# Why string theory?

I am new to String Theory. I've read that String theory is an important theory because it is a good candidate for a unified theory of all forces. It is "better" than the Standard Model of particle physics because it included gravity. So, is this the importance of String theory (to unify all forces)? Or there are other features that make it a good theory?

edit: I am not asking for a complete explanation of the theory, I'm just trying to understand its importance (conceptually, not mathematically) as a starting point to hence start exploring its details.

"Why string theory?", you ask. I can think of three main reasons, which will of course appeal to each of us differently. The order does not indicate which I consider most or least important.

### Quantum gravity

A full theory of quantum gravity - that is, a theory that both includes the concepts of general relativity and those of quantum field theory - has proven elusive so far. For some reasons why, see e.g. the questions A list of inconveniences between quantum mechanics and (general) relativity? and the more technical What is a good mathematical description of the Non-renormalizability of gravity?. It should be noted that all this "non-renormalizability" is a perturbative statement and it may well be that quantum gravity is non-perturbatively renormalizable. This hope guides the asymptotic safety programme.

Nevertheless, already perturbative non-renormalizability motivates the search for a theoretical framework in which gravity can be treated in a renormalizable matter, at best perturbatively. String theory provides such a treatment: The infinite divergences of general relativity do not appear in string theory due to a similarity between the high energy and the low energy physics - the UV divergences of quantum field theory just do not appear. See also Does the renormalization group apply to string theory?

### Restricting the landscape of possible theories, "naturalness"

Contrary to what seems to be "well-known", string theory in fact restricts its possible models more powerfully than ordinary quantum field theory. The space of all viable quantum field theories is much larger than those that can be obtained as the low-energy QFT description of string theory, where the theories not coming from a string theory model are called the "swampland". See Vafa's The String Landscape and the Swampland [arXiv link].

Furthermore, there are many deep relations between many possible models of string theory, like the dualities which led Witten and others to conjecture a hidden underlying theory called M-theory. It is worth mentioning at this point that string theory itself is only defined in a perturbative manner, and no truly non-perturbative description is known. M-theory is supposed to provide such a description, and in particular show all the known string theory variants as arising from it in different limits. To many, this is a much more elegant description of physics than a quantum field theory, where, within rather loose limits, we seem to be able to just put in any fields we like. Nothing in quantum field theory singles out the structure of the Standard Model, but notably, gauge theories (loosely) like the Standard Model appear to be generated from string theoretic models with a certain "preference". It's hard to not get a gauge theory from string theory, and generating matter content is also possible without special pleading.

### Mathematical importance

Regardless of what the status of string theory as a fundamental theory of physics is, it has proven both a rich source of motivation for mathematicians as well as providing other areas of physics with a toolbox leading to deep and new insights. Most prominent among those is probably the AdS/CFT correspondence, leading to applications of originally string theoretic methods in other fields such as condensed matter. Mirror symmetry plays a similar role for pure mathematics.

Furthermore, string theory's emphasis on geometry - most of the intricacies of the phenomenology involve looking at the exact properties of certain manifolds or more general "shapes" - means it is led to examine objects that have long been of independent interest to mathematicians working on differential or algebraic geometry and related field. This has already led to a large bidirectional flow of ideas, where again Witten is one of the most prominent figures switching rather freely between doing things of "pure" mathematical interest and investigating "physical" questions.

In the last fifty years or so, during my active career in particle physics which started in 1963, an enormous amount of elementary particle data have been gathered. These data are almost completely codified in the standard model of particle physics..

This model unifies mathematically the three disparate observed interaction between particles, (strong weak electromagnetic) into one mathematical form, and is very good in predicting future data, as the lack of new (i.e. not standard model) physics at the LHC shows.

The interaction not included in the standard model is gravitational. Any mathematical model that aims at unifying a fourth interaction in a generalized model has to be able to embed the standard model , in a sense embedding it is a necessary condition to start with any model, because the SM is encapsulation of data.

Of all the imaginative proposals for a unifying model for all four interactions, only string theories offer the geometrical/group_theoretical structure necessary for embedding the standard model naturally. Naturally means not imposing it or carrying it as an appendage. The beautiful SU(3)xSU(2)xU(1)structure of the standard model can be naturally embedded in a string theoretical model, as well as supersymmetry which many theoretical particle physicists deem inevitable as an extension of the standard model ( for mathematical theoretical reasons).

That is why string theory based models, and string theories are pursued by so many excellent theorists. It can accommodate known data.

It remains to be seen whether predictions , like supersymmetry, based on string theoretical models, will be found at the LHC or we have to wait for higher energy machines for this.

Yes, the main motivation of string theory is to give a complete, unified explanation of all known physical forces. In addition, some people like it because it's "mathematically beautiful," and others because it leads to some mathematical results that can be applied in more specific settings (e.g. in solid-state physics, or nuclear collisions), but the primary motivation is definitely to unify gravity with the Standard Model (which explains almost everything other than gravity).

One way of understanding how the idea of string theory comes in the game is by considering the need for creating a relativistic quantum theory.

After first assuming the Klein-Gordon equation ( and finding the problems of negative probabilities later to be reinterpreted as charge densities) Dirac came with his equation for spin 1/2 particles, that is for example electrons. But there are also problems considering this equation if one merely stricts himself one a particle-like theory, because the negative energies coming coming out as eigenvalues of the problem where later resolved by Dirac considering a see of negative energy particles that we cannot measure. Due to the Pauli principle, these particles forbid their positive energy counter partners to "fall" from the positive to the negative energy. But, this resolution means that a finite-particle theory has become an infinite particle theory. There are a lot of mathematics here, for how Dirac's equation solves problems with relativity, but the fact that in the end this change in our scope must come is somehow problematic, adding also the observed phenomenon of particle creation- annihilation after a certain energy threshold; and that's something in a first attempt the Dirac theory could not predict.

The main problem, as became understood, was the crucial difference as how we treated time and space. Precisely, from a special relativistic scope, time and space should be treated on equal footing. But that's not the case in our quantum mechanical formulation. Although space is an operator( even if it's an unbounded one) time is merely a variable of the functions or the vectors( or the operators respectively). How could this observation be resolved?

The first attempt would be to change the character of space to a variable also, and by that giving birth systematically to Quantum Field Theory, where valued- operators of fields with variables those of space time would be the starting point.

But another attempt seems as logical as the first. What if we didn't change space in a variable but instead we moved on promoting time to an operator. A problem of course arises immediately as to how this idea could be formulated. Quoting from Srednicki:

If time becomes an operator, what do we use as the time parameter in the Schroedinger equation? Happily, in relativistic theories, there is more than one notion of time. We can use the proper time τ of the particle (the time measured by a clock that moves with it) as the time parameter. The coordinate time T (the time measured by a stationary clock in an inertial frame) is then promoted to an operator. In the Heisenberg picture (where the state of the system is fixed, but the operators are functions of time that obey the classical equations of motion), we would have operators X μ (τ ), where X 0 = T . Relativistic quantum mechanics can indeed be developed along these lines, but it is surprisingly complicated to do so. (The many times are the problem; any monotonic function of τ is just as good a candidate as τ itself for the proper time, and this infinite redundancy of descriptions must be understood and accounted for.)... For example, once we have X μ (τ ), why not consider adding some more parameters? Then we would have, for example, X μ (σ, τ ). Classically, this would give us a continuous family of worldlines, what we might call a worldsheet, and so $X_μ (σ, τ )$ would describe a propagating string.

And thus, this is a way of introducing strings.

Hope this helps.

• This doesn't really explain why one should study string theory of all things. The "adding parameters" ploy would also indicate that there's an infinite tower of theories with increasingly many parameters, but in fact there are not - to get those "brane theories" you have to do a lot more specific things, and they also appear from string theory. – ACuriousMind Sep 1 '16 at 11:05
• @ACuriousMind It doesn't? I thought it showed one logic from which the idea of such a theory emerges. I believe that is something that allows us to understand why a theory comes into play, what's the need or necessity that potentially leads us to a formulation, modelization and theorization. And it's another think to add parameters based on a certain logic and on data and knowledge of how the models of our universe works( like SR) and it' s another think to start adding parameters just to see were that leads- it's not but, it's just not what I was referring to. – Constantine Black Sep 1 '16 at 14:41
• My issue with this logic is that this doesn't give any motivation as to what is interesting or special about exactly two parameters upon which the $X_\mu$ depend - why study $X(\tau,\sigma)$, but not $X(\tau,\sigma,\rho)$? – ACuriousMind Sep 1 '16 at 14:43
• @ACuriousMind Also, the question doesn't ask why should one study string theory, it asks for an understanding of some important features of string theory, or at least that's how I understood it. You may of course disagree about the usefulnesses of my post, no problem on that :) . Also, your comment seems to me a bit philosophical or epistemological because it regards the way we think we can acquire knowledge and how and if an infinite tower of theories is something legal in our effort of understanding or not. I do not no if the site is constructed for such discussions. – Constantine Black Sep 1 '16 at 14:46
• Why the downvote, if one can explain? – Constantine Black Sep 1 '16 at 14:50

Thanks for the answers. Besides the important and main things mentioned above, I would like to add a little note that I found interesting in String Theory. ST has a degree of uniqueness that I found important as well; the dimensionality of spacetime is emerged from calculation not imposed as in Standard Model. Also it doesn't have adjustable dimensionless parameters, just a dimensionful parameter (the string length). I just wanted to share that for the new ones curious about ST (like me), and looking forward for comments to correct or add some info.

• That was certainly believed to the true in the early days of string theory, and was in important motivation and argument in favor of string theory. Unfortunately (or fortunately, depending on your point of view), with the development of the concept of the string landscape, that's no longer necessarily believed to be true. It's true that the total number of spacetime dimensions can be derived within a given theory (26 in bosonic string theory, 10 in superstring theory, 11 in M-theory). But any number of those dimensions can be "compactified" - curled up small enough that their existence ... – tparker Sep 1 '16 at 19:00
• ... is irrelevant on human scales. Moreover, there are an enormous number of "shapes" (called Calabi-Yau manifolds) into which the extra dimensions can be curled up - at least $10^{500}$. For example, if string theory describes our universe, then presumably all but four of the spacetime dimensions are curled up smaller than our accelerators can detect, but we have absolutely no idea which Calabi-Yau shape they're curled up into. Determining which "vacuum" our universe is in is known as the "landscape problem," and is known to be NP-hard. As Don Marolf once told me ... – tparker Sep 1 '16 at 19:15
• ... "String theory has no adjustable parameters, but it does have very complicated boundary conditions ... which look an awful lot like adjustable parameters." But some people think that even though $10^{500}$ is an enormous number, the fact that it's finite makes string theory more predictive than theories with continuous free parameters, which will always have an infinite number of different solutions. – tparker Sep 1 '16 at 19:19
• oh things are a bit more complicated than I thought they are lol. But even tough the dimensions can be curled up, no one imposes "d=4" in the equations. And in all cases String theory is still unique, no? But it is not the ultimate answer. – Milou Sep 1 '16 at 19:51
• Think about it this way - once you choose a Calabi-Yau manifold "vacuum" with which to curl up the extra dimensions, then string theory becomes completely predictive. But there's no way to predict which vacuum you live in - that needs to be determined empirically. (Even this is a bit of a simplification, because it's believed that universes may be able to dynamically tunnel between different vacua. But that's getting really quite technical.) – tparker Sep 1 '16 at 20:11

An important aspect that is essential for string theory is why the worldline formalism of a quantum field theory is messy. You have vertices in the graph and that stops it from being a manifold. If you replace lines with circles you can smooth this all out into a surface that you can actually do CFT on.

• Thanks for your answer, but unfortunately I couldn't understand it. – Milou Sep 1 '16 at 4:14
• This is a commonly cited reason, but this a far too brief an answer to be useful to someone who doesn't already know what is meant here. – ACuriousMind Sep 1 '16 at 11:06