String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
Superstring Revolutions
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
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
Phenomenology String Theory Landscape
Minimal Supersymmetric Standard Model
String Phenomenology

The Sen Theorem, or Tachyon Condensation is a proof that a tachyonic ground state is unstable. Tachyons are particles with imaginary mass ( m^2<0 as opposed to bradyons, with  m^2 > 0 ) and move at superluminal speeds. In the past, it was thought that this implies that any theory with tachyons must be wrong as it stands, and the tachyons should be removed somehow, or the theory would be false.

However, it can instead be interpreted that the wrong ground state is being used, and the correct ground state is not tachyonic. 

Proof of the theoremEdit this section

If the ground state is the tachyon, a system may spontaneously produce tachyons, lowering the ground state.

 |\psi\rangle = J^-   |0\rangle

This reduces the energy of the ground state to a further negative. This would mean that this ground state wouldn't be a ground state any more, and so on. Therefore, the ground state wouldn't be stable.

Analogy with the reciprocal functionEdit this section

In Elementary Calculus, speciifically in Taylor Expansions, a function can be expanded as:

 f(x_0+h)=\sum_{k=1}^\infty \frac{h^k}{k!}        f^{(n)} (x_0)

This is known as the Taylor Series. A special, often useful, application of this occurs when we set  x_0 = 0 and  h = x :

 f(x)       =\sum_{k=1}^\infty \frac{x^k}{k!}        f^{(n)} (0)

This is known as the Maclaurin Series.

If we set  f(x)=\frac1x, then the Maclaurin Series clearly fails, as \frac1x, \mbox{ }    -\frac1{x^2}, etc. clearly are indeterminate when x=0. This could be considered as a failure of Taylor Series, but one may instead take it that we are using the wrong  x_0 to expand around. If one takes  x_0 = 1 , then this problem clearly is solved, as those derivatives are now well-defined for  x_0 = 1 . . . /

Similarly, Tachyon Condensation can instead be taken as to imply that we are only using the wrong ground state. In a different ground state, these tachyons wouldn't be there anymore.

Implications of the theoremEdit this section

Despite the remarks of the previous section, in the naive RNS String Theory, the tachyons are generally just mapped out using a GSO Projecton. The GSO Projection maps out particles with a zero fermion number; tachyons. This immediately leads to the Type IIB String Theory and the Type IIA String Theory



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