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May 5, 2024

Modification of Gravity: Motivations

- sage -

I felt that I still needed to say something about the motivations. Some of the materials have already been covered by Changhai's article on cosmological constant and Higgs mechanism.

I am not being very detail and happy to answer any questions.

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Modification of Gravity

Einstein's general relativity is an extraordinary success. There are
only a couple motivations to modify it. Let me briefly review them.

1)Overview of Darkside of the Universe.

Amazing advance of modern cosmological observations taught us that the
majority of the
energy density of our universe is dark (do not interact with photon
directly ). It appears that about 30 percent of it is made of particles
which are weakly interacting with each other. We cannot find a
candidate for those particals in the presently known list of
particles. Therefore, their identies remain a mystery. Even stranger
is that it seems that 70 percent of the is the so called dark
energy. The only known property of dark energy is that it seems to
dominate the energy density of the current universe and accelerate the
expansion. And, on the other hand, that energy density of very small
(although it is still much bigger that the energy density of the
matter which you and I are made of). The density is about 10^-12
eV^4. Or the energy scale associated with it is about 10^-3 eV.

Since both of these effects are observed through gravity, they provide
motivations for people to modify gravitational interactions.

2)Overview of Hierarchy Problem

The vacuum of quantum field theory is full of mystery. Classically,
one could set them to any value. On the other hand, there are quantum
fluctuations. In a generic quantum field theory, we expect the
spectrum extends all energy scales. Therefore, we expect fluctuations
of modes of all energy scales. The vacuum would contain all of those
fluctions. Therefore, the vacuum energy should be the sum of the
energy containing all of those fluctuations. As a result, doing it in
this naive way, the vacuum energy is divergent.

A more sophisticated modern point of view is that we do not understand
the quantum field theory to arbitrarily high energies. Actually, the
current form of theory we use would have to be modified everytime some
new energy scale is reached where some new states (or new physics )
would have to be included. Therefore, our current theory is only good
for certain energy range, or up to a cutoff of energy
scale. Therefore, the vacuum energy computed using our theory should
only include flutuations below that scale. Therefore, the value of
vacuum energy should be just, from dimensional analysis, Lambda^4,
where Lambda is the cutoff.

Now, the quantum field theory we use is the so called Standard Model
of particle physics. We through experimental data, think we understand
it to about 10^12 eV. Therefore, the vacuum energy should be about the
same size. Actually, the vacuum energy would contribute to the dark
energy I mentioned about since it would make the universe expand
faster. On the other hand, we see a clear problem here. Namely, the
contribution from vacuum energy is too big. It would be 10^48 eV^4,
which is 10^60 more than the measured value mentioned earlier. This is
a version of the so called cosmological constant problem, which is
essentially a version of a so called hierarchy problem. It stems from
the fact the the vacuum energy of the universal seems to operate on a
much lower scale than the other scales we think we know from particle
physics.

There is also another version of the hierarchy problem. The Standard
Model of particle physics relies on the a broken gauge symmetry called
electroweak symmetry. The most straight forward way of breaking it is
the so called Higgs mechanism. It requires the existence of a boson
with a mass of about 10^11 eV. Classically, again, one could pick the
mass as a free parameter. On the other hand, as Higgs particle
propagates in vacuum that is not really empty, its mass would get
contributions from all kinds of vacuum fluctuations. Again, if we just
add up all those fluctuations, the mass of the Higgs particle would be
modified to Lambda^2, where Lambda is the cutoff of the quantum field
theory contains Standard Model only. Now, what is the cutoff of the
Standard Model? Remember cut-off of a theory is supposed to be the
place where new physics supposedly come in. Standard Model already
described all the interactions we know of except gravity. The only
possibility of the next scale where new physics set in seems to be the
scale of gravity, which is determined by Newton's constant to be 10^27
eV. Therefore, following this argument, Higgs mass seems to be 10^27
eV as well, which is much much bigger than the one that is needed by
standard Model, which is 10^11 eV. Here is another hierarchy
problem. It seems to hard to understand why gravity have a much higher
energy scale than the other Standard Model physics. Or in other words,
why gravity is much weaker than others.

We see that both form of the hierarchy problem would possibly have
something to do with gravity. Therefore, they all provide motivation
to modify gravity and give different types of modifications.

> 对此，我有个想法：如果宇宙是有限的，是否意味着计算真空能量之和时，得用离散求和代替连续和（积分）？即不是对频率积分而是求离散和？
> 量子场论中的量子化，是将场的组成离散化，即场由一份一份的场量子组成，但每个场量子的能量，或频率和动量仍然是可以连续变化的。而通常量子力学中的“量 > 子化”或离散化，常常是由于粒子被局域于某个空间范围内时所带来的频率和波矢量的离散化。——当然，如果场在某个维度上被局限，则是“双重量子化”的：场由
> 一个一个的场量子组成，并且每个场量子的频率或局域空间方向上的波矢分量也是离散变化的。事实上，在卡西米尔效应中，两金属板之间的场就是这种“双重量子 > 化”的，而板子外边的场仍然是一般的量子场，因此才存在差异，有可观察效应。进去观察的总是差值而不是绝对值。
> 如果宇宙有限，求真空能量时，或许除了对粒子数求离散和，对频率也是求离散和，那样会压低。由连续到离散，有时候甚至是从发散到收敛的转变。当然“宇宙有 > 限”不是通常说的那种有限，毕竟它还是无边的。

thank you for the comment. the cosmological constant problem specifically means that we want to understand why our space-time appear to only have a very small cosmological constant. To this end, I don't think make it finite will work. Suppose we only have free field in a box, although the spectrum is discretized, the vacuum energy is still divergent. It will be interesting if you could give an example that actually convergent. By this way, just making it convergent is not necessarily enough. We have to explain why it is small as well.

making our space time finite (which is actually proposed by some groups as a way of understanding some seeming strange structure in the cosmic microwave background recently) will in some sense introduce another hierarchy problem the size of the spacetime would at least have to be the current universe which is very large (hence corresponds to a very small energy scale). This is also hard to understand since it is very different from other scales in physics.

> 宇宙有限（即空间在大尺度上有 cutoff）的情形一般来说只能影响理论的红外行为。但量子场论及宇宙学常数的计算中最令人头疼的是紫外区域，它
> 空间在大尺度上的结构并不敏感。影响理论紫外行为的手段之一是改变空间在小尺度上的结构（比如引进最小体积的概念）。但只依靠这种手段虽可以解决紫外发
> 问题，却不足以解决宇宙学常数如此微小这一观测事实。 (Comment supplied by Changhai)

二零零五年六月十五日