Background & Bosonic string theory
String theory ——The production of strong interaction
In 1970s, the amplitude of hadrons faced the problem of infinity. Thus, physicists tried to describe a point-particle as a one-dimensional string.
The first string theory was "dual resonance model", which combined S-matrix with strong interaction and interpreting it's scattering amplitude.
The particle amplitude can describe by Euler beta function and gamma function, there is a relationship between them:
The first string theory was "dual resonance model", which combined S-matrix with strong interaction and interpreting it's scattering amplitude.
The particle amplitude can describe by Euler beta function and gamma function, there is a relationship between them:
Also, particle scattering function has a slope, the corresponding illustration can show "Reggie trajectory". As a result, if one described particles as strings, the amplitude would be finite, even is renormalizable. The theory is "string theory", namely.
Fig1.1 Reggie trajectory (with Reggie slope)
The feature of string theory
Dual resonance model was developed by Veneziano, and the primary founder of string theory was Leonard Susskind. In string theory, a string swept in spacetime form a "worldsheet". We can observe and evaluate easily about the movement of the strings. Further, worldsheet is a Riemann surface actually.
Fig1.2 Leonard Susskind (the founder of string theory)
Fig1.3 String Worldsheet
General speaking, string theory is a quantum theory, and the vibration of strings corresponding to every particles of nature. However, strings are extremely small, approximately Planck length (10^-35 m), so we still can't observe them currently.
Fig1.4 Everything is consisted of strings
So we make the conclusion of string theory 's background :
1. String theory is a quantum theory in essence.
2. The various vibration of strings are corresponding to natural particles.
3. The scale of string theory is approximately Planck scale.
4. There are open strings and closed strings in the theory, and their trajectory of movement is worldsheet.
Bosonic string theory
The first string theory can only describe bosons, called "bosonic string theory". According to the evaluation, the theory requires 26-dimensional spacetime. This procession of derive involved lightcone gauge quantization, canonical quantization, Virasoro algebra, and some tools to derive the correct state and spectrum. Interestingly, the procession of evaluating the suitable dimensions of spacetime has used Riemann zeta function as below:
when s=-1, it implies:
As a consequence, the spacetime evaluated by string theory is dominated by not only the physics but also the mathematics.
The first invariant action in the theory was Nambu-Goto action:
where α' is Reggie slope, and X come from the worldsheet coordinates(X^μ) as Fig1.3.
On the other hand, the action of 2-d conformal theory(to describe string worldsheet) is Polyakov action, which shows the quantum structure of the theory :
On the other hand, the action of 2-d conformal theory(to describe string worldsheet) is Polyakov action, which shows the quantum structure of the theory :
where string tension T=1/2πα'. Polyakov action has local symmetry and global symmetry, it's classical version is Nambu-Goto action.
Moreover, giving the solution of the equation of motion, there are some possible boundary conditions as below:
1. Open string—--
(1) Neumann boundary conditions :
Moreover, giving the solution of the equation of motion, there are some possible boundary conditions as below:
1. Open string—--
(1) Neumann boundary conditions :
(2) Dirichlet boundary conditions:
2. Closed string —--
- Periodic boundary conditions:
- Periodic boundary conditions:
In the spectrum of bosonic string theory, besides the U(1) gauge bosons, there has graviton, dilaton, antisymmetric tensor, and tachyon.
- Graviton: The existence of graviton implies string theory can interpret gravity. Combining gravity and quantum theory in essence, string theory became the theory of quantum gravity.
- Dilaton:It's a scalar field, has association with the compactification of extra dimensions.
- Antisymmetric tensor: Several antisymmetric tensor gauge field in certain theory couple to higher-dimensional objects just as D-brane.
- Tachyon: In particle physics, tachyon is a hypothetical particle which always move faster than light. The tachyons exists in the theory implies the theory's spacetime structure isn't stable. As a result, eliminating the tachyon is a necessary work.
Bosonic string theory has a huge problem because of tachyon, physicists would be solving the problem with new techniques.