Little question about string theory

[jostpuur]

If a particle is point like, then point $x\in\mathbb{R}^3$ specifies the particle's spatial configuration, and the quantum mechanical wave function for the particle is

$$\Psi:\mathbb{R}^3\to\mathbb{C}$$

The spatial configuration of a closed string with fixed length L can be specified with a function

$$f:S^1\to\mathbb{R}^3$$

such that the function satisfies

$$\underset{S^1}{\int} du\;|\nabla f(u)| = L$$

Is the idea in string theory to then describe these strings with wave mappings

$$\Psi:\{f\}\to\mathbb{C}?$$

 

[jostpuur]

If this had been normal string theory, somebody would have probably already confirmed it.

Is the situation with this little like with the QFT? It is possible to describe the states of quantum fields with wave functionals, but it is not popular, and everybody wants to do everything with the operators more abstractly without explicit representations.

In string theory everything is done again with operators, although the wave mapping approach could possible work too?

(btw. If everything goes as planned (course on QFT this fall successfully), I'll be taking my first course on string theory on the spring. I'm just warming up here.)

 

[Entropia]

The idea of string theory is to describe fundamental particles as tiny, vibrating strings rather than point-like objects. These strings are thought to exist in a higher-dimensional space, and their spatial configuration is described by a function f:S^1\to\mathbb{R}^3, as mentioned in the question. This function must satisfy a certain condition, namely that the total length of the string remains fixed. This is represented by the integral in the equation provided, where the parameter u represents the points along the string's circular path.

In string theory, the quantum mechanical wave function for a closed string is represented by a functional \Psi, which maps the set of all possible string configurations (represented by the function f) to a complex number. This functional \Psi takes into account the vibrational modes of the string and how they interact with each other, giving rise to the various properties of particles that we observe.

So in short, yes, the idea in string theory is to describe these strings with wave mappings \Psi:\{f\}\to\mathbb{C}, where \{f\} represents the set of all possible string configurations. This allows for a more comprehensive understanding of fundamental particles and their behavior, as it takes into account both their spatial configuration and their vibrational modes.