commit b084f9a20cf0ad6cf7ebf2dd817dcd668733e2ac
parent 24757a9157b3ea60fb7ab6f74fbf227ef760e23d
Author: miksa <milutin@popovic.xyz>
Date: Wed, 9 Jun 2021 14:35:12 +0200
done week10.pdf
Diffstat:
3 files changed, 93 insertions(+), 4 deletions(-)
diff --git a/pdfs/week10.pdf b/pdfs/week10.pdf
Binary files differ.
diff --git a/src/week10.pdf b/src/week10.pdf
Binary files differ.
diff --git a/src/week10.tex b/src/week10.tex
@@ -168,12 +168,12 @@ Glaser}
&\;\;\;\;\;\;\;+2\text{Tr}((D_\mu\Phi)(D^\mu\Phi))
+ s\text{Tr}(\Phi^2)\\
\nonumber\\
- &\frac{1}{360}\text{Tr}(-60\DeltaF)=
+ &\frac{1}{360}\text{Tr}(-60\Delta F)=
\frac{1}{6}\Delta(Ns+4\text{Tr}(\Phi^2)).
\end{align}
Now for the cross terms of $\Omega_{\mu\nu}^E\Omega^{E\mu\nu}$ the trace
vanishes because of the anti-symmetric properties of the Riemannian
- Cruvature Tensor
+ curvature Tensor
\begin{align}
\Omega_{\mu\nu}^E\Omega^{E\mu\nu} = \Omega_{\mu\nu}^S\Omega^{S\mu\nu}
\otimes 1 - 1\otimes F_{\mu\nu}F^{\mu\nu} + 2i\Omega_{\mu\nu}^S
@@ -205,6 +205,8 @@ Glaser}
\end{proof}
\section{Fermionic Action}
+A quick reminder with what we are dealing with, the fermionic action is defined
+in the following way.
\begin{definition}
The fermionic action is defined by
\begin{align}
@@ -215,14 +217,101 @@ Glaser}
of the grading $\gamma$.
\end{definition}
+The almostcommutative Manifold we are dealing with is the following
\begin{align}
- M\times F_{ED} := \left(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes
+ &M\times F_{ED} := \left(C^\infty(M,\mathbb{C}^2),\ L^2(S)\otimes
\mathbb{C}^4,\
D_M\otimes 1 +\gamma _M \otimes D_F;\; J_M\otimes J_F,\ \gamma_M\otimes
- \gamma _F\right)
+ \gamma _F\right).\\
+ \nonumber\\
+ &\text{where:} \nonumber \\
+ &C^\infty(M,\mathbb{C}^2) = C^\infty(M) \otimes C^\infty(M)
+ &\mathcal{H} = \mathcal{H}^+ \otimes \mathcal{H}^-\\
+ &\mathcal{H} = L^2(S)^+ \otimes H_F^+ \oplus L^2(S)^- \otimes H_F^-.
\end{align}
+Where $H_F$ is separated into the particle-anitparticle states with ONB $\{e_R,
+e_L, \bar{e}_R, \bar{e}_L\}$. The ONB of $H_F^+$ is $\{e_L, \bar{e}_R\}$ and
+for $H_F^-$ we have $\{e_R, \bar{e}_L\}$. Furthermore we can decompose a spinor
+$\psi \in L^2(S)$ for each of the eigenspaces $H_F^\pm$, $\psi = \psi_R
+\psi_L$. Thus we can write for an arbitrary $\psi \in \mathcal{H}^+$
+\begin{align}
+ \psi = \chi_R \otimes e_R + \chi_L \otimes e_L + \psi_L \otimes \bar{e}_R
+ \psi_R \otimes \bar{e}_L
+\end{align}
+for $\chi_L, \psi_L \in L^2(S)^+$ and $\chi_R, \psi_R \in L^2(S)^-$.
+\begin{proposition}
+ We can define the action of the fermionic art of $M\times F_{ED}$ in the
+ following way
+ \begin{align}
+ S_f = -i\big(J_M\tilde{\chi}, \gamma(\nabla^S_\mu - i\Gamma_\mu)
+ \tilde{\Psi}\big) + \big(S_M\tilde{\chi}_L, \bar{d}\tilde{\psi}_L\big) -
+ \big(J_M\tilde{\chi}_R, d \tilde{\psi}_R\big)
+ \end{align}
+\end{proposition}
+\begin{proof}
+ We take the fluctuated Dirac operator
+ \begin{align}
+ D_\omega = D_M \otimes i + \gamma^\mu \otimes B_\mu + \gamma_M \otimes
+ D_F
+ \end{align}
+\end{proof}
+The Fermionic Action is $S_F = (J\tilde{\xi}, D_\omega\tilde{\xi})$ for a $\xi
+\in \mathcal{H}^+$, we can begin to calculate (note that we add the constant
+$\frac{1}{2}$ to the action)
+\begin{align}
+ \frac{1}{2}(J\tilde{\xi}, D_\omega\tilde{\xi}) =&\\
+ &+\frac{1}{2}(J\tilde{\xi}, (D_M \otimes i)\tilde{\xi})\label{eq:1}\\
+ &+\frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)
+ \tilde{\xi})\label{eq:2}\\
+ &+\frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes
+ D_F)\tilde{\xi})\label{eq:3}.
+\end{align}
+For equation \ref{eq:1} we calculate
+\begin{align}
+ \frac{1}{2}(J\tilde{\xi}, (D_M\otimes 1)\tilde{\xi}) &=
+ \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\psi}_L)+
+ \frac{1}{2}(J_M\tilde{\chi}_L,D_M\tilde{\psi}_R)+
+ \\&+\frac{1}{2}(J_M\tilde{\psi}_L,D_M\tilde{\psi}_R)+
+ \frac{1}{2}(J_M\tilde{\chi}_R,D_M\tilde{\chi}_L)\\
+ &= (J_M\tilde{\chi},D_M\tilde{\chi}).
+\end{align}
+For equation \ref{eq:2} we have
+\begin{align}
+ \frac{1}{2}(J\tilde{\xi}, (\gamma^\mu \otimes B_\mu)\tilde{\xi})&=
+ -\frac{1}{2}(J_M\tilde{\chi}_R, \gamma^\mu Y_\mu\tilde{\psi}_R)
+ -\frac{1}{2}(J_M\tilde{\chi}_L, \gamma^\mu Y_\mu\tilde{\psi}_R)+\\
+ &+\frac{1}{2}(J_M\tilde{\psi}_L, \gamma^\mu Y_\mu\tilde{\chi}_R)+
+ \frac{1}{2}(J_M\tilde{\psi}_R, \gamma^\mu Y_\mu\tilde{\chi}_L)=\\
+ &= -(J_M\tilde{\chi}, \gamma^\mu Y_\mu\tilde{\psi}).
+\end{align}
+For equation \ref{eq:3} we have
+\begin{align}
+ \frac{1}{2}(J\tilde{\xi}, (\gamma_M\otimes D_F)\tilde{\xi})&=
+ +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)
+ +\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)+\\
+ &+\frac{1}{2}(J_M\tilde{\chi}_L, \bar{d}\gamma_M\tilde{\chi}_L)
+ +\frac{1}{2}(J_M\tilde{\chi}_R, d\gamma_M\tilde{\chi}_R)=\\
+ &= i(J_M\tilde{\chi}, m\tilde{\psi})
+\end{align}
+Note that we obtain a complex mass parameter $d$, so we write $d:=im$ for $m\in \mathbb{R}$,
+which stands for the real mass and we obtain a nice result
+\begin{theorem}
+ The full Lagrangian of $M\times F_{ED}$ is the sum of purely gravitational
+ Lagrangian
+ \begin{align}
+ \mathcal{L}_{grav}(g_{\mu\nu})=4\mathcal{L}_M(g_{\mu\nu})
+ \mathcal{L}_\phi (g_{\mu\nu})
+ \end{align}
+ and the Lagrangian of electrodynamics
+ \begin{align}
+ \mathcal{L}_{ED} = -i\bigg\langle
+ J_M\tilde{\chi},\big(\gamma^\mu(\nabla^S_\mu - iY_\mu) -m\big)\tilde{\psi})
+ \bigg\rangle
+ +\frac{f(0)}{6\pi^2} Y_{\mu\nu}Y^{\mu\nu}.
+ \end{align}
+\end{theorem}
\end{document}
