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Some additions on cosmology

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Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
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Riccardo
2020-10-04 12:09:49 +02:00
parent 5123a92e78
commit 5a7583760e
6 changed files with 1574 additions and 636 deletions
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52
sec/app/isomorphism.tex
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@@ -3,53 +3,53 @@ In this appendix we explain the conventions used for \SU{2} and show the details
\subsection{Conventions}
We parameterise \SU{2} matrices $U$ with a vector $\vb{n} \in \R^3$ such that:
We parameterise \SU{2} matrices $U$ with a vector $\vec{n} \in \R^3$ such that:
\begin{equation}
U(\vb{n})
U(\vec{n})
=
\cos(2 \pi n)\, \1_2
+
i\, \frac{\vb{n} \cdot \vb{\sigma}}{n}\, \sin(2 \pi n),
i\, \frac{\vec{n} \cdot \vec{\sigma}}{n}\, \sin(2 \pi n),
\label{eq:su2parametrisation}
\end{equation}
where $n = \norm{\vb{n}}$ and $0 \le n \le \frac{1}{2}$.
We also identify all $\vb{n}$ when $n=\frac{1}{2}$ since in this case $U(\vb{n})= -\1_2$.
where $n = \norm{\vec{n}}$ and $0 \le n \le \frac{1}{2}$.
We also identify all $\vec{n}$ when $n=\frac{1}{2}$ since in this case $U(\vec{n})= -\1_2$.
The parametrisation is such that:
\begin{eqnarray}
U^*(\vb{n})
U^*(\vec{n})
& = &
\sigma^2\, U(\vb{n})\, \sigma^2
\sigma^2\, U(\vec{n})\, \sigma^2
=
U(\widetilde{\vb{n}}),
U(\widetilde{\vec{n}}),
\\
U^{\dagger}(\vb{n})
U^{\dagger}(\vec{n})
& = &
U^T(\widetilde{\vb{n}})
U^T(\widetilde{\vec{n}})
=
U(-\vb{n}),
U(-\vec{n}),
\\
-U(\vb{n})
-U(\vec{n})
& = &
U(\widehat{\vb{n}})
U(\widehat{\vec{n}})
\label{eq:U_props}
\end{eqnarray}
where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vb{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vb{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vb{n}}{n}$.
where $\sigma^2$ is the second Pauli matrix, $\widetilde{\vec{n}} = \qty( -n^1, n^2, -n^3 )$ and $\widehat{\vec{n}} = - \qty(\frac{1}{2} -n )\, \frac{\vec{n}}{n}$.
The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{m})$ has an explicit realisation as:
The group product of two elements $U(\vec{n} \circ \vec{m} ) = U(\vec{n})\, U(\vec{m})$ has an explicit realisation as:
\begin{equation}
\begin{split}
\cos(2 \pi \norm{\vb{n} \circ \vb{m}})
\cos(2 \pi \norm{\vec{n} \circ \vec{m}})
& =
\cos(2 \pi n)\, \cos(2 \pi m)
-
\sin(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{n} \cdot \vb{m}}{n\, m},
\sin(2 \pi n)\, \sin(2\pi m)\, \frac{\vec{n} \cdot \vec{m}}{n\, m},
\\
\sin(2 \pi \norm{\vb{n} \circ \vb{m}})\,
\frac{\vb{n} \circ \vb{m}}{\norm{\vb{n} \circ \vb{m}}}
\sin(2 \pi \norm{\vec{n} \circ \vec{m}})\,
\frac{\vec{n} \circ \vec{m}}{\norm{\vec{n} \circ \vec{m}}}
& =
\cos(2 \pi n)\, \sin(2\pi m)\, \frac{\vb{m}}{m}
\cos(2 \pi n)\, \sin(2\pi m)\, \frac{\vec{m}}{m}
+
\sin(2 \pi n)\, \cos(2\pi m)\, \frac{\vb{n}}{n}.
\sin(2 \pi n)\, \cos(2\pi m)\, \frac{\vec{n}}{n}.
\end{split}
\label{eq:product_in_SU2}
\end{equation}
@@ -58,9 +58,9 @@ The group product of two elements $U(\vb{n} \circ \vb{m} ) = U(\vb{n})\, U(\vb{
Let $I = 1,\, 2,\, 3,\, 4$ and define:
\begin{equation}
\tau_I = \qty( i\, \1_2,\, \vb{\sigma} ),
\tau_I = \qty( i\, \1_2,\, \vec{\sigma} ),
\end{equation}
where $\vb{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices.
where $\vec{\sigma} = \qty( \sigma^1,\, \sigma^2,\, \sigma^3 )$ are the Pauli matrices.
It is possible to show that:
\begin{equation}
\begin{split}
@@ -122,7 +122,7 @@ If the vector $X^I$ is real, using~\eqref{eq:tau_props} we have:
A rotation in spinor representation is defined as:
\begin{equation}
X'_{(s)} = U_{L}(\vb{n})\, X_{(s)}\, U_{R}^{\dagger}(\vb{m})
X'_{(s)} = U_{L}(\vec{n})\, X_{(s)}\, U_{R}^{\dagger}(\vec{m})
\end{equation}
and it is equivalent to:
\begin{equation}
@@ -138,9 +138,9 @@ through
\frac{1}{2}
\tr(
\qty( \tau_I )^{\dagger}\,
U_{L}(\vb{n})\,
U_{L}(\vec{n})\,
\tau_J\,
U_{R}^{\dagger}(\vb{m})
U_{R}^{\dagger}(\vec{m})
).
\end{equation}
The matrix $R$ is the $4$-dimensional rotation matrix we are looking for since: