Discreteness of space itself and 1dimensionality

[metman]

So I've been reading a book on string theory lately. It built up through relativity and quantum theory to strings. It's raised a number of questions for me, but I wanted to post 2 of them here.

1. So my understanding is that strings, being the fundamental building blocks of the universe, are indivisible and irrudicible. Nothing is smaller than them. That said, I also understand that they are not point particles; that is, they have dimension along at least one axis. This led me to wonder, is space itself discrete at some level? By that, I mean are there discrete Planck-length cubes or spheres of space, somewhat analogous to a pixel on a computer monitor, that are the smallest discrete locations a particle can occupy, or is space continuous, such that particles move through continuous, infinitely divisible space?

2. If a string is 1-dimensional, how can it be in the shape of a loop, or even ocillate in a waveform at all? Doesn't being circular or having a waveform intrinsicly imply at least 2-dimensionality? Or is the "loop" and "wave" of a string really just an illustrative device like the deformed membrane of relativity? In that case, can someone explain to me what strings really are? :)

 

[metman]

Sorry, I put this in the wrong board, could someone move it? Thanks.

 

[Civilized]

Answer to 1: the basic idea of the pixel's you describe is not compatible with one of the most well tested symmetries we know of in nature, called lorentz symmetry, this basically corresponds to the geometry of special relativity and general relativity. String theory is compatible with lorentz symmetry and this is a very important constraint that no other non-string theories of quantum gravity satisfy.

In other words, even though your pixels idea is not compatible with lorentz symmetry as found in the standard model and in string theory, at least you are in the company of other non-string models of QG.

Answer to 2: A string itself is one-dimensional, but it lives in ten dimensions where it is free to move. This is very much like the way that an ordinary piece of thread, which is approximately one-dimensional from a human scale, is free to move in all three-dimensions of ordinary space, as any seamstress will attest. If you are searching for a definition of one-dimensional, a decent working definition is "having finite length but zero area."

 

[metman]

I tried to look up Lorentz symmetry, but it was a little over my head. Is there a good place for finding a layman explanation? As to the second question, thanks a lot, that is a great way to illustrate it.

 

[Civilized]

Sorry, I don't offhand know any good physics resources for laymen.

Before understanding the Lorentz symmetry of spacetime, it will help to understand the Euclidean symmetry of plain old space.

Euclidean symmetries take the form of reflections and rotations in space. To say that the physical universe has euclidean symmetry means that the magnitude of physical quantities* is unchanged by rotations and reflections in space.

For example, the velocity of a particle has a magnitude and a direction. By doing rotations and reflections I can change the direction, but I can't change the magnitude.

*By physical quantities, what I mean precisely are vectors and tensors in space. This includes displacement, velocity, acceleration, force, momentum, etc.

Lorentz symmetry is euclidean symmetry + more, it also includes "reflections" and "rotations" in spacetime, not just space. Obviously these spacetime "reflections" and "rotations" are generalizations of the familiar symmetries which have those names, but the mathematics is what really justifies the analogy. Just as we can rotate the x spatial direction into the z spatial direction, lorentz symmetries can rotate the x spatial direction into the t time direction.

To say the universe has lorentz symmetry means that there are spacetime vectors which represent physical quantities and that the magnitude of these vectors is unchanged by apply "rotations" and "reflections."

 

[metman]

Ok, so how does that refute spatial discreteness? I'm not saying I "believe" space is discrete, I'm just interested now in why it is not, based upon the Lorentz symmetry, or anything else. My understanding of symmetry is that regardless of the angle of observation (ie changes of direction), the other aspects of the particle (velocity, mass, etc) do not change without outside interferance. However, discreteness of space doesn't seem to me to force asymmetry, it just limits the specific locations within space that the observer and the observed can occupy.

Or wait... does it have something specific to do with it being spaceTIME, not just space? But in that case, why would that not be resolvable with discrete slices of time as well?

Thanks for your posts. I appreciate it a lot.

 

[Civilized]

However, discreteness of space doesn't seem to me to force asymmetry, it just limits the specific locations within space that the observer and the observed can occupy.

In other words, in a discrete model there is a difference between "locations within space that the observer and the observed can occupy" and those that they can't, which breaks the symmetry.

...does it have something specific to do with it being spaceTIME, not just space? But in that case, why would that not be resolvable with discrete slices of time as well?

No, as an example we can see how discrete space fails to have ordinary rotational symmetry. To make things simple, consider a two dimensional space, a flat plane. Now let's compare a hexagon and a circle. The circle has exact rotational symmetry, but what about the hexagon? The answer is that the hexagon poseses only some of the rotational symmetry that the circle had, not all of it. A square has less rotational symmetry than a hexagon, and an octagon would have more rotational symmetry, and in general as we increase the number of sides on our polygon the degree of rotational symmetry increases.

The reason that we have to discuss objects with a finite number of straight sides is that in a discrete space that is all that exists. In discrete space you cannot have a circle, or any other curve, since all you can do is connect point by straight lines. From this it follows that discrete space does not respect rotational symmetry. This is exactly analogous to how discrete spacetime does not respect lorentz symmetry.

 

[metman]

Ok, that makes perfect sense. Thanks. :)

 

[jburgess04]

Thank you, Civilized. Well explained.

 

[Naty1]

Here is a thread in which I provide considerble theoretical support FOR the discretness of spacetime...note, I am NOT saying proof:

https://www.physicsforums.com/showthread.php?p=2261914&posted=1#post2261914

There are quite a few references cited for non professional reading by interested parties.

Civilized has provided a solid classical and relativistic view; quantum mechanical theories have a different view and likely the unification of relativity and quantum theory will be required to resolve the contradictions.

In addition, in recent years string theory in the eyes of some major physicsts has been shown to be highly compatible with the Holographic Principle...meaning discrete. If "it from bit" (John Wheeler) proves to be accurate, everything is even stranger than we understand! (that's likely a long way off.)
http://en.wikipedia.org/wiki/It_from_bit#Wheeler.27s_.22it_from_bit.22

Meantime there is a good chance our current understanding of space and time at the shortest scales, say Planck scale, is at best incomplete. Both quantum theory and GR breakdown so who knows??

 

[Civilized]

Civilized has provided a solid classical and relativistic view; quantum mechanical theories have a different view and likely the unification of relativity and quantum theory will be required to resolve the contradictions.

Thanks for the compliment, but I would offer the correction that currently existing quantum mechanical theories do not have a different view.

The standard model is an entirely quantum theory, and there is no doubt that spacetime is smooth and continuous in the standard model.

The mainstream extension of the standard model is string theory, and there is no doubt that spacetime is smooth and continuous in string theory.

The only theories which predict discreteness in space or time are non-standard, non-mainstream theories.

The smoothness of spacetime in all standard theories is very easy to see in even a basic technical textbook. The points of spacetime are in one-to-one correspondence with (quadruples of) real numbers, and the set of real numbers is smooth and continuous. Please Naty1, when you wish to discuss non-standard theories of discrete spacetime, do not claim that ordinary quantum mechanics, quantum field theory, the standard model, or string theory, have any aspects of discrete spacetime because they do not at all, and to claim otherwise is misinformative.

 

[Naty1]

The only theories which predict discreteness in space or time are non-standard, non-mainstream theories.

You keep saying that despite a half dozen or so reputable sources I provided in other threads...suit yourself...but the standard model of particle physics is continuous...limited tho it may be...

you know there is no experimental evidence for continuous spacetime...

 

[Civilized]

You keep saying that despite a half dozen or so reputable sources I provided in other threads

What you don't understand is that popular books on physics are mostly made of analogies and vague undefined statements. There is a big difference between popular physics, which is just for fun, and real physics which is based on mathematics and is recorded in textbooks and journal articles.

...suit yourself...but the standard model of particle physics is continuous...limited tho it may be...

At least you have finally learned that in the standard model spacetime is smooth and continuous. The standard model explains all observed phenomena aside from gravity, so I don't know anyone who considers it woefully limited.