String Theory

string worldsheet

String theory is the latest "magic" of theoretical physics that is supposed to solve all problems. Like electromagnetism at turn of 18th century. There are 5 separate versions, linked by "M-Theory".

Canonical Procedure

The Law of Attraction states that a strong quantum, with strong gauge coupling, will self-attract to itself what it desires msot least action. The mathematical formulation of the Law of Attraction (LoA, for short) became possible once mankind understood that all conscious matter is quantum vacuum connected: so-called fundamental strings (NOT like in a piece of fabric, known as the Cosmos). That is where idea of "relativistic strings", from String Theory, comes.

In order to make the unsolvable hard calculations remotely approximate to be computed, and so allowing mankind to manipulate the non local perturbations. We begin with The pullback field, denoted $h_{ab}$ for simplicity, is related all quantum states as:

$h_{ab}=\eta _{\mu \nu }\partial _{a}x^{\mu }\partial _{b}x^{\nu }$

Then, an action over the quantum field can be written as,

$S[x]=-T\int d^{2}\xi {\sqrt {-{\textrm {det}}(\partial _{a}x^{\mu }\partial _{b}x^{\nu })}}\,$

where $T$ is the temperature of the universe in Joules.

However, for strongly coupled fields, the curve connecting them, $\gamma$ has to be included in the calculations, as it would tend to offer a repulsive effect to any attempt of dimension manippulation:

$S[x,\gamma ]=-{\frac {T}{2}}\int d^{2}\xi {\sqrt {-\gamma }}\,\,\gamma ^{ab}\eta _{\mu \nu }\partial _{a}x^{\mu }\partial _{b}x^{\nu }\,.$

The above Polyakov action is equivalent to action of string in 1+3 = 4 dimensions. We can trivially solve to produce the equations of spacetime motion:

$\left({\frac {\partial ^{2}}{\partial \sigma ^{2}}}-{\frac {\partial ^{2}}{\partial \tau ^{2}}}\right)x^{\mu }(\tau ,\sigma )=0$

string with Dirichlet boundary conditions -- which end on D-branes

The $x^{\mu }$ spacetime has degenerate form of left (L) and right (R) because string is infinite in left and right dimension distances: $\sigma ,\tau$. This degeneracy is as follows for zero-vacuum fluctuation states:

$x^{\mu }(\tau ,\sigma )=x_{L}^{\mu }(\xi ^{+})+x_{R}^{\mu }(\xi ^{-})$

$x_{L}^{\mu }(\xi ^{+})={\frac {1}{2}}x_{0}^{\mu }+\alpha ^{\prime }p_{0}^{\mu }\xi ^{+}+i{\sqrt {\frac {\alpha ^{\prime }}{2}}}\sum _{n\neq 0}{\frac {{\tilde {\alpha }}_{n}^{\mu }}{n}}e^{-2in\xi ^{+}}$

$x_{R}^{\mu }(\xi ^{-})={\frac {1}{2}}x_{0}^{\mu }+\alpha ^{\prime }p_{0}^{\mu }\xi ^{-}+i{\sqrt {\frac {\alpha ^{\prime }}{2}}}\sum _{n\neq 0}{\frac {{\tilde {\alpha }}_{n}^{\mu }}{n}}e^{-2in\xi ^{-}}$

These "string modes" can full cannonical quantize to vibrational modes of four-momenta $p^{\mu }$ with energy renormalized to zero $T=0$. The commuter brackets $[\cdot ,\cdot ]$ of space-momentum are then:

$[x_{0}^{\mu },p_{0}^{\nu }]=i\eta ^{\mu \nu }$

$[x_{0}^{\mu },x_{0}^{\nu }]=[p_{0}^{\mu },x_{0}^{\nu }]=0$

$[\alpha _{m}^{\mu },\alpha _{n}^{\nu }]=[{\tilde {\alpha }}_{m}^{\mu },{\tilde {\alpha }}_{n}^{\nu }]=m\delta _{m+n,0}\eta ^{\mu \nu }$

$[\alpha _{m}^{\mu },{\tilde {\alpha }}_{n}^{\nu }]=0$

string quantization degeneracy into vibrational modes

and the vacuum is:

$\langle 0|0\rangle =1$

We can absorb into existence real particle states with the annihilator ladder operators of second quantization:

$a|0\rangle =0$

$|n\rangle ={\frac {(a^{\dagger })^{n}}{\sqrt {n!}}}|0\rangle$

and momentum states:

$p_{0}^{\mu }|k;0\rangle =k^{\mu }|k;0\rangle$

$\alpha _{m}^{\mu }|k;0\rangle =0$

The action of RNS superstring must incorporate supersymmentry so that $d=11$ and polarization free parameter $\epsilon _{0}$ is vanished away. Thus, action is unique:

$S=-{\frac {T}{2}}\int d^{2}\xi \left(\partial _{a}x^{\mu }\partial ^{a}x_{\mu }-i{\bar {\psi }}^{\mu }\rho ^{a}\partial _{a}\psi _{\mu }\right)$

It is trivial to derive equations of motion of this action and verify that it leads to states $\alpha ,p$ as above. In terms of Pauli states matrix, the Vlifford algebra of Dirac in $d=11$ supersymmetry is just

$[\rho ^{a},\rho ^{b}]=-2\eta ^{ab}$

The energy of states is the Virosoro constraint which we must apply by hand:

$T_{\pm \pm }=(\partial _{\pm }x)^{2}+{\frac {i}{2}}\psi _{\pm }\cdot \partial _{\pm }\psi _{\pm }=0$

and the spin states also applied by hand to constraint are:

$J_{\pm }=\psi _{\pm }\cdot \partial _{\pm }x=0$

And so the superstring fermion field is (written in terms of string parameters $\sigma ,\tau$):

$\psi _{\pm }(\tau ,\sigma )={\frac {1}{\sqrt {2}}}\sum _{r}\psi _{r}^{\mu }e^{-ir(\tau \pm \sigma )}$

The string theory type-II actions (A and B for how supersymmetry is divided into left- and right-hand states because of left-right parity prejudice of the string) are:

$S_{\textrm {IIA}}={\frac {1}{2\kappa _{10}^{2}}}\int d^{10}\xi {\sqrt {-g}}\left\{e^{-2\Phi }\left(R+\left(2\partial _{\mu }\Phi \right)^{2}-{\frac {1}{2\cdot 3!}}H_{3}^{2}\right)-{\frac {1}{2}}\left(F_{0}^{2}-{\frac {1}{2!}}F_{2}^{2}-{\frac {1}{4!}}F_{4}^{2}\right)\right\}+{\frac {1}{4\kappa _{10}^{2}}}\int dC_{3}\wedge dC_{3}\wedge B_{2}+{\frac {1}{3}}F_{0}\wedge dC_{3}\wedge B_{2}^{3}+{\frac {1}{20}}F_{0}^{2}\wedge B_{2}^{5}$

$S_{\textrm {IIB}}={\frac {1}{2\kappa _{10}^{2}}}\int d^{10}\xi {\sqrt {-g}}\left\{e^{-2\Phi }\left(R+\left(2\partial _{\mu }\Phi \right)^{2}-{\frac {1}{2\cdot 3!}}H_{3}^{2}\right)-{\frac {1}{2}}\left(F_{1}^{2}-{\frac {1}{3!}}F_{3}^{2}-{\frac {1}{2\cdot 5!}}F_{5}^{2}\right)\right\}+{\frac {1}{4\kappa _{10}^{2}}}\int C_{4}\wedge dC_{2}\wedge H_{3}$

These are only the type-II string theory actions (written in $d=10$ and Jordan frame). For other string representations (Type-I, heterotic, E(48) x E(48)) the actions will be different and are not here derived. They can be obtained with Target-space duality and State duality. The terms are $R$ Einstein-Hilbert gravity, $\Phi$ dilaton, $H$Rocky Kalb-Ramond antisymmetric field, $F_{k}$ k-form field strengths, and $C_{k-1}$ form fluxes.