physics22 said:
If the answer is “yes,” please do provide the equation (or equations) as I would love to see the math which I have heard is beuatiful! :) Also please provide the physical entities it refers too. Thanks! :)
This is answered in
string theory FAQ -- What are the equations of string theory? I just went and expanded a bit more there. Have a look there for the following text equipped with hyperlinks for all keywords.
So:
All local field theories in physics are prominently embodied by key equations, their equations of motion. For instance classical gravity (general relativity) is essentially the theory of Einstein's equations, quantum mechanics is governed by the Schrödinger equation, and so forth.
But perturbative string theory is not a local field theory. Instead it is an
S-matrix theory (see
What is string theory?). Therefore instead of being given by an equation that picks out the physical trajectories, it is given by a formula for how to compute scattering amplitudes. That formula is the string perturbation series: it says that the probability amplitude for ##n_{in}## asymptotic states of strings coming in (into a particle collider experiment, say), scattering, and ##n_{out}## other asymptotic string states emerging (and hitting a detector, say) is a sum over all Riemann surfaces with ##(n_{in}, n_{out})##-punctures of the ##n##-point functions of the given 2d SCFT that defines the scattering vacuum.
More in detail, a string background is equivalently a choice of 2d SCFT of central charge 15 (a “2-spectral triple”), and in terms of this the formula for the S-matrix element/scattering amplitude for a bunch of asymptotic string states ##\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}## coming in, and a bunch of states ##\psi^1_{out}, \cdots, \psi^{n_{out}}_{out}## coming out is schematically of the form
$$
S_{\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}}
\;=\;
\underset{g \in \mathbb{N}}{\sum}
\lambda^g
\underset{
{moduli \; space \; of}
\atop
{{(n_{in},n_{out}) punctured}
\atop
{{super\; Riemann \; surfaces}
\atop
{{\Sigma^{n_{in}, n_{out}}_g}
\atop
{of\; genus\; g}}}}
}{
\int
}
\left(
SCFT \; Correlator \; over \; \Sigma \; of \; states\;
{\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}}
\right)
$$
expressing the S-matrix element (scattering amplitude) shown on the left as a formal power series in the string coupling constant with coefficients the integrals over moduli space of super Riemann surfaces of the worldsheet correlators (nn-point functions) for the given incoming and outgoing string states.
With more technical details filled in, this formula reads as follows (for the bosonic string, as found in Polchinski's "String theory", volume 1, equation (5.3.9))
and for the superstring, as found in Polchinski 01, volume 2, equation (12.5.24):
This is the equation (formula) that defines perturbative string theory.
And this is of just the same form as as the Feynman perturbation series in local quantum field theory, the only difference being that the latter is more complicated: there one has to some over Feynman diagrams with labeling for all intermediate particles (virtual particles) and with some arbitrary “cutoff” to make the integrals well defined, whereas here we simply sum over all super Riemann surfaces. The different intermediate virtual particles as well as the renormalization counterterms are all taken care of by the higher string modes, encoded in the worldsheet CFT correlators.
There was a time in the 1960s, when quantum field theorists around Geoffrey Chew proposed that precisely such formulas for S-matrix elements should be exactly what defines a quantum field theory, this and nothing else. The idea was to do away with an explicit concept of spacetime and local interactions, and instead declare that all there is to be said about physics is what is seen by particles that probe the physics by scattering through it. This is an intrisically quantum approach, where there need not be any classical action functional defined in terms of spacetime geometry. Instead, all there is a formula for the outcome of scattering experiments.
Historically, this radical perspective fell out of fashion for a while with the success of QCD and the quark model in its formulation as as local field theory coming from an action functional: Yang-Mills theory.
But fashions come and go, and the original idea of Geoffrey Chew and the S-matrix approach continues to make sense in itself, and it is this form of a physical theory that perturbative string theory is an example of.
Ironically, more recently, the S-matrix-perspective also becomes fashionable again in Yang-Mills theory itself, with people noticing that scattering amplitudes at least in super Yang-Mills theory have good properties that are essentially invisible when expressing them as vast sums of Feynman diagram contributions as obtained from the action functional. For more on this see at amplituhedron.
On the other hand, there is also an analog of the second quantized field-theory-with-equations for string scattering: this is called string field theory, and this again is given by equations of motion. For instance the equations of motion of closed bosonic string field theory are of the form
$$
Q \Psi + \tfrac{1}{2} \psi \star \psi + \tfrac{1}{6} \psi \star \psi \star \psi + \cdots = 0
\,,
$$
where ##\Psi## is the string field, ##Q## is the BRST operator and ##\star## is the string field star product.
The string field ##\Psi## has infinitely many components, one for each excitation mode of the string. Its lowest excitations are the modes that correspond to massless fundamental particles, such as the graviton. Expanding the equations of motion of string field theory in mode expansions (“level expansion”) does reproduce the equations of motions of these fields as a perturbation series around a background solution and together with higher curvature corrections.
