# Tachyon Condensation

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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