|All Roads Lead to String Theory (Polchinski)|
Prior to the First Superstring Revolution
|Early History|| S-Matrix Theory|
|Bosonic String Theory|| Worldsheet|
Bosonic String Theory
String Perturbation Theory
|Supersymmetric Revolution|| Supersymmetry|
|First Superstring Revolution|| GSO Projection|
Type II String Theory
Type IIB String Theory
Type IIA String Theory
Type I String Theory
Type H String Theory
Type HO String Theory
Type HE String Theory
|Second Superstring Revolution|| T-Duality|
Horava-Witten String Theory
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
|Phenomenology|| String Theory Landscape|
Minimal Supersymmetric Standard Model
Bosonic String Theory is the original version of String Theory. It is completely determined by the Polyakov Action, and only includes bosons among it's mass spectrum. It also contains a tachyon in it's mass spectrum as the ground state.
Nambu-Goto ActionEdit this section
The Nambu-Goto Action is the action principle generally associated with Classical Bosonic String Theory. However, in principle, it is possible to use it for the fully quantised theory as well, but it is much harder. The Nambu-Goto Action can easily be obtained by looking at a Worldsheet suspended in spacetime. It can be stated as:
Where is the induced metric on the worldsheet when it is embedded in the background spacetime. This has an obvious geometrical meaning. However, due to the presence of the square root, this action is unsuitable for quantisation. For clarity, observe that under a suitable Diffeomorphism, i.e. reparameterisation, / \
Which is clearly hard to quantise.
Polyakov ActionEdit this section
The Polyakov Action is an easier-to-quantise action for the Bosonic String Theory. It can be stated as:
This Polyakov Action is not so appealing geometrically, but it is easier to quantise. It can be shown that this is classically equivalent to the Nambu-Goto Action as follows (Please click here to view properly):
|Proof that the Polyakov Action classically reduces to the Nambu-Goto Action|
We use the Euler-Lagrange Equation:
Where is the trace of the Stress-Energy-Momentum Tensor and is the induced metric on the worldsheet.
Also recall that:
From this we see that, if the intrinsic worldsheet metric satisfies the following field equation:
Virasoro AlgebraEdit this section
The L-operator in Bosonic String Theory is given by:
When , this results in ,.
The L-operators form an an algebra, known as the Virasoro Algebra. This is actually a quantisation of the Witt Algebra (also known as the Classical Virasoro Algebra, and is isomorphic to the Conformal Algebra) through a central extension.
The multiplication; of this algebra is given, by:
This algebra is actually useful in removing the negative norm states, as we will see.
This algebra also requires the Virasoro Constraints, which are stated simply as:
Imposing the Virasoro ConstrainstsEdit this section
Therefore, /. -
Mass SpectrumEdit this section
In the Classical Bosonic String Theory, the mass spectrum can be given by ; however, in the quantised Bosonic String Theory, it is given by:
Where we took because both the left- (Without the tilde) and right- (with the tilde) moving sectors are from the Bosonic String Theory.