Enough freedom if space is only 3D?

[Spinnor]

Say string theory is wrong and there are no extra space dimensions. How does a small, say well above the Plank scale, piece of space have all the freedom of the standard model? It seems string theory gives a nice geometrical picture for a bunch physics. Maybe its just neat idea and wrong, if so is there a "picture" of how a small piece of 3D space can have so much freedom found in all the different elementary fields we know? Can we picture that freedom in 3D?

What goes on at x?

Thanks for any thoughts.

 

[marcus]

A good question to ask. The standard model of particle physics is constructed on 4D Minkowski spacetime. Thus the model of how all the particles arise is based on three spatial dimensions, and one time dimension.

The standard particle model consists of fields defined on that 3+1 dimensional spacetime.

So you can well ask, how can the fields defined on familiar old 3D space (plus usual time) have enough freedom to comprise all the known particles?

I guess one way to think about it is that degrees of freedom do not all have to be spatial dimensions!

 

[Spinnor]

A good question to ask. The standard model of particle physics is constructed on 4D Minkowski spacetime. Thus the model of how all the particles arise is based on three spatial dimensions, and one time dimension.

The standard particle model consists of fields defined on that 3+1 dimensional spacetime.

So you can well ask, how can the fields defined on familiar old 3D space (plus usual time) have enough freedom to comprise all the known particles?

I guess one way to think about it is that degrees of freedom do not all have to be spatial dimensions!

How about a compromise between 4D theorists and > 4D theorists.

Lets say that where there is very little energy in the form of say particles spacetime is "mostly" 4 dimensional but where there are many particles space can change dimension, matter and energy can "puff" up space locally into a higher dimensional manifold, but can quickly revert to mostly 4 dimensional spacetime when matter is gone. For most of spacetime things are mostly quite and boring and empty and "mostly" 4D? So could we say spacetime is mostly 4D but sometimes spacetime is greater then 4D?

Marcus, could you stretch the math, so to speak, and make it work?

Thanks for any thoughts.

 

[marcus]

Say string theory is wrong and there are no extra space dimensions. How does a small...piece of space have all the freedom of the standard model?...

I'll take another shot at your original post's question.
The standard model does not have extra space dimensions. So the answer is that with 3D space you already have all the freedom of the standard model.

As far as I know, string theory being assumed wrong does not have any relevance. String theorists have not yet managed to exactly reproduce the standard model (though there was hope of this for some 20 years). So you could say that the standard model does not NEED either string theory or extra dimensions.

So assuming string to be a wrong approach, as in your post, has no practical effect. In terms of ordinary high energy particle physics, nothing is changed.

=======================

However you have an idea about space dimensionality varying with particle density, or with energy density. I don't remember seeing that idea (at least as you depict it) before.

It's an intriguing idea to speculate with. Suppose it occurred in the early universe. At the start of expansion, the initial big bang instants, a space of very high dimensionality. Dimensionality uncertain (a quantum observable) rapidly thinning out and declining to the usual 3D.

Or taking another line of speculation, what happens to dimensionality at the pit of a black hole?

A common way to measure dimensionality (and the way the mathematician Hausdorff defined it) is to look at how volume increases with radius. If V ~ r³ then space is 3D, but if V ~ r¹⁰¹ then space is 101D. If volume grows extremely rapidly with radius, then dimensionality is extremely high. And the dimensionality of space does not have to be an integer. It can vary continuously and assume a range of non-integer values.

I will let someone else take a shot at this:

Lets say that ...where there are many particles space can change dimension, matter and energy can "puff" up space locally into a higher dimensional manifold, ...
...could you stretch the math, so to speak, and make it work?

I think it's an interesting idea. Especially interesting in the context of where General Relativity breaks down at the pit of BH or the start of BB expansion. Maybe in some mathematical model of these extreme situations, dimensionality diverges or gets very very large! It reminds me of some rather speculative work by Markopoulou and others.

Also there is the varying dimensionality you get in several quantum gravity approaches. In QG you do not set up fixed brittle spatial geometry with a prearranged dimensionality. Dimensionality is something that happens. A quantum observable. A dynamic variable that can change from point to point and from one situation to the next. You get this especially clearly in the CDT work of Loll and Ambjorn. I have a link to a Loll article in my sig---from the Scientific American.

It doesn't match your idea, but at least it has the dimensionality change, so in that sense it has some kinship with your idea. You might have a look if you can find the link in my sig.

 

[arivero]

Perhaps the mother of all papers wondering about how do you fit so many information in points of the space is David Filkenstein "Space Time-Code", but it has been centuries since the last time I read his series of papers so I will not say more of it here.

One could go back to arguments about the information content needed for dynamics, the need of mass, energy, momentum etc and then go back until Newton age. An intermediate point is the first entry of extra dimensions in the game, with Klein-Gordon equation, and the paper of Klein is worthwhile. The concept of rest-mass dissappears: it is simply the momentum in the 5th dimension. Note that quantum mechanics is not really involved, just relatitivity.

$$E^2 - (p_1^2+p_2^2+p_3^2) = m_0^2$$
$$E^2 - (p_1^2+p_2^2+p_3^2+ m_0^2) = 0$$

(Incidentally, this is a good example of the serius kind of crackpottery that internet crackspots never get to bake. It even includes an episody of "Einstein being wrong" and then "Einstein acknowledges he was being bad and suggests to publish the paper")

Of course we still need particles to have energy and momentum, so the "dual space" and its composition, the "phase space", are still there.

This is not usually noticed in divulgative comments, that Kaluza-Klein theory explains electromagnetic field - which enters the metric field - AND inertial mass.