Extra Dimensions in String Theory

[Lelan Thara]

I understand time via a clock.

Arivero, what you have said, together with all else said here, makes it clear to me that the extra spatial dimensions are indeed mathematical abstractions.

I really appreciate the effort people have put in here to help me understand these extra spatial dimensions.

 

[Kea]

However, I'll guess the easiest way to determine wheter it's KK-particles that have been found, is to see whether the physics community will call them KK-particles or not...

Deal.

I guess I'm in a lose-draw situation then. Say I bet 1$. If you win I'll afford to pay you. If I win, you'll never pay me my 1000$...

Well, maybe my luck will have changed by then. One never, never knows.

 

[Hans de Vries]

..., makes it clear to me that the extra spatial dimensions are indeed mathematical abstractions.

It's all a bit of a word play I guess. Nevertheless, mathematical abstractions
or not, they are all supposed to work just like real extra dimensions. Look
around for instance in the most used introduction for string theory:

http://books.google.com/books?vid=I...thor:zwiebach&sig=F8uUPfn2kaCq4cJRP7qp4PQ-ySw
https://www.amazon.com/dp/0521831431/?tag=pfamazon01-20

A First Course in String Theory by Barton Zwiebach.
Regards, Hans

 

[EL]

what you have said, together with all else said here, makes it clear to me that the extra spatial dimensions are indeed mathematical abstractions.

If that's your conclusion, why don't you call our ordinary three spatial dimensions "mathematical abstractions" also?
What makes exactly the number "3" the most real? Don't get fooled by the fact that our brains are used to a 3-dimensional flat world.

As Hans said, this is really a word play. In principle all physics quantities are just "mathematical abstractions", but we have do draw the line for what to call "real" somewhere. IMO it's a bit of a contradiction to call extra dimensions "unreal" at the same time we call our usuall 3 spatial dimensions "real"...

 

[EL]

Deal...Well, maybe my luck will have changed by then. One never, never knows.

Maybe if I bet just 1 cent. Then at least I have a fair chance of getting my 10$ if I win?

 

[Kea]

Maybe if I bet just 1 cent.

Oh, all right, then. I'm not greedy. A win of $10 will do fine.

 

[ptalar]

How do you understand time? Via a symmetry group. You do not see time, you see "forward in the time" and "back in the time". You recognise this as the elements of translations, and then you visualize time as if it were a line, because the translation generates lines. You see it is not related to the another three independent translations you recognize, so you set it apart as a 4th dimension.

Now, if string people postulate the 26 dimensions to be generated from a group of 26 "generators of translations" with the same mathematical (ie: logical) structure than the group of 4 generator you are used too, and the new dimensions having the same scalar product than old 3 spatial ones, then you have no right to discard as "no spatial" the new ones.

A different question should happen in models where the new dimensions compactify to a different structure. But in naive Kaluza Klein compactification, it is more of the same.

What makes some dimensions compactify and others to not compactify? Is there some Darwinian logic to these dimensions that decide which dimensions compactify? Why are we in a 11 dimensionsal world with 4 dimensions expanded and 7 compactified? Is it the least energy principal where all matter seeks its lowest energy state? Could that be where we are now with 4 dimensions? Must time always be one of the surviving dimensions that do not compactify? What circumstances must exist for the other 7 dimensions to decompactify? Would we still be alive if they did decompactify? I believe Lelan is on the right track. These are all theoretical exercises. Which is needed for physics to move forward. But must be validated experimentally for the theory to be science fact. Otherwise, it becomes another religion.

 

[EL]

Oh, all right, then. I'm not greedy. A win of $10 will do fine.

Deal! Guess you'll have my cent about 2010-2020 then!

 

[arivero]

Arivero, what you have said, together with all else said here, makes it clear to me that the extra spatial dimensions are indeed mathematical abstractions.

But note they obey all the equations of the non-extra dimensions. So for a mathematician, they are in equal footing than the rest. You could say that the non extra dimensions are "mathematical abstractions thar happen to coincide with ordinary experience" while extra dimensions are "mathematical abstractions than do not coincide with ordinary experience".

A mathematician can single out time (it has a different sign in the metric matrix) but it can not tell you any difference between an extra dimension and a ordinary one. Of cours, this comes to a conflict, at the very end, with the goals of compactification.

 

[EL]

Another thing to note is that the basic structure of string theory itself does not say those 7 extra spatial dimensions need to be compactified (at least that's the impression I've got), but are treated mathematically on equal footing as our ordinary 3. However, due to the fact we havn't observed any extra dimensions, we make the conclusion they need to be compactified.

 

[arivero]

Another thing to note is that the basic structure of string theory itself does not say those 7 extra spatial dimensions need to be compactified (at least that's the impression I've got), but are treated mathematically on equal footing as our ordinary 3. However, due to the fact we havn't observed any extra dimensions, we make the conclusion they need to be compactified.

Right. As it happens:

(bosonic) string theory lives in 26 dimensions, meaning 25+1. Thus 22 extra dimensions.
Superstring theory lives in 10 dimensions, thus 6 extra dimensions.
The minimal Kaluza Klein extension for the Standard Model lives in 11 dimensions.
Supergravity has been argued to live in 11 dimensions.
M-theory is said to live in 11 dimensions. Thers is also a "F teory" living not in 10+1 but in 10+2, ie 7 extra spatial and 1 extra temporal-like dimension. It sounds fringe, but it is mainstream, or at least a mainstream subsidiary.

Your "7 extra" comes from the three last ones.

Note that besides Kaluza Klein there are some other arguments to get rid of the extra dimensions. The come by names as "Free Fermion Formulation" or "non geometric compactifications". Sci.physics.strings is an appropiate place to ask about it (but strongly moderated).

 

[Lelan Thara]

I did try to get us away from discussing "real" extra dimensions by giving an operational definition of a "spatial dimension" as "a measurement both necessary and sufficient to locate any point within an unbounded volume".

Kea has described the extra spatial dimensions as (integer) categorical dimensions, as understood within Category Theory, a subset of set theory.

El has described the extra spatial dimensions as having a metric of -1.

Arivero described the extra dimensions in terms of symmetry groups and generators of translations with the same scalar products as spatial dimensions.

None of these fit my operational definition. Nobody has tried to tell me that the extra spatial dimensions actually describe a volume that is hidden from us and unmeasureable with 3 spatial dimensions.

And this is essentially what I wanted to know.

The nice thing about being a layman is I can afford to be an iconoclast. I can say, "the emperor has no clothes", and I will not fail any courses, lose tenure or lose my job.

In that light - I've repeatedly seen the use of the word "naive" here to describe my conception of dimensions. I don't want anyone to think this offends me - I get the impression it's a standard usage amongst modern physicists, and isn't meant as a critique of me.

But I must say that what really seems naive to me is to assume that every internally consistent mathematical model must be describing something that has a physical reality.

Extra spatial dimensions will always remain unmeasureable. We will never be able to point a ruler at them.

I brought up the notion of hidden variables and where they are hidden. When I envision a quantized subatomic world, where energies and masses do not fill up the entire unbroken continuum of available values - I think, why not look for your hidden variables in the gaps between the quanta? Instead of imaginary spaces, why not look for the real, measurable energies and masses that we can't observe directly but may be able to observe indirectly, through their interactions? Why not conceive of higher dimensions in terms of energies, frequencies, wavelengths?

If the goal of quantum theories is to explain the fundamental quantum mysteries of quantization, randomness, wave-particle duality and entanglement, it seems the theories must have to function here - where we are - where the wave and particle interactions happen - not in some invisible "somewhere else".

Maybe a day will come when the mathematical formalism of higher spatial dimensions will get translated into terms other than spaces - into something measureable. But that's not a goal that's likely to happen until physicists see a good reason to do it.

That's how it looks to this layman. You all know more on these topics than I do, and I'm grateful that you shared your knowledge with me. I hope I've been able to repay the favor, in some small way, by giving you something to think about. Thanks again.

 

[arivero]

Nobody has tried to tell me that the extra spatial dimensions actually describe a volume that is hidden from us and unmeasureable with 3 spatial dimensions.

Well, but also a vertical heigth is hidden an unmeasurable with two horizontal spatial dimensions.

What you could be asking is, can I rotate my ruler from the horizontal into the vertical to measure this height, or can I not? You can, and in this sense the extra spatial dimensions are not more hidden than a vertical dimension is.

 

[Lelan Thara]

Well, but also a vertical heigth is hidden an unmeasurable with two horizontal spatial dimensions.

What you could be asking is, can I rotate my ruler from the horizontal into the vertical to measure this height, or can I not? You can, and in this sense the extra spatial dimensions are not more hidden than a vertical dimension is.

If I rotate my ruler from the horizontal to the vertical, I can locate points in a volume that I could not locate before. So the rotaion is necessary.

In an everyday volume, though, I can keep rotaing my ruler and I can no longer locate any new points that I couldn't locate before. So the three rotations are sufficient.

So what I'd have to ask you, arivero, is: do you believe the extra spatial dimensions do, in fact, describe volumes that are hidden from us and unmeasureable in 3 dimensions?

 

[EL]

Nobody has tried to tell me that the extra spatial dimensions actually describe a volume that is hidden from us and unmeasureable with 3 spatial dimensions.

My point is that this may indeed be the case! The extra dimensions may be as real as our ordinary 3, although compactified.

Extra spatial dimensions will always remain unmeasureable. We will never be able to point a ruler at them.

No no. We'll try to measure them at LHC.

Why not conceive of higher dimensions in terms of energies, frequencies, wavelengths?

This sounds more like new age stuff (no offense).
Remember that all extra dimensions in string theory are spatial.
To start with, all (including the ordinary 3) dimensions are on the same footing. It's just that all but 3 need to be compactified.

Maybe a day will come when the mathematical formalism of higher spatial dimensions will get translated into terms other than spaces - into something measureable.

Until then, I suggest we call them real.

 

[arivero]

So what I'd have to ask you, arivero, is: do you believe the extra spatial dimensions do, in fact, describe volumes that are hidden from us and unmeasureable in 3 dimensions?

One needs to read very carefully the wording of your question before answering. I think nobody in the thread *including you* has read it. Think again about my ruler example.

I believe that the extra spatial dimensions describe n-dimensional volumes that are hidden from us and unmeasureable in 3 dimensions in EXACTLY the same sense than height describes n-dimensional volumes hidden from and unmeasureable in 2 horizontal dimensions.

In fact I do not need to believe it. I can read it from the equations. There is the rotation group in 10 spatial dimensions, which any mathematician can recognise. Actually there is the full Poincare group in 10+1 dimensions. Of course the 2 dimensional rotation group does not generate the three dimensional rotation group, neither the 3 dimensional generates the 10 dimensional.

Of course (and here again one must be carefully about the wording) you are not asking me if I believe in string theory. You are asking me if I believe that the equations of string theory (actually, of the Kaluza Klein approach to string theory). And to this question I answer yes, and that I do not need to believe, only to read the equations.

You could ask me if I think that the Kaluza Klein approach to string theory describes the physical world. There my answer is no, and this is really a matter of belief, because you can not give any experimental proof neither a logical argument against it, but neither for it.

 

[Lelan Thara]

This sounds more like new age stuff (no offense).
:

No offense taken. It's a great frustration to me that New Age and neopagan types try to give an air of pseudoscience to their beliefs with all their talk of "vibrations" and "frequencies" and so on, because it make it hard to talk of such things in a scientific context and be taken seriously.

Nonetheless, I've heard it said more than once that the fundamental ingredients of reality are fields of force and waves. I can't help the fact that some distinctly unscientific people may have co-opted ideas that are scientifically important.

 

[arivero]

Why not conceive of higher dimensions in terms of energies, frequencies, wavelengths?.

Well, because such is not the case in -let me italize- the Kaluza Klein approach to string and superstring theories. It could be the case in other approaches, I can not tell. Of course you can rely on relativity and quantum principles to convert between mass, length, energy, frequency and wavelength. This is done when it is appropiate to understand the physical content of a formulation (and, as EL points out, the nomenclature has been freely borrowed and retorted to other meanings by New Age preachers. Not to be blamed, we also borrowed the concepts of energy and mass from philosophers, retorting them to other meanings. Langauge works in this way).

I do not know if other approaches to superstring theory use non spatial dimensions. I can not see how, because the equations of Einstein follow from very basic conditions on the string. But it could be that some of these basic conditions could be weakened. After all, the thing they need is a very special cancelation condition in an algebra, and they could have some other tricks to get that condition without a background space. But if they have, I haven't see such tricks described in any divulgation book.

 

[Lelan Thara]

Well, because such is not the case in -let me italize- the Kaluza Klein approach to string and superstring theories. It could be the case in other approaches, I can not tell. Of course you can rely on relativity and quantum principles to convert between mass, length, energy, frequency and wavelength. .

Now that I know that you accept extra spatial dimensions in Kaluza Klein models, but don't necessarily believe these models are descriptive of reality, I understand what you're telling me better.

...the thing they need is a very special cancelation condition in an algebra, and they could have some other tricks to get that condition without a background space.

I am totally blowing smoke now, and I admit it - but a "very special cancellation condition" sounds like something that could be modeled with destructive interference of waves, and that dimensions describing wavelengths and frequencies might actually be more useful than extra spatial dimensions.


There is a fundamental question I should have asked much earlier - how do you guys define a "space"?

I am aware that there are Minkowski spaces, Riemann spaces, Hilbert spaces, and I'm sure there must be others. I can't adequately describe all these spaces, but I know that they allow for more dimensions, or are n-dimensional.

But I have been using "space" as analogous to "volume".

Can someone tell me how a theoretical physicist defines "space"? I'm afraid I have to ask you to translate into layman's terms, or I won't get your answer.

Thanks again.

 

[EL]

Here's how Wikipedia defines space:
http://en.wikipedia.org/wiki/Space#In_physics

Space is one of the few fundamental quantities in physics, meaning that it cannot be defined via other quantities because there is nothing more fundamental known at present. Thus, similar to the definition of other fundamental quantities (like time and mass), space is defined via measurement. Currently, the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum during a time interval of 1/299792458 of a second (exact). This definition coupled with present definition of time makes our space-time to be Minkowski space and makes special relativity theory to be absolutely correct by definition.

In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinates. Special and general relativity uses space-time rather than space; space-time is modeled as a four-dimensional space (with the time axis being imaginary in special relativity and real in general relativity, and currently there are many theories which use more than four-dimensional spaces (both real and complex).

(although I'm not sure what they mean by "with the time axis being imaginary in special relativity and real in general relativity"...)

 

[Lelan Thara]

Thanks, El - I have read that Wiki entry before. The part you quoted is consistent with my conception of space, especially since it stresses that space is intrinsically defined through measurement.

I believe there is a more mathematical description of space that needs to be applied to Hilbert spaces, topological spaces and such, isn't there?

The part I'm unclear on is that it seems like once you get past the 4 necessary and sufficient dimensions, you need a different definition of space than "defined via a measurement", because you're past the point where you can make measurements.

 

[EL]

The part I'm unclear on is that it seems like once you get past the 4 necessary and sufficient dimensions, you need a different definition of space than "defined via a measurement", because you're past the point where you can make measurements.

Not sure I'm following you here. Could you elaborate a bit?

 

[Lelan Thara]

EL, what I meant was that beyond four dimensions you can't pyhsically make real-world measurements.

I did a bit of reading on topological spaces, vector spaces, inner product spaces and such, and it seems that in mathematics, "space" is defined with sets and subsets obeying certain axioms, rather than "space" as volume.

So to say the extra spatial dimensions are really spatial, but the spaces are not volumes, changes the picture.

 

[arivero]

So to say the extra spatial dimensions are really spatial, but the spaces are not volumes, changes the picture.

Hmm of course if you define that "Space" is "3-dimensional volume" it is, er, sort of a restrictive definition, and your argument becomes tautology.

 

[Lelan Thara]

Hmm of course if you define that "Space" is "3-dimensional volume" it is, er, sort of a restrictive definition, and your argument becomes tautology.

How I define space is really irrelevant. The issue at hand is how physicists define space.

And the answer, as far as I can determine, is that there is more than one definition of both "space" and "dimension". The failure to recognize this is misleading to laymen. If physicists also fail to recognize that they are using the same words to mean different things, and not recognizing the differences, their researches will suffer, IMHO.

I don't see any tautology arising from using a strict operational definition of "space" and applying it consistently. If definitions of "space" other than "volume" exist, I assume they also involve axioms that must be applied rigorously and consistently. One would hope so, anyways.

 

[arivero]

If definitions of "space" other than "volume" exist, I assume they also involve axioms that must be applied rigorously and consistently. One would hope so, anyways.

Indeed, we can for instance use some set of axioms of differential geometry. But the point to be grasped here is that the use of the word "space" as a definition is in such a way that

-an area is an "space"
-a line is an "space"
-a p-dimensional volume is an "space".

This is in distintion to layman "space" aplied only to 3 dimensional volumes.

 

[Lelan Thara]

So if I could try to sum up from that:

We see from observation that a certain number of 'degrees of freedom" are necessary to describe physical processes. Theoretical physicists can legitimately call these degrees of freedom "spatial" because they conform to the mathematical axioms that describe mathematical spaces.

And these extra degrees of freedom may all be describing processes in physical reality. But as long as they are described as an abstract - as spaces beyong our capacity to measure - we will always be left asking what these degrees of freedom really represent in the Minkowski space we are forced to function in.

 

[arivero]

So if I could try to sum up from that:

We see from observation that a certain number of 'degrees of freedom" are necessary to describe physical processes. Theoretical physicists can legitimately call these degrees of freedom "spatial" because they conform to the mathematical axioms that describe mathematical spaces.

Almost. They conform to, or they include, the mathematical axioms that describe the families of geometrical spaces, a class narrower than "mathematical spaces" and loaded with geometric meaning.

Particularly, the people of string theory, without relying in observation, builds a series of degrees of freedom that conform to the mathematical axioms that describe the families of geometrical spaces agreeing with the theory of General Relativity.

But as long as they are described as an abstract - as spaces beyong our capacity to measure - we will always be left asking what these degrees of freedom really represent in the Minkowski space we are forced to function in.

Yes but that is for the metaphysical forum. It is as telling me that the number I got in my speed ticked not really represents the concept of speed. I will argue it with the cop next time .

 

[Lelan Thara]

Almost. They conform to, or they include, the mathematical axioms that describe the families of geometrical spaces, a class narrower than "mathematical spaces" and loaded with geometric meaning.

Particularly, the people of string theory, without relying in observation, builds a series of degrees of freedom that conform to the mathematical axioms that describe the families of geometrical spaces agreeing with the theory of General Relativity. .

Should I take it from this that you don't consider geometry a branch of mathematics?

Yes but that is for the metaphysical forum. It is as telling me that the number I got in my speed ticked not really represents the concept of speed. I will argue it with the cop next time .

If I am given a set of observed numbers anad asked to create an internally consistent mathematical model of them, I can come up with any number of mathematical formalisms that will produce the observed numbers. To assume that all - or any - of those mathematical models actually describe the reality that gave rise to the observables, without even being able to conceive of the models in a physically measureable way - that, to me, is metaphysical.

You could also try telling the cop that his concept of a "speed" within a volume is naive, and he needs to divide your speed by 25 spatial dimensions. That might work.

 

[arivero]

Should I take it from this that you don't consider geometry a branch of mathematics?

No, you should understand the meaning of the word "narrower" as a technical term.