Magnetic monopoles, fine structure and String theory

[shounakbhatta]

Hello,

Can anybody please explain me how magnetic monopoles, fine structure constant and string theory are related to each other?


Thanks

 

[mitchell porter]

Can you first explain what motivates your question? Most things in physics are connected somehow, in lots of ways. If we know why you are asking, we can give an appropriate answer.

 

[shounakbhatta]

Hello Mitchell,

Surely. I was listening to a lecture of Dr.Susskind on the first introduction to String Theory. He was telling that magnetic monopoles, through they do not exists practically can be considered to exist in theory. Hence, when we increase or decrease the fine structure constant, g the magnetic monopole as well as the electric field, concerned to that of the electron also changes. Some gets lighter, some gets heavier. At a certain point of time, when g is quiet big, they just interchange (the electric field and the magnetic monopole.) So, he concludes that which one is fundamental, it is very difficult to tell.He draws an analogy, that in the same way, fundamental string, f and D1 brane also becomes lighter and heavier with the changing of g.

I want to understand a little bit more into that.

Thanks.

 

[mitchell porter]

OK, that sounds like the first 30 minutes of this lecture, where he talks about bosonization, electric-magnetic duality, and S-duality.

You could say that the first essential concept here, is the idea that a particle has a cloud of "virtual particles" surrounding it. The central object emits temporary particles, which themselves emit temporary particles, and so on.

Then the next concept is the "coupling constant", which will tell you how often the virtual particles get emitted. If the coupling constant is small, the cloud of virtual particles is sparse, and it does not extend far beyond the central object. If the coupling constant is large, the cloud of virtual particles is dense, and extends significantly beyond the central object.

Susskind is describing physical theories in which there are two complementary objects - electrically charged particle (e.g. electron) and magnetically charged particle (magnetic monopole), or fundamental string (F-string) and one-dimensional brane (D1-brane, D-string). If the coupling constant is small, then one of the objects (electron, F-string) is light and small, because its virtual cloud is small, and the other object (magnetic monopole, D1-brane) is a composite object with strong, complicated interactions. But if the coupling constant is big, the first object has a large virtual cloud, and the second object gets small and light - they behave in the opposite way.

Susskind explains this in a very informal way in the lecture. Perhaps the deeper issue is, mathematically what sort of theory is it, that exhibits this kind of symmetry between light elementary objects and heavy composite objects. Possibly electric-magnetic duality and S-duality both come from a type of symmetry in string theory, "modular invariance", which I guess is about the different ways you can place coordinates on the string... but I need to think a lot more about that.

Also, Susskind's first example, bosonization, in which the same physics can be described by fermions or by bosons, is a different thing. It's similar in that you have two perspectives, but the details are different from the later examples, and so far as I can see, bosonization must have a different ultimate cause.

 

[shounakbhatta]

Hello Mitchell,

Yes. I was listening to that only. Thank you very much for clarifying concept on magnetic monopoles and duality. I am a self learner, hence I spent night after night to get hold of the concept, at least an idea on that. I would be very glad if you can please explain me on certain things. I still have few questions, if you can please help me with that:

(1) While take of 10 dimensions, is it that String theory takes 6 dimension of Calabi-Yau manifold and 4 dimension of space time?

(2) When he gives an example of a 2-d brane, (a peanut butter sandwich) he tells that with the change in the g, the fine structure constant, the <<space expands>> and the gravitons move to that extended space. I understand that string theory is not yet detected, but still what it means? The extra dimensions, are they mathematical interpretation to accommodate, general relativity and quantum mechanics?

Thanks.

 

[mitchell porter]

I didn't reply because there are some basic ideas like, space with more than 3 dimensions, space that is a "product" of two other spaces, space that is curled back on itself or has a complicated topology... which this discussion presupposes. And I'm not sure if you actually understand those concepts yet, and I do not have the energy to write an explanation for them. Perhaps someone else will. But I will answer your questions in a way that assumes an understanding of those ideas.

(1) String theory takes place in a space-time with nine dimensions of space and one dimension of time. Mathematically, a space-time which is a product of four-dimensional space-time with a six-dimensional Calabi-Yau is just one possibility for string theory. Mathematically, string theory can also take place in a space-time in which all nine dimensions are "large" (infinite, uncompactified), or in which the compact submanifold has less or more than six dimensions. The emphasis on spaces of the form M4 x CY6 (four-dimensional space-time "times" a Calabi-Yau) is because our physical space might be like that.

(2) Susskind is actually talking about the relationship between the 9+1 dimensions of a type of string theory ("Type IIA") and the 10+1 dimensions of M-theory. The discussion of a two-dimensional peanut butter sandwich whose third dimension (depth of the peanut butter) grows to visibility was just for visualization and discussion, but in reality we are talking about the appearance of an eleventh dimension at high values of the coupling constant.

So, in the ten-dimensional Type IIA string theory, among the objects in the theory are the fundamental string, and the D0-branes, branes which are just a point (zero-dimensional), heavy objects where a string can end. Then he says that for states of Type IIA string theory in which the string coupling is large, these two objects will begin to behave in ways as if they are moving out into an eleventh dimension.

The emergent eleven-dimensional space is actually "ten-dimensional space-time times a circle", and the radius of the circle is proportional to the ten-dimensional string coupling constant. So when the circle is extremely small, the string coupling is small. When the circle is big, the string coupling would be large, i.e. interactions would be strong; but it becomes simpler to switch to a dual description. This other description is eleven-dimensional, the fundamental string (a one-dimensional object) instead becomes a donut-shape wrapped along the new "circle" direction... a 2-brane, called an M2-brane since this is M-theory... and the D0-branes that move in ten dimensions become the gravitons of eleven dimensions.

Another way to look at this is to start in eleven dimensions with M-theory. M-theory contains M2-branes, M5-branes, and a supergraviton field analogous to the metric field in general relativity. If we consider M-theory on an eleven-dimensional space in which one direction is closed like a circle, and ask what happens to the theory if we shrink that direction until it is smaller than the branes, the answer is that we get Type IIA string theory in ten dimensions: an M2-brane that was wrapped around the eleventh dimension turns into a Type IIA string, and a supergraviton turns into a D0-brane. (Also, an M2-brane that wasn't wrapped around the eleventh dimension now appears as a D2-brane, but Susskind didn't mention that.) Finally, the size of the eleventh dimension shows up as the strength of the interaction between the strings.

That the D0-branes come from the d=11 supergravity field was new to me, somehow I had missed that in my own studies, I'll have to think about it further. Much of this material was put together in a famous 1995 lecture by Witten; Susskind may also be thinking of his own work on the BFSS "matrix model" of M-theory.

 

[shounakbhatta]

Hello Mitchell,

Thank you very much for your detailed reply. Definitely, I would not expect you to explain the basic as it is a tedious process. I would read in detail your explanation and with your kind permission would ask few questions.