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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-11-10 17:32:50 +01:00
parent 334b7613c5
commit 1ffe028306
3 changed files with 11 additions and 11 deletions
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10
sec/part1/fermions.tex
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@@ -2039,7 +2039,7 @@ Using the algebra~\eqref{eq:ns-algebra} we compute the \ope of fermion fields as
\qquad \qquad
\abs{w} < \abs{z}, \abs{w} < \abs{z},
\end{equation} \end{equation}
where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}. where the operation $\no{\cdot}$ is the normal ordering with respect to the \SL{2}{\R} vacuum defined in~\eqref{eq:NS_SL2_vacuum}.
We then get the expression of the stress-energy tensor: We then get the expression of the stress-energy tensor:
\begin{equation} \begin{equation}
\begin{split} \begin{split}
@@ -2189,7 +2189,7 @@ From the usual definition of the stress-energy tensor in terms of the Virasoro g
\end{split} \end{split}
\end{equation} \end{equation}
We already hinted to the fact that the vacua state involved are not in general \SL{2}{R} invariant. We already hinted to the fact that the vacua state involved are not in general \SL{2}{\R} invariant.
In particular we can see that that the excited vacua \eexcvacket is a primary field In particular we can see that that the excited vacua \eexcvacket is a primary field
\begin{equation} \begin{equation}
\begin{split} \begin{split}
@@ -2394,7 +2394,7 @@ The last expression shows that the energy momentum tensor $\cT( z )$ is radial t
First of all we notice that the vacuum $\Gexcvacket$ is actually $\GGexcvacket$, i.e.\ it depends only on $x_{(t)}$ and $\rE_{(t)}$. First of all we notice that the vacuum $\Gexcvacket$ is actually $\GGexcvacket$, i.e.\ it depends only on $x_{(t)}$ and $\rE_{(t)}$.
We can try to interpret the previous result in the light of the usual \cft approach. We can try to interpret the previous result in the light of the usual \cft approach.
In particular we can refine the idea we discussed after~\eqref{eq:asymp_beha_Psi_on_exc_vac} that the singularity in the modes~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate} at the point $x_{(t)}$ is associated with a primary conformal operator which creates \eexcvacket with $\rE = \rE_{(t)}$. In particular we can refine the idea we discussed after~\eqref{eq:asymp_beha_Psi_on_exc_vac} that the singularity in the modes~\eqref{eq:generic-case-basis} and~\eqref{eq:generic-case-basis-conjugate} at the point $x_{(t)}$ is associated with a primary conformal operator which creates \eexcvacket with $\rE = \rE_{(t)}$.
By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{R} vacuum \regvacuum. By comparison with the stress energy tensor of an excited vacuum~\eqref{eq:T_excited_vacuum}, from the second order singularity we learn that at the points $x_{(t)}$ there is an operator which creates the excited vacuum \GGexcvacket from the \SL{2}{\R} vacuum \regvacuum.
Given the discussion in the previous section this is an excited spin field $\rS_{\rE_{(t)}}\qty( x_{(t)}) = e^{i \rE_{(t)} \phi( x_{(t)} )}$. Given the discussion in the previous section this is an excited spin field $\rS_{\rE_{(t)}}\qty( x_{(t)}) = e^{i \rE_{(t)} \phi( x_{(t)} )}$.
The first order singularities in $x_{(u)} - x_{(t)}$ are then the result of the interaction between two of the previous excited spin fields. The first order singularities in $x_{(u)} - x_{(t)}$ are then the result of the interaction between two of the previous excited spin fields.
Using the \cft operator approach we postulate that the following identification holds Using the \cft operator approach we postulate that the following identification holds
@@ -2818,7 +2818,7 @@ Therefore we have
\end{equation} \end{equation}
which can be solved by which can be solved by
\begin{equation} \begin{equation}
\ln \left\langle \left\langle
\rR\qty[% \rR\qty[%
\rS_{\rE_{(t)}}\qty(x_{(t)}) \rS_{\rE_{(t)}}\qty(x_{(t)})
\prod\limits_{\substack{u = 1 \\ u \neq t}}^N \prod\limits_{\substack{u = 1 \\ u \neq t}}^N
@@ -2828,7 +2828,7 @@ which can be solved by
= =
\cN_0 \cN_0
\qty( \qty{\rE_{(t)}} ) \qty( \qty{\rE_{(t)}} )
\prod\limits_{\substack{t = 1}{t > u}}^N \prod\limits_{\substack{t = 1 \\ t > u}}^N
\qty( x_{(u)} - x_{(t)} )^{\rE_{(u)} \rE_{(t)}}. \qty( x_{(u)} - x_{(t)} )^{\rE_{(u)} \rE_{(t)}}.
\end{equation} \end{equation}
The constant $\cN_0\qty( \qty{\rE_{(t)}} )$ which depends on the $\rE_{(t)}$ only can be fixed by using the \ope The constant $\cN_0\qty( \qty{\rE_{(t)}} )$ which depends on the $\rE_{(t)}$ only can be fixed by using the \ope

2
sec/part2/divergences.tex
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@@ -12,7 +12,7 @@ In fact in~\cite{Liu:2002:StringsTimeDependentOrbifold} the four tachyons amplit
\begin{equation} \begin{equation}
A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q ) A_4 \sim \int\limits_{q \sim \infty} \frac{\dd{q}}{\abs{q}} \ccA( q )
\end{equation} \end{equation}
where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{closed}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices). where $\ccA_{\text{closed}}( q ) \sim q^{4 - \ap \norm{\vec{p}_{\perp}}^2}$ and $\ccA_{\text{open}}( q ) \sim q^{1 - \ap \norm{\vec{p}_{\perp}}^2} \tr\qty( \liebraket{T_1}{T_2}_+ \liebraket{T_3}{T_4}_+ )$ ($T_i$ for $i = 1,\, 2,\, 3,\, 4$ are Chan-Paton matrices).
Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen. Moreover divergences in string amplitudes are not limited to four points: interestingly we show that the open string three point amplitude with two tachyons and the first massive state may be divergent when some \emph{physical} polarisations are chosen.
The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory. The true problem is therefore not related to a gravitational issue but to the non existence of the effective field theory.
In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution. In fact when we express the theory using the eigenmodes of the kinetic terms some coefficients do not exist, not even as a distribution.

10
thesis.tex
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@@ -54,11 +54,11 @@
\newcommand{\cint}[1]{\ensuremath{\oint\limits_{\ccC_{#1}}}} \newcommand{\cint}[1]{\ensuremath{\oint\limits_{\ccC_{#1}}}}
%---- states %---- states
\newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{R}}}\xspace} \newcommand{\regvacuum}{\ensuremath{\ket{0}_{\SL{2}{\R}}}\xspace}
\newcommand{\regvacuumin}{\ensuremath{\ket{0_{(\text{in})}}_{\SL{2}{R}}}\xspace} \newcommand{\regvacuumin}{\ensuremath{\ket{0_{(\text{in})}}_{\SL{2}{\R}}}\xspace}
\newcommand{\regvacuumout}{\ensuremath{\ket{0_{(\text{out})}}_{\SL{2}{R}}}\xspace} \newcommand{\regvacuumout}{\ensuremath{\ket{0_{(\text{out})}}_{\SL{2}{\R}}}\xspace}
\newcommand{\regvacuuminconj}{\ensuremath{\tensor[_{\SL{2}{R}}]{\bra{0_{(\text{in})}}}{}}\xspace} \newcommand{\regvacuuminconj}{\ensuremath{\tensor[_{\SL{2}{\R}}]{\bra{0_{(\text{in})}}}{}}\xspace}
\newcommand{\regvacuumoutconj}{\ensuremath{\tensor[_{\SL{2}{R}}]{\bra{0_{(\text{out})}}}{}}\xspace} \newcommand{\regvacuumoutconj}{\ensuremath{\tensor[_{\SL{2}{\R}}]{\bra{0_{(\text{out})}}}{}}\xspace}
\newcommand{\twsvacket}{\ensuremath{\ket{\mathrm{T}}}\xspace} \newcommand{\twsvacket}{\ensuremath{\ket{\mathrm{T}}}\xspace}
\newcommand{\twsvacbra}{\ensuremath{\bra{\mathrm{T}}}\xspace} \newcommand{\twsvacbra}{\ensuremath{\bra{\mathrm{T}}}\xspace}
\newcommand{\excvacket}{\ensuremath{\ket{T_{\rE,\, \brE}}}\xspace} \newcommand{\excvacket}{\ensuremath{\ket{T_{\rE,\, \brE}}}\xspace}