# Why 3-momenta + lorentz invariance = large energy?

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• lucas_
In summary: The relativistic dispersion relation is ##E = \sqrt{p^2 + m^2}##. The non-relativistic one is ##E = p^2 / 2m##. Both of them obviously imply that large ##p## means large ##E##.What is its equivalent in terms of the following (in quote)? What are the normal values of ##c_0##, ##c_2## and ##c_4## ?
lucas_
Why is that when there is lorentz invariance. Large 3-momentum corresponds to a large energy. And if there was no lorentz invariance. Large 3-momentum does not necessarily need to correspond to a large energy?

What has Lorentz invariance got to do with 3-momentum having large energy or not?

lucas_ said:
if there was no lorentz invariance. Large 3-momentum does not necessarily need to correspond to a large energy?

Why do you think this is true? In Newtonian mechanics, large 3-momentum does correspond to large energy.

PeterDonis said:
Why do you think this is true? In Newtonian mechanics, large 3-momentum does correspond to large energy.

This is the context:

Demystifier said:
To probe small spatial distances, one needs large 3-momenta. But if Lorentz invariance is emergent at large distances and not fundamental at small distances, then large 3-momentum does not necessarily need to correspond to a large energy. For instance, if the dispersion relation is something like
$$\omega^2=c_0+c_2{\bf k}^2+c_4{\bf k}^4+...$$
with ##c_0=m^2\geq 0##, ##c_2=1## and ##c_4<0##, then one can have small energy ##\omega## for a sufficiently large momentum ##|{\bf k}|##.

Kindly rephrase it because I don't understand the relationship between Lorentz invariance and 3-momentum having large energy or not. Thank you.

lucas_ said:
This is the context

It really, really helps, if you are asking a question based on a post in another thread, to give a link to that post, and quote from it, in your OP, instead of waiting for someone to ask for context.

What you are referring to is, as I've said in other threads, a post describing a speculative model with no evidence in its favor. Discussions of that speculative model (with references to papers in which it is published), as I have already said in other threads, belong in the Beyond the Standard Model forum, not this one.

lucas_ said:
I don't understand the relationship between Lorentz invariance and 3-momentum having large energy or not

That's because you're mixing up speculative hypotheses with actual established physics. As far as actual established physics, based on actual experimental results, is concerned, large 3-momentum does correspond to large energy.

Last edited:
“But if Lorentz invariance is emergent at large distances and not fundamental at small distances...”
The word “if” is important here. If Lorentz invariance does not apply at sufficiently small scales then the relationship between energy and three-momentum (the more of one, the more of the other) that we know and love might break down.

But that’s “if”. No experiment so far as ever come anywhere near the scale where such a thing might be observed, nor given us any reason to think it might happen.

Nugatory said:
the relationship between energy and three-momentum (the more of one, the more of the other) that we know and love might break down.

Note that it's not enough just for Lorentz invariance to no longer hold. As I pointed out earlier, even in Newtonian mechanics, large 3-momentum and large energy go together. The hypothesized dispersion relation that would cause this linkage to be violated is much more of a speculative hypothesis than just "Lorentz invariance breaks down".

PeterDonis said:
Note that it's not enough just for Lorentz invariance to no longer hold. As I pointed out earlier, even in Newtonian mechanics, large 3-momentum and large energy go together. The hypothesized dispersion relation that would cause this linkage to be violated is much more of a speculative hypothesis than just "Lorentz invariance breaks down".

What is supposed to be the normal dispersion relation formula and values that gives large momenta and large energy?

For instance, if the dispersion relation is something like
$$\omega^2=c_0+c_2{\bf k}^2+c_4{\bf k}^4+...$$
with ##c_0=m^2\geq 0##, ##c_2=1## and ##c_4<0##, then one can have small energy ##\omega## for a sufficiently large momentum ##|{\bf k}|##.

PeterDonis said:
Note that it's not enough just for Lorentz invariance to no longer hold
That’s right - perhaps the word “might” needs as much emphasis as the word “if”.

lucas_ said:
What is supposed to be the normal dispersion relation formula and values that gives large momenta and large energy?

The relativistic dispersion relation is ##E = \sqrt{p^2 + m^2}##. The non-relativistic one is ##E = p^2 / 2m##. Both of them obviously imply that large ##p## means large ##E##.

PeterDonis said:
The relativistic dispersion relation is ##E = \sqrt{p^2 + m^2}##. The non-relativistic one is ##E = p^2 / 2m##. Both of them obviously imply that large ##p## means large ##E##.

What is its equivalent in terms of the following (in quote)? What are the normal values of
##c_0##, ##c_2## and ##c_4## ? Where does for example the term
$$c_2{\bf k}^2+c_4{\bf k}^4+...$$ come from?

For instance, if the dispersion relation is something like
$$\omega^2=c_0+c_2{\bf k}^2+c_4{\bf k}^4+...$$
with ##c_0=m^2\geq 0##, ##c_2=1## and ##c_4<0##, then one can have small energy ##\omega## for a sufficiently large momentum ##|{\bf k}|##.

lucas_ said:
What is its equivalent in terms of the following (in quote)?

I have told you multiple times now that questions about the particular speculative hypothesis you are asking about need to be asked in the Beyond the Standard Model forum, not this one. There is even a thread on it in that forum; you've linked to it yourself. If you have questions about it, you can ask them there.

## 1. What is 3-momenta and how does it relate to energy?

3-momenta is a vector quantity that describes the motion of a particle in three-dimensional space. It is related to energy through the equation E = pc, where E is energy, p is 3-momenta, and c is the speed of light.

## 2. How does Lorentz invariance play a role in the relationship between 3-momenta and energy?

Lorentz invariance is a fundamental principle in the theory of special relativity, which states that the laws of physics should remain the same for all observers in uniform motion. This principle ensures that the relationship between 3-momenta and energy is consistent for all observers, regardless of their relative motion.

## 3. Why is it important to consider both 3-momenta and Lorentz invariance when studying energy?

By considering both 3-momenta and Lorentz invariance, we can accurately describe the energy of a particle in any frame of reference. This is crucial in understanding the behavior of particles at high energies, where relativistic effects become significant.

## 4. How does the combination of 3-momenta and Lorentz invariance lead to large energies?

When a particle is accelerated to high speeds, its 3-momenta increases, which in turn increases its energy. Additionally, Lorentz invariance ensures that the energy of the particle remains consistent regardless of the observer's frame of reference, allowing for large energies to be accurately described.

## 5. Can you provide an example of how 3-momenta and Lorentz invariance lead to large energies in a real-world scenario?

One example is in particle accelerators, where particles are accelerated to near the speed of light. As the particles' 3-momenta increases, so does their energy. Additionally, the principles of Lorentz invariance ensure that the energy measurements of these particles are consistent for all observers, allowing for accurate calculations of their large energies.

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