Many years ago, I convinced myself absent any real or formal background, that there should exist a smallest volume. In other words, a volume so small that if one were to get any smaller nothing physical would be able to fit inside of it. If this were true, this could have some significant physical implications.

For instance, force could not exist at this smallest volume. Matter would simply exist or not exist at each volume, since their is no region of space to propagate transfer.

Matter would be comprised of only a single type of energy, and all other forces would be merely distributions of it. So a negative charge may actually be a complete lack of this matter, and a proton would be an over distribution of it.

A lack of matter would instead be an even distribution of it.

Would there be any mathematical way to approximate the smallest possible volume?

If you are saying only the math part, yes models do exist, double special relativity, loop quantum gravity, string theory etc. Especially since the success of quantum mechanics, people have been trying to quantize everything including time.

"According to loop quantum gravity, space is made of discrete atoms each of which carries a very tiny unit of volume. … One consequence of this is that there is a smallest possible volume. This minimum volume is miniscule – about 1099 of them would fit into a thimble. If you tried to halve a region of this volume, the result would not be two regions each with half that volume. Instead, the process would create two new regions which together would have more volume than you started with. We describe this by saying that the attempt to measure a unit of volume smaller than the minimal size alters the geometry of the space in a way that allows more volume to be created."

(Smolin, Lee. Three Roads to Quantum Gravity. New York: Basic Books, 2001. p. 106)

Awesome! Thanks I will definitely buy this.

I was thinking that if that if this is true, then one could construct a brownian diffusion like model which would predict the strengths or electroweak and electrostrong forces. It would assume that what we consider an absence of matter is actually an even distribution of these smallest elements. And all other forces are simply density waves of it.

twistor59 wrote:As I said above, highly speculative, tentative and unsupported by experiment at this stage....

I understand the need for empirical conformation. However, maybe we already have it and are just looking at things the wrong way. e.g. particle wave duality...

Quantum Gravity: http://plato.stanford.edu/entries/quantum-gravity/

OnCue wrote:

twistor59 wrote:As I said above, highly speculative, tentative and unsupported by experiment at this stage....

I understand the need for empirical conformation. However, maybe we already have it and are just looking at things the wrong way. e.g. particle wave duality...

I have my thoughts on it but I'm no physicist but I think the data shows evidence for both a smooth and discrete universe, as you stated, how you look at it. For instance, those 2 photons at different energies from the same source several billion light years away that hit the Fermi satellite at the same time points t a smooth universe. Other observations point to a discrete universe. I posed a thought question to a physics friend of mine which was, "What if the universe is at least both smooth and discrete like a photon is both a wave and a particle?" I see I'm not the only one who's thought of that . I think there's an underlying reality that we need to tease out before we can truly answer the question of smallest volume, if it exists o doesn't.

Sovereign wrote:

OnCue wrote:

twistor59 wrote:As I said above, highly speculative, tentative and unsupported by experiment at this stage....

I understand the need for empirical conformation. However, maybe we already have it and are just looking at things the wrong way. e.g. particle wave duality...

I have my thoughts on it but I'm no physicist but I think the data shows evidence for both a smooth and discrete universe, as you stated, how you look at it. For instance, those 2 photons at different energies from the same source several billion light years away that hit the Fermi satellite at the same time points t a smooth universe. Other observations point to a discrete universe. I posed a thought question to a physics friend of mine which was, "What if the universe is at least both smooth and discrete like a photon is both a wave and a particle?" I see I'm not the only one who's thought of that . I think there's an underlying reality that we need to tease out before we can truly answer the question of smallest volume, if it exists o doesn't.

Well said

OnCue wrote:"If you tried to halve a region of this volume, the result would not be two regions each with half that volume. Instead, the process would create two new regions which together would have more volume than you started with. We describe this by saying that the attempt to measure a unit of volume smaller than the minimal size alters the geometry of the space in a way that allows more volume to be created."

(Smolin, Lee. Three Roads to Quantum Gravity. New York: Basic Books, 2001. p. 106)

Seems very ackward and removes one of the advantages of discrete space-time. One of the issues with uniting QM and relativity is that some non-Borel sets appear, which then allow Bannach-Tarski type issues. I.e. you get incongruent measures. If you have a discrete space time, then this issue goes away, simply because you don´t get a BTP for countably infinite sets. Smolin here seems to introduce an incongruent meaure from the get go.

My initial response to the OP is that, from a quantum mechanical perspective, the question dissolves into meaninglessness once we are talking about small enough measures. What, for instance, can be meant by "nothing physical would be able to fit inside of [the smallest volume]", when anything that small can only be described in terms of a wave function that, in the absence of infinite potential barriers (which we don't believe exist), extends throughout all of space. Indeed, even for macro objects like a pea inside a sealed tin can, there is a non-zero probability that the pea may be anywhere else in the universe, outside the tin can, so we could say the pea doesn't "fit in the can" because its wave function is not confined to the can, in the same way as we would for the infinitesimally small volume being considered.

I've never read any loop quantum gravity, but Smolin is a reputable physicist, so I can only assume that the reason he writes about the possible existence of a minimum volume is that perhaps loop quantum gravity overrides quantum mechanics under some circumstances. My understanding is that loop quantum gravity is a highly speculative and entirely hypothetical field at present, whereas QM is one of the most highly experimentally verified theories around. That is not to say that LQG may not be right, just that it's speculative.

There are three possibilities:

i. Space has fundamental parts, and these are 3-dimensional minima.
ii. Space has fundamental parts, and these are 0-dimensional points.
iii. Space doesn't have any fundamental parts, and all 3-dimensional parts of space have 3-dimensional (proper) parts ad infinitum.

Talking about space as having 'parts' appears to presuppose that space is a physical object, rather than just a mathematical model we use to express relationships between physical objects. I tend to think of space as the latter rather than the former, but there are other views.