轩轩
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在弯曲空间，两个dirac算子的对易子算出来是什么？
我们知道，在在弯曲空间，两个对矢量场的协变导数算子的对易子是riemann曲率
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《相对论通俗演义》
i will love you till the null infinity.
| 发表时间：2005-06-15, 09:10:48 |
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sage
发表文章数: 1125
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Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
by dirac operator, you mean the covariant derivative on fermions?
| 发表时间：2005-06-15, 09:13:02 |
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轩轩
发表文章数: 1352
武功等级: 易筋经
(第二重)
内力值: 567/567
Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
yes,covariant derivative on fermions.
the question i raised is relate with witten 1981
a new froof of positive mass theorem.
http://zhangxuanzhong.blog.edu.cn/
《相对论通俗演义》
i will love you till the null infinity.
| 发表时间：2005-06-15, 09:28:49 |
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sage
发表文章数: 1125
武功等级: 天山六阳掌
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Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
well, do you just want to know what it is? it is in witten's paper, equation 25.
or are you wondering how to derive it?
| 发表时间：2005-06-17, 16:37:25 |
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轩轩
发表文章数: 1352
武功等级: 易筋经
(第二重)
内力值: 567/567
Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
how to derive it
or where to find a same
http://zhangxuanzhong.blog.edu.cn/
《相对论通俗演义》
i will love you till the null infinity.
| 发表时间：2005-06-17, 21:42:19 |
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sage
发表文章数: 1125
武功等级: 天山六阳掌
(第六重)
内力值: 535/535
Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
It is not clear to me how much you know. I will assume you know something already since you are writing an article on it.
I am not sure if this is enough.
It is easy to write down covariant derivatives for a vector through affine connections.
However, we have difficulty in spinor in general relativity.
There is no way to naturally define a spinor transformation under general diffeomorphism... In other word, spinor does not carry general coordinate indices, they only carry lorentz indices, because we know that spinors naturally transform under lorentz group.
However, there is another way of thinking of general diffeomorphism, it could phrased in terms of local coordinate transformation. In this language, spinors transform naturally under local lorentz transformation. It is precisely like a local gauge transformation, where lorentz indices are now internal gauge indices. Therefore, just as local gauge theories, we have to introduce a gauge field to make the covariant derivatives. This gauge field is called spin connection
This is how we write down the covariant derivative for the spinor. I will not write the form here since it is complicated and involve a lot of upper and lower indices. You can find it in many places. The discussion here is very parallel to the discussion in the string theory book wrote by Green, Schwarz and Witten, volume two, chapter 12.
| 发表时间：2005-06-22, 22:14:07 |
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轩轩
发表文章数: 1352
武功等级: 易筋经
(第二重)
内力值: 567/567
Re: 在弯曲空间，两个dirac算子的对易子算出来是什么？
This gauge field is called spin connection
to me ,this is a new idea. i get it,thanks sage
http://zhangxuanzhong.blog.edu.cn/
《相对论通俗演义》
i will love you till the null infinity.
| 发表时间：2005-06-22, 22:25:56 |
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