I'll try outline my question in clear terms, articulating specific aspects that are its primary motivators. I'm just beginning in my exploration of physics as a student, but a persistent question that I've been grappling with is this - why does the universe manifest scale?

More precisely, what is scale, in physical terms? Is it an extension of dimension? To be clear, I recognize that there may be an explanation grounded in dimensionality; however, it seems to me that scale does not equate to dimension. In the case of our universe things are contained within a three-dimensional space (at least at the macro level), but scale implies "levels" of containment within the dimensional containment space.

The way I make sense of it is as being akin to the surface area of dimension, but is this accurate from the standpoint of physics? Has the phenomenon of scale been theoretically defined?

The most significant aspect of this for me, is why it is that physics would work "differently" at different scales. The fact that at the macro level we observe behaviors under the theoretical banner of GR/SR while at the micro level QM becomes the rule makes it seem as if scale has some sort of primacy that extends beyond space in the GR/SR sense, because it seems to be setting distinct contexts in which different physics occur. Has this been explained?

Forgive any naivety that may come across due to my inexperience with physics, and for any of the less than rigorous aspects of my questions. I'm sure certain aspects of my question are probably just due to my dearth of knowledge of physics, but I haven't been confronted with what seems like a clear answer to the essential question of just what scale is in the view of physics.

Thank you in advance for any answers and insight on this question!

  • $\begingroup$ I don't think scales have anything to do with spacetime dimensions, at least not directly. It simply says that physics are different in different scales. A more modern understanding is that there're effects only significant in high energy scale, so in lower energy scale you don't see them, and that makes physics different in this scale. This is not necessarily always true tho. Nima's favorite saying is that beyond Planck scale, these hierarchy won't manifest, you get larger and larger a black hole when probing shorter and shorter distance. $\endgroup$ – Turgon Apr 30 '19 at 5:55
  • $\begingroup$ @Turgon You have addressed the first aspect of my question, whether scale is derivable from dimension. But if it isn't, as I suspect it is the case, then why does scale exist at all? This is what I'm driving at in the second part of my question and what I haven't yet found a satisfactory answer to. But your comment about Planck scale implies to me a relationship between distance and scale. Does distance necessitate space-time or is it independent? Thanks for the comment. $\endgroup$ – Daniel C. Lucas Apr 30 '19 at 6:05

A scale is a level of analysis or observation that looks at a system on a certain length (or time) scale, ignoring phenomena much smaller and larger than this length or time. Note that this is something we humans do, not the world! However, it is very useful since physics at different scales does look fairly different.

One way of looking at why physics is different on different scales is to note that different forces and interactions have particular length scales. The most obvious are the strong and weak nuclear forces, but this is also true for van der Waals forces and surface tension in liquids. The reason for the range of the nuclear forces can be explained using the Yukawa potential. Surface tension occurs because of microscopic interactions (intermolecular forces) and hence acts on larger scales up to the scale where other typical forces like gravity overwhelm it. The relative "mix" of forces at different scales hence tends to vary.

The same is true for interactions: the mean-free path of molecules is one length scale that is different from their physical size. Physics below the molecular size is dominated by the quantum interactions, above it the molecule is largely a classical system, and above the mean-free path molecules can be treated diffusing statistically.

In some sense dimensionality affects this since it affects how strongly different forces fall off with distance. On very large scales electromagnetism and gravity, the forces with unbounded range, tends to dominate and they fall off as $1/r^{D-1}$. Dipole forces (magnetic fields and tidal forces) fall off as $1/r^D$, so beyond a certain scale (dependent on how intense the dipoles typically are) they will not be relevant.

Things that remain the same when you rescale the problem are often useful or interesting, hence the interest in the renormalization group that is used to develop physical theories for things that are unchanged by rescaling. Many of these phenomena show non-integer dimensionality $D$ and are hence fractal.


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