Basic Scheme and Fundamental Equations of String Theory

[masteralien]

TL;DR Summary

What is the basic scheme and equation of String Theory, Classical Mechanics has F=ma or Lagrange/Hamilton for the trajectory, QM the Schrodinger Equation for the Wavefunction, GR the Einstein Field Equation for the Metric, Electromagnetism has Maxwell’s Equations for the EM Fields, QFT The Dyson Schwinger Equations for the Correlation Functions what is the equivalent for String Theory.

In Classical Mechanics you solve for a particle’s position through F=ma or the Lagrangian and Hamiltonian Methods. Quantum Mechanics you solve for the wavefunction or density matrix either the Schrodinger or Von Neumann Equations respectively or use Operator and Path Integral Methods, Electromagnetism has Maxwell’s Equations for the EM Fields through direct solution or Jefimenko’s Equations for the dynamic case and Biot Savart/Coulomb’s law for the static case.

General Relativity has the Einstein Field Equations for the Metric. In Quantum Field Theory one computes n point Correlation Functions through the Schwinger Dyson Equations, Feynman Diagrams and Rules in Position Space, S Matrix/LSZ formula (Momentum Space Diagrams), or Path Integrals.


What is the equivalent procedure/equations in String Theory or M theory. Does String Theory even have a “Main Equation” and a quantity to compute. What is the main Differential Equation governing String Theory and what are the important quantities to compute and through what methods does one use, presumably its the behavior of Strings but is it like how the String Wavefunction evolves or its Correlation Functions and their evolution in spacetime like in QFT. Im just curious and have no experience in String Theory and want to know whats the main equation and procedure of String Theory.

 

[Vanadium 50]

I don't think such a thing exists. String theory is not a single theory - it is a framework upon which to build theories.

I also disagree with your premise. As you yourself say, there are at least three different formulations of Classical Mechanics. So there is no single "master equation" there either.

 

[masteralien]

I don't think such a thing exists. String theory is not a single theory - it is a framework upon which to build theories.

I also disagree with your premise. As you yourself say, there are at least three different formulations of Classical Mechanics. So there is no single "master equation" there either.

Well technically in Classical Mechanics because Hamilton’s Equations, Lagrange’s Equations, and F=ma are equivalent you do get the same EOM so there is technically a master equation the EOM for the particle. Technically the master equation would be F=ma as its the most direct with Lagrange’s and Hamilton’s Equations alternative and more efficient ways of getting F=ma but you do get F=ma in some form at the end.


Also you say no such equation exists is there a reason for why. Also when you say no such equation exists are you saying no overarching equation for any string theory or no equations exist for specific string theories.

Also whats the standard procedure of String Theory and quantities computed then. Like what do you actually compute in String Theory.

 

[PeterDonis]

Moderator's note: Thread moved to the Beyond the Standard Models forum.

 

[PeterDonis]

@masteralien you gave this thread an "A" level. That indicates a graduate level background in the subject matter. Which in turn means that you should at least be familiar with the basic literature on string theory. Are you? And if you are, hasn't that at least given a basic answer to your questions?

 

[masteralien]

@masteralien you gave this thread an "A" level. That indicates a graduate level background in the subject matter. Which in turn means that you should at least be familiar with the basic literature on string theory. Are you? And if you are, hasn't that at least given a basic answer to your questions?

I am not familiar with String Theory. I know only up to QFT basics. I just was curious to know if there’s a main String Theory Equation. Also not every graduate is familiar with or even knows String Theory as its a very small part of any physics program usually. And my question was also whats the standard procedure of String Theory what quantities do you compute and what are the key methods. Vanadium didnt answer that part only the part about a main equation

 

[masteralien]

@masteralien you gave this thread an "A" level. That indicates a graduate level background in the subject matter. Which in turn means that you should at least be familiar with the basic literature on string theory. Are you? And if you are, hasn't that at least given a basic answer to your questions?

I gave A because the question is advanced level as it involves String Theory

 

[fresh_42]

Vanadium didnt answer that part only the part about a main equation

Would a theorem on highest-weight Fock space representations of Virasoro algebras qualify as an answer?

 

[masteralien]

Would a theorem on highest-weight Fock space representations of Virasoro algebras qualify as an answer?

Sure but I was asking more so about observables and quantities you compute like how in QFT you compute correlation functions with Feynman Diagrams. Is there a similar thing in String Theory.

Also does String Theory have a notion like a one particle density or position space probability density function like in QM.

 

[PeterDonis]

I gave A because the question is advanced level as it involves String Theory

Ok, but...

I am not familiar with String Theory.

If you have a graduate level background in physics (which you should if you are going to start an "A" level thread), what's stopping you from becoming familiar with string theory, by reading at least some of the literature?

 

[mitchell porter]

The archetypal string theory calculation is still the perturbative expansion of the string S-matrix for a given background, in terms of a conformal field theory on Riemann surfaces. This can be found in any textbook. But it is presumed that there is a nonperturbative theory that includes branes as well.

 

[masteralien]

Ok, but...


If you have a graduate level background in physics (which you should if you are going to start an "A" level thread), what's stopping you from becoming familiar with string theory, by reading at least some of the literature?

Idk have other things to do at the moment and thought I could ask this question to get a basic idea

 

[masteralien]

The archetypal string theory calculation is still the perturbative expansion of the string S-matrix for a given background, in terms of a conformal field theory on Riemann surfaces. This can be found in any textbook. But it is presumed that there is a nonperturbative theory that includes branes as well.

What about Bound States in String Theory also does String Theory have a notion of a “position space probability density” or some kind of equivalent

 

[mitchell porter]

What about Bound States in String Theory also does String Theory have a notion of a “position space probability density” or some kind of equivalent

Bound states are strings stuck to branes or branes stuck to branes. Position space probability density is the same as in QFT, except that one is talking about the position of the string's center of mass.

 

[masteralien]

Bound states are strings stuck to branes or branes stuck to branes. Position space probability density is the same as in QFT, except that one is talking about the position of the string's center of mass.

What about the Hydrogen Atom can that be modeled with String Theory

 

[Vanadium 50]

What about the Hydrogen Atom can that be modeled with String Theory

Are you sure you want an A-level answer? It sounds like your background is maybe I.

 

[masteralien]

Are you sure you want an A-level answer? It sounds like your background is maybe I.

Yes I would like an A level answer

 

[berkeman]

What about Bound States in String Theory? Also does String Theory have a notion of a “position space probability density” or some kind of equivalent?

What about the Hydrogen Atom? Can that be modeled with String Theory?

Fixed those for you. It makes it a lot easier for others to read what you are writing if you use punctuation and capitalization. Just sayin'

 

[mitchell porter]

What about the Hydrogen Atom can that be modeled with String Theory

We could try :-) If I was doing it, I'd just try to imitate a field-theory model of the hydrogen atom using stringy ingredients. E.g. model an electron bound to a proton, as a light string bound to a heavy D0-brane.

 

[masteralien]

We could try :-) If I was doing it, I'd just try to imitate a field-theory model of the hydrogen atom using stringy ingredients. E.g. model an electron bound to a proton, as a light string bound to a heavy D0-brane.

Interesting is this solvable analytically

 

[Vanadium 50]

I would like an A level answer

You asked for it.

Using string theory is utterly and completely inappropriate for solving the hydrogen atom. String theory operates at the Planck scale or at least the GUT scale. This is at least 25 orders of magnitude larger than the scale of atomic binding. It would be like trying to do weather forecasting by tracking the motion of every particle in the atmosphere - just plain silly.

In a complete string theory - which we do not have - one would solve for the electrons and quarks, use a complete theory of QCD - which we don't have either - to bind the quarks into protons, and then use ordinary QM to solve for the hydrogen atom.

 

[masteralien]

You asked for it.

Using string theory is utterly and completely inappropriate for solving the hydrogen atom. String theory operates at the Planck scale or at least the GUT scale. This is at least 25 orders of magnitude larger than the scale of atomic binding. It would be like trying to do weather forecasting by tracking the motion of every particle in the atmosphere - just plain silly.

In a complete string theory - which we do not have - one would solve for the electrons and quarks, use a complete theory of QCD - which we don't have either - to bind the quarks into protons, and then use ordinary QM to solve for the hydrogen atom.

I understand its silly just was curious to see, as QFT has the Bethe Salpeter Equation for Hydrogen Bound States does String Theory have something similar. Also even if its inappropriate it can maybe be an interesting exercise.

Also you say we dont have a complete String Theory how comes I hear M Theory was supposedly the thing which completed String Theory could you elaborate this seems interesting.

 

[berkeman]

Also you say we dont have a complete String Theory how comes I hear M Theory was supposedly the thing which completed String Theory could you elaborate this seems interesting.

Did you seriously not read my post #18? Do you not see the lightning bolts in my hand?

 

[mitchell porter]

A comment on @Vanadium 50 #21.

Certainly the path to the actual hydrogen atom via string theory requires first finding a string vacuum (if there is one) in which all the particles have the right masses, etc, then building a proton out of quarks (the Sakai-Sugimoto model gives us an idea of how this could work in string theory), then describing bound states of an electron and a proton.

However, there is a more elemental question that can be tackled right away: can we describe something like a hydrogen atom in string theory, e.g. a bound state of a light and heavy particle that are oppositely charged under some U(1) field? A toy model of a stringy hydrogen atom, so to speak. This is the spirit in which I was replying to @masteralien.

In general, string theory is just quantum field theory but with a few extra features. So the path to answering a question like this is to first address the question in quantum field theory, and then try to understand the extra considerations demanded by string theory. For example, stringy calculations are easier with supersymmetry, so one might start by considering the "hydrogen-like bound states" of supersymmetric QED, and then what they look like in a stringy implementation of QED.

But also remember that the description of the hydrogen atom in QFT is itself somewhat heuristic! The same will therefore be true of a stringy description. Quantum field theory is still a work in progress in many ways, and string theory even more so.

 

[masteralien]

A comment on @Vanadium 50 #21.

Certainly the path to the actual hydrogen atom via string theory requires first finding a string vacuum (if there is one) in which all the particles have the right masses, etc, then building a proton out of quarks (the Sakai-Sugimoto model gives us an idea of how this could work in string theory), then describing bound states of an electron and a proton.

However, there is a more elemental question that can be tackled right away: can we describe something like a hydrogen atom in string theory, e.g. a bound state of a light and heavy particle that are oppositely charged under some U(1) field? A toy model of a stringy hydrogen atom, so to speak. This is the spirit in which I was replying to @masteralien.

In general, string theory is just quantum field theory but with a few extra features. So the path to answering a question like this is to first address the question in quantum field theory, and then try to understand the extra considerations demanded by string theory. For example, stringy calculations are easier with supersymmetry, so one might start by considering the "hydrogen-like bound states" of supersymmetric QED, and then what they look like in a stringy implementation of QED.

But also remember that the description of the hydrogen atom in QFT is itself somewhat heuristic! The same will therefore be true of a stringy description. Quantum field theory is still a work in progress in many ways, and string theory even more so.

What do you mean QFT is still a work in progress I thought we have the Standard Model and QED all sorted out?

Could you elaborate a bit on why and how QFT is a work in progress I thought we had it figured out?

 

[mitchell porter]

Both the mathematical foundations and the methods of calculation are not completely figured out.

 

[jedishrfu]

The original question of the OP has been answered, so now's a good time to close this thread.

For further reading, there are several basic String Theory books:
- String Theory for Dummies
- The Complete Idiots Guide to String Theory

https://www.amazon.com/String-Theory-Dummies-Andrew-Zimmerman/dp/1119888972?tag=pfamazon01-20

https://www.dummies.com/article/aca...string-theory-for-dummies-cheat-sheet-209405/

https://www.amazon.com/Complete-Idiots-Guide-String-Theory/dp/1592577024?tag=pfamazon01-20