Strings: One Dimensional? How?

[Hypercube]

The strings of the string theory are often thought to be one dimensional, but how that that be physically possible? If the strings were one dimensional that would imply that their width and length is zero, and having any dimensional *length*(for lack of a better term) of zero is not physically possible. So why are strings thought to be one dimensional?

[ajb]

Another question for you is "why are particles zero dimensional?"

 

They are defined that way and classically there is no problem defining particles to be 0-d or strings of 1-d.

 

Of course, quantum mechanically this is smeared out.

[BenTheMan]

Nothing is impossible! Not if you can imagine it! That's what being a scientist is all about!

 

You just can't IMAGINE a one-dimensional thing. Aparently, however, you have no problem imagining four dimensional things...

[Hypercube]

First of all just because the strings are classified as one dimensional does not mean that they really are. It is an accepted physical law that something cannot have no width, length or depth.

[BenTheMan]

It is an accepted physical law that something cannot have no width, length or depth.

 

Accepted? By who?

[someguy]

First of all just because the strings are classified as one dimensional does not mean that they really are. It is an accepted physical law that something cannot have no width, length or depth.

 

do you mean that they view that "one dimensional" string as being for example of dimension nX, nY, aZ, for n->0 and 'a' any real number?

 

and i was for some reason under the impression that these strings were somehow coiled. wouldn't a 1 dimensional object need to be by definition a perfectly straight object? if i could see a one dimensional string making a coil i would feel compelled to call it a three dimensional object, but i could sort of also see how the string itself may only have length... so i guess i would just go cross eyed.

 

I suppose part of my problem is i can't imagine how a one dimensional object can exist in 4 dimensions. or even a two dimensional object or even a three dimensional object. I can't really imagine a one dimensional universe either but that's besides the point. so this might be a weird question but, is this string meant to exist in its own one dimension? or is it in fact existing like a one dimensional coil in our 4 dimensions?

[BenTheMan]

and i was for some reason under the impression that these strings were somehow coiled. wouldn't a 1 dimensional object need to be by definition a perfectly straight object?

 

Think of a dimension as a coordinate. Think of a circle, with no inside, of constant radius. Exactly one coordinate defines a place on the circle. A circle can be deformed if you like, and whatever shape you end up with (so long as you don't break the circle, or put any kinks in it) will be isomorphic (big word, I know---I think it's the right one to use!) to a circle.

 

The other one dimensional topology (another big word) is a line segment. Now, any one dimensional object with no kinks or cusps, and which isn't closed is isomorphic to a straight line. This means that you can take your one dimensional object and straighten it out and make it a straight line.

 

``Isomorphic to'' means (loosely, of course) that you can take one thing and reform it (without breaking it or creating a hole in it) to something else. ``Topology'' defines (loosely, again!) the class of surfaces for which this can be done. For example, an oval has the topology of a circle, because the two are isomorphic. (ajb, who knows much more math than I do, is probably cringing because I am giving very specific examples. The field of Topology is very broad, and I only understand what I need of it. Rest assured that there is not much more that I know about this subject!)

 

A better example is a three dimensional lump of clay. So long as it doesn't have any holes in it, a three dimensional lump of clay is isomorphic to a sphere. Now, as long as you don't break the clay (i.e. put holes into it), you can flatten it and shape it---for example, you've seen the Venus de Milo---well, she is made from a lump of clay (in a sense!), and has no holes. So the Venus de Milo is isomorphic to a sphere.

 

But suppose your lump of clay DOES have a hole in it. Then it is isomorphic to a donut, or more specifically, a torus. Now you can take any shape with one hole in it, and turn it into a donut, or vice versa. Take Michelangelo's David. His arms touch his body in two places, so if he were made of a lump of clay, one could reshape it into a donut with TWO holes (called a Riemann surface of genus 2).

 

I suppose part of my problem is i can't imagine how a one dimensional object can exist in 4 dimensions. or even a two dimensional object or even a three dimensional object.

 

Think of coordinates man! You have no problem drawing a parabola on a set of coordinate axes, right? What you are doing is drawing a one dimensional shape (the parabola) in a two dimensional plane. Dimension = coordinate.

 

Mathematically, you have expressed a function [math]f(x) = x^2[/math] which maps a straight line (the real numbers) to a parabola. All you have really done is take something which is a straight line (the real number axis) and bent it a little bit into a parabola. So, I would say, a straight line is isomorphic to a parabola.

 

so this might be a weird question but, is this string meant to exist in its own one dimension? or is it in fact existing like a one dimensional coil in our 4 dimensions?

 

Think of the parabola example above. A string lives in our four (plus six!) dimensions---the string itself is one dimensional, and over time, it sweeps out a two dimensional surface (just like if you take a coin and spin it fast enough, it looks like a three dimensional sphere).

 

Hope this wasn't too confusing. Oh yeah, and if a guy called Farsight replies, ignore him. He does not understand these concepts and will mislead you. It is, of course, for you to decide.

[Sayonara]

Hope this wasn't too confusing. Oh yeah, and if a guy called Farsight replies, ignore him. He does not understand these concepts and will mislead you. It is, of course, for you to decide.

 

You are right; it is for the OP to decide. So please refrain from making disparaging remarks about other members.

[BenTheMan]

Sayonara---

 

The problem with these internet discussion fora is that credibility is determined in an ad hoc manner. Farsight (and other internet crackpots) can only confuse people who don't know any better. If you'd like me to, I can link to Farsight's posts and let you decide if he knows what he is talking about.

 

Barring that, I will continue to warn people that he is an idiot, in threads where he is likely to comment. If you really want to, you can give me warnings or infractions, or go around edting my posts.

[Sayonara]

Sayonara---

The problem with these internet discussion fora is that credibility is determined in an ad hoc manner. Farsight (and other internet crackpots) can only confuse people who don't know any better. If you'd like me to, I can link to Farsight's posts and let you decide if he knows what he is talking about.

You must deal only with the arguments made, not the person who is making them. You can leave that to the staff.

 

You agreed to conduct yourself in line with our rules when you joined the site, the very first of which prohibits attacks on other members.

 

Barring that, I will continue to warn people that he is an idiot, in threads where he is likely to comment. If you really want to, you can give me warnings or infractions, or go around edting my posts.

I have no intention of chasing around after you like a fool, and neither will any of the other staff.

 

It would be much simpler to bar you from the physics sub-forum, but rest assured that if you harass other members you will rack up those infraction points very quickly.

[Phi for All]

If you'd like me to, I can link to Farsight's posts and let you decide if he knows what he is talking about.

Good advice, but we don't need the links. The membership can read what is posted whenever it's posted and make their decisions. This will happen quite nicely without your ad hominem approach.

 

Barring that, I will continue to warn people that he is an idiot, in threads where he is likely to comment.

You will not. It's counterproductive to your cause because you are breaking the rules and forcing everyone, in fairness, to give more credence to an argument that may not deserve it (people go out of their way to defend an underdog). You are making yourself look like some self-appointed militia-of-one which also undermines your cause. And lastly, warning people about Farsight in threads *before* he comments (however likely he is to post there) is defamation and again puts you in the black hat.

 

You need to trust that the membership and Staff are smart enough to see through anyone who doesn't know what they're talking about. You need to trust that the evidence, untainted by fallacious arguments, will speak for itself.

[Sayonara]

And do please remember that there is a "report this post" feature, which can be used in instances of sheer crackpottery.

 

Note that it is not to be used for any post one happens to disagree with (this actually happens sometimes; it is quite depressing).

[BenTheMan]

Now that we've thouroughly addressed how bad of a person I am, how about my beautiful explanation of dimension?

[ajb]

At the risk of side tracking the original post;

 

unfortunately Ben, this forum is full of people who do not understand basic physics/maths and crackpots. More than once have I thought about never posting again. However, I do answer posts about sensible well posed physics questions that I feel I can answer and get some satisfaction out of helping people who help themselves. As for the others, I do not care for them and try not to post rubbish or get involved in arguments.

 

I do worry about people who do not know physics and start to listen to the crackpots instead of using reputable sources. I have seen this in supposedly "reputable" seminars held by a local society (that's all I will say right now on that) as well as online.

 

All we can do is hope that people listen to the right people, but what are the chances of that when the crackpot ideas seem so more "understandable" than mainstream physics which is by all accounts a difficult subject?

[BenTheMan]

I do worry about people who do not know physics and start to listen to the crackpots instead of using reputable sources. I have seen this in supposedly "reputable" seminars held by a local society (that's all I will say right now on that) as well as online.

 

This is what originally made me join an internet discussion forum. I found a guy who didn't believe in quarks! Ok, maybe you have something intelligent to say, but when I asked him how he explained confinement he didn't know! So you want to get rid of quarks, AND you have no acceptable alternative theory other than ``well, it just seems too complicated''.

 

This is the problem. People who dismiss modern physics out of hand and people who worship Einstein as a god... Not only are they ignorant, they convince other people to be ignorant, too. And then those people go out and vote and raise ignorant childeren.

 

The problem with these websites is that people don't think about the responses. They only read the original posts, become convinced that GR is wrong, and then leave a comment like ``I've always thought something like this must be true...'' End of story. Someone who WANTS to understand nature isn't the problem---they will participate in discussions and try to understand. But a casual observer won't invest that much time.

 

Scientists seem to have a real problem with Intelligent Design, and rightly so. But what I don't understand is why such crackpottery is tolerated in a self-proclaimed ``Science'' forum. Because if evolution is ``just a theory'', then so is GR, and so is Quantum Mechanics, and so is QCD.

 

Suppose hypercube had asked a question like ``how do we know that the earth is 4 billion years old'', and my response had been ``Look at the uranium isotope ratios. And if so-and-so says something about the Earth being 6000 years old, they are an idiot and don't listen to them.'' Would the moderators have admonished me and threated me with a ban? Would I have been called a ``physics vigilante''?

[ajb]

This is the problem. People who dismiss modern physics out of hand and people who worship Einstein as a god... Not only are they ignorant, they convince other people to be ignorant, too. And then those people go out and vote and raise ignorant childeren.

 

I agree, but overall it is not a fight you are going to win on here. Sad, but true.

[Xerxes]

Ben, although I agree with most of what you say, I find this a little over the top, in fact rather offensive.

Not only are they ignorant, they convince other people to be ignorant, too. And then those people go out and vote and raise ignorant children.

That raises issues a so-called science forum is not qualified to deal with. So leave it aside.

 

But.... you wanted a response to your first post on the original question. Obviously I don't know, not being a physicist, but I will say this:

 

You start by saying

Think of a dimension as a coordinate.

You mean, of course, a set of coordinates. And next:

and whatever shape you end up with (so long as you don't break the circle, or put any kinks in it) will be isomorphic (big word, I know---I think it's the right one to use!) to a circle.

Two problems here: although I am anticipating slightly, the topological property you refer to is not "isomorphism" but "homeomorphism". I can explain if you want.

 

Moreover, topological spaces can be homeomorphic regardless of what you call "kinks and cusps"; as you rightly say, the only criterion is the number of "holes" they have (this is the connectedness property, well there's more to it than that), kinks and cusps are OK.

 

Neither can I convince myself of the truth of this

``Topology'' defines (loosely, again!) the class of surfaces for which this can be done.

Topology comes in two flavours - point-set topology and algebraic topology. Neither of these has anything to do with surfaces. Maybe you're thinking of manifolds here? And in any case, the word "topology" is widely abused; it is, technically, a set T of subsets of some set S which has certain defined properties, one of which posits the existence of the topological space {S,T}, on the other hand, "topology is used to refer to the study of such spaces. And I thought mathematicians liked precision!

 

Oh, and finally, you say

a three dimensional lump of clay is isomorphic to a sphere.

As the OP was about dimensions, you might have been wise to specify the 3-sphere; unless otherwise specified, the sphere is taken to be 2-dimensional.

[someguy]

Think of a dimension as a coordinate. Think of a circle, with no inside, of constant radius. Exactly one coordinate defines a place on the circle. A circle can be deformed if you like, and whatever shape you end up with (so long as you don't break the circle, or put any kinks in it) will be isomorphic (big word, I know---I think it's the right one to use!) to a circle.

 

The other one dimensional topology (another big word) is a line segment. Now, any one dimensional object with no kinks or cusps, and which isn't closed is isomorphic to a straight line. This means that you can take your one dimensional object and straighten it out and make it a straight line.

 

``Isomorphic to'' means (loosely, of course) that you can take one thing and reform it (without breaking it or creating a hole in it) to something else. ``Topology'' defines (loosely, again!) the class of surfaces for which this can be done. For example, an oval hdas the topology of a circle, because the two are isomorphic. (ajb, who knows much more math than I do, is probably cringing because I am giving very specific examples. The field of Topology is very broad, and I only understand what I need of it. Rest assured that there is not much more that I know about this subject!)

 

A better example is a three dimensional lump of clay. So long as it doesn't have any holes in it, a three dimensional lump of clay is isomorphic to a sphere. Now, as long as you don't break the clay (i.e. put holes into it), you can flatten it and shape it---for example, you've seen the Venus de Milo---well, she is made from a lump of clay (in a sense!), and has no holes. So the Venus de Milo is isomorphic to a sphere.

 

But suppose your lump of clay DOES have a hole in it. Then it is isomorphic to a donut, or more specifically, a torus. Now you can take any shape with one hole in it, and turn it into a donut, or vice versa. Take Michelangelo's David. His arms touch his body in two places, so if he were made of a lump of clay, one could reshape it into a donut with TWO holes (called a Riemann surface of genus 2).

 

 

 

Think of coordinates man! You have no problem drawing a parabola on a set of coordinate axes, right? What you are doing is drawing a one dimensional shape (the parabola) in a two dimensional plane. Dimension = coordinate.

 

Mathematically, you have expressed a function [math]f(x) = x^2[/math] which maps a straight line (the real numbers) to a parabola. All you have really done is take something which is a straight line (the real number axis) and bent it a little bit into a parabola. So, I would say, a straight line is isomorphic to a parabola.

 

 

 

Think of the parabola example above. A string lives in our four (plus six!) dimensions---the string itself is one dimensional, and over time, it sweeps out a two dimensional surface (just like if you take a coin and spin it fast enough, it looks like a three dimensional sphere).

 

Hope this wasn't too confusing. Oh yeah, and if a guy called Farsight replies, ignore him. He does not understand these concepts and will mislead you. It is, of course, for you to decide.

 

I see what you mean, still can't really imagine a string properly. I can imagine any of these shapes as a plotted function but not in reality. I was trying to imagine a 2 dimensional object since this post though and i sort of thought.. well a wave is sort of a two dimensional object isn't it? (3 if you count time). granted it causes the moving of 3 dimensional objects but the wave itself could be considered to have no thickness could it not?

 

taking your coin thing how fast would it need to spin in order for you to call it a 3 dimensional object? for some reason i was under the impression that that was the speed of light. does it actually sweep out to a two dimensional surface? or is it just so close the difference is negligible? that actually makes sense to me about string theory. (still not the whole 6 dimensional part) but it seems to me that nearly every kind of energy in the universe is some kind of motion. and since matter is such a concentrated amount of energy it would kind of just make sense that it is made up of smaller and smaller things moving really fast, even maybe things of lesser dimension but due to their moving become more dimensional. i don't see why it "needs" to be that way but i kinda like that idea about string theory, even though i still have a little trouble wrapping my head around the idea that something can exist as being of less dimensions than 4 (unless you count waves like i was saying before, and i'm not sure really what i think of that yet).

[BenTheMan]

Xerxes---

 

You mean, of course, a set of coordinates.

 

Unless I am mistaken, 1 dimension = 1 coordinate.

 

Two problems here: although I am anticipating slightly, the topological property you refer to is not "isomorphism" but "homeomorphism". I can explain if you want.

 

No, thank you. I am not a mathematician, so I probably wouldn't get very much out of it. I will look these words up in my copy of Nakahara if I ever need to use them rigorously.

 

Moreover, topological spaces can be homeomorphic regardless of what you call "kinks and cusps"; as you rightly say, the only criterion is the number of "holes" they have (this is the connectedness property, well there's more to it than that), kinks and cusps are OK.

 

Thank you for clearing this up. I had imagined that kinks and cusps break one of the rules of a manifold, specifically that it is not everywhere smooth and differentiable.

 

Maybe you're thinking of manifolds here? And in any case, the word "topology" is widely abused; it is, technically, a set T of subsets of some set S which has certain defined properties, one of which posits the existence of the topological space {S,T}, on the other hand, "topology is used to refer to the study of such spaces. And I thought mathematicians liked precision!

 

Again, I am no mathematician. Personally, I find that when explaining things like this to laymen, mathematical rigor is a pretty useless thing. I can understand soapboxes, I've a few of my own, so thank you for the good definition.

 

As the OP was about dimensions, you might have been wise to specify the 3-sphere; unless otherwise specified, the sphere is taken to be 2-dimensional.

 

Again, rigor. Most laypeople that I've known imagine a sphere as three dimensional (which some people tell me is a 3-ball), NOT two dimensional. I won't quibble over definitions, and if you think that you can answer Hypercube's question in a better way, I invite you to do just that.

[BenTheMan]

someguy---

 

These things are quite difficult to picture. I am a bit surprised that you aren't asking about 11 dimensions! Either way, the way I think about dimensions is coordinates---if you only need one coordinate, then it is one dimensional.

 

taking your coin thing how fast would it need to spin in order for you to call it a 3 dimensional object?

 

I think you may have misunderstood me a bit. What I am talking about is something called a ``world-volume''. The easiest example is a point, which is zero dimensional---no length, no depth, no height...just a dot. If we let that dot move, it moves along a one dimensional line, called its world-line. In other words, if you plotted the point's position versus time, you would have a line. Also notice that one coordinate is no longer enough to describe the point---you have to know where it is in time too.

 

The same with a string. Suppose you have a one dimensional string. You can put one coordinate on the string, so you know where you are along the string. But this is not longer good enough---you must also know WHEN you are...so you need two coordinates---one along the string, and one time coordinate.

[Xerxes]

First this: Ben: I concede the tone in my last was somewhat haughty. I apologize, I hope you're not terminally miffed. Now, this is less trivial than it might at first appear.

If the strings were one dimensional that would imply that their width and length is zero, and having any dimensional *length*(for lack of a better term) of zero is not physically possible.

 

OK, the answer to your question is not "that's just the way it is", nor is it "it's a mathematical abstraction". Rather it is this: an object is said to be one-dimensional if the notions of width and length have no meaning. Let me try and explain.

 

First we need to distinguish at least two different "sorts" of dimension. Consider a circle. You need a sheet of paper on which to draw it, right?; this sheet is a 2-dimensional object, or "space" as you might say; the circle "lives" in a 2-space. Likewise, a sphere lives in 3-space. Is there a rule? Let's call the space an object lives in it's ambient space. Let's further say that any object in this space can be uniquely located by reference to a coordinate set whose cardinality is defined to be the dimension of the space. Let's take this as the definition of the dimension of an ambient space. Fine so far, I trust.

 

Let's now consider the objects in some n-space, and apply the same definition of dimension. How many "numbers", or better, parameters, do I need to uniquely specify a point on a line? A circle? One in both cases, obviously; these are 1-dimensional objects. How about a sphere? Two, latitude and longitude; this is 2-dimensional. And so on. (you can call these parameters coordinates if you want, like Ben did, I'm not sure it's standard, though). The pattern seems to be that the dimension of an ambient space is always greater than that of its embedded objects. There are theorems out there to this effect, you don't need to worry about them, though.

 

The reason I rambled on about ambient spaces is this: the equation that describes the unit circle, x² + y² = 1, for example, is a function on the ambient space, not on the object itself. Hence my opening comment: it makes no more sense to talk about the length and width of a 1-dimensional object, zero or otherwise, than it does to talk about north of the North Pole; it doesn't parse.

 

And finally, in this overlong post. (This is the sort of thing that causes my mates throw beer at me): the circle is "properly" referred to as the 1-sphere, the sphere as the 2-sphere, the ball (aka "solid" sphere) as the 3-sphere, etc.

[BenTheMan]

Xerxes---

 

Not at all. I do have a habit of stepping on mathematical definitions (as many physicsts do), and mathematicians always seem to hate that. For example, one of the post docs I am working with was lamenting about how to describe representation to his mother. I told him that I explained it to my mother using dogs. We all have ideas of what dogs are, and what they look like. But when I draw one on a piece of paper, my dog will ostensibly look different from your dog. Sure they have some of the same characteristics, but they are certainly different. Just like my representaiton of Dirac matrices look different from yours---that's ok, they still DO the same thing, they still ACT on the same space, they just LOOK different. He wasn't so amused.

 

Either way, thank you for clearing up some of my misconceptions.

[Xerxes]

For example, one of the post docs I am working with was lamenting about how to describe representation to his mother.

No shit? I have trouble explaining it to myself, and still I'm not sure I get it. Ah well. But oddly enough, this is a subject I am now ploughing (plowing?) through, alone. I have Fulton & Harris, and I know a bit of Lie Theory, but I'm finding it really, really tough.

 

Maybe you'd like to start a pedagogical thread on reps?

[BenTheMan]

Maybe you'd like to start a pedagogical thread on reps?

 

God no. You saw how good my pedagogical thread on topology was:)

 

In physics we have this process called ``dimensional regularization''. Basically, you take an integral that has terrible UV behavior (i.e. logarithmic or worse divergences as energy goes to infinity) and write it as an integral in a different number of dimensions---[math]4 - \epsilon[/math]. Then you evaluate it, take the limit as [math]\epsilon[/math] goes to zero, and you're left with a finite piece and an infinite piece. You throw the infinite piece away (effectively) and keep the finite piece, which can be used (quite successfuly) to predict experiments.

 

This is very much how I view math. There's some I understand and some I don't...I only keep the piece that's easy to understand because it's all I need anyway.

 

Talk to Jim:

http://www.scienceforums.net/forum/showthread.php?t=27168

[someguy]

bentheman I don't know 11 dimensions seems too crazy for me yet and i don't even really understand properly how they are meant to be mathematically. I even have some trouble with 5. understanding 11 dimensions and what that means in reality just seems way beyond me right now. however one dimension or two dimensions or even three if they exist should and i think must be not only describable as known things (like not in a math sense, just like we say the 4th dimension is time rather than plotting the 4th dimension in a graph) but we should be able to find examples or recreate examples or know exactly why we can't but sub-atomically we could. But i'm still wrestling with the idea that waves are two dimensional, so maybe that's one down and two to go.