M-Theory

Friday, December 07, 2018

M-Theory - More Dimensions, Supergravity and Unification

After reading the posts on string theory on this blog, you may have started to think that string theory is an amazing theory that does not have many problems, and will certainly turn out to be correct once computers and experiments can help verify some things.
One major problem we have not discussed yet is that there are actually five different versions of string theory...

There are 5 versions of string theory which all incorporate supersymmetry differently. Since the equations in string theory are so complex, it is not possible to eliminate any of the five theories at the moment as well.

Perturbation Theory

Perturbation theory is a mathematical method for finding an approximate solution to a problem.
You start by solving a simpler problem to get you in the ball park, then slowly improving your solution by adding more detail.
If you were asked to predict the motion of the Earth, the way you could do this using perturbation theory is by starting by considering how the sun affects the Earth's motion. This would give you a rough estimate as the sun's gravity has the largest influence on the Earth's gravity. You should probably add the moon's effect next since that has the next biggest affect on the Earth. Next you would add mars, Jupiter etc....
Each time you add another planet, your prediction for the motion of the Earth would become slightly more refined and eventually, the effects would be negligible and so adding more gravitational influences would be pointless.

This method is not always applicable however. If you have a trinary star system where the gravitational effects are all of equal, the method fails. You would start by considering the effect of Star A on star B but then when you add your improvement, your improvement would be just as significant as your ball park estimate. The improvement would also impact how star A moves significantly. If you tried to improve again by taking into account how star A moves differently, This would effect star C a lot as well. This kind of loop continues forever and your estimate doesn't improve.

A Trinary Star System

The key with perturbation is that the initial estimate has to be able to get you close enough to the target.
The way I think of perturbation is like the sum of geometric series to infinity. In the first example, the series converges to a value because the terms get smaller each time but in the second example, the series diverges because the terms are all the same or getting bigger. The second example would be like trying to add up 1 + 2 + 4 + 8 + .... (all the way to infinity).

String Interactions

The uncertainty principles in quantum mechanics means that when two strings interact by joining and splitting, while joint, as long as the time is very small, a virtual string pair can be created and annihilated. This formation and annihilation of a virtual string pair can occur any number of times. If it happens once, it is called a one-loop process, if it happens twice it is a two-loop process etc...

This means to exactly calculate quantities associated with this interaction, you need to consider all the possible number of loop processes. This is infinite and the calculations increase in difficulty as the loop number increases and so the perturbation framework is the only way to go about this. This relies on the the 0 loop process being the dominant factor and the subsequent loop processes being less and less significant.

Are We In The Ballpark?

The string coupling constant determines whether we are. This number determines the probability that quantum foam will go crazy and yield virtual strings.

A high string coupling constant means fluctuations are more likely and so the string is likely to split many times(high in this case means a string coupling constant greater than 1).

A low constant (less than 1) means that we are unlikely to see splits.

So what's the value of the string coupling constant?

The conclusions we reach in string theory are only valid if the string coupling constant is less than one. The precise value of the constant impacts masses and charges of string vibration, so it is very important.

Attempts have been made to solve an equation which should tell us the string coupling constant but in all 5 versions of string theory, it reduces to SCC x 0 = 0. This means the SCC (string coupling constant) can take on any value to be a solution to the equation.

What Does This Mean?

1. The equations allow many solutions and so we lose. We in essence get an upgraded standard model but fail to reach a theory of everything. We have to simply accept the most fundamental laws of physics instead of being able to actually explain why time only goes forward, why we have 3 spacial dimensions, why there are 4 fundamental forces etc...

2. The flexibility shows a flaw in our reasoning. Maybe the SCC is greater than 1 and so we are simply way off the mark.

3. Maybe because we have approximate equations, there are many solutions. If one day we manage to get exact equations, the solution(s) will be a lot more restrictive.

Duality

The idea of duality in physics can be very important as we have seen. Mirror symmetry is an example of this as it involves two geometrically distant Calabi-Yau spaces that have the same laws of physics. The radius - 1/radius symmetry is another example of this duality.

In 1995, Edward Witten suggested a new type of duality which gave birth to M Theory. He suggested that the 5 string theories are different ways of describing the same thing.

The reason why they all seemed different is because we only looked at them when they were weakly coupled (SCC < 1). Since we rely on perturbative methods, we have been unable to answer what happens if the SCC > 1. If we could answer these questions however, then according to Witten, we would have noticed these dualities.

He suggested that along with a 6th theory, the strong coupling behavior of any string theory has a dual description in terms of a weak coupling.

BPS States

If you have a problem in physics, such as which 3 particles are in a box, it is nearly impossible to answer since there are many different combinations of particles to choose from. Supersymmetry constrains these kind of problems by giving clues such as, the maximum mass of the box is ___. This means when SCC > 1, one can use this kind of idea to find something called BPS states. This is the minimum mass for a chosen value of charge, e.g, we might say if the charge of a particle is -1, the minimum mass is the mass of an electron. The key idea here is that this does not use perturbation and so it can be used when SCC > 1.

M-Theory

Take Type 1 theory and analyse it when the SCC < 1 using perturbation. When SCC > 1, look at the BPS states in the theory. Witten found that the strong coupling of type 1 agrees with the weak coupling of Heterotic-O.
This means Type 1 and Heterotic-0 are actually dual and have a strong-weak duality.

This means we can always use the SCC < 1 theory to analyse physics using perturbation. For example, if we want to look at Type 1 theory when the SCC > 1, simply look at Heterotic-O when the SCC < 1.This has not been proven rigorously yet, but the perfect alignment that we see very strongly suggests that this duality is correct.

Type IIB is also self-dual. This means if SCC > 1, we can change it to the reciprocal 1/SCC. The reciprocal value of the SCC will be less than one and so we can use our perturbative framework.

Another duality is Type IIA and TypeIIB
and Het O and Het E. They get exchanged by the R - 1/R relationship we discovered earlier. The physics in a universe with radius R in Type IIA is identical to the physics in a universe with radius 1/R using type IIB theory.

Supergravity

Before string theory, there was a theory called supergravity which attempted to unify quantum field theory and general relativity. It incorporated supersymmetry because supersymmetry meant that fermions and bosons cancelled which calmed the quantum foam. The theory failed, however was closest to success when 11 spacial dimension were incorporated, 4 extended space-time dimensions and 7 curled spacial dimensions. When low energies are used, the extended nature of strings can not be probed and so point particles can be used to approximate them. String theory at low energies therefore should be an exact version of a quantum field theory. It turns out that 10D supergravity is the closest approximation of string theory. There are 4 10D supergravity theories, 3 are low energy approximations of Type IIA, Type IIB and Heterotic-E. The 4th is an approximation of both Type I and Heterotic-O.

Edward Witten also claimed that Type IIA had a low energy approximation of 11D supergravity by analyzing properties at all the string coupling constants (using BPS States when SCC > 1).

An Eleventh Dimension?

Heterotic-E string theory actually has an 11D description.
When SCC < 1, it looks 10D but when the SCC > 1 and increases, an additional dimension becomes visible for the string. The string turns from a 1D loop into a 2D membrane with a width controlled by the SCC. When we used perturbation, we assumed the SCC < 1 and so we never saw this additional dimension. This made strings always act 1D. A key point here is that string won't be able to vibrate in this new dimension however so things don't get too crazy.

This is the same for Type IIA strings where the 11th dimension is controlled by the SCC. In this case however, the string expands to become a tube instead of a 2D membrane. At low energies, this theory is approximated by 11D supergravity, but at high energy what happens? Who knows?
If the SCC is high, what's physics like? Remember all our previous assumptions were based on a SCC < 1.

M Theory

Elephant and 3 blind men is a brilliant analogy to describe M-Theory. The different parts of M-Theory are like different parts of the elephant. Scientists have been holding different parts of the elephant and claiming "it's a wall", "it's a snake", "it's a spear" etc... What M-Theory is suggesting, is that they are all actually part of one bigger unified theory and that string theory was much richer than we ever thought.

Forces

In an earlier post, we established that supersymmetry allowed 3 of the 4 forces to have the same strength in string theory. You can make gravity that same strength by carefully altering Calabi-Yau shapes. This is a little dodgy however as it is like looking at the answer booklet on a homework and writing in the answer instead of showing you got there through obvious logical steps. Edward Witten in 1995 showed that the strength of gravity can change and can meet the other 3 forces by changing the string coupling constant. We do not know the value of this parameter and so it leaves a good chance that in a final theory of everything, the four fundamental forces will be united.

More Dimensions

Even more dimensions can get put in as BPS states have 3D, 4D and up to 9D strings. In this case, physics would get really crazy. This is because the mass of the extended string (apart from 1D) is inversely proportional to the SCC. With a low SCC, strings are massive and therefore require a lot of energy. This means higher dimensional branes have small effects on the physics we see around us. If the SCC is high however, the higher dimensional branes could have a visible impact on physics.

With M-Theory, string theory becomes much more complex than anyone would've imagined. The string coupling constant seems like an important parameter that needs to be determined. If physicists manage to determine this, it will tell us which part of the diagram our universe is in.

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