How long is a piece of string?

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I have been meaning to write an opinion piece titled: Has a generation of theory been stolen by the strings? when it came to my attention we have never had a post outlining the basics of string theory. To ensure a more natural ordering for those who do follow RTU on a regular basis, today we will look at some of the central ideas in string theory and why it is has been such a luring prospect for some of the most intelligent minds on the planet. Before we begin I should manage expectations of what I will offer – I intend to explain in simple terms what string theory is, which will mean some statements may leave the reader with some questions – for example if I tell you that the equations of relativity can be found within string theory this isn’t quite as satisfying as showing you, but it’s also much easier for you to read and me to write. What credence you assign these statements will depend on who you are, but we will stay strictly in the domain of blogging.

If you do want something more rigorous, but still non-mathematical the best resource I can offer is The Little Book of String Theory – from the URL I believe this to be a legally hosted copy. For a more mathematical introduction, this is a good resource – although very few (me included) will be ready to fully follow this, as the content is advanced postgraduate level and beyond.

Genesis

Grand ideas of extra dimensions are not really contemporary, nor are they always wrong. Einstein’s experimentally verified theory of relativity was forward thinking in it’s masterful addition of time as a fourth dimension.  Less popular however, is the work of a German mathematician by the name of Theodor Kaluza, who many would argue took this idea much further. He was sitting in his house, marveling at the addition of extra dimensions in Einstein’s theory of relativity, and quite reasonably asked if it worked once, why not twice? In the early 20th century little was known of the weak and strong nuclear forces, so Kaluza thought about the electromagnetic force which was largely understood. He imagined a world with not three but four spacial dimensions, along with the time dimension, poked around a little and marveled as the equations of electromagnetism fell out of the bottom. He celebrated, for in his mind he had just found the unifying theory of Physics – the maths seemed to be all there.

Anyone who has read any popular Physics, or indeed this blog will know that there is no 5-dimensional Kaluza theorem which we hold as the final jewel of modern physics. The theory was fraught with difficulty, with much of the known constants of the universe computationally incorrect under Kaluza’s theorem. By the late 1940s after a lot of work by many individuals (including Einstein), the theory largely went away and unification went quiet.

Then comes string theory – which on the surface may not seem like a rebirth of Kaluza’s logic; but it most certainly is. The source of the river of thought is an age old question; take a cube of matter and divide it down and down until we get to the smallest divisible unit – is there such an thing, and if so what is it? These fundamental constituents of reality are not made of anything, they just are. String theory takes the idea that the whole universe before your eyes can be made up of small microscopic strings which vibrate under tension and, depending on the mode of the vibration produces different particles which we had previously labelled fundamental. If you stopped reading here you would most certainly understand the most basic premise of string theory, but miss the richness of why it’s so appealing.

A cosmic orchestra

If you have read any popular science literature in relation to string theory, you will have seen some form of reference to music – the symphony of the universe, or something to that effect. In general terms, we can define a string as anything that is longer than it is wide. In answer to the question, how long is a piece of string?, the predictions of string theory have a string at around the Planck length – a millionth of a billionth of a billionth of a centimeter. The size of a string is so unthinkably small, that for now we only ever consider them in the realms of theory. Indeed it was only recently we were able to probe the atom – the scale difference we are talking about here is akin to expanding an atom to the size of the earth, with a tree representing a string. In numerical terms this is around the Planck length, 10−35 m, which makes it very hard to test.

String theory proposes that in the same way point-like particles are modeled in quantum field theory, strings may also be modeled. We can allow these strings to be open, like some sort of cosmic worm, or closed into a loop. We are perfectly comfortable with the fact that a macroscopic string, such as a guitar string can vibrate at different frequencies and in doing so produce different notes – there are many factors that can impact this, the tension in the string being a key one. In a wholly analogous way, we can model the fundamental strings as vibrating in patterns, but instead of producing musical notes, the vibrations they make produce the different particles around us – something like this, if we could see just one in very slow motion. They are vibrating at a certain energy range, giving all of the richness you see before you.

string-red

This makes string theory a truly unifying theory, since when we talk about different particles we are actually just talking about different manifestations of the same thing – we are talking about the movement of one fundamental object (or rather billions and billions of them!). There is something pleasing to the scientist about explanations that involve oscillatory motion, which I can’t quite put my finger on – but it is certainly there.

The first difference that should jump out when comparing the music of a guitar to a string in space-time is the apparent absence of tension. If we took a guitar sting that was not attached to anything at either end and plucked that, we would quite rightfully not expect very much to happen. So are the strings of string theory under tension, or is this where the analogy breaks down? Well no it does not – strings too have tension, however unlike the guitar string it is an intrinsic tension within the string. This is one area where for the sake of brevity, we must leave as is but when you imagine them in your head they are indeed taught.

Branes

I thought about omitting this section – I think that in a basic understanding of string theory it isn’t wholly essential, but given you cannot read much further into the subject without encountering the word, it is included for completeness. A brane is a wonderful construction, and it comes from the word membrane. Up to now, when we have been talking about strings we have been taking about one dimensional strings – objects which have length. If we want to do clever mathematics, we need ways to generalize our ideas – this is where branes come in. Branes allow us to work with point-particles, which we can consider as a brane of dimension 0, strings which as discussed have dimension 1 or indeed any level of dimensions – we generalize a p-brane as a brane in dimension p. Whilst a brane may feel like a mathematical construction, they are modeled as real objects in the physical universe, possessing mass and charge.

The reason branes are an important part of string theory is in part a consequence of the open strings we discussed earlier. It is required that the two endpoints of these open strings lie on the end of a brane (called a D-brane) in order to exist – as such we can eliminate an image of open strings having random start and end points, and replace it with the idea of a string stretched between two membranes. Do remember that here we are talking about open strings – in the case of closed strings, as shown in the earlier image we do not require such constructions. The image below is a representation of this, with the branes on either side represented in two dimensions; what we would typically think of as a membrane.

800px-D3-brane_et_D2-brane

It is the study of branes that demonstrates the versatility of string theory – and therein the appeal. What seemingly is a theory of wiggling one dimensional objects is actually a description of the geometry of space-time, in a far richer manner than we ever could have thought possible. We can model from 0 dimensions as far as the imagination will allow, and then further. The question remains however, is this reality?

Extra dimensions

Now we come back to the logic of Kaluza – applying extra hidden dimensionality to arrive at a more complete description of the universe. The mathematics of string theory was largely carved out by arguably the most intelligent man to have ever lived, Edward Witten (who still lives by the way). When you are working on an idea which involves pure theory (because it can’t be properly tested yet) you need to rely on your mathematics. There are certain fundamental features of mathematics which we know to be “true”, along with certain mathematical relationships which we know to describe the world in which we reside. When something happens that contradicts our reality, say some physical quantity tends to infinity, we know that something isn’t right and we need to fix it. Mathematics can work as a diagnostic tool for any theory to identify any bugs it may have. String theory does work just fine – providing that you allow the world around us to have 11 dimensions -10 of space and 1 of time. There is always a catch. If you have seen different numbers quoted, this is for M-theory, the leading brand of string theory.

It is clear that if there are indeed more than three spacial dimensions and one time dimension then these extra dimensions are rather subtle – we do not experience them in our everyday lives. The most popular explanation for this is that the dimensions are very small, so small that it would be impossible for us to ever detect them in our day to day life – much like a telephone wire may look one dimensional from afar. This is clearly a stretch for the imagination; and it is virtually (if not totally) impossible for a human being to picture this scenario, without picturing it in three spacial dimensions, which isn’t really the point. Without studying the maths, you have to take it at face value that the extra dimensions are needed – your interpretation of this depends on who you are. For some of us, if the mathematics is hinting at something our perception of reality needs altering; for others if the mathematics isn’t mirroring reality then the math is broken. The jury is out for now I am afraid.

One thing is for sure – string theory, be it the final theory or not, is the most promising unified theory we have ever had so it demands a certain level of respect. The logo to this site is actually intimately linked with string theory – this is a Calabi-Yau manifold, which before applications in string theory was a purely topological construction. Compactification is the construction of four dimensional space-times with additional “hidden” dimensions – the one that works best with our understanding of the universe is this very manifold, which in string theory is taken to have 6 dimensions (i.e. the 6 hidden dimensions to give the 10 spatial dimensions mentioned earlier). If this is the true reality, the truth might look a little bit less like Instagram and more like this.

manifold

Should we get comfortable with this idea, then it is thought this may unlock the answers to some of the deepest questions out there. There are a handful of numbers, constant wherever you go in the universe that more or less characterize the world around us. It is why they say the universe is finely tuned – if you were to change any of these numbers by very slight amounts there could be no universe in its current form. We are wonderfully good a finding and using these numbers, but up to now we cannot find any logical (or illogical) reasoning to say why they are the way they are. Whilst we may have to accept the answer they just are, many hope that the interaction with higher dimensions, presently unavailable to us that will unlock the secrets in the way the universe behaves.

Where next

String theory needs to go somewhere. It has had a long and largely successful history with string (and superstring) revolutions – so why have we not written the final physics book and sat back and relaxed? Well the reason is we can’t just accept the totally alien universe offered up by string theory, just because it offers glimpses of a final theory. One of the ideas to test the theory is to try to use the LHC in Geneva to break a fundamental law of physics – the conservation of energy. If we can create a collision in which we have less energy after than we did before we know something funky would have happened – at that point a very luring explanation would be that the energy had leaked into higher dimensions, so it was not so much lost but rather unaccounted for.

It is important to remember we don’t have the full theory behind string theory. We have a really clever idea, in which we have fleshed out the mathematics and found we can model what we know about the world in a way that was far more successful than anything we have ever done before. It hints at many things which make us uncomfortable – and we are left to decide if we should continue to chip away at redefining reality, or look for something that feels a little more comfortable.

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