 19:28:16 |
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kanex
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
Make myself clear: I mean free Lagrangian.
江畔何人初见月`江月何年初照人`
| 发表时间：2005-03-26, 01:01:14 |
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sage
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
Make myself clear: I mean free Lagrangian.
=========================================
It is still not clear what you want. take spin 1 as an example:
by 'free langrangian', do you mean free from interaction with matter or completely free? the lagrangian for a massless spin-1 gauge field without interactions with matter is trivial,
starting from a vector potential A, define field strength F=dA. then the lagrangian is just
1/4 F^2.
however, except for U(1) gauge theory, this is not a free lagrangian. For non-abelian gauge field, this term includes self-interaction of the gauge fields.
| 发表时间：2005-03-27, 13:59:15 |
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kanex
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
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。
先不考虑自然界中spin 1/2和spin 1粒子的千丝万缕的联系。
江畔何人初见月`江月何年初照人`
| 发表时间：2005-03-28, 11:24:42 |
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sage
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Re: 丢失了不少帖子，重新问QFT问题，望昌海兄也看看
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.
先不考虑自然界中spin 1/2和spin 1粒子的千丝万缕的联系。
| 发表时间：2005-03-28, 20:18:35 |
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龙