# How long is a piece of string?

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.

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.

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.

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.

## 43 responses to “How long is a piece of string?”

1. Good intro and can’t help but ask: are branes entangled with brains, or does something else tangle up our minds? 😉

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• Thank you very much! Branes I suppose would be mixed up in just about anything!

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2. I enjoyed your excellent simplification it was about the right level for me. As for resting comfortably about reality let’s hope one day it will happen.

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• Thank you very much! I am glad you enjoyed, let us hope so!

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3. Excellent,as we’ve come to expect, I also share Mr Barker’s interest on a possible mind/matter integration in relation to string/M theory. I must confess I can’t keep up with theoretical physics, I was under the impression, clearly incorrectly, that string theory had been found suspect……..but then I remember when the Everett Wheeler many worlds theory of quantum reality was in vogue !!!
Thank you Joseph, your effort is much appreciated.

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• Thank you very much! And there are plenty of things suspect with string theory so you are not wrong… indeed it most certainly has not been accepted as “the” theory that paints the final picture – but it wouldn’t be quite correct to say it’s totally wrong (in my opinion). But yes opinions are moving!

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4. Appreciate the well written explanation. Gives me a lot to chew on. But, IMO, …we can never rest comfortably because there is no absolute truth: we can only discover what we can perceive, through are senses or our instruments, and they are infinitely limited in from of the infinite universe. In fact, that’s why I just blogged about!

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• How do you know there is no absolute truth. We have already greatly extended our senses , hearing what we could not hear , seeing what we cannot see.

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• Thank you very much! I am glad you enjoyed. It is certainly a big question – is there a final prize or an infinite chase. I don’t think anyone can “know” the answer, unless they hold the final prize – since you cannot say for certain something is infinite, you can just say it has not yet ended. To me, both prospects are just as exciting in 2017!

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• Both prospects are just as exciting? Perhaps. I am agnostic on the question of whether or not there is an absolute reality, but I certainly hope not!

What would our noble species of stick-poking great ape do with itself if it’s curiosity and thirst for discovery were ever satisfied once and for all?

No, I must come down firmly on the side of the monk Kenko, who around 1330 wrote, “The most precious thing in life is its uncertainty,”

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5. …but just because. IMO, we’ll never find the absolute truth doesn’t mean we should stop looking! Hell no!

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• Isn’t the statement “there is no absolute truth” an absolute one?

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• I think the most we can say is – we have not yet found an absolute truth, and I suspect that there may not be one. We can’t know for sure unless we’ve discovered how all physics operates.

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6. Back in 1939 when I was 13 I got interested in a popularization of Einstein’s four dimensional universe but found it tough to visualize. Around that time a short story in Astounding Science Fiction by Robert Heinlein “He Built a Crooked House” (see https://www.reddit.com/r/heinlein/comments/16v550/and_he_built_a_crooked_house_short_story_pdf/?st=j0wdwim0&sh=58679702 )
got me interested in tesseracts ( https://www.youtube.com/watch?v=t-WyreE9ZkI ) but to think in four dimensions was not the easiest exercise. I was having trouble with three dimensions when I got the idea of inventing a band saw blade with twice the life of a standard blade by using a mobius strip as a blade with teeth along the single edge but how to constrain that vicious ribbon within a usable machine escaped me. Perhaps someone with a livelier imagination can solve these multidimensional concepts visually but I must admit, at least so far, it’s beyond me. There is a much longer but not so concise discussion of these very theoretical concepts at
https://www.youtube.com/watch?v=h9MS9i-CdfY It does not really clarify much that is impenetrable to me but might be of some interest.

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• A very interesting few links which I will be sure to check out! A mobius strip saw – an interesting concept indeed. I see the issue though, there would be areas where the teeth would certainly snag where you don’t want them to. That’s one I am going to think over. Thank you also for the documentary – I am looking to learn more about hidden dimensions so it will be of great interest!

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7. “The most popular explanation for this is that the dimensions are very small.”

This explanation, along with “coiled up,” is commonly given by Green and the other string advocates. It seems a desperate expedient. I doubt anyone would have conceived of dimensions being small or large without the need to explain their not being experienced. Is the concept of small dimensions logically valid?

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• Yes that’s sort of what I was getting at when I was saying it is almost impossible not to imagine hidden dimensions just being in three dimensions anyhow. The best (and only) way we really have to do it is by using mathematical structures – which to many are not all that satisfying!

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8. Well written. ‘Ve read it to the end. Looking forward one like that quantum gravity.

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• Thank you very much!

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9. Okay, I’m sure what I’m about to say is ridiculous, nevertheless, here goes: You mention that energy lost at the subatomic level might be recovered at a higher dimension. That triggered the thought of emergence. And that triggered the thought of a God emerging from the natural complexity of the universe. Please keep in mind that if questioned, I will deny that I ever said this. 🙂

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10. That’s quite a difficult post you’ve got there. Not easy to master or understand. It’s hard to believe that there are more than 4 dimensions.
If this string theory is real what could we do with 10 dimensions? Or what could exist in those 6 extras according to you ?

P.S Please let me know if you’ve found anything about those numbers.

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• Sorry I keep forgetting your numbers! It’s been a pretty crazy time with work/uni!

We have no idea to be truthful what could happen in those extra dimensions, but it is thought quite logically we would be unable to detect things fully with three dimensional means in higher dimensions. It is possible that the higher dimensions are responsible for a lot of the quantum mechanical phenomena we find so weird. Remember of course this is pure brainstorming… We just can’t know the answer

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• Yes I thought that you had forgotten, it’s ok, well whenever you have time…..

I suppose time will tell us what those dimensions are for and if they really exist 🙂

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• If you want, I can repost those numbers with more details

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• Ok, there is this ancient book (whose title isn’t so important) some of its chapters start with a few separate letters which are not an abbreviation and are not linked to the chapter itself (not until now at least). So I thought that these letters must be numbers (but referring to what?).

Some of them are repeated in different chapters, the number of times repeated is between the brackets.

Statistics:
30 chapters
14 different letters (set between 1 to 5 letters)

1, 30, 40 (6 times) “multiplying by 6 it would be 6, 180, 240”
1, 30, 40, 90
1, 30, 200 (5 times) “5, 150, 1000”
1, 30, 40, 200
20, 5, 10, 70, 90
9, 5
9, 60, 40 (2 times) “18, 120, 80”
9, 60
10, 60
90
8, 40 (7 times) “56, 280”
70, 60, 100
100
50

Thank you for your time, feel free to ask if you need anything.

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• This is just a reminder in case you have forgotten, did you find anything yet?

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• Can I just check with you – what makes you so sure there is a link in them? There certainly isn’t anything obvious to suggest one, is there something which makes you sure there is one?

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• Thank you for your reply, I’m not sure there is a link, but I suppose there might be one because “actually it’s 29 chapters not 30, sorry” some of these chapters are consecutive, and these letters which repeat themselves seems too curious to be coincidental.

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• I thought that it might have to do with circles and degrees maybe? I don’t know

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• So what’s up guys? Have you collided with antimatter particles on you way home?

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11. Why a string instead of a dot? Isn’t a round shape more natural? And wouldn’t a string be composed of dots? (dot: the smallest part of the smallest part).

Wouldn’t a string would suggest dots linked by yet another force?

Is a dimension like another world, as in sci-fi? Or is it just something else being measured, like length, height, width, spin, flavor, etc.?

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• A 0 dimensional string is a dot (in the point particle sense). To be honest, the string idea didn’t come from selecting the most logical shape – it just came from applying these little lines instead of particles on a length scale we believe would be fundamental. Usually when we say “what if the world was made of little cubes” and we model it nothing actually works or looks like the world. With string theory, not only does it work but it heals many issues in Physics – at the trade off from higher dimensions. So it is not so much that strings are logical… more it seems implausible that the should be such agreement to reality without a deeper truth, whatever that may be.

And I think the higher dimensions would have a similar feel to length, height and widith. The Shape of Inner Space is a wonderful read on the subject, written by Yau (as in Calibai-Yau manifold)

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12. Interesting topic Joseph. It reminds me of my neighbor in Los Alamos asking me for years to draw him a picture of the 10th dimension. We’re still digesting the concept of a 4th dimension in the visual arts – it is quite a few steps up the ladder of knowledge.
How do you integrate “time” in a flat two-dimensional representation – that is unless you completely rewire the make-up of human perception?

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13. Thank you Joseph for the abridged explanation – explained a number of things for me.

I question the statement: “..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) ” Surely in a Universe of 10 spatial and 1 of Time there are actually 7 hidden spatial dimensions (11 total minus our ‘known’ 4, the 1 of time being shared)?

On a different point, our known 3-dimensional space requires each of the 3 dimensions to be at exactly 90 degrees to the other two. Could the other 7 out of 10 dimensions not be in parallel with any of our 3? But separated out of phase or in some other way that makes them indistinguishable to our senses from the regular ones?

Finally why can we stand still in our 3 dimensions but not stand still in time? Why can we only travel (and not rest) in one direction on the time ‘axis’?

love.

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14. Thank you for such a fascinating, apparently competent, and well-written overview of string theory, Joseph! I feel grateful to you for it.

This is quite an insignificant point, yet involves a pet peeve of mine: I don’t think it’s usually necessary for blog authors to defend their decision to leave out information for the sake of easy comprehension. It suffices, I think, for the author to merely mention that he or she is not attempting a comprehensive treatment of the topic. Moreover, should someone attempt to shame the author for, say, “being superficial”, the real shame is on them for failing to grasp that not every discussion of some matter needs to be comprehensive in order to be of value.

Having gotten my pet peeve out of the way, I would ask a philosophical question. Suppose we came up with a model of strings that predicted every “key number” in the universe. In such circumstances, would you be prepared to say strings were real? That is, that the model mirrored a reality? Just curious to understand your thinking, but not wishing to open up a debate.

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15. This is my favourite saying, whenever anyone asks me to quantify something. I’ll have to come back to this page later, when I have more time.

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• Thank you very much! Glad you visited

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