大家对Doubly Special Relativity如何看？又名为Deformed special relativity

我简要说明一下：

自从2001年有人在Phys. Lett. A上发表一篇文章以来，这个领域似乎火起来了，在英国的Nature杂志上都已经出了几篇这类文章。

其基本思想是：我们知道在伽利略相对性基础上加上光速不变原理，就变成了爱因斯坦相对性。如果再加上Planck长度不变性，就得到Doubly Special Relativity。即假定在任何惯性参照系中，时空的最小尺度——Planck长度， 是不变的，即运动方向上的Lorentz长度收缩，在Planck长度那里为零。

这样得到的结果是：粒子的色散关系E^2=p^2+m^2中，右边增加一项跟Planck长度相关的项；光速跟光子波长有关——当然这些只有在Planck标度下效应才明显。人们试图进一步从数学结构来寻找该理论的合理性，例如发现它对应数变形的彭加勒代数。

人们试图用该理论打开一条通往量子引力理论的新途径，试图解释一些宇宙学上的奇特观察结果，试图用它来重新考察重整化，等等。

我在看<比光速还快>的那本科普书中有讲到类似的东西,不知是否是一回事?那是该书作者与Lee Smolin合作的一个成果,他们主要的观点是:量子引力在普朗克尺度起作用,在不同惯性参照系下看应该是一致的,所以大于普朗克尺度的应该在不同惯性参照系下都是大于普朗克尺度的,小于普朗克尺度的应该在不同惯性参照系下都是小于普朗克尺度的,等于普朗克尺度的应该在不同惯性参照系下都是等于普朗克尺度的.由这个观点出发,他们得到了与星空兄上面提到的类似结论.当时我感觉这个想法与Einstein当年的光速不变性有异曲同工之妙,挺物理的,所以VSL(变光速理论)也不是一点没道理的.

Let me say first that I am not an expert on this subject at all. Therefore, what I have said below is only my personal impression. Whoever really interested in this subject should go through original papers or listen to bigger experts.

There has been a lot of effort in studying the Lorentz violating
effect. A lot of those approaches necessarily involve deforming the
dispersion relation of the Special Relativity

E^2=p^2+m^2. (p is spatial momentum, of course)

Notice that candidate of quantum gravity (unfortunately we have only
one which barely qualifies this title: string theory) naturally
modifies dispersion relation into

E^2 = p^2 + m^2 + f( k^2 / M^2)

where k is the four momentum and M is some fundamental scale
(presumably the string scale). This is not Lorentz violation, since
the UV theory (string theory in this case), is manifestly Lorentz
invariant. We expect those things to be there, from an effective
field theory point of view.

A true Lorentz violating deformation will have the form

E^2 = p^2 + m^2 + g(E,p)

where g is some generic (but not Lorentz inv. function of E and
p). Since the Lorentz violation has not been observed, such effects
are constrained to be small. Depending on the specific model, the
deformation part above is either small because it has a small number
in front of it, or suppressed by some large scale, such as the Planck
scale.

There are many ways to introduce Lorentz violation. One systematical
way is through the so called spurion analysis motivated by spontaneous
symmetry breaking. It envisions that the fundamental theory is Lorentz
invariant, and this symmetry is spontaneously broken. In field theory,
we do this by introducing spurions. We first pretend that spurions are
fields which transform in some representation of the symmetry we are
interested in. We proceed, under this assumption, to write down the
most general invariant Lagrangian. Then, we give spurion a vev and the
symmetry is spontaneously broken.

For example, we introduce a_mu (a 4-vector) as a spurion. We could
then write, with a scalar field

a_mu \phi^* \partial^mu \phi

which is Lorentz invariant if a_mu transforms as a Lorentz vector. Now
we give a_mu a vev, i.e., making it a constant vector. Then, this is a
Lorentz symmetry breaking term in the Lagrangian.

We could, in this way, introduce many terms of Lorentz violating terms
this way. Many of them are renormalizable, which means their
coefficient must be small. The others are
non-renormalizable. Therefore, they should at least be suppressed by a
high scale.

Now, we are breaking a global symmetry spontaneously. You should be
asking what happens to the Goldstone. There is very nice story there
which has just been worked out a couple of years ago. This is called
ghost condensation. however, I will not go into details about that here.

This, in my point of view, is the only healthy and well-motivated way
to introduce Lorentz violation, so far.

However, there are also other ways of motivating and organizing the
introduction of Lorentz violation. All of them involving saying
something special about energy (and/or momentum in a lorentz violating
way, of course).

Doubly special relativity is one.

It says that Planck Energy must be invariant in nature. Therefore, we
must extend Lorentz transformation incorporating a new constant, the
Planck constant.

Apart from philosophical reasons, I don't see a motivation for doing
that. The supporters of DSR like to say that this is just like going
from Galilei transformation to Lorentz transformation by introducing a
new constant (speed of light). However, notice that we have a very
good motivation for including the speed of light. After all, there is
Maxwell theory telling us speed of light is a constant. There is no
indication why Plank energy must be a constant.

However, let's admit for now that it is nontheless possible and see its
consequences.

There are many attempts trying to just write various forms of
dispersion relations and transformation laws such that Planck Energy
is invariant. It is possible to do so.

However, such transformation laws are generically problematic, due to
an argument (still has not been fully addressed by the supporters) by
Unruh et al (gr-qc/0308049). Here is the gist of that argument.

We know that linear Lorentz transformation (the one we all know),
denoted as L here, has two fix points in energy, 0 and infinity.
We write DSR transformation as L*F (where F is the modification).
Since E_planck is invariant under L*F, F must map E_planck to
infinity. A polynomial F cannot do it. Therefore, we must have
non-polynomial behavior for F. Therefore, we also
expect the dispersion relation, which is invariant under DSR, also
have non-polynomial behavior in E. Therefore, the theory is not a
finite expansion in powers of derivatives. Therefore, it is
non-local. Moreover, since the non-locality will be integrated over
the world-line, it is not small (could be much larger than the Planck
length). This is problematic.

There are other criticisms of the DSR. Some original proposed DSR
transformations turn out to be just a change of base of the normal
Lorentz transformation (this has a very fancy mathematical name which
I will not mention). Therefore, it is not really that revolutionary.

Since we don't have a clear definition of energy here (E can be
transformed into E^2 in some frame), one could wonder whether an
unambiguous choice is possible. The answer is not clear, but maybe
possible with some work.

So far, I have been only talking about particles worldlines and DSR
transformation on it. One might want to make a field theoretical
Lagrangian which indeed has such dispersion relations. We are looking
for a derivative (probably infinite) expansion with a scale. This fits
of characteristic of the effective Largrangian from non-commutative
geometry. Therefore, a lot of people jumped on and wrote papers on
this subject. So far, it seems to me that not much has been acheived
(in spite of a lot of highly mathematical papers). Even this is done,
it is not clear how much more theoretical foundation it delivers.

Despite some claims, DSR has no root in quantum gravity yet. (Well,
expect loop quantum gravity claims they can do it. However, LQG is not
even a theory yet.)

All in all, with all these mess, I did not see a clean prediction from
DSR (even assume it is somehow healthy). All I see is just some
modification of dispersion relation. If it is not crazy, it will
probably just be some non-renormalizable operators suppressed by
Planck scale which are already there in many earlier consideration
anyway.

谢谢semi兄和sage兄的回复。semi兄提到的那个人，是该理论的重要创始人之一，但不是最原始的创始人。

sage兄的回复信息量很大，抵得上看好多篇文献，很多地方评论击中要害，谢谢你！

除了sage兄回复的，我的个人感觉是：
1）先从美学的角度看，狭义相对论被这样修改之后，理论模样变得很丑陋。
2）我最近在一个非对易几何论文专辑中，看到Planck长度的来源，还真是把测不准关系跟Schwarzschild黑洞半径理论相结合的产物（过去我自己推得这样一个物理解释，以为自己给Planck长度找到了一个物理解释，殊不知Planck长度本来就是这样来的）。因此，Planck长度应该是我们能够测量到的最小尺度：再小下去的话，足够大的能量涨落在足够小的空间范围内产生，以致于微观的Schwarzschild 黑洞产生。这样一来，我觉得Planck长度只是我们能够测量到的最小尺度，而不是时空本身的最小尺度——当然，你可以说，测量不到的就是没有的，或者是没有意义的，但这可能是一个哲学观点（如同量子力学的哥本哈根解释），如果这样，那我们这个宇宙视界之外的其他（平行）宇宙如果存在，由于我们不可观察，对我们而言就等同于不存在。更激进地，对瞎子而言，月亮是不存在的。
因此，试图把Planck长度置于光速那样的不变地位，在物理上也是不合理的。