## LATEX

### جواب على سؤال

\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)=v^{\mu}(x)~,~{\rm Flat}.
\end{eqnarray}

\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)+\Delta x^{\rho}\Gamma_{\rho\sigma}^{\mu}\tilde{v}^{\sigma}=v^{\mu}(x),~{\rm Curved}.
\end{eqnarray}

\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)=v^{\mu}(x)-\Delta x^{\rho}\Gamma_{\rho\sigma}^{\mu}v^{\sigma}\iff \tilde{v}^{\mu}(x)+\Delta x^{\rho}\nabla_{\rho}\tilde{v}^{\mu}(x)=v^{\mu}(x).
\end{eqnarray}

Under the variation of the dynamical variable which is here the metric $g_{\mu\nu}\longrightarrow g_{\mu\nu}+\delta g_{\mu\nu}$ we obtain the variation of the action:
\begin{eqnarray}
\delta S_{\rm HE}
&=&\int d^4 x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R).
\end{eqnarray}
The principle of least action $\delta S_{\rm HE}=0$ gives immediately Einstein's equations (in vacuum) as Euler-Lagrange equations of motion for the metric:
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=0.
\end{eqnarray}
Inclusion of matter fields will lead to Einstein's equations with a non-vanishing energy-momentum tensor $T_{\mu\nu}$, viz
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}.
\end{eqnarray}

Parallel transport and covariant derivative: \begin{eqnarray} \oint \partial_{\alpha}V^{\mu}\neq 0. \end{eqnarray} \begin{eqnarray} \oint \nabla_{\alpha}V^{\mu}=0. \end{eqnarray} \begin{eqnarray} x^{\mu}\longrightarrow x^{\prime\mu}~:~\nabla_{\alpha}V^{\mu}\longrightarrow \nabla_{\alpha}^{\prime}V^{\prime\mu}=\frac{\partial x^{\beta}}{\partial x^{\prime\alpha}}\frac{\partial x^{\prime \mu}}{\partial x^{\nu}}\nabla_{\beta}V^{\nu}.
\end{eqnarray} \begin{eqnarray} \nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma_{\alpha\mu}^{\nu}V^{\alpha}. \end{eqnarray} \begin{eqnarray} \nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}-\Gamma_{\mu\nu}^{\alpha}\omega_{\alpha}. \end{eqnarray} Metric is covariantly constant and the affine connection (Christoffel symbols): \begin{eqnarray} &&\nabla_{\mu}g_{\alpha\beta}=0=\partial_{\mu}g_{\alpha\beta}-\Gamma_{\mu\alpha}^{\rho}g_{\rho\beta}-\Gamma_{\mu\beta}^{\rho}g_{\alpha\rho}=0\nonumber\\ &&\Rightarrow \Gamma_{\mu\nu}^{\alpha}=\frac{1}{2}g^{\alpha\beta}\big(\partial_{\mu}g_{\nu\beta}+\partial_{\nu}g_{\mu\beta}-\partial_{\beta}g_{\mu\nu}\big). \end{eqnarray}

Scalars:
\begin{eqnarray}
f \longrightarrow f^{\prime}=f.
\end{eqnarray}
Vectors:
\begin{eqnarray}
S_HE{\mu}\longrightarrow V^{\prime\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}V^{\nu}.
\end{eqnarray}
One-forms (dual vectors):
\begin{eqnarray}
\omega_{\mu}\longrightarrow \omega^{\prime}_{\mu}=\frac{\partial x^{\prime}_{\mu}}{\partial x_{\nu}}\omega_{\nu}.
\end{eqnarray}

شكرًا على هذا الطرح الرائع، لديَّ عدّة اسئلة بروف يدري.

1- ما هو مفهوم الزمكان من خلال الأوتار الفائقة؟
2- اذا كانت الأوتار تصنّع كل شيء بذبذباتها، فهل تصنع المكان؟ و إن كانت كذلك، فهل الوتر الواحد يصنع مكانه، أم كيف؟
3- من خلال نظرية الوتر الفائق و محاول تكميم القوى الأربعة، نعلم ان الفوتونات هي تكم للقوى الكهرطيسية مثلما ان الكرافيتونات هي تكميم لقوى الجاذبية..
الفوتون الواحد يستجيب للشحنة الكهربائية وهو بنفسه غير مشحون، و الكرافيتونات لا تستجيب للشحنة الكهربائية بينما تستجيب للطاقة و الكُتلة.. و يحمل كل كرافيتون طاقة لذلك يستجيب كل كرافيتونين لبعضهما و يؤثّران على بعضيهما:
أ- كيف لا يحصل انهيار للمكان من خلال تجاذب كل الكرافيتونات مع بعضها البعض و جعل الكون كله عبارة عن ثقب اسود؟
ب- اذا كان كل كرافيتون له طاقة، فهذا يعني ان له مجال، و تكميم مجال كل كرافيتون هو كرافيتون أخر، وهذا يضعنا في دوامة أن كل كرافيتون ام يصنع مليارات من الكرافيتونات البنات بشكل طولي، فلماذا لا يكون هنالك انهيار، بسبب التجاذب الذي تشكّله الكتلة الكلية لكل تلك الكرافيتونات المُتكدّسة على بعضها؟
4- ماذا يعني ان الفوتونات تخضع لإعادة الإستنظام بينما الكرافيتونات لا تفعل ذلك؟
5- هل تخضع موجات الجاذبية لتأثير دوبلر؟؟؟
مع كل الشكر و التقدير لحضرتك بروف.

### الرجل الذى بارك الله له فى علمه و عمله و عمره و زوجته

الفضاء-زمن يحدد للمادة كيف يجب ان تتحرك و المادة تحدد للفضاء-زمن كيف يجب ان ينحنى.
ويلر Wheeler احد اعظم فيزيائى القرن العشرين فى واحد من اقواله البليغة المعبرة عن احد آراءه الثاقبة.
و ويلر هو استاذ فايمان Feynman و ثورن Thorn وهما من الحائزين على نوبل.
و استاذ بكينشتاين Bekenstein (صاحب القانون مع هاوكينغ Hawking الخاص بترموديناميك الثقب الاسود).
لكن مصطلح (الثقب الاسود black hole) هو من اختراع ويلر و ليس من اختراع بيكنشتاين أو هاوكينغ. و كذا مصطلح (الثقب الدودى wormhole).
و و يلر هو ايضا استاذ والد Wald (المختص الاول فى النسبية العامة و صاحب الكتاب الاول فى النسبية العامة) و هو استاذ ايفريت Everett (صاحب تفسير عديد-العوالم manyworld interpretation للميكانيك الكمومى).
بل ويلر هو صاحب (تجربة الاختيار المؤجل delayed choice experiment) احد اعظم تجارب الميكانيك الكمومى المحققة تجريبيا و التى تتحدى جميع الفلسفة الارسطية و لا نفعت معها فى فهمها لا الكانطية المتسامية ولا الهيومية الحسية وحتى فلسفة الخيال لابن عربى قد تعجز حيالها.
و و يلر هو صاحب مصطلح (الشيء من البت it from bit) و هى تعبر عن نظرة تذهب او تتعدى فلسفة الجوهر substance (وان المادة مشكلة من ذرات اساسية او ان الفضاء-زمن هو متقطع) و تتعدى ايضا فلسفة الطريقة process philosophy التى تعتبر الزمن و التغير هو الاساس و ليس الجوهر و المادة.
ف (الشيء من البت) تتعدى كل ذلك لانها تفترض ان كل شيء لا هو مادة و لا هو قوة بل هو معلومة (وهى كلمة البت) او بالاحرى فان كل شيء علم (وهى نظرة ابن عربى بشكل فيزيائى فى ان الاعيان الثابتة فى علم الله هى اساس الوجود فى الخيال الاكبر).
و ويلر هو صاحب مصطلح البت الكمومى او qbit الذى هو اساس المعلوماتية الكمومية (وربما العلم الكمومى).
و و يلر هو ايضا استاذ ميسنر Misner (احد الثلاثة الذى وضعوا الصياغة الهاميلتونية ال ADM للنسبية العامة) و استاذ انراه Enruh (صاحب تأثير انراه فى النسبية العامة الذى هو اساس فى فهم اشعاع الثقب الاسود) و هو ايضا استاذ كلوادر Klauder (صاحب الحالات المتماسكة coherent states فى الميكانيك الكمومى) و استاذ وايتمان Wightman (مؤسس الصياغة البديهية axiomatic formulation لنظرية الحقل الكمومى).
فهو استاذ عدد معتبر جدا من الفيزيائيين الكبار.
و ويلر هو صاحب مصطلح (مصفوفة التصادم scattering matrix) احد اهم مصطلحات و مفاهيم نظرية الحقل الكمومى.
و ويلر هو من فسر الانشطار النووى nuclear fission و كيفية تشكل النجوم و هو صاحب مصطلح (الرغوة السبينية spin foam) التى هى ليست الا الصياغة عبر تكاملات الطريق path integrals لفايمان لنظرية الثقالة الكمومية الحلقية loop quantum gravity.
وهو ايضا مخترع مصطلح (الفضاء-الممتاز superspace) وهو اساسى فى جميع النظريات التى تستعمل التناظر الممتاز supersymmetry و على رأسها نظرية الاوتار الممتازة. فكلمة (ممتازة super و بعضهم يعربها فائقة) ترجع كلها الى قكرة تمديد الفضاء-زمن العادى بمتغيرات او احداثيات فرميونية fermionic (تتصرف مثل الالكترونات و الكواركات).
و ويلر كن اول من قدم مع تلميذه فايمان نظرية تريد تفسير السهم فى الزمن عن طريقة تفسير السهم فى الاشعاع وهى نظرية الممتص absorber theory.
و ويلر هو صاحب معادلة ويلر-دى ويتر Wheeler-Dewitt equation التى تصف دالة حالة الحقل الثقالى وهو ايضا مخترع مصطلح (الهندسة الديناميكية geometrodynamics) التى تريد تكميم الحقل الثقالى كحقل معيارى gauge field مثلما فعلنا ذلك فى الحقل النووى القوى المعروف تحت اسم (الديناميك اللونى chromodynamics).
فدرجات الحرية فى النسبية العامة هى المترية metric و الانحناء curvature و غيرهما من المفاهيم الهندسية اما درجات الحرية فى الحقل النووى فهى الشحنات اللونية color charges (التى هى تعميم للشحنات الكهربائية).
و نظرية الثقالة الكمومية الحلقية يمكن اعتبارها على انها الوريث الشرعى للهندسة الديناميكية.
و من افكاره العبقرية هى فرضية (كون الالكترون-الواحد) one-electron universe.
وهى تنص على ان كل هذا الكون لا يحتوى الا على الكترون واحد و ما نراه الكترونات عديدة هى تجليات لهذا الالكترون الوحيد. و هذا ما اسميه شخصيا نظرية وحدة الوجود المادية.
ولديه تحوير عبقرى آخر على المبدأ الانطروبيكى او المبدأ البشرى يسمى المبدأ الانطروبيى التشاركى participatory enthropic principle الذى هو فى رايى التفسير العلمى الوحيد (الموجود اليوم) الذى يعطى تفسيرا لوجود الوعى و العقل فى هذا العهد بالضبط من عمر الكون و على هذه الارض بالضبط (اذن هو محاولة تخفيف رأى كوبرنيكوس Copernicus فى أن الارض لا هى محورية و لا هى مركزية).
فهذا رجل عبقرى باتم معنى الكلمة مدرسة لوحده باتم معنى الكلمة فهو كما ترون مخترع مبدع للفيزياء و للغة الفيزيائية و للفيزيائيين على قدر سواء قل نظيره صراحة لا ينكر ذلك لا عالم و لا جاهل.
وهو قد عاش 98 سنة (اذن الله قد بارك له فى عمره) وهو كان احد اعمدة مشروع مانهاتن و مر عبر عدة جامعات امريكية عريقة (اذن بارك الله له ايضا فى عمله الى اقصى الحدود) و كان متزوجا من امرأة واحدة لمدة 72 سنة (اذن بارك الله له ايضا فى الزوجة و كان له منها ثلاثة اولاد).

## Exercise $1$:

Question $1$: Show that if $\nabla$ and $\tilde{\nabla}$ are two different covariant derivatives on the spacetime manifold and $\omega$ is some cotangent dual field then \begin{eqnarray} \nabla_{\mu}\omega_{\nu}=\tilde{\nabla}\omega_{\nu}-C_{\mu\nu}^{\gamma}\omega_{\gamma}.\nonumber \end{eqnarray} Question $2$: By using the torsion free condition show that $C_{\mu\nu}^{\gamma}=C_{\nu\mu}^{\gamma}$.
Question $3$: By demanding that the inner product of two vectors $v^{\mu}$ and $w^{\mu}$ is invariant under parallel transport show that the metric must be covariantly constant.
Question $4$: Determine the form of the tensor $C_{\mu\nu}^{\gamma}$ in terms of the metric $g_{\mu\nu}$.
Question $5$: By noting that the action of $\nabla_a\nabla_b-\nabla_b\nabla_a$ on tangent dual vectors is equivalent to the action of a tensor of type (1,3) determine the Riemann curvature tensor $R_{abc}^d$ in terms of the Christoffel symbols $\Gamma_{ab}^{c}$.

## Solution $1$:

Question $1$

The covariant derivative acting on scalars must be consistent with tangent vectors being directional derivatives. Indeed, for all $f\in {\cal F}$ and $t^{\mu}\in V_p$ we must have
\begin{eqnarray}
t^{\mu}\nabla_{\mu} f=t(f)\equiv t^{\mu}\partial_{\mu}f.
\end{eqnarray}
In other words, if $\nabla$ and $\tilde{\nabla}$ be two covariant derivative operators, then their action on scalar functions must coincide, viz
\begin{eqnarray}
t^{\mu}\nabla_{\mu} f=t^{\mu}\tilde{\nabla}_{\mu} f=t(f).
\end{eqnarray}
We compute now the difference $\tilde{\nabla}_{\mu} (f\omega_{\nu})-{\nabla}_{\mu} (f\omega_{\nu})$ where $\omega$ is some cotangent dual vector. We have
\begin{eqnarray}
\tilde{\nabla}_{\mu} (f\omega_{\nu})-{\nabla}_{\mu} (f\omega_{\nu})&=&\tilde{\nabla}_{\mu}f.\omega_{\nu}+f\tilde{\nabla}_{\mu}\omega_{\nu}-\nabla_{\mu}f.\omega_{\nu}-f\nabla_{\mu}\omega_{\nu}\nonumber\\
&=&f(\tilde{\nabla}_{\mu}\omega_{\nu}-\nabla_{\mu}\omega_{\nu}).\label{100}
\end{eqnarray}
We use without proof the following result. Let $\omega_{\nu}^{'}$ be the value of the cotangent dual vector $\omega_{\nu}$ at a nearby point $p^{'}$, i.e. $\omega_{\nu}^{'}-\omega_{\nu}$ is zero at $p$. Since the cotangent dual vector $\omega_{\nu}$ is a smooth function on the manifold, then for each $p^{'}\in M$, there must exist smooth functions $f_{(\alpha)}$ which vanish at the point $p$ and cotangent dual vectors $\mu_{\nu}^{(\alpha)}$ such that
\begin{eqnarray}
\omega_{\nu}^{'}-\omega_{\nu}=\sum_{\alpha}f_{(\alpha)}\mu_{\nu}^{(\alpha)}.
\end{eqnarray}
We compute immediately
\begin{eqnarray}
\tilde{\nabla}_{\mu}(\omega_{\nu}^{'}-\omega_{\nu})-{\nabla}_{\mu}(\omega_{\nu}^{'}-\omega_{\nu})=\sum_{\alpha}f_{(\alpha)}(\tilde{\nabla}_{\mu}\mu_{\nu}^{(\alpha)}-{\nabla}_{\mu}\mu_{\nu}^{(\alpha)}).
\end{eqnarray}
This is $0$ since by assumption $f_{(\alpha)}$ vanishes at $p$. Hence we get the result
\begin{eqnarray}
\tilde{\nabla}_{\mu}\omega_{\nu}^{'}-\nabla_{\mu}\omega_{\nu}^{'}=\tilde{\nabla}_{\mu}\omega_{\nu}-\nabla_{\mu}\omega_{\nu}.
\end{eqnarray}
In other words, the difference $\tilde{\nabla}_{\mu}\omega_{\nu}-\nabla_{\mu}\omega_{\nu}$ depends only on the value of $\omega_{\nu}$ at the point $p$ although both  $\tilde{\nabla}_{\mu}\omega_{\nu}$ and $\nabla_{\mu}\omega_{\nu}$ depend on how $\omega_{\nu}$ changes as we go away from the point $p$ since they are derivatives. Putting this differently we say that the operator  $\tilde{\nabla}_{\mu}-\nabla_{\mu}$ is a linear map which takes cotangent dual vectors at a point $p$ into tensors, of type $(0,2)$, at $p$ and not into tensor fields defined in a neighborhood of $p$. We write
\begin{eqnarray}
{\nabla}_{\mu}\omega_{\nu}=\tilde{\nabla}_{\mu}\omega_{\nu}-C^{\gamma}~_{\mu\nu}\omega_{\gamma}.\label{104}
\end{eqnarray}
The tensor $C^{\gamma}~_{\mu\nu}$ stands for the map  $\tilde{\nabla}_{\mu}-\nabla_{\mu}$ and it is clearly a tensor of type $(1,2)$. By setting $\omega_{\mu}=\nabla_{\mu}f=\tilde{\nabla}_{\mu}f$ we get ${\nabla}_{\mu}\nabla_{\nu}f=\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}f-C^{\gamma}~_{\mu\nu}\nabla_{\gamma}f$.

Question $2$

By employing now the torsion free condition $\nabla_{\mu}\nabla_{\nu} f=\nabla_{\nu}\nabla_{\mu} f$ we get immediately
\begin{eqnarray}
C^{\gamma}~_{\mu\nu}=C^{\gamma}~_{\nu\mu}.
\end{eqnarray}

Question $3$

Let $C$ be a curve with a tangent vector $t^{\mu}$.  Let $v^{\mu}$ be some tangent vector defined at each point on the curve. The vector $v^{\mu}$ is parallelly transported along the curve $C$ iff
\begin{eqnarray}
t^{\mu}\nabla_{\mu}v^{\nu}|_{\rm curve}=0.
\end{eqnarray}
If $t$ is the parameter along the curve $C$ then $t^{\mu}=dx^{\mu}/dt$ are the components of the vector $t^{\mu}$ in the coordinate basis. The parallel transport condition  reads explicitly
\begin{eqnarray}
\frac{dv^{\nu}}{dt}+\Gamma^{\nu}~_{\mu\lambda}t^{\mu}v^{\lambda}=0.\label{am}
\end{eqnarray}
By demanding that the inner product of two vectors $v^{\mu}$ and $w^{\mu}$ is invariant under parallel transport we obtain, for all curves and all vectors,  the condition
\begin{eqnarray}
t^{\mu}\nabla_{\mu}(g_{\alpha\beta}v^{\alpha}w^{\beta})=0\Rightarrow \nabla_{\mu} g_{\alpha\beta}=0.
\end{eqnarray}
Thus given a metric $g_{\mu\nu}$ on a manifold $M$ the most natural covariant derivative operator is the one under which the metric is covariantly  constant.

Question $4$:

There exists a unique covariant derivative operator $\nabla_{\mu}$ which satisfies $\nabla_{\mu}g_{\alpha\beta}=0$. The proof goes as follows. We know that $\nabla_{\mu}g_{\alpha \beta}$ is given by (from question $1$)
\begin{eqnarray}
\nabla_{\mu} g_{\alpha \beta}=\tilde{\nabla}_{\mu}g_{\alpha \beta}-C^{\gamma}~_{\mu \alpha}g_{\gamma\beta}-C^{\gamma}~_{\mu \beta}g_{\alpha \gamma}.
\end{eqnarray}
By imposing  $\nabla_{\mu}g_{\alpha \beta}=0$ we get
\begin{eqnarray}
\tilde{\nabla}_{\mu} g_{\alpha \beta}=C^{\gamma}~_{\mu \alpha}g_{\gamma\beta}+C^{\gamma}~_{\mu \beta}g_{\alpha \gamma}.
\end{eqnarray}
Equivalently
\begin{eqnarray}
\tilde{\nabla}_{\alpha}g_{\mu \beta}=C^{\gamma}~_{\alpha \mu}g_{\gamma\beta}+C^{\gamma}~_{\alpha \beta}g_{\mu \gamma}.
\end{eqnarray}
\begin{eqnarray}
\tilde{\nabla}_{\beta}g_{\mu \alpha}=C^{\gamma}~_{\mu \beta}g_{\gamma\alpha }+C^{\gamma}~_{\alpha \beta}g_{\mu \gamma}.
\end{eqnarray}
Immediately, we conclude that
\begin{eqnarray}
\tilde{\nabla}_{\mu}g_{\alpha \beta}+\tilde{\nabla}_{\alpha}g_{\mu \beta}-\tilde{\nabla}_{\beta}g_{\mu \alpha}=2C^{\gamma}~_{\mu \alpha}g_{\gamma\beta}.
\end{eqnarray}
In other words,
\begin{eqnarray}
C^{\gamma}~_{\mu \alpha}=\frac{1}{2}g^{\gamma\beta}(\tilde{\nabla}_{\mu}g_{\alpha\beta}+\tilde{\nabla}_{\alpha}g_{\mu \beta}-\tilde{\nabla}_{\beta}g_{\mu \alpha}).
\end{eqnarray}
This choice of $C^{\gamma}~_{\mu\alpha}$ which solves $\nabla_{\mu}g_{\alpha\beta}=0$ is unique. In other words, the corresponding covariant derivative operator is unique.

The most important case corresponds to the choice $\tilde{\nabla}_a=\partial_a$ for which case $C^c~_{ab}$ is denoted $\Gamma^c~_{ab}$ and is called the Christoffel symbol.

Question $5$:

Since the action of $\nabla_a\nabla_b-\nabla_b\nabla_a$ on tangent dual vectors is equivalent to the action of a tensor of type $(1,3)$. Thus we can write
\begin{eqnarray}
(\nabla_a\nabla_b-\nabla_b\nabla_a)\omega_c=R_{abc}~^d\omega_d.
\end{eqnarray}
The tensor $R_{abc}~^d$ is precisely the  Riemann curvature tensor. We compute explicitly

\begin{eqnarray}
\nabla_a\nabla_b\omega_c&=&\nabla_a(\partial_b\omega_c-\Gamma^d~_{bc}\omega_d)\nonumber\\
&=&\partial_a(\partial_b\omega_c-\Gamma^d~_{bc}\omega_d)-\Gamma^e~_{ab}(\partial_e\omega_c-\Gamma^d~_{ec}\omega_d)-\Gamma^e~_{ac}(\partial_b\omega_e-\Gamma^d~_{be}\omega_d)\nonumber\\
&=&\partial_a\partial_b\omega_c-\partial_a\Gamma^d~_{bc}.\omega_d-\Gamma^d~_{bc}\partial_a\omega_d-\Gamma^e~_{ab}\partial_e\omega_c+\Gamma^e~_{ab}\Gamma^d~_{ec}\omega_d-\Gamma^e~_{ac}\partial_b\omega_e+\Gamma^e~_{ac}\Gamma^d~_{be}\omega_d.\nonumber\\
\end{eqnarray}
Thus
\begin{eqnarray}
(\nabla_a\nabla_b-\nabla_b\nabla_a)\omega_c
&=&\bigg(\partial_b\Gamma^d~_{ac}-\partial_a\Gamma^d~_{bc}+\Gamma^e~_{ac}\Gamma^d~_{be}-\Gamma^a~_{bc}\Gamma^d~_{ae}\bigg)\omega_d.
\end{eqnarray}
We get then the components
\begin{eqnarray}
R_{abc}~^d
&=&\partial_b\Gamma^d~_{ac}-\partial_a\Gamma^d~_{bc}+\Gamma^e~_{ac}\Gamma^d~_{be}-\Gamma^e~_{bc}\Gamma^d~_{ae}.
\end{eqnarray}

## Exercise $2$:

Show that $dV=\sqrt{-g}d^4x$ is a scalar quantity under diffeomorphisms.

## Solution $2$:

Let us recall the familiar Levi-Civita symbol in $n$ dimensions defined by
\begin{eqnarray}
\tilde{\epsilon}_{\mu_1...\mu_n}&=&+1~{\rm even}~{\rm permutation}\nonumber\\
&=&-1~{\rm odd}~~{\rm permutation}\nonumber\\
&=&~~0~{\rm otherwise}.
\end{eqnarray}
This is a symbol and not a tensor since it does not change under coordinate transformations. The determinant of a matrix $M$ can be given by the formula
\begin{eqnarray}
\tilde{\epsilon}_{\nu_1^{}...\nu_n^{}}{\rm det} M&=&\tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}M^{\mu_1^{}}~_{\nu_1^{}}...M^{\mu_n^{}}~_{\nu_n^{}}.
\end{eqnarray}
By choosing $M^{\mu}~_{\nu^{}}=\partial x^{\mu}/\partial y^{\nu^{}}$ we get the transformation law
\begin{eqnarray}
\tilde{\epsilon}_{\nu_1^{}...\nu_n^{}}&=&{\rm det}\frac{\partial y^{}}{\partial x}~ \tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}\frac{\partial x^{\mu_1}}{\partial y^{\nu_1^{}}}...\frac{\partial x^{\mu_n}}{\partial y^{\nu_n^{}}}.
\end{eqnarray}
In other words $\tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}$ is not a tensor because of the determinant appearing in this equation. This is an example of a tensor density. Another example of a tensor density is ${\rm det} g$. Indeed, from the tensor transformation law of the metric $g^{'}_{\alpha\beta}=g_{\mu\nu}(\partial x^{\mu}/\partial y^{\alpha})( \partial x^{\nu}/\partial y^{\beta})$ we can show in a straightforward way that
\begin{eqnarray}
{\rm det} g^{'}&=&({\rm det}\frac{\partial y^{}}{\partial x})^{-2}~ {\rm det} g.
\end{eqnarray}
The actual Levi-Civita tensor can then be defined by
\begin{eqnarray}
\epsilon_{\mu_1...\mu_n}=\sqrt{{\rm det} g^{}}~\tilde{\epsilon}_{\mu_1...\mu_n}.
\end{eqnarray}
Next under a coordinate transformation $x\longrightarrow y$ the volume element transforms as
\begin{eqnarray}
d^nx\longrightarrow d^ny={\rm det}\frac{\partial y}{\partial x}~d^nx.\label{ele}
\end{eqnarray}
In other words the volume element transforms as a tensor density and not as a tensor. We verify this important point in our language as follows. We write
\begin{eqnarray}
d^nx&=&dx^0\wedge dx^1\wedge ...\wedge dx^{n-1}\nonumber\\
&=&\frac{1}{n!}\tilde{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}.\label{sdf}
\end{eqnarray}
Recall that a differential $p-$form is a $(0,p)$ tensor which is completely antisymmetric. For example scalars are $0-$forms and dual cotangent vectors are $1-$forms. The  Levi-Civita tensor $\epsilon_{\mu_1...\mu_n}$ is a $4-$form. The differentials $dx^{\mu}$ appearing in the second line of equation  (\ref{sdf}) are $1-$forms and hence under a coordinate transformation $x\longrightarrow y$ we have $dx^{\mu}\longrightarrow dy^{\mu}=dx^{\nu}\partial y^{\mu}/\partial x^{\nu}$.  By using this  transformation law we can immediately show that $dx^n$ transforms to $d^ny$ exactly as in equation (\ref{ele}).

It is not difficult to see now that an invariant volume element can be given by the $n-$form defined by the equation
\begin{eqnarray}
dV=\sqrt{{\rm det} g}~ d^nx.
\end{eqnarray}
We can show that
\begin{eqnarray}
dV&=&\frac{1}{n!}\sqrt{{\rm det} g}~\tilde{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}\nonumber\\
&=&\frac{1}{n!}{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}\nonumber\\
&=&{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\otimes ...\otimes dx^{\mu_n}\nonumber\\
&=&\epsilon.
\end{eqnarray}
In other words the  invariant volume element is precisely the  Levi-Civita tensor. In the case of Lorentzian signature we replace ${\rm det} g$ with  $-{\rm det} g$.

## Exercise $3$:

- Derive equation (1) and then derive the vacuum Einstein’s equations (2).
- Add matter fields to the Hilbert-Einstein action and derive the Einstein’s equations (3).
- Determine the energy-momentum tensor $T_{\mu\nu}$.
- Calculate the energy-momentum tensor for scalar and electromagnetic fields.
- Calculate the energy-momentum tensor for the cosmological constant and determine the corresponding form of the Einstein’s equations.

## Solution $3$:

We compute
\begin{eqnarray}
\delta S
&=&\int d^n x\delta \sqrt{-{\rm det} g}~g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}.\nonumber\\
\end{eqnarray}
We have
\begin{eqnarray}
\delta R_{\mu\nu}&=&\delta R_{\mu\rho\nu}~^{\rho}\nonumber\\
&=&\partial_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\partial_{\mu}\delta\Gamma^{\rho}~_{\rho\nu}+\delta (\Gamma^{\lambda}~_{\mu\nu}\Gamma^{\rho}~_{\rho\lambda}-\Gamma^{\lambda}~_{\rho\nu}\Gamma^{\rho}~_{\mu\lambda})\nonumber\\
&=&(\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\Gamma^{\rho}~_{\rho\lambda}\delta\Gamma^{\lambda}~_{\mu\nu}+\Gamma^{\lambda}~_{\rho\mu}\delta\Gamma^{\rho}~_{\lambda\nu}+\Gamma^{\lambda}~_{\rho\nu}\delta \Gamma^{\rho}~_{\lambda\mu})-(\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}-\Gamma^{\rho}~_{\mu\lambda}\delta\Gamma^{\lambda}~_{\rho\nu}+\Gamma^{\lambda}~_{\mu\rho}\delta\Gamma^{\rho}~_{\lambda\nu}\nonumber\\
&+&\Gamma^{\lambda}~_{\mu\nu}\delta \Gamma^{\rho}~_{\rho\lambda})+\delta (\Gamma^{\lambda}~_{\mu\nu}\Gamma^{\rho}~_{\rho\lambda}-\Gamma^{\lambda}~_{\rho\nu}\Gamma^{\rho}~_{\mu\lambda})\nonumber\\
&=&\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}.
\end{eqnarray}
In the second line of the above equation we have used the fact that $\delta \Gamma^{\rho}~_{\mu\nu}$ is a tensor since it is the difference of two connections. Thus

\begin{eqnarray}
\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}&=&\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\bigg(\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}\bigg)\nonumber\\
&=&\int d^n x \sqrt{-{\rm det} g}~\nabla_{\rho}\bigg(g^{\mu\nu}\delta \Gamma^{\rho}~_{\mu\nu}-g^{\rho\nu}\delta \Gamma^{\mu}~_{\mu\nu}\bigg).
\end{eqnarray}
We compute also (with $\delta g_{\mu\nu}=-g_{\mu\alpha}g_{\nu\beta}\delta g^{\alpha \beta}$)
\begin{eqnarray}
\delta\Gamma^{\rho}~_{\mu\nu}&=&\frac{1}{2}g^{\rho\lambda}\bigg(\nabla_{\mu}\delta g_{\nu\lambda}+\nabla_{\nu}\delta g_{\mu\lambda}-\nabla_{\lambda}\delta g_{\mu\nu}\bigg)\nonumber\\
&=&-\frac{1}{2}\bigg(g_{\nu\lambda}\nabla_{\mu}\delta g^{\lambda\rho}+g_{\mu\lambda}\nabla_{\nu}\delta g^{\lambda\rho}-g_{\mu\alpha}g_{\nu\beta}\nabla^{\rho}\delta g^{\alpha\beta}\bigg).
\end{eqnarray}
Thus
\begin{eqnarray}
\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}
&=&\int d^n x \sqrt{-{\rm det} g}~\nabla_{\rho}\bigg(g_{\mu\nu}\nabla^{\rho}\delta g^{\mu\nu}-\nabla_{\mu}\delta g^{\mu\rho}\bigg).
\end{eqnarray}
By Stokes's theorem this integral is equal to the integral over the boundary of spacetime of the expression $g_{\mu\nu}\nabla^{\rho}\delta g^{\mu\nu}-\nabla_{\mu}\delta g^{\mu\rho}$ which is $0$ if we assume that the metric and its first derivatives are held fixed on the boundary. The variation of the action reduces to
\begin{eqnarray}
\delta S
&=&\int d^n x\delta \sqrt{-{\rm det} g}~g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}R_{\mu\nu}.
\end{eqnarray}
Next we use the result
\begin{eqnarray}
\delta \sqrt{-{\rm det} g}=-\frac{1}{2}\sqrt{-{\rm det} g}~ g_{\mu\nu}\delta g^{\mu\nu}.
\end{eqnarray}
Hence
\begin{eqnarray}
\delta S
&=&\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R).
\end{eqnarray}
This will obviously lead to Einstein's equations in vacuum  which is partially our goal. We want also to include the effect of matter  which requires considering the more general actions of the form
\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^nx~\sqrt{-{\rm det}g}~R+S_M.\label{HE}
\end{eqnarray}
\begin{eqnarray}
S_M=\int d^nx~\sqrt{-{\rm det}g}~\hat{\cal L}_M.
\end{eqnarray}
The variation of the action becomes
\begin{eqnarray}
\delta S
&=&\frac{1}{16\pi G}\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\delta S_M\nonumber\\
&=&\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}\bigg[\frac{1}{16\pi G}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}} \bigg].
\end{eqnarray}
In other words
\begin{eqnarray}
\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S}{\delta g^{\mu\nu}} &=&\frac{1}{16\pi G}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.
\end{eqnarray}
Einstein's equations are therefore given by
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}.
\end{eqnarray}
The stress-energy-momentum tensor must therefore be defined by the equation
\begin{eqnarray}
T_{\mu\nu}=-\frac{2}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.
\end{eqnarray}
As a first example we consider the action of a scalar field in curved spacetime given by
\begin{eqnarray}
S_{\phi}=\int d^nx \sqrt{-{\rm det} g}~\bigg[-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi)\bigg].
\end{eqnarray}
The corresponding stress-energy-momentum tensor is calculated to be given by
\begin{eqnarray}
T_{\mu\nu}^{(\phi)}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\nabla_{\rho}\phi\nabla_{\sigma}\phi-g_{\mu\nu}V(\phi).
\end{eqnarray}
As a second example we consider the action of the  electromagnetic field in curved spacetime given by
\begin{eqnarray}
S_{A}=\int d^nx \sqrt{-{\rm det} g}~\bigg[-\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\bigg].
\end{eqnarray}
In this case the  stress-energy-momentum tensor is calculated to be given by
\begin{eqnarray}
T_{\mu\nu}^{(A)}=F^{\mu\lambda}F^{\nu}~_{\lambda}-\frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}.
\end{eqnarray}
The cosmological constant is one of the simplest matter action that one can add to the Hilbert-Einstein action. It is given by
\begin{eqnarray}
S_{\rm cc}=-\frac{1}{8\pi G}\int d^4x\sqrt{-{\rm det}g}\Lambda.
\end{eqnarray}
In this case the energy-momentum tensor and the Einstein equations read
\begin{eqnarray}
T_{\mu\nu}=-\frac{\Lambda}{8\pi G}g_{\mu\nu}.
\end{eqnarray}
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0.
\end{eqnarray}

### حول التكميم القانونى للنسبية العامة

وبعد قضاء عطلة العيد فى حجر بل عزل شبه كلى لا ندرى اسبابه الحقيقية و ندرى فقط سببه المعلن رسميا وهو مرض الكورونا صرفت العطلة (مع هذا مختارا و لست مضطرا) مركزا تماما على مسألة التكميم القانونى للنسبية العامة و كتبت فيها اربعة منشورات تقنية (لكن مبسطة بالنسبة للطالب الدارس و الاستاذ المتخصص) ثلاثة منشورات جاهزة فى الروابط ادناه و الاخير ما زال قيد التحضير.
أهم شيء ركزت عليه هو الحسابات المفصلية التى قدمتها جاهزة للفحص و كذا المعانى الفيزيائية وراء كل هذه الامور.
الهدف هو الوصول الى فهم الثقالة الحلقية loop quantum gravity فهما دقيقا و سترون انها ليست الا نظرية معيارية للرابطية gauge theory of the connection اذن هى تقع بالضبط فى نفس مضمار نظرية الوتر الممتاز.
وعكس ادعاءات ثيمان thiemann فى المرجع ادناه فاننا يمكن ان نفهم و بدقة شديدة نظرية ثقالة الحلقية و نستطيع ان نحكم عليها فى الاخير حكم المختص و ليس حكم الدخيل.
وحكمى عليها شخصيا ليس حكما سلبيا بل هو حكم ايجابى جدا فهى اكدت لى الكثير من القناعات التى اكتسبتها من دراساتى الطويلة فى الفيزياء النظرية و بالخصوص قضية الثقالة الكمومية.
لهذا فاننى لا ادرى لماذا يعتقد ثيمان ان جميع من هو متأثر بفيزياء الجسيمات و نظرية الوتر سيكون حكمه سلبيا عليها و لهذا اتهمهم بانهم لا يستطيعون فهمها الا فهم الدخيل عليها و ليس فهم المختص والفرضية الاولى منه خاطئة و النتيجة التى ذهب اليها اخطأ.
اذن نظرية الوتر و نظرية الثقالة الكمومية نجحوا جميعا (نجاحا نسبيا) فى تكميم الثقالة و النسبية العامة لانهم اعادوا صياغة النسبية العامة كنظرية معيارية gauge theory.
و قد كانت هذه هى عبقرية ايشتيكار Ashtekar حينما ركز على الزمرة SU(2) او بالاحرى الزمرة SL(2,C) للرابطيات الثنائية-الذاتية self-dual connections التى بداخل زمرة لورنتز للدورانات و تحويلات لورنتز الخاصة بالنسبية الخاصة.
بعده فان باربيرو Barbero و ايميرزى Immirzi عرفوا كيف يقومون بالتدوير الحقيقى (يشبه جدا تدوير ويك Wick rotation بل ايشتيكار يسميه تدوير ويك) للرابطية المركبة التى اكتشفها ايشتيكار و تحويل النسبية العامة فعلا الى زمرة SU(2) حقيقية.
واذكر ان ايميرزى و قد كنت اعرفه شخصيا و قد عملت معه (وهو ايطالى من اصل لبنانى كما ذكر لى) كان يمقت جدا نسبة وسيط ايميرزى Immirzi parameter اليه (وهو الوسيط الذى يدخل فى تدوير ويك لرابطية ايشتيكار) لأنه على ما يبدو لم يكن مقتنعا به (لكن لم أسأله ابدا عن هذا الموضوع لهذا فاننى لست متأكدا من الاسباب التى تقف وراء موقفه).
ثم بعد ذلك جاء روفيلى Rovelli (وله ميتافيزيقا قوية جدا بخصوص كل هذه الامور و ليس فقط فيزياء) و كذا سمولين Smolin و بينوا كيف تتحول حلقات ويلسون Wilson loops لهذه النظرية المعيارية الى التمثيل الحلقى loop representation للثقالة الكمومية و لان الزمرة هى SU(2) فان الذى يدخل فى التمثيل و الحساب هو قيم العزم الحركى الذاتى أو السبين spin ومنها نحصل على منحنيات السبين spin graphs التى تمثل الفضاء-زمن و التى كان قد اقترحها بنروز Penrose مستقلا تماما منذ السبعينات كتمثيل لهندسة الفضاء-زمن.
هذه المنحنيات (منحنيات السبين) هى بالضبط حالات الجملة و هى حالات ذاتية eigenstates لمؤثرات المساحة و الحجم التى عرفها روفيلى و سمولين وهى تؤثر فى فضاء هيلبرت Hilbert space للحالات الفيزيائية للجملة.
منحينات السبين تشكل اذن اساس لفضاء هيلبرت الفيزيائى.
و هذه ربما أهم نتيجة على الاطلاق لانها تؤكد على تقطيع discretization بنية الفضاء-زمن على مستوى ابعاد بلانك وعلى ان الفضاء-زمن فى النهاية ينبعث emerges من لا شيء فقط نفترض استقلال الخلفية background independence و الصمود الديفيومورزفزمى diffeomorphism invariance.
أهم شيء هنا من الناحيتين الرياضية و الفيزيائية هو فضاء هيلبرت الذى لا يمكن تعريفه بدقة الا بعد حل قيود النظرية (وهذا فعله اولا بشكل دقيق ثيمان thiemann).
لان النظرية الثقالية هنا قد تحولت الى نظرية معيارية فهى تتميز بقيد يسمى قيد غوس Gauss constraint (وهو نفسه قانون غوس للكهرباء) و لان النظرية الثقالية تتميز بالتناظر تحت تحويلات الديفيوموريفزم diffeomorphism transformations (اى التناظر تحت تأثير التحويلات العامة للاحداثيات general coordinate transformations) فهى تتميز ايضا بقيدين آخرين هما قيدى الهامليتونية Hamiltonian constraint و قيد الديفيومورفيزم diffeomorphism constraint.
حل هذه القيود هو الذى يؤدى بنا الى تعريف فضاء هلبرت و اكتشاف ان حلقات ويلسن او بالاحرى منحنيات السبين تقدم تمثيل هندسة الفضاء-زمن على مستويات بلانك الصغيرة جدا.
لفهم كل هذه الامور بشكل دقيق علينا فهم جزء معتبر جدا من النسبية العامة لخصته فى المنشورات الثلاثة ادناه.
فعل هيلبرت و اينشتاين للنسبية العامة
صياغة الرباعيات للنسبية العامة
الصياغة الهاميلتونية للنسبية العامة و الديناميك الهندسى

This is the third post of a series of four posts concerned with the canonical quantization of general relativity.

Spacetime is naturally assumed to be globally hyperbolic which means that it is diffeomorphic to the direct product $\mathbb{R}\times \Sigma$ where $\Sigma$ is a three-dimensional smooth manifold.

We consider then a foliation of the spacetime manifold given by the spatial Cauchy  hypersurfaces $\Sigma_t$ of constant time $t$.  Let $n^{\mu}$ be the unit normal vector field to the hypersurfaces  $\Sigma_t$ which is given explicitly by
\begin{eqnarray}
n^{\mu}=-N\frac{\partial t}{\partial x_{\mu}}~,~n_{\mu}n^{\mu}=-1.
\end{eqnarray}
The normalization $N$ is the lapse function which measures the rate of change of the proper time with respect to the coordinate time $t$ as one moves normally to the hypersurfaces $\Sigma_t$. It  is given explicitly by
\begin{eqnarray}
N=-g_{\mu\nu}t^{\mu}n^{\nu}.\label{lapse}
\end{eqnarray}
The time flow in this foliated spacetime will be given by a vector field $t^{\mu}$ which satisfies $t^{\mu}\nabla_{\mu}t=1$, i.e.
\begin{eqnarray}
t^{\mu}=\frac{\partial x^{\mu}}{\partial t}.
\end{eqnarray}
We decompose $t^{\mu}$ into its normal and tangential parts with respect to the hypersurface  $\Sigma_t$ as
\begin{eqnarray}
t^{\mu}=Nn^{\mu}+N^{\mu}~,~N^{\mu}=N^ie_i^{\mu}.
\end{eqnarray}
The $e_i^{\mu}$ are  tangent vectors to the hypersurface  $\Sigma_t$ given by
\begin{eqnarray}
e_i^{\mu}=\frac{\partial x^{\mu}}{\partial y^i}.
\end{eqnarray}
The $y^i$ are coordinates on the hypersurface $\Sigma_t$, i.e. the coordinates $x^{\mu}$ are split as $x^{\mu}\longrightarrow y^{\mu}=(t,y^i)$, and $N^i$ is the so-called shift vector which measures the shift of the local spatial coordinate system as one moves normally to the hypersurfaces $\Sigma_t$. It is  given by
\begin{eqnarray}
N^{\mu}=h^{\mu}~_{\nu}t^{\nu}.\label{shift}
\end{eqnarray}
The three-dimensional metric $h_{ij}$ is the induced metric on the  hypersurface $\Sigma_t$ given explicitly by
\begin{eqnarray}
h_{ij}=g_{\mu\nu}e_i^{\mu}e_j^{\nu}=h_{\mu\nu}e_i^{\mu}e_j^{\nu}~,~h_{\mu\nu}=g_{\mu\nu}+n_{\nu}n_{\nu}.
\end{eqnarray}
We define the inverse metric $h^{ij}$ in the usual way, viz $h_{ij}h^{jk}=\delta_i^k$.  More precisely, we compute (using also $h^{\mu\nu}=g^{\mu\nu}-n^{\mu\nu}$) the result $h_{\mu\nu}h^{\nu\alpha}=\delta_{\mu}^{\alpha}+n_{\mu}n^{\alpha}$.

We compute immediately that
\begin{eqnarray}
dx^{\mu}&=&\frac{\partial x^{\mu}}{\partial t} dt+\frac{\partial x^{\mu}}{\partial y^i}dy^i\nonumber\\
&=&t^{\mu}dt+e_i^{\mu}dy^i\nonumber\\
&=&(N dt) n^{\mu}+(dy^i+N^i dt) e_i^{\mu}.
\end{eqnarray}
Also
\begin{eqnarray}
ds^2&=&g_{\mu\nu}dx^{\mu}dx^{\nu}\nonumber\\
&=&g_{\mu\nu}\bigg[N^2dt^2n^{\mu}n^{\nu}+(dy^i+N^idt)(dy^j+N^jdt)e_i^{\mu}e_j^{\nu}\bigg]\nonumber\\
&=&-N^2dt^2+h_{ij}(dy^i+N^idt)(dy^j+N^jdt).\label{4dmetric1}
\end{eqnarray}
The ADM metric is then given explicitly by (with $N_i=h_{ij}N^j$)
\begin{eqnarray}
g_{\mu\nu}=\left( \begin{array}{cc}
-N^2+N^iN_i &  N_j \\
N_i & h_{ij}
\end{array} \right).
\end{eqnarray}
The inverse ADM metric $g^{\mu\nu}$ is then given by
\begin{eqnarray}
g^{\mu\nu}=\left( \begin{array}{cc}
-\frac{1}{N^2} &  \frac{1}{N^2}N^j \\
\frac{1}{N^2}N^i & h^{ij}-\frac{1}{N^2}N^iN^j
\end{array} \right).
\end{eqnarray}
We conclude that all information about the original four-dimensional metric $g_{\mu\nu}$ is contained in the lapse function $N$, the shift vector $N^i$ and the three-dimensional metric $h_{ij}$. The lapse and the shift $N$ and $N^i$ are not dynamical variables but only Lagrange multipliers yielding under their respective variation the so-called Hamiltonian and diffeomorphism constraints which satisfy a Dirac algebra of first class constraints. A particular choice of $N$ and $N^i$, i.e. a particular choice of  foliation plays the role of a gauge fixing condition (called the time gauge) for the diffeomorphism group. In other words, invariance under general coordinate transformations which form the group of diffeomorphism  is not lost but only fixed and in fact the diffeomorphism invariance of the theory is still precisely encoded in the Dirac algebra of the first class constraints of the theory.

Next we would like to rewrite the Hilbert-Einstein Lagrangian density  in terms of the three-dimensional quantities $N$,  $N^i$ and $h_{ij}$ and then compute the Hamiltonian density.

First we compute
\begin{eqnarray}
\sqrt{-g}d^4x=N\sqrt{h}d^4y.
\end{eqnarray}
A central object in the discussion of how the hypersurfaces $\Sigma_t$ are embedded in the four-dimensional spacetime manifold ${\cal M}$ is the extrinsic curvature $K_{\mu\nu}$. This is given essentially by $1)$ comparing the normal vector $n_{\mu}$ at a point $p$ and the parallel transport of the normal vector $n_{\mu}$ at a nearby point $q$ along a geodesic connecting $q$ to $p$  on the hypersurface $\Sigma_t$ and then $2)$ projecting the result onto the hypersurface $\Sigma_t$. The first part is clearly given by the covariant derivative whereas the projection is done through the three-dimensional metric tensor. Hence the extrinsic curvature must be defined by
\begin{eqnarray}
K_{\mu\nu}&=&-h_{\mu}^{\alpha}h_{\nu}^{\beta}\nabla_{\alpha}n_{\beta}\nonumber\\
&=&-h_{\mu}^{\alpha}\nabla_{\alpha}n_{\nu}.
\end{eqnarray}
In the second line of the above equation we have used $n^{\beta}\nabla_{\alpha}n_{\beta}=0$ and $\nabla_{\alpha}g_{\mu\nu}=0$. We can check that $K_{\mu\nu}$ is symmetric and tangent, viz
\begin{eqnarray}
\end{eqnarray}
The next goal is to compute in terms of the three-dimensional quantities the scalar curvature $R$.  We start from (where $G$ is the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-Rg_{\mu\nu}/2$)
\begin{eqnarray}
R&=&-Rg_{\mu\nu}n^{\mu}n^{\nu}\nonumber\\
&=&-2(R_{\mu\nu}-G_{\mu\nu})n^{\mu}n^{\nu}\nonumber\\
&=&-2R_{\mu\nu}n^{\mu}n^{\nu}+R_{\mu\nu\alpha\beta}h^{\mu\alpha}h^{\nu\beta}.
\end{eqnarray}
We compute
\begin{eqnarray}
R_{\mu\nu\alpha\beta}h^{\mu\alpha}h^{\nu\beta}&=&h_{\beta\rho}R_{\mu\nu\alpha}~^{\rho}h^{\mu\alpha}h^{\nu\beta}\nonumber\\
&=&g^{\beta\eta}g^{\kappa\sigma}\big(h_{\kappa}^{\mu}h_{\eta}^{\nu}h_{\sigma}^{\alpha}R_{\mu\nu\alpha}~^{\rho}h_{\rho}^{\theta}\big)h_{\theta\beta}\nonumber\\
&=&g^{\beta\eta}g^{\kappa\sigma}\big(^{(3)}R_{\kappa\eta\sigma}~^{\theta}+K_{\kappa\sigma}K_{\eta}^{\theta}-K_{\eta\sigma}K_{\kappa}^{\theta}\big)h_{\theta\beta}\nonumber\\
&=&g^{\kappa\sigma}\big(^{(3)}R_{\kappa\eta\sigma}~^{\theta}+K_{\kappa\sigma}K_{\eta}^{\theta}-K_{\eta\sigma}K_{\kappa}^{\theta}\big)h_{\theta}^{\eta}\nonumber\\
&=&^{(3)}R+K^2-K_{\mu\nu}K^{\mu\nu}.
\end{eqnarray}
In the third line we have used the first Gauss–Codacci relation. Next we compute
\begin{eqnarray}
R_{\mu\nu}n^{\mu }n^{\nu}
&=&\nabla_{\mu}(Kn^{\mu}+n^{\nu}\nabla_{\nu}n^{\mu})-K_{\mu\nu}K^{\mu\nu}+K^2.
\end{eqnarray}
The first term is a total divergence and hence it can be neglected. We get then the so-called ADM Lagrangian density
\begin{eqnarray}
\end{eqnarray}
In the above equation $K^2=(h_{\mu\nu}K^{\mu\nu})^2$. The extrinsic curvature $K_{\mu\nu}$ is the covariant analogue of the time derivative of the metric. Indeed, by using the concept of the Lie derivative we can show after some more steps that  (where $D_{\mu}$ is the three-dimensional covariant derivative)
\begin{eqnarray}
K_{\mu\nu}
&=&-\frac{1}{2N}\big(\dot{h}_{\mu\nu}-D_{\mu}N_{\nu}-D_{\nu}N_{\mu}\big).
\end{eqnarray}
It is straightforward now to compute the conjugate momentum  $\Pi^{\mu\nu}$ corresponding to the metric $h_{\mu\nu}$. We find
\begin{eqnarray}
&=&-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu}).
\end{eqnarray}
From this identity we can show that $h_{\mu\nu}\Pi^{\mu\nu}=2\sqrt{h}h_{\mu\nu}K^{\mu\nu}$.

The ADM Hamiltonian density is then given by
\begin{eqnarray}
&=&-\sqrt{h}N^{(3)}R+2D_{\mu}N_{\nu}.\Pi^{\mu\nu}+\frac{N}{\sqrt{h}}(\Pi_{\mu\nu}\Pi^{\mu\nu}-\frac{1}{2}\Pi^2)\nonumber\\
&=&\sqrt{h}N\bigg[-^{(3)}R+\frac{\Pi_{\mu\nu}\Pi^{\mu\nu}}{h}-\frac{\Pi^2}{2h}\bigg]+\sqrt{h}N_{\nu}\bigg[-2D_{\mu}\bigg(\frac{\Pi^{\mu\nu}}{\sqrt{h}}\bigg)\bigg]\nonumber\\
&=&\sqrt{h}NH_0+\sqrt{h}N_{i}H^{i}.
\end{eqnarray}
In the last two lines we have dropped a total divergence since it only leads to a  boundary term which is assumed to be negligible for large spatial surfaces encompassing spacetime. Indeed, the corresponding ADM Lagrangian density takes  the form
\begin{eqnarray}
&=&\dot{h}_{\mu\nu}\Pi^{\mu\nu}-\sqrt{h}NH_0-\sqrt{h}N_{i}H^{i}.
\end{eqnarray}
The Hamiltonian is then obtained by integrating the Hamiltonian density over the hypersurface $\Sigma_t$. We get

\begin{eqnarray}
&=& \int d^3y \sqrt{h}NH_0+\int d^3y \sqrt{h}N_{i}H^{i}\nonumber\\
&=&H(N)+D(\vec{N}).
\end{eqnarray}
Finally by varying the Lagrangian density with respect to the lapse function $N$ and the shift vector $N^{\mu}$ we obtain the Hamiltonian and diffeomorphsim first class constraints given respectively by
\begin{eqnarray}
H_0\equiv -^{(3)}R+\frac{\Pi_{\mu\nu}\Pi^{\mu\nu}}{h}-\frac{\Pi^2}{2h}=0.
\end{eqnarray}
And
\begin{eqnarray}
H_i\equiv -2D_{\mu}\bigg(\frac{\Pi^{\mu\nu}}{\sqrt{h}}\bigg)=0.
\end{eqnarray}
This vanishing should be properly understood not as identically vanishing but as weakly vanishing in the sense of Dirac, i. e. it vansihes only on physical states not any state. We have then the constraints

\begin{eqnarray}
H(N)\simeq 0.
\end{eqnarray}
And
\begin{eqnarray}
D(\vec{N})\simeq 0.
\end{eqnarray}
The Hamiltonian constraint  $H(N)\simeq 0$ constrains the Hamiltonian (and it generates the time flow of the theory which connects different hypersurfaces $\Sigma_t$) whereas the diffeomorphism constraint $D(\vec{N})\simeq 0$ constrains the momentum of the theory (and generates diffeomorphism transformations on the hypersurfaces $\Sigma_t$ themselves). These constraints are first class which means that they do close under the Poisson brackets, i.e. their Dirac algebra is given by \cite{Thiemann:2006cf}
\begin{eqnarray}
&&\{D(\vec{N}),D(\vec{N}^{\prime})\}=8\pi GD({\cal L}_{\vec{N}}\vec{N}^{\prime})\nonumber\\
&&\{D(\vec{N}),H({N}^{\prime})\}=8\pi G H({\cal L}_{\vec{N}}{N}^{\prime})\nonumber\\
&&\{H({N}),H({N}^{\prime})\}=8\pi G D(q^{-1}(NdN^{\prime}-N^{\prime}dN)).
\end{eqnarray}
In the above equation  ${\cal L}$ is the Lie derivative and $q$ is the pullback metric, i.e. $q\equiv h$. This algebra is universal in the sense that it encodes in a precise sense the diffeomorphism invariance of the theory (despite the explicit choice of the foliation, i.e. the explicit choice of the lapse function $N$ and the shift vector $N^i$ which should only be viewed as a gauge fixing choice).

In fact any theory characterized by invariance under general coordinate transformations will contain Hamiltonian and diffeomorphism constraints satisfying precisley the above Dirac algebra (this always comes about from the arbitrary nature of the foliation and the associated arbirary choice of the lapse function $N$ and shift vector $N^i$  which necessarily appear as Lagrange multipliers in the action with singular Legender transformation).

We remark that the Hamiltonian vansihes also weakly which is also another universal property of theories with diffeomorphism invariance which is the fact that there is no Hamiltonian in the dynamics of these theories (since there is no time really!!)  but only Hamiltonian constraint.

In summay, the phase space of the Hamiltonian formulation of general relativity consists  therefore of all pairs $(h_{ij},\Pi^{kl})$ where the extrinsic curvature $K^{kl}$ stands in place of the momentum $\Pi^{kl}$ through the relation $\Pi_{\mu\nu}=-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu})$. The fundamental Poisson bracket is given by
\begin{eqnarray}
\{h_{ij}(t,\vec{y}_1),\Pi^{kl}(t,\vec{y}_2)\}= \delta^3(\vec{y}_1-{y}_2)\delta_{i}^k\delta_j^l.
\end{eqnarray}
The starting point of the canonical quantization program of geometrodynamics is then the commutation relations
\begin{eqnarray}
[\hat{h}_{ij}(t,\vec{y}_1),\hat{\Pi}^{kl}(t,\vec{y}_2)]= i\hbar \delta^3(\vec{y}_1-\delta{y}_2)\delta_{i}^k\delta_j^l.
\end{eqnarray}
The operators  $\hat{h}_{ij}(t,\vec{y}_1)$ and $\hat{\Pi}^{kl}(t,\vec{y}_2)$ are defined on physical states $\Psi(h_{ij})$ by
\begin{eqnarray}
&&\hat{h}_{ij}\Psi(h_{ij})={h}_{ij}\Psi(h_{ij})\nonumber\\
&&\hat{\Pi}^{kl}\Psi(h_{ij})=-i\hbar\frac{\delta}{\delta h_{kl}}\Psi(h_{ij}).
\end{eqnarray}
The physical states $\Psi(h_{ij})$ are thoses states in the Hilbert space which are annihilated by the Hamiltonian and diffeomorphism constraints which are also implemented as  operators, namely

\begin{eqnarray}
\hat{H}(N)\Psi(h_{ij})= 0.
\end{eqnarray}
And
\begin{eqnarray}
\hat{D}(\vec{N})\Psi(h_{ij})= 0.
\end{eqnarray}

### The vielbein formalism

This is the second post of a series of four posts concerned with the canonical quantization of general relativity.

## The vielbein formalism

First the vielbein field is essentially the square root of the metric. Physically, it gives the local orientations $\xi^m$ of freely falling frames in the gravitational field associated with the metric $g_{\mu\nu}$, i.e.  $\xi^m$  are orientations of the local inertial frames with respect to the  coordinate axes  $x^{\mu}$ of the curved spacetime manifold, viz
\begin{eqnarray}
e_{\mu}^m=\frac{\partial \xi^m}{\partial x^{\mu}}.
\end{eqnarray}
The metric is the square toot of the vielbein which means that we have ($\eta$ being the flat metric with signature $-1$, $+1$, $+1$, $+1$)
\begin{eqnarray}
g_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}.\label{fundamental}
\end{eqnarray}
The inverse of $e_{\mu}^m$ is denoted by $e_m^{\mu}$, viz $e_m^{\mu}e_{\mu}^n=\eta_m^n$ and $e_{\mu}^me_m^{\nu}=\eta_{\mu}^{\nu}$. Thus, we also have $g^{\mu\nu}=e^{\mu}_m e^{\nu}_n\eta^{mn}$.

The fundamental equation  (\ref{fundamental}) can be derived in a straightforward way from the Clifford algebra of the Dirac matrices in the curved spacetime manifold which is of the usual form but only with the replacement $\eta\longrightarrow g$, i.e.
\begin{eqnarray}
\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu\nu}. \label{clifford}
\end{eqnarray}
Indeed, equation (\ref{fundamental}) is obtained by replacing in the Clifford algebra (\ref{clifford}) the ansatz (where $\gamma^m$ are the flat spacetime  Dirac matrices)
\begin{eqnarray}
\gamma^{\mu}=\gamma^m e_m^{\mu}.
\end{eqnarray}
Two solutions $e_{\mu}^m$ and $f_{\mu}^m$ of (\ref{fundamental}) are related by a local Lorentz transformation $\Lambda\in SO(1,3)$ (where $SO(1,3)$ is the restricted Lorentz group with antisymmetric generators) as follows
\begin{eqnarray}
f_{\mu}^m=\Lambda_k^me_{\mu}^k~,~\Lambda_k^m\eta_{mn}(\Lambda^T)_l^n=\eta_{kl}.
\end{eqnarray}
Obviously, the index $\mu$ refers to the curved coordinates $x^{\mu}$ of the manifold whereas the index $m$ refers to the flat coordinates $\xi^m$. A vector field $v$ will then have components $v^{\mu}$  in the system $x^{\mu}$ and components $v^m$ in the system $\xi^m$. They are related by means of the vielbein, i.e. $v^{\mu}=v^me_m^{\mu}$ and $v^m=v^{\mu}e_{\mu}^m$.

The covariant derivative with respect to the curved index $\mu$ is defined by means of the parallel transport with respect of the affine connection $\Gamma$ given by the Christoffel symbols, viz
\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)=v^{\mu}(x)-\Delta x^{\rho}\Gamma_{\rho\sigma}^{\mu}v^{\sigma}\iff \tilde{v}^{\mu}(x)+\Delta x^{\rho}\nabla_{\rho}\tilde{v}^{\mu}(x)=v^{\mu}(x).
\end{eqnarray}
Similarly, the covariant derivative with respect to the flat index $m$ is defined by means of the parallel transport with respect of the spin connection $\omega$, viz
\begin{eqnarray}
\tilde{v}^{m}(x+\Delta x)=v^{m}(x)-\Delta x^{\rho}\omega_{\rho n }^{m}v^{n}\iff \tilde{v}^{m}(x)+\Delta x^{\rho}\nabla_{\rho}\tilde{v}^{m}(x)=v^{m}(x).
\end{eqnarray}
Since we are dealing with the parallel transport of the same vector we must have the property that the parallel transport of the curved index from $x$ to $x+\Delta x$ followed by its the projection to a flat index equal to the projection at $x$ of the curved index to a flat index followed by the parallel transport of the flat index from $x$ to $x+\Delta x$, viz
\begin{eqnarray}
\tilde{v}^{\mu}(x+\Delta x)e_{\mu}^m(x+\Delta x)=\tilde{v}^{m}(x+\Delta x).
\end{eqnarray}
This gives immediately the so-called vielbein postulate which states that the vielbein $e_{\mu}^m$ is covariantly constant with respect to both connections affine $\Gamma$ (which acts on the curved index $\mu$) and spin $\omega$ (which acts on the flat index $m$), i.e.
\begin{eqnarray}
{\cal D}_{\rho}e_{\mu}^m&\equiv &\partial_{\rho}e_{\mu}^m-\Gamma_{\rho\mu}^{\nu}e_{\nu}^m+\omega_{\rho n}^me_{\mu}^n\nonumber\\
&=&0.
\end{eqnarray}
This equation gives us the spin connection $\omega$ (which defines the spinor bundle) in terms of the affine connection  $\Gamma$ (which defines the tangent/co-tangent bundle) or vice versa. Clearly, the spin connection encodes the local Lorentz (special coordinate transformations) invariance of the theory whereas the affine connection encodes diffeomorphism (general coordinate transformations) invariance.

Now by requiring that the length of the vector $v$ to be invariant under the parallel transport we get by using the flat and the curved indices respectively the two equivalent results that the spin connection is antisymmetric (by using the components $v^m$) and the metric is covariantly constant (by using the components $v^{\mu}$), viz
\begin{eqnarray}
\omega_{\mu}^{mn}=-\omega_{\mu}^{nm}~,~\omega_{\mu}^{mn}=\omega_{\mu k}^m\eta^{kn}.
\end{eqnarray}
\begin{eqnarray}
{\nabla}_{\rho}g_{\mu\nu}\equiv \partial_{\rho}g_{\mu\nu}-\Gamma_{\rho\mu \nu}-\Gamma_{\rho\nu \mu}~,~\Gamma_{\rho\mu \nu}=\Gamma_{\rho\mu}^{\lambda}g_{\lambda\nu}.
\end{eqnarray}
The spin connection will play the role of a local $SO(1,3)$ gauge field representing local Lorentz invariance and hence this connection must be antisymmetric by construction and as a consequence the invariance of the length and the covariant constancy of the metric follow naturally from a local symmetry principle.

The Riemann curvature tensor can be computed from the commutator of two covariant derivatives. The action of this commutator on the vielbein field should vanish identically. We have then
\begin{eqnarray}
[{\cal D}_{\rho},{\cal D}_{\sigma}]e_{\mu}^m=0.
\end{eqnarray}
From this equation we obtain the result that the Riemann curvature tensor in terms of $\Gamma$ is equal to the Riemann curvature tensor in terms of $\omega$, viz
\begin{eqnarray}
R_{\rho\sigma\mu}^{\nu}(\Gamma)e_{\nu}^{m}=R_{\rho\sigma n}^m(\omega)e_{\mu}^n.
\end{eqnarray}
Where
\begin{eqnarray}
R_{\rho\sigma\mu}^{\nu}(\Gamma)=\partial_{\rho}\Gamma_{\sigma\mu}^{\nu}+\Gamma_{\rho\tau}^{\nu}\Gamma_{\sigma\mu}^{\tau}-(\rho\leftrightarrow\sigma).
\end{eqnarray}
\begin{eqnarray}
R_{\rho\sigma n}^{m}(\omega)=\partial_{\rho}\omega_{\sigma n}^{m}+\omega_{\rho k}^{m}\omega_{\sigma n}^{k}-(\rho\leftrightarrow\sigma).
\end{eqnarray}Hence we get
\begin{eqnarray}
R_{\rho\sigma \mu\tau}(\Gamma)=R_{\rho\sigma \mu\tau}(\omega)~,~R_{\rho\sigma \mu\tau}(\Gamma)=-R_{\rho\sigma \tau\mu}(\Gamma).
\end{eqnarray}
The action of general relativity written in terms of the metric $g$  (the Hilbert-Einstein action) is therefore the same as the action written in terms of the vielbein field $e$ and the spin connection $\omega$ (the Palatini action).

Another important property satisfied by the affine connection $\Gamma$ in general relativity is torsionless (the Christoffel symbol $\Gamma_{\mu\nu}^{\rho}$ is symmetric in its two lower indices $\mu$ and $\nu$). In general torsion is generated from the vielbein field ($T=de+\omega\wedge e$) in the same way that curvature is generated from the spin connection ($R=d\omega+\omega\wedge\omega$).

Also, similarly to the fact that the curvature tensor measures the gap if one parallel transports a vector parallel to itself along a closed curved the torsion tensor measures the gap if one parallel transports one vector along another one minus the other way around.

As we have said the affine Levi-Civita connection defines the tangent bundle which is the associated vector bundle corresponding to the $O(1,3)$ bundle of orthonormal frames. In fact the affine connection is induced from the  connection  on the $O(1,3)$ bundle of orthonormal frames and since the spacetime manifold is orientable this connection can be restricted to the $SO(1,3)$ bundle of orthonormal frames and then lifted to a connection on the corresponding spinor bundle.  Thus, the spin connection is indeed more fundamental than the affine connection in every respect.

Finally, we note that spinors provide a representation of the Lorentz group $SO(1,3)$. Thus, they transform covariantly under Lorentz transformations with the covariant derivative given explicitly in terms of the spin connection $\omega$  by (with $\gamma_{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$)
\begin{eqnarray}
D_{\mu}\psi=\partial_{\mu}\psi+\frac{1}{4}\omega_{\mu}^{mn}\gamma_{mn}\psi.
\end{eqnarray}
The covariant Dirac action in a curved spacetime manifold (discovered by Wigner in $1929$) is therefore given by (with $e=\sqrt{-{\rm det} (g_{\mu\nu})}={\rm det}(e_{\mu}^m)$)
\begin{eqnarray}
{\cal L}_D=-\frac{e}{2}\bar{\psi}\gamma^{\mu}D_{\mu}\psi.
\end{eqnarray}

### The Hilbert-Einstein action

This is the first post of a series of four posts concerned with the canonical quantization of general relativity.

## Equivalence principle

The equivalence principle states that spacetime must be a manifold, i.e,  locally it must look like flat Minkowski spacetime. Thus, curvature can be cancelled by freely falling observers in the gravitational field associated with the metric and as a consequence the spacetime manifold will be seen as flat by these observes.

Indeed, by choosing the so-called normal coordinates near any point of the spacetime manifold it is seen that the trajectories of freely falling objects looks locally like straight lines. Effectively, the manifold is approximated there by its tangent vector space. The difference between the metric $g_{\mu\nu}$ and the Minkowski metric $\eta_{\mu\nu}$ vanish to first order while the difference at second order is characterized by the so-called Riemann curvature tensor $R_{\mu\alpha\nu\beta}$. Explicitly, we have
\begin{eqnarray}
g_{\mu\nu}=\eta_{\mu\nu}-\frac{1}{3}R_{\mu\alpha\nu\beta}x^{\alpha}x^{\beta}+...
\end{eqnarray}

## Review of General Relativity

We consider a Riemannian (curved) manifold ${\cal M}$ with a metric $g_{\mu\nu}$. A coordinates transformation is given by
\begin{eqnarray}
x^{\mu}\longrightarrow x^{'\mu}=x^{'\mu}(x).
\end{eqnarray}
The vectors and one-forms on the manifold are quantities which are defined to transform under the above coordinates transformation respectively as follows
\begin{eqnarray}
V^{'\mu}=\frac{\partial x^{'\mu}}{\partial x^{\nu}}V^{\nu}.\label{contr}
\end{eqnarray}
\begin{eqnarray}
V^{'}_{\mu}=\frac{\partial x^{\nu}}{\partial x^{'\mu}}V_{\nu}.\label{cov}
\end{eqnarray}
The spaces of vectors and one-forms are the tangent and co-tangent bundles.

A tensor is a quantity with multiple indices (covariant and contravariant) transforming in a similar way, i.e. any contravariant index is transforming as (\ref{contr}) and any covariant index is transforming as (\ref{cov}). For example, the metric $g_{\mu\nu}$ is a second rank symmetric tensor which transforms as
\begin{eqnarray}
g^{'}_{\mu\nu}(x^{'})=\frac{\partial x^{\alpha}}{\partial x^{'\mu}}\frac{\partial x^{\beta}}{\partial x^{'\nu}}g_{\alpha\beta}(x).
\end{eqnarray}
The interval $ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}$ is therefore invariant. In fact, all scalar quantities are invariant under coordinate transformations. For example, the volume element $d^4x\sqrt{-{\rm det}g}$ is a scalar under coordinate transformation.

The derivative of a tensor does not transform as a tensor. However, the so-called covariant derivative of a tensor will transform as a tensor. The covariant derivatives of vectors and one-forms are given by
\begin{eqnarray}
\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma_{\alpha\mu}^{\nu}V^{\alpha}.
\end{eqnarray}
\begin{eqnarray}
\nabla_{\mu}V_{\nu}=\partial_{\mu}V_{\nu}-\Gamma_{\mu\nu}^{\alpha}V_{\alpha}.
\end{eqnarray}
These transform indeed as tensors as one can easily check. Generalization to tensors is obvious. The Christoffel symbols $\Gamma_{\mu\nu}^{\alpha}$ are given in terms of the metric $g_{\mu\nu}$ by
\begin{eqnarray}
\Gamma_{\mu\nu}^{\alpha}=\frac{1}{2}g^{\alpha\beta}\big(\partial_{\mu}g_{\nu\beta}+\partial_{\nu}g_{\mu\beta}-\partial_{\beta}g_{\mu\nu}\big).
\end{eqnarray}
There exists a unique covariant derivative, and thus a unique choice of Christoffel symbols,  for which the metric is covariantly constant, viz
\begin{eqnarray}
\nabla_{\mu}g_{\alpha\beta}=0.
\end{eqnarray}
The straightest possible lines on the curved manifolds are given by the geodesics. A geodesic is a curve whose tangent vector is parallel transported along itself.  It is given explicitly by the Newton's second law on the curved manifold
\begin{eqnarray}
\frac{d^2x^{\mu}}{d\lambda}+\Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}=0.
\end{eqnarray}
The $\lambda$ is an affine parameter along the curve. The timelike geodesics define the trajectories of freely falling particles in the gravitational field encoded in the curvature of the Riemannian manifold.

The Riemann curvature tensor $R_{\mu\nu\beta}^{\alpha}$ is defined in terms of the covariant derivative by
\begin{eqnarray}
(\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})t^{\alpha}=-R_{\mu\nu\rho}^{\alpha}t^{\rho}.
\end{eqnarray}
It is given explicitly by
\begin{eqnarray}
R_{\mu\nu\rho}^{\alpha}=\partial_{\nu}\Gamma_{\mu\rho}^{\alpha}-\partial_{\rho}\Gamma_{\mu\nu}^{\alpha}+\Gamma_{\sigma\nu}^{\alpha}\Gamma_{\mu\rho}^{\sigma}-\Gamma_{\rho\sigma}^{\alpha}\Gamma_{\mu\nu}^{\sigma}.
\end{eqnarray}
We define the Ricci tensor $R_{\mu\nu}$ and the Ricci scalar $R$ by the equations
\begin{eqnarray}
R=g^{\mu\nu}R_{\mu\nu}.
\end{eqnarray}
\begin{eqnarray}
R_{\mu\nu}=R_{\mu\alpha\nu}^{\alpha}.
\end{eqnarray}
The Einstein's equations for general relativity reads (with $T_{\mu\nu}$ being the energy-momentum tensor and $G$ is the Newton's constant)
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi GT_{\mu\nu}.
\end{eqnarray}

## The Hilbert-Einstein Action

The dynamical variable is obviously the metric $g_{\mu\nu}$. The goal is to construct an action principle from which the Einstein's equations follow as the Euler-Lagrange equations of motion for the metric. This action principle will read as
\begin{eqnarray}
S=\int d^nx~{\cal L}(g).\label{dig}
\end{eqnarray}
The first problem with this way of writing is that both $d^nx$ and ${\cal L}$ are tensor densities rather than tensors. We digress briefly to explain this important different.

Let us recall the familiar Levi-Civita symbol in $n$ dimensions defined by
\begin{eqnarray}
\tilde{\epsilon}_{\mu_1...\mu_n}&=&+1~{\rm even}~{\rm permutation}\nonumber\\
&=&-1~{\rm odd}~~{\rm permutation}\nonumber\\
&=&~~0~{\rm otherwise}.
\end{eqnarray}
This is a symbol and not a tensor since it does not change under coordinate transformations. The determinant of a matrix $M$ can be given by the formula
\begin{eqnarray}
\tilde{\epsilon}_{\nu_1^{}...\nu_n^{}}{\rm det} M&=&\tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}M^{\mu_1^{}}~_{\nu_1^{}}...M^{\mu_n^{}}~_{\nu_n^{}}.
\end{eqnarray}
By choosing $M^{\mu}~_{\nu^{}}=\partial x^{\mu}/\partial y^{\nu^{}}$ we get the transformation law
\begin{eqnarray}
\tilde{\epsilon}_{\nu_1^{}...\nu_n^{}}&=&{\rm det}\frac{\partial y^{}}{\partial x}~ \tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}\frac{\partial x^{\mu_1}}{\partial y^{\nu_1^{}}}...\frac{\partial x^{\mu_n}}{\partial y^{\nu_n^{}}}.
\end{eqnarray}
In other words $\tilde{\epsilon}_{\mu_1^{}...\mu_n^{}}$ is not a tensor because of the determinant appearing in this equation. This is an example of a tensor density. Another example of a tensor density is ${\rm det} g$. Indeed from the tensor transformation law of the metric $g^{'}_{\alpha\beta}=g_{\mu\nu}(\partial x^{\mu}/\partial y^{\alpha})( \partial x^{\nu}/\partial y^{\beta})$ we can show in a straightforward way that
\begin{eqnarray}
{\rm det} g^{'}&=&({\rm det}\frac{\partial y^{}}{\partial x})^{-2}~ {\rm det} g.
\end{eqnarray}
The actual Levi-Civita tensor can then be defined by
\begin{eqnarray}
\epsilon_{\mu_1...\mu_n}=\sqrt{{\rm det} g^{}}~\tilde{\epsilon}_{\mu_1...\mu_n}.
\end{eqnarray}
Next under a coordinate transformation $x\longrightarrow y$ the volume element transforms as
\begin{eqnarray}
d^nx\longrightarrow d^ny={\rm det}\frac{\partial y}{\partial x}~d^nx.\label{ele}
\end{eqnarray}
In other words the volume element transforms as a tensor density and not as a tensor. We verify this important point in our language as follows. We write
\begin{eqnarray}
d^nx&=&dx^0\wedge dx^1\wedge ...\wedge dx^{n-1}\nonumber\\
&=&\frac{1}{n!}\tilde{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}.\label{sdf}
\end{eqnarray}
Recall that a differential $p-$form is a $(0,p)$ tensor which is completely antisymmetric. For example scalars are $0-$forms and dual cotangent vectors are $1-$forms. The  Levi-Civita tensor $\epsilon_{\mu_1...\mu_n}$ is a $4-$form. The differentials $dx^{\mu}$ appearing in the second line of equation  (\ref{sdf}) are $1-$forms and hence under a coordinate transformation $x\longrightarrow y$ we have $dx^{\mu}\longrightarrow dy^{\mu}=dx^{\nu}\partial y^{\mu}/\partial x^{\nu}$.  By using this  transformation law we can immediately show that $dx^n$ transforms to $d^ny$ exactly as in equation (\ref{ele}).

It is not difficult to see now that an invariant volume element can be given by the $n-$form defined by the equation
\begin{eqnarray}
dV=\sqrt{{\rm det} g}~ d^nx.
\end{eqnarray}
We can show that
\begin{eqnarray}
dV&=&\frac{1}{n!}\sqrt{{\rm det} g}~\tilde{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}\nonumber\\
&=&\frac{1}{n!}{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\wedge ...\wedge dx^{\mu_n}\nonumber\\
&=&{\epsilon}_{\mu_1...\mu_n}dx^{\mu_1}\otimes ...\otimes dx^{\mu_n}\nonumber\\
&=&\epsilon.
\end{eqnarray}
In other words the  invariant volume element is precisely the  Levi-Civita tensor. In the case of Lorentzian signature we replace ${\rm det} g$ with  $-{\rm det} g$.

We go back now to equation (\ref{dig}) and rewrite it as
\begin{eqnarray}
S&=&\int d^nx~{\cal L}(g)\nonumber\\
&=&\int d^n x\sqrt{-{\rm det} g}~\hat{\cal L}(g).
\end{eqnarray}
Clearly ${\cal L}=\sqrt{-{\rm det} g}~\hat{\cal L}$. Since the invariant volume element $d^n x\sqrt{-{\rm det} g}$ is a scalar the function $\hat{\cal L}$ must also be a scalar and as such can be identified with the Lagrangian density.

We use the result that the only independent scalar quantity which is constructed from the metric and which is at most second order in its derivatives is the Ricci scalar $R$. In other words the simplest choice for the Lagrangian density $\hat{\cal L}$ is
\begin{eqnarray}
\hat{\cal L}(g)=R.
\end{eqnarray}
The corresponding action is called the Hilbert-Einstein action. We compute
\begin{eqnarray}
\delta S
&=&\int d^n x\delta \sqrt{-{\rm det} g}~g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}.\nonumber\\
\end{eqnarray}
We have
\begin{eqnarray}
\delta R_{\mu\nu}&=&\delta R_{\mu\rho\nu}~^{\rho}\nonumber\\
&=&\partial_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\partial_{\mu}\delta\Gamma^{\rho}~_{\rho\nu}+\delta (\Gamma^{\lambda}~_{\mu\nu}\Gamma^{\rho}~_{\rho\lambda}-\Gamma^{\lambda}~_{\rho\nu}\Gamma^{\rho}~_{\mu\lambda})\nonumber\\
&=&(\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\Gamma^{\rho}~_{\rho\lambda}\delta\Gamma^{\lambda}~_{\mu\nu}+\Gamma^{\lambda}~_{\rho\mu}\delta\Gamma^{\rho}~_{\lambda\nu}+\Gamma^{\lambda}~_{\rho\nu}\delta \Gamma^{\rho}~_{\lambda\mu})-(\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}-\Gamma^{\rho}~_{\mu\lambda}\delta\Gamma^{\lambda}~_{\rho\nu}+\Gamma^{\lambda}~_{\mu\rho}\delta\Gamma^{\rho}~_{\lambda\nu}\nonumber\\
&+&\Gamma^{\lambda}~_{\mu\nu}\delta \Gamma^{\rho}~_{\rho\lambda})+\delta (\Gamma^{\lambda}~_{\mu\nu}\Gamma^{\rho}~_{\rho\lambda}-\Gamma^{\lambda}~_{\rho\nu}\Gamma^{\rho}~_{\mu\lambda})\nonumber\\
&=&\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}.
\end{eqnarray}
In the second line of the above equation we have used the fact that $\delta \Gamma^{\rho}~_{\mu\nu}$ is a tensor since it is the difference of two connections. Thus

\begin{eqnarray}
\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}&=&\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\bigg(\nabla_{\rho}\delta \Gamma^{\rho}~_{\mu\nu}-\nabla_{\mu}\delta \Gamma^{\rho}~_{\rho\nu}\bigg)\nonumber\\
&=&\int d^n x \sqrt{-{\rm det} g}~\nabla_{\rho}\bigg(g^{\mu\nu}\delta \Gamma^{\rho}~_{\mu\nu}-g^{\rho\nu}\delta \Gamma^{\mu}~_{\mu\nu}\bigg).
\end{eqnarray}
We compute also (with $\delta g_{\mu\nu}=-g_{\mu\alpha}g_{\nu\beta}\delta g^{\alpha \beta}$)
\begin{eqnarray}
\delta\Gamma^{\rho}~_{\mu\nu}&=&\frac{1}{2}g^{\rho\lambda}\bigg(\nabla_{\mu}\delta g_{\nu\lambda}+\nabla_{\nu}\delta g_{\mu\lambda}-\nabla_{\lambda}\delta g_{\mu\nu}\bigg)\nonumber\\
&=&-\frac{1}{2}\bigg(g_{\nu\lambda}\nabla_{\mu}\delta g^{\lambda\rho}+g_{\mu\lambda}\nabla_{\nu}\delta g^{\lambda\rho}-g_{\mu\alpha}g_{\nu\beta}\nabla^{\rho}\delta g^{\alpha\beta}\bigg).
\end{eqnarray}
Thus
\begin{eqnarray}
\int d^n x \sqrt{-{\rm det} g}~g^{\mu\nu}\delta R_{\mu\nu}
&=&\int d^n x \sqrt{-{\rm det} g}~\nabla_{\rho}\bigg(g_{\mu\nu}\nabla^{\rho}\delta g^{\mu\nu}-\nabla_{\mu}\delta g^{\mu\rho}\bigg).
\end{eqnarray}
By Stokes's theorem this integral is equal to the integral over the boundary of spacetime of the expression $g_{\mu\nu}\nabla^{\rho}\delta g^{\mu\nu}-\nabla_{\mu}\delta g^{\mu\rho}$ which is $0$ if we assume that the metric and its first derivatives are held fixed on the boundary. The variation of the action reduces to
\begin{eqnarray}
\delta S
&=&\int d^n x\delta \sqrt{-{\rm det} g}~g^{\mu\nu}R_{\mu\nu}+\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}R_{\mu\nu}.
\end{eqnarray}
Next we use the result
\begin{eqnarray}
\delta \sqrt{-{\rm det} g}=-\frac{1}{2}\sqrt{-{\rm det} g}~ g_{\mu\nu}\delta g^{\mu\nu}.
\end{eqnarray}
Hence
\begin{eqnarray}
\delta S
&=&\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R).
\end{eqnarray}
This will obviously lead to Einstein's equations in vacuum  which is partially our goal. We want also to include the effect of matter  which requires considering the more general actions of the form
\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^nx~\sqrt{-{\rm det}g}~R+S_M.\label{HE}
\end{eqnarray}
\begin{eqnarray}
S_M=\int d^nx~\sqrt{-{\rm det}g}~\hat{\cal L}_M.
\end{eqnarray}
The variation of the action becomes
\begin{eqnarray}
\delta S
&=&\frac{1}{16\pi G}\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\delta S_M\nonumber\\
&=&\int d^n x \sqrt{-{\rm det} g}~\delta g^{\mu\nu}\bigg[\frac{1}{16\pi G}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}} \bigg].
\end{eqnarray}
In other words
\begin{eqnarray}
\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S}{\delta g^{\mu\nu}} &=&\frac{1}{16\pi G}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)+\frac{1}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.
\end{eqnarray}
Einstein's equations are therefore given by
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G T_{\mu\nu}.
\end{eqnarray}
The stress-energy-momentum tensor must therefore be defined by the equation
\begin{eqnarray}
T_{\mu\nu}=-\frac{2}{\sqrt{-{\rm det} g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.
\end{eqnarray}
As a first example we consider the action of a scalar field in curved spacetime given by
\begin{eqnarray}
S_{\phi}=\int d^nx \sqrt{-{\rm det} g}~\bigg[-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-V(\phi)\bigg].
\end{eqnarray}
The corresponding stress-energy-momentum tensor is calculated to be given by
\begin{eqnarray}
T_{\mu\nu}^{(\phi)}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\nabla_{\rho}\phi\nabla_{\sigma}\phi-g_{\mu\nu}V(\phi).
\end{eqnarray}
As a second example we consider the action of the  electromagnetic field in curved spacetime given by
\begin{eqnarray}
S_{A}=\int d^nx \sqrt{-{\rm det} g}~\bigg[-\frac{1}{4}g^{\mu\nu}g^{\alpha\beta}F_{\mu\nu}F_{\alpha\beta}\bigg].
\end{eqnarray}
In this case the  stress-energy-momentum tensor is calculated to be given by
\begin{eqnarray}
T_{\mu\nu}^{(A)}=F^{\mu\lambda}F^{\nu}~_{\lambda}-\frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}.
\end{eqnarray}
The cosmological constant is one of the simplest matter action that one can add to the Hilbert-Einstein action. It is given by
\begin{eqnarray}
S_{\rm cc}=-\frac{1}{8\pi G}\int d^4x\sqrt{-{\rm det}g}\Lambda.
\end{eqnarray}
In this case the energy-momentum tensor and the Einstein equations read
\begin{eqnarray}
T_{\mu\nu}=-\frac{\Lambda}{8\pi G}g_{\mu\nu}.
\end{eqnarray}
\begin{eqnarray}
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\Lambda g_{\mu\nu}=0.
\end{eqnarray}

### On the canonical quantization of general relativity

General relativity can be formulated either using the metric tensor $g_{\mu\nu}$ (and implicitly the affine Levi-Civita connection defined in terms of Christoffel symbols $\Gamma_{\mu\nu}^{\rho}$) or in terms of the  vielbein field or tetrad $e_{\mu}^m$ and the spin connection $\omega_{\mu}^{mn}$. The two formulations  are equivalent but contrary to widespread belief the formulation based on the spin connection is more fundamental for two reasons. First, fundamental matter fields in Nature are described by chiral fermions (leptons and quarks) which when propagating in a curved spacetime feels the metric only through the vielbein field.  The second reason is the fact that the most successful canonical quantization of general relativity today (loop quantum gravity) uses as canonical conjugate variables the (densitized) vielbein field from one hand and the (self-dual part of the) spin connection from the other hand as the canonical momentum and the configuration variable respectively (and not the $3-$dimensional metric $h_{ij}$ or euivalently the vielbein field or triads $e_{i}^m$ and the extrinsic curvature $K_{ij}$ used in the classic ADM formulation). It will be seen that general relativity can then be reformulated as a complex $SU(2)$ gauge theory of the self-dual spin connections and as a consequence general relativity becomes a dynamical theory for three-dimensional connections and not for three-dimensional geometries and this makes it embeddable into Yang-Mills gauge theory.

## The Palatini action and Ashtekar variables

The Hilbert-Einstein action of general relativity expressed in terms of the $4-$dimensional metric $g$ is equivalent to the Palatini action expressed in terms of the vielbein field $e$ and the spin connection $\omega$. In the first formulation the affine connection is used implicitly since it is determined by the metric tensor whereas in the second formulation the spin connection is used explicitly since it is an independent dynamical variable. The Palatini action is of the form ($G$ being Newton's constant)
\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^4x\epsilon_{mnkl}\tilde{\eta}^{\mu\nu\alpha\beta}e_{\mu}^me_{\nu}^nR_{\alpha\beta}^{kl}.\label{pal1}
\end{eqnarray}
The indices $m,n,...$ are internal indices associated with the local $SO(1,3)$ Lorentz group whereas the indices $\mu,\nu,...$ are external indices associated with spacetime (and consequently with the local diffeomorphism group of general coordinate transformations).  The tensor $\tilde{\eta}$ is the Levi-Civita tensor density corresponding to the curved indices $\mu,\nu,...$ whereas $\epsilon$ is the flat Levi-Civita symbol. The curvature (from the previous results) is defined by the relation
\begin{eqnarray}
R_{\rho\sigma\mu\nu}=R_{\rho\sigma n}^me_{\mu}^ne_m^{\alpha}g_{\alpha\nu}.
\end{eqnarray}
Now we use the result (with $g={\rm det}(g_{\mu\nu})$)
\begin{eqnarray}
\tilde{\eta}^{\alpha\beta\mu\nu}\epsilon_{klmn}e_{\mu}^me_{\nu}^n=\frac{\sqrt{-g}}{2}(e_k^{\alpha}e_l^{\beta}-e_k^{\beta}e_l^{\alpha}).
\end{eqnarray}
The Palatini action becomes  then (recall that the Ricci curvature tensor and the Ricci scalar are defined by $R_{\mu\nu}=R^{\rho}_{\mu\rho\nu}$ and $R=g^{\mu\nu}R_{\mu\nu}$)

\begin{eqnarray}
S&=&\frac{1}{16\pi G}\int \sqrt{-g} d^4x e_k^{\alpha}e_l^{\beta}R_{\alpha\beta}^{kl}\nonumber\\
&=&\frac{1}{16\pi G}\int \sqrt{-g} d^4x R.
\end{eqnarray}
This is the Hilbert-Einstein action.

Alternatively, quantization based on the Palatini action gives immediately geometrodynamics of Wheeler, DeWitt and others. Indeed, the conjugate momentum associated with the spin connection $\omega_{\mu}^{mn}$ is found to be given by $\Pi^{\mu}_{mn}=\tilde{\eta}^{\mu\nu\alpha}\epsilon_{mnkl}e_{\nu}^ke_{\alpha}^l$. The theory has thus an additional  (second class) constraint consisting in the fact that the momentum is decomposable as a product of two  vielbein fields. By solving the second class constraint  (using Dirac's formalism) we obtain new canonical variables in which the spin connection is lost as a dynamical variable and we end up again with geometrodynamics \cite{Ashtekar:1988sw}.

The revolutionary solution  provided by Ashtekar (see \cite{Ashtekar} for a modern review and for the original references) consists in insisting that the Palatini action is the correct starting point but with the additional twist that the spin connection must be self-dual. In other words, we must replace in the Palatini action the real $SO(1,3)$ spin connection $\omega_{\mu}^{mn}$ by the complex  self-dual connection $A_{\mu}^{mn}$defined by
\begin{eqnarray}
A_{\mu}^{mn}=\frac{1}{2G}(\omega_{\mu}^{mn}-\frac{i}{2}\epsilon^{mn}~_{kl}\omega_{\mu}^{kl}).
\end{eqnarray}
The complex connection $A_{\mu}^{mn}$ is self-dual because it satisfies the self-dual condition

\begin{eqnarray}
iA_{\mu}^{mn}=\frac{1}{2}\epsilon^{mn}~_{kl}A_{\mu}^{kl}.\label{sdc}
\end{eqnarray}The Palatini action becomes

\begin{eqnarray}
S=\frac{1}{16\pi G}\int d^4x\epsilon_{mnkl}\tilde{\eta}^{\mu\nu\alpha\beta}e_{\mu}^me_{\nu}^nF_{\alpha\beta}^{kl}.\label{pal2}
\end{eqnarray}
$F$ is the curvature tensor of the self-dual connection $A$. Thus, it must be given by
\begin{eqnarray}
F_{\alpha\beta m}^{n}=\partial_{\alpha}A_{\beta m}^n-\partial_{\beta}A_{\alpha m}^n+G^4A_{\alpha m}^kA_{\beta k}^n-G^4A_{\beta m}^kA_{\alpha k}^n.
\end{eqnarray}
We can check that the classical equations of motion derived from the self-dual Palatini action (\ref{pal2}) are exactly equivalent to the classical equations of motion derived from the original Palatini action (\ref{pal1}). In particular, the variation of the action (\ref{pal2}) with respect to the connection $A_{\mu}^{mn}$ gives as equation of motion the result that $A_{\mu}^{mn}$ is the (self-dual part of the) spin connection $\omega_{\mu}^{mn}$ which is compatible with the vielbein field $e_{\mu}^m$, i.e. it is determined  by  the condition ${\cal D}_{\rho}e_{\mu}^m=0$. Hence,  the connection  $A_{\mu}^{mn}$ is completely determined by the vielbein field $e_{\mu}^m$. On the other hand, the variation of the action (\ref{pal2}) with respect to the  vielbein field $e_{\mu}^m$ gives as equation of motion the result that the spacetime metric $g_{\mu\nu}=e_{\mu}^me_{\nu}^{n}\eta_{mn}$ solves Einstein's equations.

Thus the classical equations of motion derived from the self-dual Palatini action (\ref{pal2}) are exactly equivalent to the classical equations of motion derived from the standard Palatini action (\ref{pal1}). But this does not mean that the two actions are identical. Indeed, the difference between (\ref{pal2}) and (\ref{pal1}) is an imaginary term which is not a pure divergence but reproduces as a correction to the equation of motion the first Bianchi identity (the trace of the dual of the Riemann tensor vanishes) which thus holds automatically. This imaginary term leads however under the Legendre transform of the self-dual Palatini action to a different conjugate momentum which is linear instead of being quadratic in the vielbein field and that makes the self-dual Palatini action (\ref{pal2}) distinctly different and quite superior to the standard Palatini action (\ref{pal1}) which is nothing else but the Hilbert-Einstein action. See \cite{Ashtekar} and references therein.

## General relativity as an SU(2) gauge theory of self-dual spin connections

We start with the Hilbert-Einstein action  with variable given only by the metric tensor $g_{\mu\nu}$ (the affine Levi-Civita connection $\Gamma_{\mu\nu}^{\rho}$ is not an independent variable here). Then by performing a Legendre transform and an ADM analysis the canonically conjugate variables are from the one hand the three-dimensional metric $h_{\mu\nu}$ or equivalently the vielbein fields (or triads) $e_{\mu}^{m}$ and from the other hand we have the corresponding conjugate momentum $\Pi_{\mu\nu}$ defined in terms of the extrinsic curvature $K_{\mu\nu}$ by the relation $\Pi_{\mu\nu}=-\sqrt{h}(K_{\mu\nu}-Kh_{\mu\nu})$.  We are here only dealing with geometrodynamics where only first class constraints are involved.

If we start on the other hand with the standard Palatini action (\ref{pal1}) with variables given by the real vielbein field   $e_{\mu}^m$ and the real spin connection $\omega_{\mu}^{mn}$ then the canonically conjugate variables are the spin connection $\omega_{\mu}^{mn}$ and the conjugate momentum  $\Pi^{\mu}_{mn}=\tilde{\eta}^{\mu\nu\alpha}\epsilon_{mnkl}e_{\nu}^ke_{\alpha}^l$. By solving the second class constraint  (the momentum is decomposable as a product of two  vielbein fields) we obtain new canonical variables in which the spin connection is lost as a dynamical variable and we end up again with geometrodynamics.

We also recall that the vielbein field  $e_{\mu}^m$ is covariantly constant with respect to both the spin connection $\omega_{\mu}^{mn}$ and the Levi-Civita connection $\Gamma_{\mu\nu}^{\alpha}$, viz
\begin{eqnarray}
{\cal D}_{\rho}e_{\mu}^m&=&{\partial}_{\rho}e_{\mu}^m-{\Gamma}_{\rho\mu}^{\nu}e_{\nu}^m+\omega_{\rho n}^{m}e_{\mu}^n\nonumber\\
&=& 0.
\end{eqnarray}
This compatibility condition gives the spin connection in terms of the Levi-Civita connection and the  vielbein field.

However, the theory formulated in terms of the Ashtekar variables given by a self-dual spin connection $A_{\mu}^{mn}$ which is necessarily complex and a real vielbein field  (or tetrads) $e_{\mu}^m$ with an action given by the self-dual Palatini action (\ref{pal2}) is equivalent to a complex $SU(2)$ gauge theory of the self-dual spin connections . Indeed, after Legendre transform the canonically conjugate variables are found to be the self-dual spin connection $A_{\mu}^{mn}$ with a corresponding conjugate momentum  $\Pi^{\mu}_{mn}$ which is also self-dual and furthermore is proportional to a single vielbein field not two and hence second class constraints are avoided.

Here it is technically simpler to start with complex general relativity since the connection is necessarily complex. So we start with a complex vielbein field $e_{\mu}^m$ and a complex self-dual $SO(1,3)$ connection $A_{\mu}^{mn}$ with an action given by the Palatini action (\ref{pal2}). After Legendre transform we get as our canonically conjugate variables the connection $A_{\mu}^{mn}$ and the conjugate momentum $\Pi^{\mu}_{mn}$ which are both in the self-dual part of the complexified $so(1,3)$ Lie algebra.

The original spin connection $\omega_{\mu}^{mn}$ is a real $SO(1,3)$ connection and recall that $SL(2,\mathbb{C})$ is the universal cover of $SO(1,3)$. The self-dual connection $A_{\mu}^{mn}$  belongs however to the complexified group $SO(1,3)_{\mathbb{C}}$.  We have the Lie algebra isomorphisms
\begin{eqnarray}
so(1,3)_{\mathbb{C}}=so(4)_{\mathbb{C}}=so(3)_{\mathbb{C}}\oplus so(3)_{\mathbb{C}}.
\end{eqnarray}
The first $so(3)_{\mathbb{C}}$ factor represents self-dual (chiral, right-handed) fields whereas the second factor represents anti-self-dual (anti-chiral, left-handed) fields. The connection $A_{\mu}^{mn}$ is a complex  connection (thus belonging to $so(1,3)_{\mathbb{C}}$ not to  $so(1,3)$ like the connection $\omega_{\mu}^{mn}$)  which is also self-dual (thus it belongs to the first factor $so(3)_{\mathbb{C}}$).

We are therefore dealing with an $so(3)_{\mathbb{C}}-$valued one-form and since the universal cover of $SO(3)$ is $SU(2)$ the connection $A_{\mu}^{mn}$ is in fact an $su(2)_{\mathbb{C}}-$valued one-form.

Using the above isomorphism between the self-dual subalgebra of the complexified Lie algebra $so(1,3)_{\mathbb{C}}$ and the complexfied Lie algebra $so(3)_{\mathbb{C}}$ we can map the  canonically conjugate variables $A_{\mu}^{mn}$ and $\Pi^{\mu}_{mn}$ to the $so(3)_{\mathbb{C}}-$valued fields $A_{\mu}^n$ and $\Pi_{\mu}^n$ given respectively by

\begin{eqnarray}
A_{\mu}^m=\frac{1}{2}A_{\mu kl}\epsilon^{klm}~,~\Pi_{\mu}^m=\frac{1}{2}\Pi_{\mu kl}\epsilon^{klm}.
\end{eqnarray}
The self-dual connection $A_{\mu}^m$ is also called the chiral spin connection. As it turns out, the canonical momentum $\Pi_{\mu}^m$ is precisely the  densitized vielbein field  (or triad) given by
\begin{eqnarray}
\Pi_{\mu}^m=\tilde{e}_{\mu}^m=\sqrt{h}e_{\mu}^m.
\end{eqnarray}
The Ashtekar variables are precisley the densitized triad $\tilde{e}_{\mu}^m$ and the self-dual connection $A_{\mu}^m$.  The self-dual Palatini action in terms of these variables takes the form
\begin{eqnarray}
S=\int d^4x \bigg(-2i\tilde{e}_{\mu}^m{\cal L}_tA_m^{\mu}-2i(t^{\mu}A_{\mu}^m)G_m+2iN^{\mu}{\cal V}_{\mu}+\frac{N}{\sqrt{h}}{\cal S}\bigg). \label{pal3}
\end{eqnarray}
The quantities $G_m$, ${\cal V}_{\mu}$ and ${\cal S}$ are explicitly given by
\begin{eqnarray}
G_m={\cal D}_{\mu}\tilde{e}_m^{\mu}~,~{\cal V}_{\mu}=\tilde{e}_n^{\nu}F_{\mu\nu}^n~,~{\cal S}=\epsilon_{ijk}\tilde{e}_i^{\mu}\tilde{e}_j^{\nu}F_{\mu\nu}^k.
\end{eqnarray}
The curvature $F_{\mu\nu}^k$ of the gauge field $A_{\mu}^k$ is explicitly given by
\begin{eqnarray}
F_{\mu\nu}^l=\partial_{\mu}A_{\nu}^l-\partial_{\nu}A_{\mu}^l+G\epsilon_{lmn}A_{\mu}^mA_{\nu}^n.
\end{eqnarray}
This shows explicitly that we are indeed dealing with an $SU(2)$ gauge theory.

In the first term of the self-dual Palatini action (\ref{pal3}) the operator ${\cal L}_t$ is the Lie derivative along the time direction and hence ${\cal L}_tA_m^{\mu}$ is the covariant time derivative of the field configuration $A_m^{\mu}$ along the vector field $t^{\mu}=Nn^{\mu}+N^{\mu}$ which defines the spacetime foliation with hypersurfaces $\Sigma_t$ whose normal vector field is given by $n^{\mu}$ ($N$ and $N^{\mu}$ are then the lapse function and the shift vector).

From the first term in the action (\ref{pal3}) which is then of the form $p\dot{q}$ we can immediately conclude that the densitized triad $\tilde{e}_{\mu}^m$ is precisely the conjugate momentum $p$ associated with the self-dual connection $A_{\mu}^m$ which acts exactly as the configuration variable $q$. Indeed, the fundamental Poisson brackets are of the form
\begin{eqnarray}
\{A_{\mu}^m(x), \tilde{e}^{\nu}_n(y)\}=\frac{i}{2}\delta_m^n\delta_{\mu}^{\nu}\delta^3(x-y).
\end{eqnarray}
In summary, we have gone from the ADM variables consisting of the three-dimensional metric $h_{\mu\nu}$ (or equivalently the densitized triads $\tilde{e}_{\mu}^m$)  and the canonical momentum $\Pi_{\mu\nu}$ (or equivalently the extrinsic curvature $K_{\mu}^m$ defined by $K_{\mu}^m=K_{\mu\nu}e^{\nu m}$) to the complex Ashtekar variables consisting of $\tilde{e}_{\mu}^m$ and the connection $A_{\mu}^m$. The relation between the self-dual connection $A_{\mu}^m$ and the original variables $\tilde{e}_{\mu}^m$ and $K_{\mu}^m$ is given explicitly by
\begin{eqnarray}
GA_{\mu}^m=\Gamma_{\mu}^m-iK_{\mu}^m.\label{fundamental}
\end{eqnarray}
The spin connection $\Gamma_{\mu}^m$ which is compatible with the densitized triads $\tilde{e}_{\mu}^m$ is given obviously by the relation
\begin{eqnarray}
\Gamma_{\mu}^m=\frac{1}{2}\omega_{\mu kl}\epsilon^{klm}.
\end{eqnarray}
See  \cite{Pullin:1993fw} and references therein.

At the end of all this we will naturally need to return to real (Lorentzian) general relativity and thus one must impose reality conditions. In terms of the geometrodynamic variables these reality conditions are simply the requirements that the three-dimensional metric $h_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}$ and the  extrinsic curvature $K_{\mu\nu}$ must be real. Let us emphasize here that the self-dual spin connection $A_{\mu}^{mn}$ given by equation (\ref{sdc}) is necessarily complex (since the spacetime manifold is Lorentzian) and thus the reality conditions will not alter this fact. But, in Euclidean signature the self-dual connections are necessarily real and thus the reality conditions which are needed to be imposed on complex general relativity to recover the real phase space are the requirements  that the triads must be real and the connections must also be real. In Lorentzian signature the connections will remain complex after imposing the reality conditions (which will cause other problems for the integration measure in the quantum theory).

The reality conditions can also be understood in a more illuminating way as follows.

We start with real general relativity, i.e. real  vielbein field and real spin connection in the Palatini action. After Legendre transform we can take as our variables the densitized triads $\tilde{e}_{\mu}^m$ (instead of the three-dimensional metric $h_{\mu\nu}$) and the extrinsic curvature $e_{\mu}^m$ (instead of the momentum $\Pi_{\mu\nu}$).

On the real phase space $(q,p)\equiv (\tilde{e}_{\mu}^m,K_{\mu}^m)$ we perform then a complex canonical transformation which takes us to the complex Ashtekar variables $(q,z)$ where $z$ is the complex coordinate given by $z=f(q)-ip$. Explicitly, $f(q)$ is the spin connection $\Gamma_{\mu}^m$ which is determined by the densitized triads $\tilde{e}_{\mu}^m$ and $z$ is  precisely the self-dual connection $GA_{\mu}^m=\Gamma_{\mu}^m-iK_{\mu}^m$. As we have seen $\tilde{e}_{\mu}^m$ and $A_{\mu}^m$ are canonically conjugate to each other where the densitized triads are what play the role of the conjugate momentum in the  Ashtekar variables $(q,z)$ contrary to their role in the original real coordinates $(q,p)$, i.e.  $(q^A,p^A)\equiv (A_{\mu}^m, \tilde{e}_{\mu}^m)$.

The reality conditions are now given by the requirement that the three-dimensional metric  $h_{\mu\nu}=e_{\mu}^me_{\nu}^n\eta_{mn}$ is real and the requirement that $GA_{\mu}^m-\Gamma_{\mu}^m$ is pure imaginary.

In the Palatini action (\ref{pal3}) which is written in terms of Ashtekar variables the second, third and fourth terms lead to the constraints. Indeed, the lpase function $N$, the shift vector $N^{\mu}$ and the component of the connection $A_{\mu}$ along the time direction, i.e. $t^{\mu}A_{\mu}^m$ are all Legendre multipliers and the variation of the action with respect to them will lead to the constraints

\begin{eqnarray}
G_m={\cal D}_{\mu}\tilde{e}_m^{\mu}=0.\label{const1}
\end{eqnarray}
\begin{eqnarray}
{\cal V}_{\mu}=\tilde{e}_n^{\nu}F_{\mu\nu}^n=0.\label{const2}
\end{eqnarray}
\begin{eqnarray}
{\cal S}=\epsilon_{ijk}\tilde{e}_i^{\mu}\tilde{e}_j^{\nu}F_{\mu\nu}^k=0. \label{const3}
\end{eqnarray}
These seven first class constraints are simple polynomials in  the basic variables (as opposed to what happens in geometrodynamics).  And, they reduce the $9$ degrees of freedom of $A_{\mu}^i$ to the two degrees of freedom of the graviton.

The constraints (\ref{const2}) and (\ref{const3}) are the diffeomorphism and Hamiltonian constraints found in geometrodynamics which generate respectively spatial diffeomorphisms on each surface $\Sigma_t$ and time evolution between different surfaces $\Sigma_t$ and $\Sigma_{t+\delta t}$.

The first constraint (\ref{const1}) is the so-called Gauss constraint and it represents Gauss law in this gauge theory and  generates local $SO(3)$ invariance of the triads. It arises from the fact that the time component of the connection $A_{\mu}$ is not a dynamical field.  We are really dealing with an $SU(2)$ gauge theory on the the three-dimensional surfaces $\Sigma_t$ with gauge field $A_{\mu}^m$ and since $\tilde{e}^{\mu}_m$ is the conjugate momentum it will act as the electric field $E^{\mu}_m$ with a quantized flux leading to quantized geometry and discretized spacetime  (in the form of discrete spectra of areas and volumes) and also leads to the absence of gravitational singularities.

Thus, the Gauss constraint and the diffeomorphism constraints generate the local invariance group which is the semi-direct product of the local $SO(3)$ rotation group of the triads and the spatial diffeomorphism group on $\Sigma_t$. On the other hand, the scalar constraint is of the form $G^{\alpha\beta}p_{\alpha}p_{\beta}=0$ where $G$ is the supermetric and thus this constraint generates null geodesics motion in the configuration space of the connection.

Similarly to the constraints, the Hamiltonian and the equations of motion are all low order polynomials in the basic variables of Ashtekar.

## The real SU(2) gauge theory

The self-dual or chiral connection $A_{\mu}^m$ in Lorentzian signature  is a complex $SU(2)$ gauge field which means in particular that the corresponding holonomies or Wilson loops (which define the obeservables of the quantum gauge theory) are non-compact, i.e. they belong to a non-compact subgroup $SL(2,\mathbb{C})_{\rm sd}$ of $SL(2,\mathbb{C})$ generated by the self-dual part of the Lie algebra $sl(2,\mathbb{C})$. As a consequence the path integrals defining the quantum theory are ill defined and require a regularization in the form of a Wick rotation in the internal space which sends the non-compact group $SL(2,\mathbb{C})_{\rm sd}$ to the compact group $SU(2)$, i.e. the chiral connection $A_{\mu}^m$ given by (\ref{fundamental}) is replaced with a real $SU(2)$ gauge field given by

\begin{eqnarray}
GA_{\mu}^m=\Gamma_{\mu}^m+\beta K_{\mu}^m.\label{fundamental1}
\end{eqnarray}
In other words, we replace the "-i" in (\ref{fundamental})  with a real parameter $\beta$ in (\ref{fundamental1}) called the Barbero-Immirzi parameter \cite{Barbero:1994ap,Immirzi:1996di}. The fundamental Poisson brackets become
\begin{eqnarray}
\{A_{\mu}^m(x), \tilde{e}^{\nu}_n(y)\}=-\frac{\beta}{2}\delta_m^n\delta_{\mu}^{\nu}\delta^3(x-y).
\end{eqnarray}
The choice of the parameter does not alter the Gauss and  the spatial diffeomorphism constraints. But the Hamiltonian (which is a linear combination of the constraints) acquires an additional term, viz
\begin{eqnarray}
H=\frac{\epsilon_{lmn}\tilde{e}^{\mu}_m\tilde{e}^{\nu}_nF_{\mu\nu}^l}{\sqrt{h}}+2\frac{\beta^2+1}{\beta^2}\frac{\tilde{e}^{\mu}_m\tilde{e}^{\nu}_n-\tilde{e}^{\mu}_n\tilde{e}^{\nu}_m}{\sqrt{h}}(A_{\mu}^m-\Gamma_{\mu}^m)(A_{\nu}^n-\Gamma_{\nu}^n).
\end{eqnarray}
The second term vanishes for $\beta=\pm 1$. This Hamiltonian was simplified by Thiemann. See for example his book \cite{Thiemann:2007zz}.

## Loop representation and spin networks

We have now a real $SU(2)$ gauge theory with a connection or gauge field $A_{\mu}^m$ living on a three-dimensional surface $\Sigma_t$. The classical configuration space (the space of all connections   $A_{\mu}^m$ ) is denoted by ${\cal A}$ while the quantum configuration space is denoted by $\bar{\cal A}$ which is an extension of ${\cal A}$.

Each connection $A_{\mu}^m$ defines a holonomy $h_{\alpha}[A]$ along any oriented path on the surface $\Sigma_t$, i.e. $\alpha: [s_0,s_1]=[0,1]\longrightarrow \Sigma_t$ with an affine parameter $s$ by the relation \cite{Loll:1993yz}
\begin{eqnarray}
h_{\alpha}[A]=U(s_1,s_0)={\cal P}\exp\bigg(-\int_{0}^{1}ds\dot{\alpha}^{\mu}(s)A_{\mu}^m(\alpha(s))T_m\bigg).
\end{eqnarray}
The ${\cal P}$ is the usual path ordering operation  (operators with larger values of $s$ are placed to the left of the operators with to smaller values of $s$) and $T_m$ are the usual generators of $SU(2)$ which satisfy the Lie algebra
\begin{eqnarray}
[T_m,T_n]=i\epsilon_{mnl}T_l.
\end{eqnarray}
The set of all holonomies $h_{\alpha}[A]$ define the quantum configuration space $\bar{\cal A}$ in the same way that the set of all connections $A_{\mu}^m$ define the classical configuration space ${\cal A}$. A holonomy (called generalized connection in \cite{Huggett:1998sz}) is a map on the space of all paths in $\Sigma_t$ which assigns an element of the group $SU(2)$ to each path $\alpha(t)$ (in contrast to the connection which is a map on the hypersurface $\Sigma_t$ which assigns an element of the Lie algebra $su(2)$ to each point on the surface).

The holonomy defines the parallel transport of a spinor in the background of the configuration $A_{\mu}^m$ along the curve $\alpha(t)$ between the start point $\alpha(0)$ and the end point $\alpha(1)$ and it measures the accumulated phase difference between the initial and final values of the spinor at the two points. More explicitly, under gauge transformations $g$ we must have
\begin{eqnarray}
h_{\alpha}\longrightarrow h_{\alpha}^{\prime}=g^{-1}(\alpha(1))h_{\alpha}g(\alpha(0)).
\end{eqnarray}
This defines the so-called generalized gauge transformations \cite{Huggett:1998sz} which act on holonomies only at the end points of paths in contrast to ordinary gauge transformations which act on connections at every point of $\Sigma_t$. They are generated by the Gauss gauge constraint which can then be solved explicitly by using only Wilson loops which are holonomies traced out around closed paths (loops), viz
\begin{eqnarray}
W_{\alpha}[A]=Tr h_{\alpha}[A].\label{WL}
\end{eqnarray}
These are gauge invariant by construction.

If we denote the set of all  generalized gauge transformations by $\bar{\cal G}$ then the gauge invariant quantum configuration space must be given by the quotient $\bar{\cal A}/\bar{\cal G}$. This space can be viewed as a projective limit of a family  of compact, smooth and finite dimensional configuration spaces $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$.

Each configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is  labelled by a path or graph $\alpha$ characterized by $N$ edges and $V$ vertices (each edge $e$ starts at  a vertex $v_{e_0}$ and ends at a vertex $v_{e_1}$). From the one hand, the space $\bar{\cal A}_{\alpha}$ is the space of generalized connections or holonomies over the graph $\alpha$ which consists of the mappings which assign to each edge of the graph an element of the group $SU(2)$, i.e. $\bar{\cal A}_{\alpha}$ is isomorphic to $SU(2)^N$. From the other hand, the space $\bar{\cal G}_{\alpha}$ is the space of generalized gauge transformations over the graph $\alpha$ which consists of all mappings which assign to each vertex an element of $SU(2)$. More precisely, the action of a given generalized gauge transformation $g$ on an edge $e$ of the graph $\alpha$ is given by $h_{\alpha}(e)\longrightarrow h_{\alpha}^{\prime}(e)=g^{-1}(v_{e_1})h_{\alpha}(e)g(v_{e_0})$. Hence the space $\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^V$ and as a consequence the  configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ is isomorphic to $SU(2)^{N-V}$.

The  Hilbert space of state vectors on the  configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$ (which is compact and finite dimensional) is precisely the space of square-integrable functions ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ where the measure is obviously induced by the usual the Haar measure on $SU(2)$. Elementary quanta of geometry are elements of ${\cal H}_{\alpha}$ and they are of the form \cite{Huggett:1998sz}
\begin{eqnarray}
\Psi_{\alpha}(h_{\alpha})=\psi(h_{\alpha}(e_1),...,h_{\alpha}(e_N))~,~\psi\in SU(2)^N.
\end{eqnarray}
In summary, the members of the family  $\{\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}\}$ are all compact and finite dimensional spaces with corresponding Hilbert spaces ${\cal H}_{\alpha}=L^2(\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha})$ and as a consequence the projective limit $\bar{\cal A}/\bar{\cal G}$ which is also compact admits a regular Borel measure allowing the construction of a corresponding Hilbert space of square-integrable functions ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ where the measure is induced by the usual the Haar measure on $SU(2)$.

The Hilbert space  ${\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$ admits a more interesting decomposition as a direct sum of  finite dimensional orthogonal Hilbert spaces   ${\cal H}_{\alpha,\vec{j}}$ characterized together with the graph $\alpha$ by a vector $\vec{j}$ of half-integers, i.e. $\vec{j}=(j_1,j_2,...,j_N)$ where the integer $j_i$ represents the irreducible representation of $SU(2)$ which labels the edge $i$ of the graph $\alpha$. We have then \cite{Ashtekar:2014kba}
\begin{eqnarray}
{\cal H}=L^2(\bar{\cal A}/\bar{\cal G})=\oplus_{\alpha,\vec{j}} {\cal H}_{\alpha,\vec{j}}.
\end{eqnarray}
${\cal H}_{\alpha,\vec{j}}$ are Hilbert spaces of spin network  states on the  configuration space $\bar{\cal A}_{\alpha}/\bar{\cal G}_{\alpha}$. A spin network state $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ is then an element of ${\cal H}_{\alpha,\vec{j}}$ which is characterized by $1)$ the graph $\alpha$, $2)$ the $N$ irreducible representations $j_i$ associated with the edges of the graph and $3)$ the $V$ intertwining operators $i_i$ associated with the vertices of the graph.

More precisely,  let $\rho_e$ be the irreducible representation associated  with the edge $e$, i.e. $\rho_e$ is the homomorphism $\rho_e:SU(2)\longrightarrow {\rm End}(V_e)$ where ${\rm End}(V_e)$ is the group of endomorphisms of a vector space $V_e$. Then let $S(v)$ be the set of edges with the vertex $v$ as a source (start point) and let $T(v)$ be the set of edges with the vertex $v$ as a target (end point). An intertwining operator $I_v$ is a linear endomorphism between the two vector spaces $\Oplus_{e\in S(v)}V_e$ and $\Oplus_{e\in T(v)}V_e$. The intertwining operator $I_v$ can also be understood as an invariant element of the representation $\Oplus_{e\in S(v)}V_e\otimes \Oplus_{e\in T(v)}V_e^{\star}$. Thus, $\vec{i}$ is a labelling of the edges of the graph $\alpha$ by irreducible representations of $SU(2)$. Similarly, $\vec{i}$ is a labelling of the vertices of the graph $\alpha$ by intertwining operators from the tensor product of incoming representations to the tensor product of the outgoing representations \cite{Baez:1994hx}.

For example, in Penrose spin networks  \cite{Penrose} we consider trivalent graphs labelled by spins $j$ satisfying the rule that if the spins of the edges at a given  vertex are $j_1$, $j_2$ an d$j_3$ then these three spins must satisfy the rules of conservation of angular momentum, i.e. the Clebsch-Gordon condition $|j_1-j_2|\leq j_3\le j_1+j_2$ must hold. This Clebsch-Gordon condition  is a necessary and sufficient  condition for the existence of  intertwining operators from $j_1\otimes j_2$ to $j_3$.

The set of all spin network states with all possible graphs $\alpha$,  all assignements $\vec{j}$ of irreducible representations of $SU(2)$ to the edges, and all assignements $\vec{i}$ of intertwining operators to the vertices form the Hilbert space  $\cal H}=L^2(\bar{\cal A}/\bar{\cal G})$.

The spin network states  $\Psi_{\alpha,\vec{j},\vec{i}}(A)$ are eigenvectors  of area and volume operators in the hypersurfaces $\Sigma_t$ with discrete spectra \cite{Rovelli:1995ac}.

For example, the area of a two-dimensional surface $\Sigma$ with induced metric $h_{ab}^{(2)}$ in the three-dimensional hypersurface $\Sigma_t$ with  induced metric $h_{ab}$ is given by
\begin{eqnarray}
A_{\Sigma}=\int dx^1dx^2\sqrt{{\rm det} h^{(2)}}.
\end{eqnarray}
The metric $h_{ab}^{(2)}$ is the induced metric on the surface $\Sigma$
which can be expressed in terms of the metric $h_{ab}$, then in terms of the  densitized triads $\tilde{e}_m^{\mu}$ using the relation $\tilde{e}_m^{\mu}\tilde{e}^{\nu m}=h^{\mu\nu}{\rm det} h$. In the quantum theory the densitized triads become operators $\hat{\tilde{e}}_m^{\mu}$ which act (up to a numerical factor) as $\delta/\delta A_{\mu}^m$ and hence the area $A_{\Sigma}$  becomes an area operator $\hat{A}_{\Sigma}$ which admits the Wilson loops (\ref{WL}) as eigenvectors, viz \cite{Gambini:2011zz}

\begin{eqnarray}
A_{\Sigma}W_{\alpha}[A]=8\pi l_{\rm P}^2\beta \sum_I\sqrt{j_I(J_I+1)}W_{\alpha}[A].
\end{eqnarray}
The sum is over all edges of the Wilson loop $\alpha$ that intersect  the surface $\Sigma$. The surface areas of $\Sigma$ is then given by $8\pi l_P^2\beta$ \sum_I\sqrt{j_I(J_I+1)}$where$j_I$is the spin associated with the edge$I$,$\beta$is the Immirzi parameter and$l_P\$ is the Planck length.

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