# Extra Dimensions in String Theory

• Lelan Thara
In summary, the conversation revolves around the concept of "higher spatial dimensions" in string theory and whether they are just mathematical abstractions or have a physical existence. Some physicists believe that these dimensions could be measured in a simple way, while others argue that they are just a result of mathematical symmetries in the theory. The idea of using category theory to understand these dimensions is also mentioned. Ultimately, the question remains as to whether these dimensions are real or just a theoretical construct.
Lelan Thara
Hello, folks - this is my first post here. I am hoping some of you will be kind enough to help me understand something I have struggled with for a long time.

A brief background:

My lifelong interest in science fiction led me to start reading about relativity and quantum physics around 20 years ago. However, I am stricly a layman with no mathematical formalism. What knowledge I have of these topics comes mostly from reading authors like Paul Davies, Michio Kaku. John Gribbin, Steven Hawking and other popularizers of physics.

When I read such books, I am always acutely aware that the language of physicists is math, and that I am getting my information from "translators". So when I run into concepts that I have trouble accepting, I always wonder whether the problem is the concept - or the translation.

So my question has to do with the "higher spatial dimensions" of string theory.

I am aware that a "dimension" to a mathematician is not necessarily the same thing as a "dimension" to a classical physicist, or anyone using dimensions in the real world. A mathematician's dimensions can be abstractions, but a physical dimension is a usable means of measurement.

If you'll forgive a simple example - a mathematician can draw two crosses - two sets of axes - on a sheet of paper. He can label one set of axes, "x,y", and one set of axes "a, b", and say that he has "four spatial dimensions" - right?

But obviously the surface of his sheet of paper still remains two dimensional. Two measurements at right angles to each other are "necessary and sufficient" to locate any point on the paper.

My question is - are the "higher spatial dimensions" of string theory mathematical abstractions only? Do physicists actually believe that these extra dimensions are describing some unseen place, some space, that can't be measured and located within the standard three dimensions of length. width, and depth?

What exactly is a "higher spatial dimension" to a theoretical physicist?

I have clues that the literal existence of these higher spatial dimensions is intended to be taken seriously. I've heard them described as "compactified", and the analogy of the interior of a garden hose, which looks one dimensional from a distance but reveals other dimensions close up. I can only accept this as a mathematical abstraction. It's the same as saying "small measurements are in a different dimension than big measurements." A mathematician could say, "I define anything bigger than a meter as dimension X and smaller than a meter as dimension Y" - but to suggest that creates a new physical space, or a higher dimension as laymen think of such things, makes no sense to me.

I've heard that the extra spatial dimensions are curled up into manifolds. A dimension in the real world is a means of measurement, not a place, not a thing. How can a means of measurement be curled up? If I curl up a tape measure, I can still locate any point on that tape measure with straight yardsticks in the standard three dimensions of length, width, and depth.

The whole idea of "higher spatial dimensions" and "hyperspace" just sounds so Flatland to me. It seems the exact same as Flatlanders talking about "higher area dimensions" or "hyperplanes".

Sorry for revealing my bias. Can anyone explain whether these higher spatial dimensions are meant to be taken literally - what they are - how they describe any space that can't be measured in 3 dimensions?

(Just to let you know, I'm aware of time as a dimension and am leaving it out of my comments as a convenience.)

If anyone can help me understand this better, in layman's language, I would be extremely grateful. Thanks in advance!

Lelan Thara said:
My question is - are the "higher spatial dimensions" of string theory mathematical abstractions only? Do physicists actually believe that these extra dimensions are describing some unseen place, some space, that can't be measured and located within the standard three dimensions of length. width, and depth?

That's a very good question, Lelan. Some physicists persist in believing that it is more than a mathematical abstraction, and take the extra dimensions literally to be measurable in the simple way that you imagine. They are wrong, of course.

What exactly is a "higher spatial dimension" to a theoretical physicist?

There are many types of abstract dimension. The best answer we know at present is the notion of (integral) categorical dimension, as in Category Theory. This is a powerful extension of ordinary set theoretic mathematics. In fact, sets are examples of pointlike (zero dimensional) structures. String theory is currently in the process of being rewritten in the language of category theory, but many people still find this very mysterious because they have not accepted the necessity of examining the underlying physical principles of the theory.

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Extra Dimentions

As far as I understand it the extra dimentions in string theory arise when you go from the a the classic description of string theory to the quantum description. It comes about because the theory must obey certain symmetry principles which force higher dimentions into the theory. Usually this is called the "conformal anomally".
Since as of yet experimental evidence for string theory is lacking, it is unknown whether these dimentions are real. Although because there are many string theories, some give x dimentions, some gives y dimentiions physicists are wondering whether the number of space time dimentions is even significant at all.

What Kea said though is very interesting. Can you refer me to some introductory article on the subject?

Kea said:
Some physicists persist in believing that it is more than a mathematical abstraction, and take the extra dimensions literally to be measurable in the simple way that you imagine. They are wrong, of course.
Why are they of course wrong? It's fully possible the extra dimensions are "real". Of course they have to be curled up, since otherwise we would already have measured them. The scale of the extra dimensions is though not as constrained as one first may guess, it is still posible they will be several microns in radius. These spatial extra dimensions are one of the things LHC at CERN will be able to look for.

Kea - thanks for giving me a direction to look in. I'll do searches on categorical dimensions and category theory and see if that helps me understand better.

Gouranja - ahve you ever considered the possibility that the lack of experimental evience may actually be connected to the fact that the theory is conceptualized in terms of extra spatial dimensions that can't be physically measured?

El - if the extra dimensions are real, how do you conceive of them? Are they "somewhere else"? Can I point a ruler at them?
If these extra dimensions are in the particles (excuse me, strings) that make up my body, my desk and so on, they must be in the same physical place that I am in. So why would I not be able to locate them within the standard 3 spatial dimensions?

Also, how do you conceive of a "curled up dimension" if a dimension is a means of measurement? I refer you back to my example of a curled up tape measure, every point of which can still be located in the standard 3 dimensions. I can see how, abstractly, a mathematician can define a curled frame of reference and call it a dimension - but on a physical level, I don't understand how an extra curled axis takes me outside of anything I can measure with 3 dimensions.

Thanks you, all.

Lelan Thara said:
El - if the extra dimensions are real, how do you conceive of them? Are they "somewhere else"? Can I point a ruler at them?
If they exist, they are "everywhere", in the same sense that the usual 3 dimensions are "everywhere". However they are compactified (as long as we are not considering braneworlds, but let's skip that for the moment) meaning they are curled up into tiny balls with tiny radii.
An analogy I like to think of is the following:
Consider a 1-dimensional world; an infinitely long line, with a creature living in it. The creature can only move in (and percieve) the directions backwards and forwards. Now add another dimension so that the universe becomes an infinitely long cylinder with a radius much smaller than the size of the creature. Question is, will the creature really be able to differ between the universes? Will it be able to determine wheter it is living in a 1d-universe, or a 2d-universe with one of the dimensions curled up (the direction around the cylinder)?
For example, distances between points on a macroscopic scale (that is at the scale of the creature) doesn't depend very much on which "curled up dimension"-coordinates the points have.
Then we also have the quantum effects. According to quantum machanics particle momenta along the curled up dimension can only take on certain quantized values. The smaller the compactification radius, the larger the smallest allowed momenta will be. Hence a particle need to be very energetic to be allowed to move in the new dimension. Since the creature will just see that the energy of the particle has increased, while the momentum along the large dimension is still the same, it will interpret this as a new much heavier particle, a so called Kaluza-Klein particle (which indeed is a proposed dark matter candiate).
This was an example of a universal extra dimension, which all particles are allowed to propagate in.
The other, more string theroy inspired alternative, is that only gravity is allowed to propagate in the extra dimension, while standard model particles are confineded to our 3d-world. This would turn up as a deviation from the 1/r^2 law for gravity at distances depending on the radius of the extra dimension.

Hope I didn't make a to large mess out of it. Here's some more nice reading:
http://superstringtheory.com/experm/exper5.html
http://superstringtheory.com/experm/exper5a.html

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Thanks, El - I looked at your links. I have to admit there is much in them I don't understand, but I do feel I'm getting closer to understanding what the reality of these extra spatial dimensions is.

And they do sound to me like mathematical abstractions.

I don't reject that these mathematical models may be describing something real. But the reality they seem to be describing is a complex interaction of forces that can be made more manageable in a mathematical way by introducing abstract dimensions.

I don't see anything in what you posted, or in your links, that leads me to think I couldn't locate any of these forces in the three standard spatial dimensions.

My feeling, as a layman reading about these things, is that extra dimensions are indeed necessary to explain quantum reality. But I suspect that a day will come when we will define these dimensions as something other than spatial.

I can accept the idea of invisible forces interacting with the observable world to create quantum reality. But I can't see why something being invisible means that it is not locatable within an unbounded volume defined by length, width and depth.

A compactified dimension that is too small to observe just says to me that you need smaller units of measure within the standard 3 dimensions.

Take the "garden hose analogy". Imagine we're talking about a real garden hose, observed from orbit. Yes, the hose would look like a one-dimensional line to an astronaut in orbit - but if the astronaut returns to Earth, goes to the hose and measures its diameter with a ruler, would you seriously contend that our astronaut had discovered an "extra spatial dimension"?

I can accept that abstract mathematics can define small measurements as a different dimension than big measurements. But in the physical world of real measurement, no matter how small or large a measurement is, I can locate it in an unbounded volume with only 3 - and no less than 3 - spatial dimensions. Provided, of course, that my measuring device has sufficiently small or sufficiently large calibrations.

I have a ruler on my desk that measures 32nds of an inch. My car measures distances in miles. Is my car in a different dimension than my ruler?

Lelan Thara said:
... However, I am stricly a layman with no mathematical formalism. What knowledge I have of these topics comes mostly from reading authors like Paul Davies, ...

You might appreciate Lee Smolin's book "The Trouble with Physics".

He is a good explainer and skilled at translating the content of equations into ordinary English.

But I suspect that a day will come when we will define these dimensions as something other than spatial.

there is indeed a growing expectation on the part of various people in several different lines of research (including some in string) that space, as a continuum with some fixed integer number of dimensions, will prove not to be fundamental but will instead turn out to be a kind of macroscopic illusion emerging from a more fundamental microscopic web of relationships.

many of the people looking for a more fundamental basis for space time and matter call what they are working on "background independent" approaches because they do not assume at the start some fixed background continuum with a fixed number of dimension and a pre-structured geometry.

To get a sense of the various "background independent" approaches people are working on, take a look at Dan Oriti's short essay at Christine Dantas' blog.

http://christinedantas.blogspot.com/2006/11/one-year-of-cbi-invited-post-daniele.html

Dan is talking about a new book that Cambridge University Press will be bringing out (planned for early next year). The book is called
Approaches to Quantum Gravity: towards a new understanding of space, time and matter

It includes some string theory but it is mainly about 3 or 4 newer approaches which are more explicitly background indpendent and aim at a more fundamental picture.

Oriti is the editor of the book which has chapters by some 20 or so people and also Q&A discussion---going to be a great book I think.

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Lelan Thara said:
A compactified dimension that is too small to observe just says to me that you need smaller units of measure within the standard 3 dimensions.

Yes but what is your ruler? The only "ruler" available to you is other particles, via collision experiments. If extra dimentions exsits then they would affect your ruler as well as the so called extra dimention you are trying to measure, so this problem is not at all trivial.

Also the lack of experimental evidence for these dimentions is because we don't have enough energy to to perform these experiments and "probe" the the particles at such small distances.

marcus!
Thanks for all the good links.
I have been promoting a model which would fit the bill for Daniele Oriti.
See my journal for details.

I am looking forward to his book
"Approaches to Quantum Gravity: towards a new understanding of space, time and matter"
jal

Lelan Thara said:
And they do sound to me like mathematical abstractions.
The question of what is "real" or not is really kind of moot. I know this has been discussed several times here in connection to when people are arguing about wheter "virtual particles" really are there or not, and I think ZapperZ would be the right person to add some useful comments about this.
What I mean by a "real" extra dimension is that it is on equal footing with our usual 3.

I don't see anything in what you posted, or in your links, that leads me to think I couldn't locate any of these forces in the three standard spatial dimensions.
Well, if I have a model with a "real" extra dimension and calculate what effects this would have on different experiments, and then go on to measure exactly those effects, I would call them "real". Of course you could always say that its just that our 3d-world obeys different laws than we thought it was, and call my extra dimension an abstractation. But how could you then call anything "real"?

But I suspect that a day will come when we will define these dimensions as something other than spatial.
As marcus commented, it may be that dimensions in general will be defined in terms of something else. Who knows?

Take the "garden hose analogy". Imagine we're talking about a real garden hose, observed from orbit. Yes, the hose would look like a one-dimensional line to an astronaut in orbit - but if the astronaut returns to Earth, goes to the hose and measures its diameter with a ruler, would you seriously contend that our astronaut had discovered an "extra spatial dimension"?
I've never liked those "from far a telephone wire looks 1-dimensional but when you come closer you see it is 2-dimensional"-analogies, since I don't think they explain the essence of what a compact extra dimension is, but mearly gives rise to confusion. At least I find the analogy I gave much easier.

I have a ruler on my desk that measures 32nds of an inch. My car measures distances in miles. Is my car in a different dimension than my ruler?
No, not at all. It's nothing like that I'm trying to say.

EL - by asking whether the extra spatial dimensions are "real" or "mathematical abstractions", I'm afraid I've posed my question poorly. I'm aware that on some level even the 3 standard spatial dimensions in Euclidean space are a "mathematical abstraction".

So I hope you don't mind if I try to put the question more specifically.

If I can go back to my example of a mathematician drawing axes on a sheet of paper - let's say he draws ten sets of axes instead of two. Let's say he has a perfectly valid reason for doing this, and everything he plots on these axes represents something "real". Assuming all his axes extend to the edges of the page, he can pick any set of axes and locate any point on the paper, including all points he's plotted on his other 8 axes. So two dimensions are what I'm calling "real" - what I should be calling "necessary and sufficient" - and the other 8 are what I'm calling "mathematical abstractions".

So what I'm asking is - can all points within the 9, 10, or 25 spatial dimensions in string theories be located within a volume defined by 3 spatial dimensions? Are 3 spatial dimensions still "necessary and sufficient" to locate any point within the multiple extra dimensions?

Or are the extra dimensions really intended to describe volumes where 3 dimensions are not sufficient to locate all points in physical space?

To put it crudely, are the extra dimensions describing "somewhere else", or are they a more elaborate description of the normal way we think of 3-dimensional volume?

Thanks again - I feel I am learning something here.

Lelan Thara said:
So what I'm asking is - can all points within the 9, 10, or 25 spatial dimensions in string theories be located within a volume defined by 3 spatial dimensions?

Hi Lelan

One would like to think of the 3 spatial dimensions as an emergent classical geometry from the appropriate abstract picture, which I claim uses categorical dimension rigorously in place of the literal string dimensions. These categorical dimensions have some correspondence with the number of 'particles' (but this cannot be well defined right here) that people mention as a means of defining rulers. It is correct to expect no observations of naive extra dimensions, but one should equally be reluctant to impose the usual 3 dimensions a priori upon the quantum theory.

I apologise if unmathematical references for category theory in physics are hard to come by, but philosophers, for instance, have certainly written a great deal about these ideas.

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Lelan Thara said:
Are 3 spatial dimensions still "necessary and sufficient" to locate any point within the multiple extra dimensions?
I would say no. If there's a universal compactified extra dimension then you need a fourth coordinate to locate a point in this 4d-space.
However, this fourth coordinate will only be of importance for physics at scales comparable with the compactification radius of the extra dimension.

To put it crudely, are the extra dimensions describing "somewhere else", or are they a more elaborate description of the normal way we think of 3-dimensional volume?
Well, again. That really depends on what you define as "real". I would call them "real", but wheter I say they are "somewhere else" or not depends on what kind of extra dimensions they are.
If they are such that only gravity can propagate inside them, then maybe it's ok to say they are "somewhere else" since we can't put any matter or photons etc there. That is, we are stuck to our 3d-sheet which is embedded in the extra dimensions.
But if they are universal I'm not sure I can agree they are "somewhere else".

Btw, please correct me if I've got something wrong.

Kea, EL - one of my original questions was if "physicists actually believe" in the literal truth of higher spatial dimensions. From you two, it seems the best answer would be, "some physicists do believe in literal, higher spatial dimensions, some don't".

A few more questions: - do any of the extra dimensions have names? Are they identified with any familiar principles? For instance, is there a "dimension of angular momentum" or anything like that?

I know in general terms that strings are thought to vibrate in these extra dimensions. If I think of a real-world vibrating object like a guitar string, I can imagine establishing a plane perpendicular to the string, and establishing a circular reference in that plane, a kind of protractor, to measure the angles at which the string vibrates. Could I then call those angles "spatial dimensions"? And is that analogous to any of the extra spatial dimensions in string theory?

I'm trying to get a clearer idea of what makes the extra dimensions "spatial". Do you have to be functioning in a higher-dimensional space to begin with, like a Hilbert space, to recognize these dimensions as spatial?

(BTW, I don't really understand HIlbert spaces, other than thinking they are not restricted to 3 dimensions.)

Kea, I expect your answer would be that the extra dimensions aren't spatial. EL, (and anyone else who's interested) I'm curious what you would say about the above questions.

Kea said:
I apologise if unmathematical references for category theory in physics are hard to come by, but philosophers, for instance, have certainly written a great deal about these ideas.

No need to apologize. Just want to say I've found your answers both helpful and elegant, and I appreciate that you are not talking down to me.

EL said:
Btw, please correct me if I've got something wrong.

I assume this wasn't directed at me. I'm here to admit the limits of my knowledge, and I'm unlikely to know if you get something wrong.

Everybody else - please correct EL if he's got it wrong.

Slightly on topic, this afternoon I was going for some sun corner in the Cam river and I changed mind at trinity and entered into the library, with the result I got the cold but not the sun, in the corner of physics. But I happenned to read one of the oldest articles of Witten, a Nuclear Physics B on Kaluza Klein and 11 Dimensional Gravity.

He comes to 11 dimensions via a way Lelan Thara could enjoy: he asks for the minimal dimension a space must have in order to admit a decent action of G=SU(3)xSU(2)xU(1). This is done by looking the highest interesting subgrup H and quotienting G/H. The conclusion is that the dimension is 7, and so he starts to study Kaluza Klein theories in 4+7=11 dimensions (KK being defined broadly as the art of considering that these extra 7 dimensions are real and then they suffer the Einstein Equations).This early paper can explain why witten become so interested on jumping from 10 to 11 in string theory. Also it explains the interest on Dirac zero modes, because he argues that the 11 dimensional Dirac zero modes are the 4 dimensional particles; the eigenvalue of the 4 dimensional Dirac operator should be here minus the eigenvalue of the 7 dimensional operator, if you buy the argument.

Now, one wonder, how is that Connes has got to build a respetable action of G in a zero dimensional space. Well, know we know it is a 6 mod 8 dimensional space, but it is still zero dim from spectral point of view. And were it 7 mod 8, one could think that it has met Witten arguments.

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Lelan Thara said:
Everybody else - please correct EL if he's got it wrong.

You mean, if He...?

Lelan Thara said:
Kea, I expect your answer would be that the extra dimensions aren't spatial.

Yes, in the sense that extra classical large scale dimensions should not be observed. The categorical abstraction quantises the notion of point itself by framing it in a relational language which is a bit like drawing complex networks of causal connections not in real space but in terms of logic.

Lelan Thara said:
I'm trying to get a clearer idea of what makes the extra dimensions "spatial".
This is determined by the metric (I don't know how familiar you are with relativity though). Normaly we can write the invariant spacetime interval ds in terms of a time interval dt and three space intervalls dx, dy and dz, according to ds^2 = dt^2 - dx^2 - dy^2 - dz^2. If we add an extra dimension, call it the w-direction, and say that ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - dw^2, it is a spatial dimension since it, as the other 3 spatial dimensions, has a minus sign infront. The extra dimensions which you need to add in string theory in order to make the theory consistant are all spatial, while the number of temporal dimensions are still 1.
In principle you could of course also try adding more temporal dimensions, but this messes things up too much (I think).

arivero said:
You mean, if He...?
What do you mean by this?

Kea said:
Yes, in the sense that extra classical large scale dimensions should not be observed.
Of course we can not have universal extra dimensions without compactifying them enough, since then we would already have noticed them. Also in the case of extra dimensions in which only gravity is allowed to propagate, the compactification needs to be effective enough to not contradict the down to a mm-scale checks of the 1/r^2 law of gravity.
Only in some braneworld theories it looks like you can have some non-compactified extra dimension, but I hardly know about that subject.
However, I cannot see why there couldn't be "real" extra dimensions with small enough compactifiaction radii to avoid current experimental constraints?

EL said:
However, I cannot see why there couldn't be "real" extra dimensions with small enough compactifiaction radii to avoid current experimental constraints?

Because the theory is not elegant, and there is absolutely no reason to expect observations of these curled up dimensions, which by definition are classical in their extension, albeit small. In fact, the whole compactification program can be eliminated in the matrix model ribbon graph approach to M-theory.

Kea said:
Because the theory is not elegant,
Why? Is a 4+1 dimensional spacetime less elegant than a 3+1?
(And, anyway, why must the world be elegant?)

and there is absolutely no reason to expect observations of these curled up dimensions,
There are plenty of experimental consequences araising from compactified extra dimensions, as mensioned above.

EL said:
Why? Is a 4+1 dimensional spacetime less elegant than a 3+1?

We're not talking about starting with a 3+1D classical spacetime. That's the point. How can a quantum gravitational theory describing mass numbers possibly arise from such a severe restriction on the geometry? Perhaps I should also point out that here at PF we have had a number of interesting discussions about mass quantum numbers.

Kea. My question is simply: what makes the number of "real" spatial dimensions to be exactly 3? Why can't there as well be a 4th one on the same footing (although compactified)?

EL said:
My question is simply: what makes the number of "real" spatial dimensions to be exactly 3?

We require the theory to reproduce GR with 3+1 dimensions in some limit. There is no physical theory that I know of which requires 4+1 dimensions to be recovered. The graviton has only been correctly understood to date, AFAIK, from the measurement geometry point of view (see Brannen's homepage http://www.brannenworks.com/ ) from which lepton masses may be derived.

Dimension 3 is very special from the categorical point of view. For instance, at the tricategorical level one is forced to abandon ordinary Cartesian product in favour of the correct universal product. One is then forced to work in an $\infty$-dimensional framework. The recovery of real spatial dimensions is of course fairly involved, but roughly speaking the 3 comes about because of the special properties of the new products.

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Kea said:
We require the theory to reproduce GR with 3+1 dimensions in some limit.
Yes, but so does GR in 4+1 dimensions also, with one dimension compactified.

There is no physical theory that I know of which requires 4+1 dimensions to be recovered.
That I can agree with, but this doesn't rule out the existence of more "real" dimensions.

Let's say they find KK-particles at LHC, wouldn't you then call the extra dimension of the KK-theory "real"?

There are a number of reasons why superstrings should live in tendimensional spacetime : getting rid of the conformal anomaly, realising spacetime supersymmetry by worldsheet supersymmetry etc...

One important fact is that the isometry group of a D=10 massless boson is SO(8), which is a very special lie-group (the only one where the spinor representation has the same number of components as the TWO vectorial ones). I personally think that there is something profound in here that we have not quite understood.

Usually string theorists think of the extra spacelike dimensions as being real ones, although from the low energy viewpoint where spacetime looks fourdimensional you could regard it as an abstract internal space of gauge transformations. In short, it depends in the viewpoint you take.

EL said:
Let's say they find KK-particles at LHC...

Are you a betting person? I could do with a few extra dollars...

Dimitri Terryn said:
I personally think that there is something profound in here that we have not quite understood.

I do not disagree.

Dimitri Terryn said:
Usually string theorists think of the extra spacelike dimensions as being real ones, although from the low energy viewpoint where spacetime looks fourdimensional you could regard it as an abstract internal space of gauge transformations. In short, it depends in the viewpoint you take.
I fully agree with what you're saying.
Everything depends on what you decide to call "real" and not.
(One could as well argue that e.g. "electrons" and "photons" etc are not "real", but only exist in our models of the microscopic world.)

Kea said:
Are you a betting person? I could do with a few extra dollars...
No, I'm not saying I think they will be discovered, I'm just saying they have not been ruled out yet.
My question is though: would you call such extra dimension "real"?

Lisa Randall, in her book Warped Passages, takes at least one higher dimension very seriously. Her model is a bulk and boundary model; the standard model forces ( which of course includes all our biology) propagate in the boundary, identified with our 4-D spacetime, but the particles we see are projections or shadows of particle moving in the higher dimensional bulk (she has a precise mathematical description of this). She has calculated that the LHC ought to see clear signs if this model is correct.

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