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

#### 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?

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#### PeterDonis

Mentor
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.

#### lucas_

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

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.

#### PeterDonis

Mentor
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.

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.

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#### Nugatory

Mentor
“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.

#### PeterDonis

Mentor
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".

#### lucas_

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 $|{\bf k}|$.

#### Nugatory

Mentor
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”.

#### PeterDonis

Mentor
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 $E$.

#### lucas_

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 $|{\bf k}|$.

#### PeterDonis

Mentor
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.

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