Explaining Equation: L = \Phi+ [i \partial\tau - H]\Phi + \Phi+ * \Phi * \Phi

[Alcubierre]

Hello,

I did not know where to post this, so my apologies in advance and move it if necessary. But could anyone explain this equation to me?

L = $\Phi$⁺ [i $\partial$_$\tau$ - H]$\Phi$ + $\Phi$⁺ * $\Phi$ * $\Phi$

 

[fzero]

That is the Lagrangian for open string field theory in the light-cone gauge. Compare with equations 2.15 and 3.7b in http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?198708063 That is a technical paper that assumes the reader is familiar with quantum field theory. Since you appear to be a high-school student, I'm not quite sure how to even explain the terms in language that you'd be familiar with. Even though that precise equation doesn't appear, the wikipedia entry http://en.wikipedia.org/wiki/String_field_theory tries to explain the motivation, along with some more technical details, of string field theory.

Referring to the figures at http://en.wikipedia.org/wiki/String_field_theory#Light-cone_string_field_theory, the operator $\Phi$ is the string field, which creates the string state at a given point in space, represented by the cylinder. The first term describes how this state evolves in time. The 2nd, cubic, term describes interactions, which look like the splitting and joining indicated along the dotted lines in the second diagram on that page.

 

[Alcubierre]

You're correct, I am just a curious high school student. But thank you for the explanation, I was actually wondering what the operators were all about. What kind of math goes into solving such equations? And a follow-up, are the symbols used in the equations consistent throughout the theory or do they vary between researchers? I know that seems unreasonable to ask but I have read papers in which they use stars ("Introduction to String Theory," Hoof, Gerard 't, 2004) and it seems, for lack of a better word, random, to me.

 

[fzero]

You're correct, I am just a curious high school student. But thank you for the explanation, I was actually wondering what the operators were all about. What kind of math goes into solving such equations?

I've never really studied string field theory in any detail, but the math subjects that go into defining that formula involve functional analysis, differential geometry and group theory. It's possible to get a bit more exotic and introduce some concepts from noncommutative geometry to describe the equations as well. Of course, as in many cases in physics, the physical principles are actually a bit more profound than the mathematics. You really need to understand quantum field theory well to really capture the meaning, which is itself built on a solid background of classical mechanics, quantum mechanics and electrodynamics.

And a follow-up, are the symbols used in the equations consistent throughout the theory or do they vary between researchers? I know that seems unreasonable to ask but I have read papers in which they use stars ("Introduction to String Theory," Hoof, Gerard 't, 2004) and it seems, for lack of a better word, random, to me.

The symbols and notation are sometimes correlated with the age of the author, or at least with what era they learned the subject (and the sources they learned from). In 't Hooft's case, he was probably familiar with the subject in the 70s and 80s, so his perspective includes some of the notation and pedagogy that was common then. If you took a look at the string theory notes by David Tong, who is younger by around a couple of decades or so, you will probably find some differences (like an emphasis on conformal field theory techniques), but other things will be similar. It can sometimes be challenging for a student, but by the time you start to really understand a subject, you can understand other types of notation that have the same meaning.

Now, I took a quick look at my copy of 't Hooft's notes and don't see any mention of string field theory, so I'm not sure which "stars" you're referring to. Sometimes a star denotes complex conjugation, but here it's being used to define a specific type of product between string fields.

 

[Morgoth]

I don't know if that's it all about, just pointing something out:
Ψ⁺ is used even in Dirac's fields to denote not exactly the complex conjugation, but there exist γ⁰

if you find it difficult to understand these things, it's better to start studying more "easy" quantum physics. I use " because it is not easy. People tend to dream of the big ideas, but forget that all those big ideas are built over some basic ones, that can of course be really complex. As is Quantum Theory, and in expansion the whole QFT. There is no reason to start studying books and literature, if you have not seen simpler concepts. If things were that simple, we wouldn't need great minds to introduce us from classical to quantum mechanics, and later on in quantum field theory, etc... We would immediately have gone in QFT and finito. I wonder if a high school student can really understand what is for example the Lagrangian (even classically)...

 

[Alcubierre]

I was watching a video with Michio Kaku and that equation came up and I wanted clarification, that's all. I do have an idea of what Lagrangian is, without looking it up, and I know it is something to do with conservation of momentum and energy in systems and in which conditions energy and momentum are conserved. But, as to your statement, I am starting with the "easy" quantum physics, I just came across that equation in a video for the laymen.