ajb

It is a quanum mechanical matrix element. Without knowing what [Math]J[/Math] or [Math]Q[/Math] is, that is all I can say. So yes, it does mean something.

This is true! But then how big is the blackboard and how big does the professor write?

It is radical in the sence that it is different to quantum theory of point like particles. The one thing it does seem to is include gravity. In the lower lying states spin-2 gravitons are present, so string theory naturally includes gravity. This I think is one of it's biggest selling points. That and the fact that one can also include gauge theories and potentially the standard model. It is possible that during inflation fundamental strings were "blown up" to the macroscopic level. These strings could be detected astronomically. However, I don't think it is clear how one could tell the difference between a cosmic string and a fundamental string. So, in principle maybe one day we will see a string! Well, so far there is no experimental observations that go against the standard model. So in that respect we are "finished". However, there are problems with the standard model from a theoretical and philosphical point of view. It is these that need attention. (Also not to mention calculational difficulties). Yes, people should think for themselves, but science works by adding to current knowlegde. You would be very foolish if you went back to the "very begining" every time you tried to understand something. By this I mean one always works on top of ideas and constructions that other scientists have developed. Currently, string theory looks like the best candidate for a unified theory. Also, to my knowledge there are no theoretical results from other approaches that go against string theory calculations. Things like black hole entropy and discrete space-time on small scales all agree (maybe up to some numerical factors ect.). Also of verry recient interests are gravity-gauge dualities such as the AdS-CFT correspondence. It is now possible to do calculations in QCD using gravity and string theory! What I would say, is don't knock string theory untill you have had a go at string theory. Although I do not work on string theory directly, I know about it's sucesses and failures. It is these failures that allow research on the subject to continue. On a personal note, I think that we will undertand a lot more about string theory when we have a good knowledge of string field theory. I think this is partly why string theory is difficult at the moment. Most approaches are based on first quantisation and not second quantisation. If string field theory can be developed then one might be able to use geometric ideas to get at string interactions. This I think would be a break through. At the moment adding interaction to first quantised strings is done "by hand". It is this that I feel makes things unclear.

Why do you find that amusing?

I agree with this. Not that I work on string theory directly, I know plenty of people that do. What I would say is that is is a little frustrating that very little phenomenology has come from string theory. I have met Szabo before, he seems a nice guy. I have also sopken to one of his students before.

If you know a little quantum mechanics and special relativity then you should be able to cope with the books I suggest. If you look at http://arxiv.org/abs/hep-th/0207142 you can get a free version of Szabo's book.

The questions you pose are real research questions. On the extra dimensions there are two solutions to this. Either they are compactifed, that is curled up so small that our space-time looks 4 dimensional or we live on some 4-d extended object known as a brane. (or maybe some combination of the two ideas). As of yet, there is no good explanation as to why we live in 4 dimensions. Depending on your mathematical background I suggest you read some introductory books on the subject. I suggest "An Introduction to String Theory and D-Brane Dynamics" by Richard J. Szabo or "A First Course in String Theory" by Barton Zwiebach. Szabo has an online version of his book. Search on SPIRES to find it.

In physics we are almost always interested in how things change was we vary some parameters. Often this would be time. Unless we are dealing with things that don't change in time, I cannot see how we could get rid of it, i.e. have no dynamics. Of course in relativity we mix up space and time, so we can think of space and time as playing the same role. This is ok. We can even think of generalising mechanics to have more that one time, that works. But no time, I just can't see how that would work. To Swanshot; is that strictly true what you said? I mean, even on [math]\mathbb{R}^{2}[/math] thought of as a (trivial) fibre bundle, that is one copy for time and one for space there exists diffeomorphisms that are not bundle automorphisms and hence destroy the distinction between space and time.

Thats great fredrik. I thought our solutions would be the same, but I did not check.

I asked mathematica and it gave me a horrible answer involving the productlog. As the productlog is an inverse function some of the solutions may be missing. anyway, here is my answer [math]y(t) = -C2 -C2 \; Productlog \left( \frac{e^{-1 - \frac{t-C1}{C2}}}{C2}\right)[/math] As to how to get to that answer I don't know. Best of luck working it out!

I would echo the earlier statements. I can think of 4 main areas of research, (I don't list QFT, strings or quantum gravity as they go further than what I would call QM) 1) Fundamental; using functional analysis to set up QM. This is very mathematical. 2) Applied; solving the Schrodinger equation either exactly or numerically. Could be part of condensed matter or atomic physics research. 3)Quantum information; quantum computing and similar. 4)QM as a "test" of quantum field theory; QM can be thought of as 1-d quantum field theory, so one can test ideas here. Supersymmetric and PT-symmetric QM are examples of this. I am sure there are other areas of research...

To my knowledge, general relativity does not "easily" allow for a topological change of space-time. By this I mean cut, puncture, twist or glue space-time. Having a Lorentzian metric puts some restrictions on the topology and as such any topological change must be consistent with that. This is is what makes it difficult. In the context of quantum gravity, it is suggest that not only do we need to consider fluctuations in the geometry, but also topology. Because of this you will find that most papers about space-time topological change will be in quantum gravity. see http://arxiv.org/abs/gr-qc/9406053, for references.

First thing, General Relativity tells you the local geometry of space-time, but not the global topology (i.e. it's global shape). Sometimes you can make statments about the topology via local geometry, eg the sphere for example has a constant local curvature and so you can make this a global notion. Generally, one can have many manifolds with the same local geometry but are very different topologically. (consider compactification for example). Thus the topology of the universe is not fixed by GR.

I don't know what you are asking here. What is true, is that gravity can be viewed as "two copies" of a standard gauge theory. This deep fact allows one to relate caluclations in gravity and gauge theory. You should look up things like MHV amplitudes and the Kawai, Lewellen, and Tye relations. These things relate tree-level amplitudes in gravity and gauge theory. It has also been shown how to go further that tree-level. Thsi stuff is not very familiar to me, so you will have to do your own research here.

I think that most scientists tend to steer clear of such questions as it could damage the reputation. As such not much real science has been done in this area.

The answer is YES. Look up the AdS-CFT correspondence.

Go away and learn some QFT and you will know what I mean.

I suggest you read Andrew Liddle's book An Introduction to Modern Cosmology. It is a good introduction to modern cosmology. It doesn't answer all the questions you have posed, but it will be a good place for you to start.

I don't understand what you are saying. First you said that Tachyons have imaginary mass then you say they have negative mass. I think that Tachyons have imaginary "rest mass". (unless you modify the mass-shell condition as I suggest). This is not a problem as we can never measure the rest mass of a Tachyon. This is becasue as we move slower than light we can never share a rest frame with something moving faster than light. The same is true for photons, we can never share a rest frame with them and so we cannot have a good notion of rest mass for a photon, i.e. we have the "strange" m = 0. So I would not get too hung up on the notion of an imaginary mass. We never observe it. The "mass" of a tachyon should be considered as a parameter in the theory and you should be very careful about attatching a physical notion to it. (This is what one should do when dealing with QFT anyway as we have issues of renormalisation to contend with before we can attach any physical significance to parameters). Which is what I said.

I should also say that some of the original papers on the subject can be found on George Sudarshan's website. He was one of the first people to investigate faster than light particles from a modern view point.

One way to avoid the issue of an imaginary mass is to modify the mass shell condition [math]E^{2} + m^{2}= p^{2}[/math] By doing so we now have a positive mass squared and so no issue about imaginary mass. Doing this does not effect any of the properties of tachyons. In quantum field theory tachyons are usually associated with a poor choice of vacuum when doing pertubation theory. That is they can be removed from the spectrum by making another choice. In string theories, this approach is more difficult and is the subject of "taychon condensation". Tachyons as well as generically being unstable, would produce some disturbing effects notably if they are charged. As they interact with the electromagnetic field they lose energy due to Cherenkov radiation. As they lose energy they accelerate and lose more energy in a run away process! This process could produce infinite energy! The same thing would happen in a gravitational field. So, it looks like QFT should forbid tachyons even if relativity does not.

String theory was originally discovered by trying to describe hadron physics and not as a unification scheme. When gauge theory took over string theory fell out of fasion. It was ressurected when it was shown that the string spectrum must contain a graviton-like state. Also, it was show that such theories are good quantum theories, i.e. they suff no anomalies (provided we work in higer dimensions). For details pick up any good book on string theory. I suggest BUSSTEPP Lectures on String Theory by Richard Szabo, which has a good but short historical overview. On a personal not, I notice a lot of "string bashing" going on from people who openly state they don't understand much about string theory. Is this just jumping on the bandwagon?

I think the method is to concider how much mass is lost due to the nuclear reactions in the star per unit time. This you can estimate this from the luminosity using [math] E = m c^{2}[/math]. Let the rate of mass loss be [math]R[/math]. It has units of mass/time. You need to concider that the nuclear reaction is 70% efficeint when calculating this. Let the mass of fuel be [math]M[/math], which you know to be 10% of the total mass. If you evaluate [math]T = \frac{M}{R}[/math] you will have an estimate of the lifetime of the star. I will let you work out the numbers, but be careful of units. Another method is to use the so called Mass-Luminosity relation. [math]L = m^{3.5} [/math]. As an example, the Sun's lifetime of 10 billion years. Hope that helps.

I think interest in string theory will remain constant or even increase, but this is due to all the string bashing of late. String theory itself will probably be a less popular theory as viewed by the general public.

Calculate the Jacobian factor when changing from cartesian coordinates to polar coordinates and you will get the factors you desire.