Philosophy of String Theory

The Philosophy of String Theory or just String Theory analyzes the philosophy behind a mathematical model used in particle physics, that purports to be an all-encompassing theory of the universe, unifying the forces of nature, including gravity, in a single quantum mechanical framework.

String theory unifies Einstein’s theory of general relativity with quantum mechanics. Moreover, it does so in a manner that retains the explicit connection with both quantum theory and the low-energy description of spacetime.

Formalism
The Einstein-Hilbert action is given by
 * $$\color{white} S_{EH}=\dfrac{1}{16\pi G_N}\int d^4x\sqrt{-g}\mathcal{R}.$$

Newton’s constant $$\color{white}G_{N}$$ can be written as $$\color{white}8\pi G_N=\dfrac{\hbar c}{M_{pl}^2}.$$ We will work with standard units $$\color{white}\hbar=c=1$$ and metric signature $$\color{white}\eta_{\mu\nu}=\mathrm{diag}(-1,+1,+1,\ldots,+1).$$

Non-Renormalizable Gravity
Quantum field theories with irrelevant couplings are typically ill-behaved at high energies, rendering the theory ill-defined. Gravity is no exception. Theories of this type are called non-renormalizable, which means that the divergences that appear in the Feynman diagram expansion cannot be absorbed by a finite number of counterterms. In pure Einstein gravity, the symmetries of the theory are enough to ensure that the one-loop S-matrix is finite. The first divergence occurs at two-loops and requires the introduction of a counterterm of the form,
 * $$\color{white} \Gamma\sim\dfrac{1}{\epsilon}\dfrac{1}{M^4_{pl}}\int d^4x\sqrt{-g}\mathcal{R}^{\mu\nu}_{\rho\sigma}\mathcal{R}^{\rho\sigma}_{\lambda\kappa}\mathcal{R}^{\lambda\kappa}_{\mu\nu}$$

with $$ \color{white} \epsilon=4-D.$$ All indications point towards the fact that this is the first in an infinite number of necessary counterterms. Coupling gravity to matter requires an interaction term of the form,
 * $$\color{white}S_{i n t}=\int d^{4}x\frac{1}{M_{p l}}h_{\mu\nu}T^{\mu\nu}+{\cal O}(h^{2}).$$

This makes the situation marginally worse, with the first divergence now appearing at one-loop. When the momentum $$\color{white} k $$ running in the loop is large, the Feynman diagram is badly divergent: it scales as $$\color{white} \frac{1}{M_{pl}^{4}}\int^{\infty}d^{4}k.$$ Non-renormalizable theories are commonplace in physics, they are typically viewed as effective field theories, valid only up to some energy scale $$\color{white} Λ$$. One deals with the divergences by simply admitting ignorance beyond this scale and treating $$\color{white} Λ$$ as a cut-off on any momentum integral.

Singularities
In general relativity we typically think about the geometry as a whole, rather than bastardizing the Einstein-Hilbert action and discussing perturbations around flat space. In this language, the question of high-energy physics turns into one of short distance physics. Classical general relativity is not to be trusted in regions where the curvature of spacetime approaches the Planck scale and ultimately becomes singular. A quantum theory of gravity should resolve these singularities.

[[file:Wikipedia.png]] Wikipedia

 * String Theory
 * M-Theory

MIT OpenCourseWare

 * String Theory For Undergraduates