This is to continue the discussion with Ji4 Hou4 Feng1

我以为粒子才是唯一的实在, 场是数学模型, 呵呵. sage 兄有空议论一下?

In terms of physics, there is no big difference between these two statements. Yes, we do start by saying there is particle. A local and Lorentz invariant quantum theory directly leads to quantum fields. On the other hand, if you start from local quantum fields, the most elementary and natural excitations of it are particles. In this sense, it is equivalent to say we have particle or we have fields. It is different, I think, from just a mathematical model, which implies there could be many other possibilities.

Beyond this point, it probably depends on what we mean by reality. This is beyond my ability.

他用了一个三色圆盘来解释规范对称, 当时并没有读懂, 所以现在已经忘记他具体怎么说的了.

顺便问一下, Weinberg 从无质量自旋为1 的粒子出发, 导出规范对称.

This is a proof of we need gauge symmetry to talk about massless spin-1 particles. The basic idea is that a generic Lorentz vector (spin-1) has 4 polarizations. However, to preserve unitarity, the contribution of longitudinal component must cancel the contribution of time-like component. Therefore, there are only 2 physical degree of freedom, the transverse polarizations. That is, a generic consistent theory of massless vectors must have two "wasted" degrees of freedom. How do I make sure that we always end-up with only 2-degrees of freedom? Gauge symmetry is exactly the requirement to make a consistent theory of massless vectors have redundancies, i.e., half of the degrees of freedom are redundant which can be chosen by hand and not fixed by dynamics.

这可以算作 "自旋为 1 的粒子的理论一定是规范场论" 的证明吗?

We need gauge symmetry only if the spin-1 particle is massless. Or, to be more precise, only when the mass of the spin-1 particle is far below the cut-off. Or to be even more precise, if the cut-off the theory is larger than M_V/g (M_V is the mass of the vector, g is the coupling).


他的引力书上对自旋2的无质量粒子做了同样的事情, 可以算作 "自旋为2 的粒子的
理论一定是广义相对论" 的证明吗?

Again, very similar to spin-1 case. And we need diffeomorphism only if the spin-2 particle is massless. The only difference here is that in the case of massive spin-2, the dependence of cut-off on mass is different and quite complicated.

我自然是说无质量整数自旋的粒子.

我记得李淼在他的科普里说,

"弦论的低能有效理论是规范理论和广义相对论现在看来并不奇怪, 因为有以下定理: 包含自旋1(无质量)粒子的可重整化相互作用理论一定是规范理论; 包含自旋2(无质量)粒子的可重整化相互作用理论一定是广义相对论. 但是这个证明是一级一级地证明的, 很难看到其中的物理原理. "

所以我想问的是, Weinberg 从粒子出发构造场论, 在无质量自旋1的时候有些麻烦, 从而必须引入规范对称. 这可以作为李淼提到的这个 "定理" 的证明吗?

"弦论的低能有效理论是规范理论和广义相对论现在看来并不奇怪, 因为有以下定理: 包含自旋1(无质量)粒子的可重整化相互作用理论一定是规范理论;

Not because of renormalizability, but because of unitarity.

包含自旋2(无质量)粒子的可重整化相互作用理论一定是广义相对论.

Strange statement. GR is not renormalizable.

It is no big deal that the effective theory of spin-2 zero modes is GR. However, it is somewhat interesting that strings contain both spin-2 and spin-1 as excitations, thus provides a hope for a unified description.

但是这个证明是一级一级地证明的, 很难看到其中的物理原理. "

Why? The physics is very clear.

所以我想问的是, Weinberg 从粒子出发构造场论, 在无质量自旋1的时候有些麻烦, 从而必须引入规范对称. 这可以作为李淼提到的这个 "定理" 的证明吗?

And, Weinberg provides such a proof.

哦. 不好意思. 那不是李淼的原话, 是我凭记忆写的, 弱智了.

哦. 不好意思. 那不是李淼的原话, 是我凭记忆写的, 弱智了.
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I see. I hope that I have answered you question. :-)

当然, 有些断言因为我还是个门外汉, 不是那么理解. 不过你的回答肯定了我自己的想法. 我是觉得这个问题好像没有那么复杂.