The Catsters

_[The Catsters](http://www.simonwillerton.staff.shef.ac.uk/TheCatsters/)_ was a group at Sheffield University, around [[Eugenia Cheng]] and [[Simon Willerton]], who created a collection of pedagogical videos on topics of [[category theory]].

* [The Catsters Channel](http://www.youtube.com/user/TheCatsters) on YouTube

* [Table of contents](http://www.simonwillerton.staff.shef.ac.uk/TheCatsters/)

Eg. on [[monads]]:

* [[The Catsters]], *Monads*, short video lectures (2007) [1:[YT](https://www.youtube.com/watch?v=9fohXBj2UEI&list=PL0E91279846EC843E&index=1), 2:[YT](https://www.youtube.com/watch?v=Si6_oG7ZdK4&list=PL0E91279846EC843E&index=2), 3:[YT](https://www.youtube.com/watch?v=eBQnysX7oLI&list=PL0E91279846EC843E&index=3), 3A:[YT](https://www.youtube.com/watch?v=uYY5c1kkoIo&list=PL0E91279846EC843E&index=4), 4:[YT](https://www.youtube.com/watch?v=Cm-O_ZWEIGY&list=PL0E91279846EC843E&index=5), 5/6:[YT](https://www.youtube.com/watch?v=g-SCYArh5RY&list=PL0E91279846EC843E&index=6)]

category: reference


This entry is about the article

* [[Yvonne Choquet-Bruhat]],
  _The Cauchy Problem in Classical Supergravity_
  Letters in Mathematical Physics 7 (1983) 459-467. 0377

## Summarry

Here is the quintessence of the article.

### Terminology

The article considers super fields valued in a "single, arbitrary, large enough" [[Grassmann algebra]] in the sense exposed in the book

* Bryce de Witt, Supermanifolds, Cambridge Monographs on Mathematical Physics, 1984, 1992

In the following we shall not quite follow this, but make use of the observation that this really means that we are working over the [[sheaf topos]] on the [[site]] of [[superpoint]]s, as described at [[super ∞-groupoid]]. This changes nothing about the actual computations and formulas, but somewhat strenghtens the conceptual background.

### Field configurations

Let $\mathfrak{siso}(d,1)$ be the [[super Poincare Lie algebra]] in some dimension. The field [[configuration space]] of [[supergravity]] is the [[groupoid of Lie-algebra valued forms]] $\Omega^1(X, \mathfrak{siso})$ on a given [[spacetime]] manifold $X$. This is the [[super-groupoid]] that assigns to a given [[superpoint]] $\mathbb{R}^{0|q}$ with function algebra the [[Grassmann algebra]] $\Lambda_q$ the ordinary [[groupoid of Lie-algebra valued forms]] of the ordinary [[Lie algebra]] $(\mathfrak{g} \otimes \Lambda_q)_{even}$:

$$
 \Omega^1(X, \mathfrak{g}) : \mathbb{R}^{0|q} \mapsto
  \Omega^1(X, (\mathfrak{g} \otimes \Lambda_q)_{even})
  \,.
$$

Its [[object]]s are the field configurations, its [[morphism]]s the [[gauge transformation]]s.  The collection of objects over $\mathbb{R}^{0|q}$ is the collection of [[dg-algebra]]s homomorphims

$$
  Hom_{dgAlg}(W(\mathfrak{g} \otimes \Lambda_q)_{even}), \Omega^\bullet(X))
  \,,
$$

where on the left we have the [[Weil algebra]] of $(\mathfrak{g} \otimes \Lambda_q)_{even}$. The Weil algebra is a [[free construction|free]] dg-algebra on the generators

* $\{e^a \otimes (\Lamba_q^*)_{even} \}$

* $\{\omega^{a b} \otimes (\Lamba_q^*)_{even} \}$

* $\{\psi^{\alpha} \otimes (\Lamba_q^*)_{odd} \}$ .

Accordingly a field configuration over $\mathbb{R}^{0|q}$ is

1. the [[graviton]] given by

   1. the [[vielbein]] $\{E^a_\mu : X \to (\Lambda_q)_{even}\}$;

   1. the [[spin connection]] $\{\Omega_\mu{}^{a b} : X \to (\Lambda_q)_{even}\}$;

1. the [[gravitino]] given by $\Psi_\mu^\alpha : X \to (\Lambda_q)_{odd}$.

### The Cauchy problem

The [[Euler-Lagrange equation]]s for these fields are schematically of the form

$$
  R_{\mu \nu}^{a b } + F = 0
$$

$$
  D \Psi = 0
  \,.
$$

These are over each $\mathbb{R}^{0|q}$ equations in $C^\infty(X, \Lambda_q)$. In degree 0 in the Grassmann generators this is just the ordinary Einstein equaitons of [[gravity]] for the 0-Grassmann degree component of the fields. This is a well defined and causal Cauchy problem.

The observation now is: the equations can be solved by [[induction]] over the total number of Grassmann generators. In each step, the problem is a well-defined causal Cauchy problem in a version of ordinary [[gravity]] coupled to a bunch of extra fields.

category: reference


This entry provides links for the book

* [[Andreas Kriegl]], [[Peter Michor]]:

  \linebreak

  **The Convenient Setting of Global Analysis**

  \linebreak

  Mathematical Surveys and Monographs

  Volume 53

  AMS (1997)

  ISBN:978-0-8218-0780-4

  [ams:surv-53](https://bookstore.ams.org/surv-53)

  [pdf](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf)

on [[differential geometry]] in terms of [[Frölicher spaces]] and [[diffeological spaces]], in particular also on [[infinite-dimensional manifolds]].

category: reference

[[!redirects The Convenient Setting Of Global Analysis]]
[[!redirects the Convenient Setting Of Global Analysis]]
[[!redirects Convenient Setting Of Global Analysis]]
[[!redirects The Convenient Setting of Global Analysis]]
[[!redirects the Convenient Setting of Global Analysis]]
[[!redirects Convenient Setting of Global Analysis]]
[[!redirects The convenient setting of global analysis]]
[[!redirects Convenient setting of global analysis]]
[[!redirects the convenient setting of global analysis]]
[[!redirects convenient setting of global analysis]]


The Dirac Electron

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

The purpose of this page is to explain - using appropriate mathematical terminology - Dirac's theory of coupling particles with spin to a [[Yang-Mills field|Yang-Mills]] [[gauge field]].

We proceed in three steps: first we recall relevant facts about the gauge field itself, then we discuss charged particles in gauge fields, and finally we add spin.

## Yang-Mills Theory

(see: the main article about [[Yang-Mills theory]])

Under a _[[spacetime]]_ we understand a [[smooth manifold|smooth]], oriented, [[pseudo-Riemannian manifold]].

+-- {: .un_def}
###### Definition

A **[[Yang-Mills theory]]** over a [[spacetime]] $M$ is:

* A [[Lie group]] $G$, called the _gauge group_, together with an $\mathrm{Ad}$-invariant scalar product $\kappa: \mathfrak{g} \times \mathfrak{g} \to \R$ on its [[Lie algebra]] $\mathfrak{g}$.

* A $G$-[[principal bundle]] $P$ over $M$.

A _gauge field_ is a [[connection on a bundle|connection]] $\omega \in \Omega^1(P,\mathfrak{g})$ on $P$. The [[action functional]] is

$$
S_{YM}(\omega) := \frac{1}{2} \int_M \| F_{\omega} \|_\kappa^2.
$$

=--

+-- {: .un_remark}
###### Remark

Above we have used the following notation:

* $F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P))$ is the curvature of $\omega$.

* $\mathrm{Ad}(P) := P \times_{\mathrm{Ad}} \mathfrak{g}$ is the
  [[adjoint bundle]].

* $\| \psi \|_{\kappa}^2 := \psi \wedge_{\kappa} \star \psi   \in \Omega^n(M)$. In general, if $U,V,W$ are vector spaces, $\varphi \in \Omega^p(M,V)$, $\psi\in\Omega^q(M,W)$ and $f: V \times W \to U$ is a linear map, we have $\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U)$.

* $\star$ is the [[Hodge-star operator]] determined by the [[metric]] on $M$.

=--

+-- {: .un_remark}
###### Remark

The [[Euler-Lagrange equation]]s determined by the above action together with the [[Bianchi identity]] are called _[[Yang-Mills equations]]_:

$$
\mathrm{D}^{\omega}\star F_{\omega} = 0
\quad\text{ and }\quad
\mathrm{D}^{\omega}F_{\omega}=0,
$$

where $\mathrm{D}^\omega$ denotes the [[exterior covariant derivative|covariant derivative]].

=--

+-- {: .un_definition}
###### Definition

A _gauge transformation_ is a smooth [[bundle]] morphism $g: P \to P$.

=--

+-- {: .un_remark}
###### Remark

Let $g: P \to P$ be a gauge transformation.

* If $\omega$ is a connection on $P$, then $g^{*}\omega$ is another connection on $P$.

* One can identify $g$ with a smooth map $\tilde g: P \to G$, namely by $g=r_{\tilde g}$, i.e. $g(p) = p \cdot \tilde g(p)$ for all $p\in P$.

* The pullback along a gauge transformation restricts to an [[automorphism]] of $\Omega^k_{\rho}(P,V)$. In terms of the associated map $\tilde g$, we have
$$
g^{*}\psi = \rho(\tilde g^{-1},\psi)\text{.}
$$

=--

+-- {: .un_theorem}
###### Theorem

The Yang-Mills action functional $S_{YM}$ is gauge-invariant, i.e.
$$
S_{YM}(g^{*}\omega) = S_{YM}(\omega)
$$
for all gauge transformations $g:P \to P$.

=--

+-- {: .proof}
###### Proof

We have $g^{*}\omega  = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta$ and $g^{*}\Omega = \mathrm{Ad}_{\tilde g}^{-1}(\Omega)$.
Under the isomorphism $\Omega^k(P,\mathrm{Ad}) \cong \Omega^2(M,\mathrm{Ad}(P))$ this corresponds to $F_{g^{*}\omega} = \mathrm{Ad}_{g} (F_{\omega})$. Since the bilinear form $\kappa$ is $\mathrm{Ad}$-invariant by assumption,
$$
\| F_{g^{*}\omega} \|
= \mathrm{Ad}_{g} (F_{\omega}) \wedge_{\kappa} \mathrm{Ad}_{g} (F_{\omega})
= F_{\omega} \wedge_{\kappa} F_{\omega}
= \| F_{\omega}\|.
$$

=--

+-- {: .un_example}
###### Example

Let $M$ be a [[spacetime]]. A classical [[electromagnetic field]] theory over $M$ is a Yang-Mills Theory over $M$ with gauge group $G=U(1)$. In more detail:

* for an electromagnetic field theory given by a $U(1)$-bundle $P$ over $M$, we have $\mathrm{Ad}(P) \cong P \times \R$, so that $\Omega^k(M,\mathrm{Ad}(P)) \cong \Omega^k(M)$ and $\Omega^k_{\mathrm{Ad}}(P,\mathfrak{g})\cong \Omega^k_{\mathrm{Ad}}(P)$. In particular, $F_{\omega} \in \Omega^2(M)$.

* Since $U(1)$ is [[abelian group|abelian]], $\mathrm{d}(\mathrm{Ad}) = 0$ and so $\mathrm{D}^{\omega}=\mathrm{d}$ on $\Omega^{k}_{\mathrm{Ad}}(P)$.

* Thus, the Yang-Mills equations reduce to  Maxwell's equations for an electromagnetic field on $M$:
$$
\mathrm{d}\star F_{\omega} = 0
\quad\text{ and }\quad
\mathrm{d}F_{\omega}=0
\text{.}
$$

=--

## General Matter Fields

+-- {: .un_def}
###### Definition

Let $G$ be a gauge group. A _matter type for  $G$_ is a tuple $(V,h,\rho,f)$ consisting of:

* a finite-dimensional real [[vector space]] $V$ called the _internal state space_.

* a [[scalar product]] $h: V \times V \to \R$.

* a [[representation]] $\rho: G \times V \to V$ that is [[isometry|isometric]] with respect to $h$ i.e. $h(\rho(g)(v),\rho(g)(w)) = h(v,w)$.

* a [[smooth function]] $f: V \to \R$ that is $\rho$-invariant, i.e. $f(\rho(g)(v))=f(v)$.

=--

+-- {: .un_def}
###### Definition

Let $P$ be a principal $G$-bundle over $M$, and let $\mathcal{T} =(V,h,\rho,f)$ be a matter type for $G$. A _field for $P$ of type $\mathcal{T}$_ is a smooth [[section]] $\phi: M \to P \times_{\rho} V$. Its  [[action functional]] is

$$
S_{\mathcal{T}}(\omega,\phi) := \int_M  \;\|\, \mathrm{D}^{\omega}(\phi)\, \|^2_{h}  +  \star (f \circ \phi) \text{.}
$$

=--

+-- {: .un_theorem}
###### Theorem

The action functional $S_{\mathcal{T}}$ is gauge invariant, i.e.
$$
S_{\mathcal{T}}(g^{*}\omega,g^{*}\phi) = S_{\mathcal{T}}(\omega,\phi)
$$
for all gauge transformations $g:P \to P$.

=--

+-- {: .proof}
###### Proof

One calculates that $g^{*}\omega = \mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta$, where $\tilde g:P \to G$ is the smooth map associated to $g$ via $g(p) = p\cdot  \tilde g(p)$. Further, $g^{*}\psi = \rho(\tilde g^{-1})(\psi)$. A computation shows
$$
\mathrm{d}(\rho(\tilde g^{-1})(\psi)) = \rho(\tilde g^{-1})(\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1})(\psi)\text{,}
$$
where $\mathrm{d}\rho: \mathfrak{g} \otimes V \to V$.
Now we compute

$\quad\quad\mathrm{d}^{g^{*}\omega}(g^{*}\psi)$

$\quad\quad\quad\quad= \mathrm{d}\rho(\tilde g^{-1},\psi) + g^{*}\omega \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)$

$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi) + \tilde g^{*}\bar\theta\wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi) + (\mathrm{Ad}_{\tilde g}^{-1}(\omega) - \tilde g^{*}\bar\theta) \wedge_{\mathrm{d}\rho} \rho(\tilde g^{-1},\psi)$

$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}\psi)  + \rho(\tilde g^{-1},\omega\wedge_{\mathrm{d}\rho} \psi)$

$\quad\quad\quad\quad= \rho(\tilde g^{-1},\mathrm{d}^{\omega}\psi)\text{.}$

Since $h$ is invariant, the invariance of the first term follows. The invariance of the second term is clear.

=--

+-- {: .un_remark}
###### Remark

One can either keep a connection $\omega$ fixed and consider

$$
S_{\omega}(\phi) := S_{\mathcal{T}}(\omega,\phi)
$$

as a matter field in an "external gauge field", or consider the combined action functional

$$
S(\omega,\phi) := S_{Y\!M}(\omega) + S_{\mathcal{T}}(\omega,\phi)\text{.}
$$

=--

+-- {: .un_example}
###### Example
**(Scalar particle in an external, trivial gauge field)**

We consider $G=\left \lbrace e \right \rbrace$, $P=M$, so that necessarily $\omega=0$. A _scalar field_ is field for $M$ of matter type $(\R,h,\id,f)$ where $f(x) := -\frac{1}{2}m^2x^2$ and $h(x,y) = x y$. The action functional is
$$
S(0,\phi) =\frac{1}{2} \int_M  \| \mathrm{d}\phi \|_h^2 +\star m^{2}\phi^2\text{.}
$$
The Euler-Lagrange equation is the _[[Klein-Gordon equation]]_
$$
(\triangle + m^2)\phi = 0\text{,}
$$
where $\triangle := \delta \circ \mathrm{d}: \Omega^k(M) \to \Omega^{k}(M)$ is the Laplace operator and $\delta := \star \mathrm{d} \star$ is the exterior coderivative.

=--

+-- {: .un_example}
###### Example
**(Charged particle in an  electromagnetic field, e.g. a $\pi^{-}$-meson)**

Let $P$ be a $U(1)$-principal bundle over $M$. A _field of charge $n \in \Z$_ is a field for $P$ of matter type $(\C,h,\rho_n,f)$, where $\rho_n: U(1) \times \C \to \C$ is defined by $\rho_n(z,z') := z^n z'$  and $f(z) := -\frac{1}{2}m^2|z|^2$. The action functional is
$$
S(\omega,\phi) =\frac{1}{2} \int_M  \| \mathrm{D}^{\omega}\phi \|^2 +\star m^{2} \| \phi \| ^2\text{.}
$$
The Euler-Lagrange equation is _covariant Klein-Gordon equation_
$$
(\triangle^{\omega} +m^2) \phi = 0\text{,}
$$
where $\triangle^{\omega} :=\delta^{\omega}\circ \mathrm{D}^{\omega}$ is the _covariant Laplace operator_ and $\delta^{\omega} := \star \mathrm{D}^{\omega} \star$ is the _exterior covariant coderivative_.

=--

## Matter Fields with Spin

The [[Klein-Gordon equation]]s found above are -- unlike the [[Schrödinger equation]] -- of second order on time. Dirac's motivation was to find a first order equation which upon iteration yields the Klein-Gordon equation.
We first discuss free [[spinor]]s (where free means that they are not coupled to an [[electromagnetic field]], but still feel the "[[gravity]]" of the [[spacetime]] [[manifold]]), and then add the coupling.

### Free Spinors

+-- {: .un_example}
###### Example

We recall some facts about [[Clifford algebra]]s and the [[spin group]].

* We denote by $C(p,q)$ the Clifford algebra on $\R^{p,q}$, i.e. the quotient of the tensor algebra of $\R^{p+q}$ by the ideal generated by $v \otimes w + w \otimes v + 2 \left \langle v,w  \right \rangle\cdot 1$, where $\left \langle -,-  \right \rangle$ is the Minkowski scalar product of signature $(p,q)$.

* The map $v \mapsto -v$ extends to an anti-automorphism $\alpha: C(p,q) \to C(p,q)$, whose eigenspace decomposition yields the usual $\Z_2$-grading on $C(p,q)$.

* We have $\dim C(p,q) = 2^{p+q}$.

* The Clifford algebra inherits a [[bilinear form]] $H(v,w) := (v^{tr}w)_0$, where $()^{tr}$ is the anti-automorphism of the tensor algebra that reverts the order of [[tensor product]]s, and $()_0$ denotes the degree 0 part.

* We denote by $SO(p,q)$ the group of linear maps $\R^{p+q}\to \R^{p+q}$ that preserve the product $\left \langle -,-  \right \rangle$.
We define
$$
Spin(p,q) := \left \lbrace v_{1}\cdot ... \cdot v_{2r}\in C(p,q)\mid v_i \in \R^{p+q}, \| v_i \|=1,r \in \N \right \rbrace \text{.}
$$
Then, we define a group homomorphism
$\Lambda: Spin(p,q) \to SO(p,q)$
by $\Lambda(\varphi)v = \alpha(\varphi) v \varphi^{-1}$. This gives a central extension
$$
1 \to \Z_2 \to Spin(p,q) \to SO(p,q) \to 1\text{.}
$$

* We denote by $\C(p,q) := C(p,q) \otimes_{\R} \C$ the complexification of the Clifford  algebra. The bilinear form $H$ on $C(p,q)$ extends to a sesquilinear form $H$ on $\C(p,q)$ defined by $H(v,w)=(v^{tr}\bar w)_0$.

* Multiplication in $\C(p,q)$ restricts to an action of $\spin{p,q}$ on $\C(p,q)$. One can decompose $\C(p,q)$ into $k$ copies of a subrepresentation $\Sigma$:
$$
\C(p,q) = \Sigma \oplus ... \oplus  \Sigma \text{.}
$$

* The representation $\Sigma$ is isometric with respect to $H$.
If $p+q$ is odd, $k=2^{(p+q-1)/2}$, and $\Sigma$ is irreducible. If $p+q$ is even, $k=2^{(p+q)/2}$, and $\Sigma = \Sigma^{+} \oplus \Sigma^{-}$ with $\Sigma^{\pm}$ irreducible and  $\dim_{\C}\Sigma^{\pm}=2^{(p+q-2)/2}$.

=--

+-- {: .un_remark}
###### Remark

We also need some facts about [[spin structure]]s.

* Let $M$ be a [[spacetime]] with [[pseudo-Riemannian metric]] of [[signature]] $(p,q)$. We denote by $FM$ be the principal $SO(p,q)$-bundle over $M$ of orthonormal frames, the _[[frame bundle]]_.

* A _[[spin structure]]_ on $M$ is a principal $Spin(p,q)$-bundle $SM$ over $M$ together with a bundle morphism $\lambda: SM \to FM$ such that $\lambda(X\cdot\varphi) = \lambda(X)\cdot\Lambda(\varphi)$ for all $X\in SM$ and all $\varphi\in Spin(p,q)$.

* Let $\theta \in \Omega^1(FM,\mathfrak{so}(p,q))$ be the Levi-Cevita connection on $FM$. Then,
$$
\Theta := \mathrm{d}\Lambda^{-1}(\lambda^{*}\theta) \in \Omega^1(SM,\mathfrak{spin}(p,q))
$$
is a connection on $SM$.

=--

+-- {: .un_remark}
###### Remark

Finally, we recall the definition of the [[Dirac operator]].

* Let $\rho: C(p,q) \times V \to V$ be a [[representation]], with $V \subset \C(p,q)$. Note that in the above realization of the group $Spin(p,q)$, the representation $\rho$ restricts to a representation of $Spin(p,q)$.
The _spinor bundle_ is the vector bundle $V M := SM \times_{\rho} V$.

* _Clifford multiplication_ is a map
$$
TM \otimes V M \to V M: u \otimes s \mapsto u \cdot s\text{.}
$$
It is defined as follows. We write $s=(X,v)$ with $X\in SM$ and $v\in V$. We consider the orthonormal frame $\alpha_X := \lambda(X): TM \to \R^n$. Then, $\alpha_X(u) \in \C(p,q)$ and
$$
u \cdot s := (X,\alpha_X(u)\cdot v)\text{.}
$$
One can show using above-listed properties of the Clifford algebra that this definition does not depend on the choice of the representative $(X,v)$.

* The _[[Dirac operator]]_ is
$$
D: \Gamma(M,V M) \to \Gamma(M,V M) : \psi \mapsto \sum_{i} e_i \cdot  \mathrm{D}^{\Theta}\psi (e_i)
$$
where $e_i\in TM$ runs over a local orthonormal basis.

=--

+-- {: .un_def}
###### Definition

Let $M$ be a spacetime with spin structure $SM$, and considered as a  $Spin(p,q)$ as a Yang-Mills theory over $M$. A _free spinor_ is a field for $SM$ of type $(V,h,\rho,f)$, where $V \subset \C(p,q)$, the scalar product $h$ is
$$
h(v,w):=\frac{1}{2}(H(v,w) + H(w,v))\text{,}
$$
and $\rho$ is the restriction of the multiplication in $\C(p,q)$ to $Spin(p,q)$.
The action functional is
$$
S(\psi) := \int_M D \psi \wedge_{h} \star \psi  + \star f \circ \psi\text{.}
$$

=--

+-- {: .un_def}
###### Definition

The Euler-Lagrange equation determined by the action functional $S(\psi)$ is the _Dirac equation_
$$
D\psi + \mathrm{i}m\psi = 0\text{.}
$$

=--

+-- {: .un_def}
###### Definition
**(Weyl spinors)**

We assume spacetime to have even dimension.
_[[Weyl spinor]]s_ have $V=\Sigma^{\pm}$, with the sign corresponding to left/right-handed spinors. Thus, $\dim_{\C}(V)=2$.
Further $f=0$ (they are massless). In the standard model, neutrinos are left-handed Weyl spinors.

=--

+-- {: .un_def}
###### Definition
**(Dirac spinors)**

We assume spacetime to have signature $(1,3)$. [[Dirac spinor]]s have $V=\Sigma^{+} \oplus \Sigma^{-}$, so that $\dim_{\C}(V)=4$. The function $f$ is taken to be $f(v)=-mh(v,v)$. In the standard model, electrons are Dirac spinors.

=--

+-- {: .un_remark}
###### Remark

In the physical literature, the picture is slightly different:
The representation space $V$ of a spinor is not a subspace of the Clifford algebra, but rather $\C^n$. One can think about this as a further  association of $\C^n$ to the Clifford bundle $VM$ using a representation of $\C(p,q)$ on $\C^n$. Below we describe this in the case of the electron, i.e. $(p,q)=(1,3)$ and $V= \Sigma^{+} \oplus \Sigma^{-}$. Another difference is here that instead of $Spin(1,3)$, physicists often use the (non-canonically) isomorphic group $SL(2,\C)$.

* One starts with the following representation  $\gamma: \C(1,3) \to \mathrm{Gl}(4,\C)$. Consider the $\R$-linear map
$$
\gamma: \R^4 \to \mathrm{Gl}(4,\C): v \mapsto \begin{pmatrix} 0 & v' \\
v'' & 0
\end{pmatrix}\text{,}
$$
where
$$
v' := \begin{pmatrix} v_0 + v_3 & v_1 - iv_2 \\
v_1 + iv_2 & v_0 - v_3 \\
\end{pmatrix}
\quad\text{ and }\quad
v'' = \begin{pmatrix} v_0-v_3 & -v_1 + iv_2 \\
-v_1-iv_2 & v_0+v_3 \\
\end{pmatrix}\text{.}
$$
It satisfies $\gamma(v) \cdot \gamma(v)= - \left \langle v,v  \right \rangle I_4$. The Clifford algebra has a universal property that implies that $\gamma$ extends uniquely to a representation of $\C(1,3)$.
The images of the standard basis vectors $e_0,...,e_3$ are often called _$\gamma$-matrices_, $\gamma_k := \gamma(e_k)$.

* The restriction of the representation $\gamma$ to $Spin(1,3)$ is a representation $\rho: Spin(1,3) \times \C^4 \to \C^4$. It splits into a direct sum of two representations equivalent to $\Sigma^+$ and $\Sigma^-$. Using $\rho\circ\alpha = \alpha$, one checks using the above definition of the group homomorphism $\Lambda: Spin(p,q) \to SO(p,q)$ that
$$
\gamma(\Lambda(\varphi)v) = \rho(\varphi)\gamma(v)\rho(\varphi)^{-1}\text{.}
$$

* The above mentioned identification between $Spin(1,3)$ and $SL(2,\C)$ is
$$
Spin(1,3) \to SL(2,\C) : v_1 \cdot...\cdot v_{2r} \mapsto v_1'v_2''v_3'\cdot...\cdot v_{2r}''\text{.}
$$
Under this isomorphism, $\rho$ becomes
$$
\rho: SL(2,\C) \to \mathrm{Gl}(4,\C) : A \mapsto \begin{pmatrix}A & 0 \\
0 & A^{*-1}
\end{pmatrix}\text{.}
$$
Under this identification, the splitting of $\rho$ into a direct sum yields the defining representation, often called $D^{(1/2,0)}$ and its conjugate, often called $D^{(0,1/2)}$.

* Finally, the bilinear form $H$ becomes
$$
H(v,w) := v^{\mathrm{tr}}\gamma_0\bar w.
$$

=--

+-- {: .un_remark}
###### Remark

If $M=\R^{1,3}$ one can take the trivial spin structure $SM=M \times SL(2,\C)$. It has a canonical global section, so that a spinor $\psi$ can be identified with a map $\psi: M \to \C^4$. The Dirac operator is now
$D \psi = \gamma^{i}\partial_i \psi$, where $\gamma^i := \eta^{ik}\gamma_k$. Now, the Dirac equation is
$$
\gamma^i\partial_i \psi + \mathrm{i} m\psi = 0\text{.}
$$
Let's go back to Dirac's orginial motivation. Dirac was looking for a first order differential equation
$$
\alpha^k\partial_k\psi +im\psi=0
$$
for functions $\psi: \R^{4} \to \C^n$, whose solutions are automatically solutions of the Klein-Gordon equation
$$
(\triangle + m^2)\psi = 0\text{,}
$$
where $\triangle= \partial^k\partial_k$. If $\psi$ is a solution to the first equation,
$$
\alpha^{j}\partial_j\alpha^k\partial_k \psi = \alpha^{j}\partial_j(-im\psi) = - m^2\psi\text{.}
$$
This is the Klein-Gordon equation, if $\alpha^{i}\alpha^{j}=\eta^{ij}$. This can only be satisfied for matrices, so that better $\alpha^{i}\alpha^{j}=\eta^{ij}I_n$. Since $\eta^{ij}I_n$ is a symmetric matrix, this can be written as
$$
\frac{1}{2}(\alpha^{i}\alpha^j + \alpha^j\alpha^i)= \eta^{ij}\text{.}
$$
The smallest matrices satisfying this relation are the above "gamma matrices". If $m=0$, there is a solution in dimension two, the "Pauli matrices".

=--

+-- {: .un_remark}
###### Remark

For a general spacetime $M$, and unlike in the previous remark, $D^2 \neq \triangle$. Rather, $D^2$ is given by the _Lichnerowiz formula_ which has in fact been proved first by Schrödinger. So, Dirac's motivation actually fails for "curved spacetimes".

=--

### Charged Spinors

+-- {: .un_remark}
###### Remark
**(Bundle Splicing)**

* Consider principal $G_k$-bundles $P_k$ over $M$, for $k=1,2$. The fibre product $P_1 \times_M P_2$ is a principal $(G_1 \times G_2)$-bundle over $M$, denoted $P_1 \circ P_2$.

* If $\omega_1$ and $\omega_2$ are connections on $P_1$ and $P_2$, respectively, then
$$
\omega_1 \circ \omega_2 := \mathrm{pr}_1^{*}\omega_1 \oplus \mathrm{pr}_2^{*}\omega_2 \in \Omega^1(P_1 \circ P_2,\mathfrak{g}_1 \oplus \mathfrak{g}_2)
$$
is a connection on $P_1 \circ P_2$.

* Suppose $V$ is a vector space and $\rho_k: G_k \to \mathrm{Gl}(V)$ are representations, such that $\rho_1(g_1) \circ \rho_2(g_2) = \rho_2(g_2) \circ \rho_1(g_1)$ for all $g_1\in G_1$ and $g_2 \in G_2$. Then, $\rho_1 \times \rho_2$ is a representation of $G_1 \times G_2$ on $V$.

=--

+-- {: .un_remark}
###### Remark

Let $M$ be a spacetime with spin structure $SM$ and let $P$ be a Yang-Mills theory over $M$ with gauge group $G$. For $\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V)$ a representation with $V \subset \C(p,q)$, and $\rho_P: G \to \mathrm{Gl}(V)$ a commuting representation, the associated bundle $VP := (SM \circ P) \times_{(\rho_{SM} \times \rho_P)} V$ still has a Clifford multiplication.
For $\omega$ a connection on $P$, one can define a Dirac operator
$$
D^{\omega}: \Gamma(M,\Sigma M \circ P) \to \Gamma(M,\Sigma M \circ P):\psi \mapsto \sum_{i} e_i \cdot \mathrm{D}^{\Theta \circ \omega}(\psi)(e_i)\text{.}
$$

=--

+-- {: .un_def}
###### Definition

Let $M$ be a spacetime with spin structure, let $P$ be a Yang-Mills theory with gauge group $G$ over $M$, and let $\rho_P$ be a representation of $G$ on $V$ commuting with $\rho_{SM}$. A _charged spinor_ is a field for $SM \circ P$ of type $(V,h,\rho_{SM} \times \rho_P,f)$, where $V \subset \C(p,q)$ and $H$ is given as before.
Its action functional is
$$
S(\omega, \psi) := \int_M D^{\omega} \psi \wedge_{h} \star \psi    + \star f \circ \psi\text{.}
$$

=--

+-- {: .un_def}
###### Definition
**(Spinor in an electromagnetic field)**

Here, $\rho_{SM}: Spin(p,q) \to \mathrm{Gl}(V)$ is some representation, and $\rho_P: U(1) \to \mathrm{Gl}(V)$ is given by complex multiplication with $z^{n}$, where $n\in \Z$ is the charge of the spinor. Obviously $\rho_{SM}$ and $\rho_P$ commute. The Euler-Lagrange equation is
$$
D^{\omega}\psi + im\psi = 0\text{.}
$$
If $M=\R^{1,3}$ one can take $SM=M \times SL(2,\C)$. The canonical global section identifies $\psi$ with a smooth function $\psi: \R^{3,1} \to \C^4$ and the connection $\omega$ with a 1-form with components $A_{i}$. Then,
$$
D^{\omega}\psi = \gamma^{i}(\partial_{i} + A_i)\psi\text{.}
$$
This gives the "[[Dirac equation]]" one usually finds in a textbook.

=--

## Related concepts

* [[fermionic path integral]]

* [[minimal coupling]]

[[!include standard model of fundamental physics - table]]

## References

Useful literature on this topic is:

* [[Christian Bär]], _Introduction to Spin Geometry_, Oberwolfach Reports 53 (2006), p. 3135-3136.

* D. Bleecker, _Gauge Theory and Variational Principles_, Addison-Weasley, 1981.

* H. Blaine Lawson Jr. , Marie-Louise Michelson, _Spin geometry_, Princeton Univ. Press, 1989.

* G. L. Naber, _Topology, Geometry and Gauge Fields_, Springer, 1999.

[[!redirects Dirac electron]]
[[!redirects spinors in Yang-Mills theory]]


the first few orientals

The first few [[orientals]] look as follows (cf. [Street 1987](oriental#Street87), [p. 8](https://ncatlab.org/nlab/files/Street-AlgebraOfOrientedSimplexes.pdf#page=8)):

\begin{tikzcd}[
  column sep=20pt
]
  O^1
  \;=\;
  &[+10pt]
  0
  \ar[rr]
  &&
  1
\end{tikzcd}

\begin{tikzcd}[
  column sep=20pt,
  row sep=10pt
]
  &[+10pt]
  &
  1
  \ar[ddr]
  \ar[dd, Rightarrow, shorten=9pt]
  \\
  O^2 \;=\;
  &
  \\
  &
  0
  \ar[uur]
  \ar[rr]
  &{}&
  2
\end{tikzcd}

\begin{tikzcd}[
  column sep=20pt,
  row sep=20pt
]
  &[+10pt]
  1
  \ar[rr]
  \ar[
    Rightarrow,
    dr,
    shorten=6pt
  ]
    &&
  2
  \ar[dd]
  &&
  1
  \ar[rr]
  \ar[ddrr]
  &&
  2
  \ar[dd]
  \ar[
    dl,
    Rightarrow,
    shorten=6pt
  ]
  \\
  O^3
  \;=\;
  &
  &
  {}
  \ar[
    Rightarrow,
    dr,
    shorten=6pt
  ]
  &
  {}
  \ar[
    Rightarrow,
    rr,
    shorten=10pt
  ]
  \ar[
    -,
    rr,
    shorten=10pt
  ]
  &
  {}
  &
  {}
  &
  {}
  \ar[
    dl,
    Rightarrow,
    shorten=6pt
  ]
  \\
  &
  0
  \ar[uu]
  \ar[rr]
  \ar[uurr]
    &&
  3
  &&
  0
  \ar[uu]
  \ar[rr]
  &&
  3
\end{tikzcd}

\begin{tikzpicture}
  \draw (-8,0) node {
    $O^4 \;=\;$
  };

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  \draw[-]
    (-1.2,.025) to (1.2,.025);
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    (-1.2,-.025) to (1.2,-.025);
  \draw[-]
    (-1.2,-.075) to (1.2,-.075);
  \draw[->, line width=1.4pt]
    (1.27,0) to (1.28,0);
  \end{scope}

  \begin{scope}[yshift=3.2cm]
  \draw[-]
    (-.5,.05) to (.5,.05);
  \draw[-]
    (-.5,-.05) to (.5,-.05);
  \draw[-]
    (-.5,0) to (.56,0);
  \draw[->, line width=.7pt]
    (.55,0) to (.56,0);
  \end{scope}

  \begin{scope}[rotate={-1*360/5}]
  \begin{scope}[yshift=3.2cm]
  \draw[-]
    (-.5,.05) to (.5,.05);
  \draw[-]
    (-.5,-.05) to (.5,-.05);
  \draw[-]
    (-.5,0) to (.56,0);
  \draw[->, line width=.7pt]
    (.55,0) to (.56,0);
  \end{scope}
  \end{scope}

  \begin{scope}[rotate={-2*360/5}]
  \begin{scope}[xscale=-1, yshift=3.2cm]
  \draw[-]
    (-.5,.05) to (.5,.05);
  \draw[-]
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  \draw[-]
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  \draw[->, line width=.7pt]
    (.55,0) to (.56,0);
  \end{scope}
  \end{scope}

  \begin{scope}[rotate={-3*360/5}]
  \begin{scope}[xscale=-1,yshift=3.2cm]
  \draw[-]
    (-.5,.05) to (.5,.05);
  \draw[-]
    (-.5,-.05) to (.5,-.05);
  \draw[-]
    (-.5,0) to (.56,0);
  \draw[->, line width=.7pt]
    (.55,0) to (.56,0);
  \end{scope}
  \end{scope}

  \begin{scope}[rotate={-4*360/5}]
  \begin{scope}[yshift=3.2cm]
  \draw[-]
    (-.5,.05) to (.5,.05);
  \draw[-]
    (-.5,-.05) to (.5,-.05);
  \draw[-]
    (-.5,0) to (.56,0);
  \draw[->, line width=.7pt]
    (.55,0) to (.56,0);
  \end{scope}
  \end{scope}

  \begin{scope}[
    scale=.75,
    shift={({(0+.75)*360/5}:5.6)}
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    ({(0+.25)*360/5}:2.1)
    node {$3$};
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    node {$0$};
  \draw
    ({(4+.25)*360/5}:2.1)
    node {$4$};

  \draw[->]
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    to
    ({(0+.25+.05)*360/5}:1.8);
  \draw[->]
    ({(2+.25+.05)*360/5}:1.8)
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    ({(4+.25-.05)*360/5}:1.8);

  \draw[<-]
    ({(0+.1+.25)*360/5}:2)
    to
    ({(1-.1+.25)*360/5}:2);
  \draw[<-]
    ({(1+.1+.25)*360/5}:2)
    to
    ({(2-.1+.25)*360/5}:2);
  \draw[<-]
    ({(2+.1+.25)*360/5}:2)
    to
    ({(3-.1+.25)*360/5}:2);
  \draw[<-]
    ({(3+.1+.25)*360/5}:2)
    to
    ({(4-.1+.25)*360/5}:2);
  \draw[<-]
    ({(4+.1+.25)*360/5}:2)
    to
    ({(5-.1+.25)*360/5}:2);

  \end{scope}

  \begin{scope}[
    scale=.75,
    shift={({(1+.75)*360/5}:5.6)}
  ]

  \draw
    ({(0+.25)*360/5}:2.1)
    node {$3$};
  \draw
    ({(1+.25)*360/5}:2.1)
    node {$2$};
  \draw
    ({(2+.25)*360/5}:2.1)
    node {$1$};
  \draw
    ({(3+.25)*360/5}:2.1)
    node {$0$};
  \draw
    ({(4+.25)*360/5}:2.1)
    node {$4$};

  \draw[->]
    ({(2+.25-.05)*360/5}:1.8)
    to
    ({(0+.25+.05)*360/5}:1.8);
  \draw[->]
    ({(3+.25+.05)*360/5}:1.8)
    to
    ({(0+.25-.05)*360/5}:1.8);

  \draw[<-]
    ({(0+.1+.25)*360/5}:2)
    to
    ({(1-.1+.25)*360/5}:2);
  \draw[<-]
    ({(1+.1+.25)*360/5}:2)
    to
    ({(2-.1+.25)*360/5}:2);
  \draw[<-]
    ({(2+.1+.25)*360/5}:2)
    to
    ({(3-.1+.25)*360/5}:2);
  \draw[<-]
    ({(3+.1+.25)*360/5}:2)
    to
    ({(4-.1+.25)*360/5}:2);
  \draw[<-]
    ({(4+.1+.25)*360/5}:2)
    to
    ({(5-.1+.25)*360/5}:2);

  \end{scope}

  \begin{scope}[
    scale=.75,
    shift={({(2+.75)*360/5}:5.6)}
  ]

  \draw
    ({(0+.25)*360/5}:2.1)
    node {$3$};
  \draw
    ({(1+.25)*360/5}:2.1)
    node {$2$};
  \draw
    ({(2+.25)*360/5}:2.1)
    node {$1$};
  \draw
    ({(3+.25)*360/5}:2.1)
    node {$0$};
  \draw
    ({(4+.25)*360/5}:2.1)
    node {$4$};

  \draw[->]
    ({(3+.25+.05)*360/5}:1.8)
    to
    ({(0+.25-.05)*360/5}:1.8);
  \draw[->]
    ({(3+.25-.05)*360/5}:1.8)
    to
    ({(1+.25+.05)*360/5}:1.8);

  \draw[<-]
    ({(0+.1+.25)*360/5}:2)
    to
    ({(1-.1+.25)*360/5}:2);
  \draw[<-]
    ({(1+.1+.25)*360/5}:2)
    to
    ({(2-.1+.25)*360/5}:2);
  \draw[<-]
    ({(2+.1+.25)*360/5}:2)
    to
    ({(3-.1+.25)*360/5}:2);
  \draw[<-]
    ({(3+.1+.25)*360/5}:2)
    to
    ({(4-.1+.25)*360/5}:2);
  \draw[<-]
    ({(4+.1+.25)*360/5}:2)
    to
    ({(5-.1+.25)*360/5}:2);

  \end{scope}

  \begin{scope}[
    scale=.75,
    shift={({(3+.75)*360/5}:5.6)}
  ]

  \draw
    ({(0+.25)*360/5}:2.1)
    node {$3$};
  \draw
    ({(1+.25)*360/5}:2.1)
    node {$2$};
  \draw
    ({(2+.25)*360/5}:2.1)
    node {$1$};
  \draw
    ({(3+.25)*360/5}:2.1)
    node {$0$};
  \draw
    ({(4+.25)*360/5}:2.1)
    node {$4$};

  \draw[->]
    ({(1+.25-.05)*360/5}:1.8)
    to
    ({(4+.25+.05)*360/5}:1.8);
  \draw[->]
    ({(3+.25-.05)*360/5}:1.8)
    to
    ({(1+.25+.05)*360/5}:1.8);

  \draw[<-]
    ({(0+.1+.25)*360/5}:2)
    to
    ({(1-.1+.25)*360/5}:2);
  \draw[<-]
    ({(1+.1+.25)*360/5}:2)
    to
    ({(2-.1+.25)*360/5}:2);
  \draw[<-]
    ({(2+.1+.25)*360/5}:2)
    to
    ({(3-.1+.25)*360/5}:2);
  \draw[<-]
    ({(3+.1+.25)*360/5}:2)
    to
    ({(4-.1+.25)*360/5}:2);
  \draw[<-]
    ({(4+.1+.25)*360/5}:2)
    to
    ({(5-.1+.25)*360/5}:2);

  \end{scope}

  \begin{scope}[
    scale=.75,
    shift={({(4+.75)*360/5}:5.6)}
  ]

  \draw
    ({(0+.25)*360/5}:2.1)
    node {$3$};
  \draw
    ({(1+.25)*360/5}:2.1)
    node {$2$};
  \draw
    ({(2+.25)*360/5}:2.1)
    node {$1$};
  \draw
    ({(3+.25)*360/5}:2.1)
    node {$0$};
  \draw
    ({(4+.25)*360/5}:2.1)
    node {$4$};

  \draw[->]
    ({(1+.25-.05)*360/5}:1.8)
    to
    ({(4+.25+.05)*360/5}:1.8);
  \draw[->]
    ({(2+.25+.05)*360/5}:1.8)
    to
    ({(4+.25-.05)*360/5}:1.8);

  \draw[<-]
    ({(0+.1+.25)*360/5}:2)
    to
    ({(1-.1+.25)*360/5}:2);
  \draw[<-]
    ({(1+.1+.25)*360/5}:2)
    to
    ({(2-.1+.25)*360/5}:2);
  \draw[<-]
    ({(2+.1+.25)*360/5}:2)
    to
    ({(3-.1+.25)*360/5}:2);
  \draw[<-]
    ({(3+.1+.25)*360/5}:2)
    to
    ({(4-.1+.25)*360/5}:2);
  \draw[<-]
    ({(4+.1+.25)*360/5}:2)
    to
    ({(5-.1+.25)*360/5}:2);

  \end{scope}

\end{tikzpicture}


the fundamental group and Galois theory

Let $o$ be a Dedekind domain, let $K:=Quot(o)$ denote its quotient field,  let $L/K$ be a finite separable field extension of degree $n$, let $O\supset o$ be the integral closure of $o$ in $L$. Then $O$ is in particular a Dedekind domain

Let for

$$o\stackrel{i}{\to} O\to L$$

$f:=Spec(i):Spec(O)\to Spec(o)$ be the induced map between the [[nLab:spectrum|ring spectra]] called *ramified covering*

Let $p\in Spec(o)$ be a maximal prime ideal. Then the ideal $pO$ in $O$ has a unique product decomposition

$$pO=P_1^{e_1}\dots P_r^{e_r}$$

with different $P_i\in Spec(O)$. It is custom to introduce some classifying vocabulary depending on the kind of this decomposition and the degree of the field extension $f_i:=[O/P_i:o/p]$: the $e_i$ are called *ramification indices* and the $f_i$ are called *inertia degrees* (see e.g. the [German wikipedia](http://de.wikipedia.org/wiki/Verzweigung_(Algebra) for more details). These satisfy the *fundamental identity* $\Sigma_i e_i f_i =n$ and every point $p\in Spec(O)$ has $\le n$ preimages. If $p$ has $\lt n$ preimages then $p$ is called *ramified*.

Let now $L/K$ be [[nLab:Galois theory|galois]] then the [[nLab:Galois group|Galois automorphisms]] $\sigma\in Gal(L/K):=Aut_K(L)$ (i.e. the automorphisms of $L$ which restrict to the identity on $K$) induce automorphisms of [[nLab:scheme|schemes]] $Spec(\sigma)$ and (since $\sigma$ fixes $o$) the diagram

$$\array{
Spec(O)&\stackrel{\Spec(\sigma)}{\to}&\Spec(O)
\\
\downarrow^f &\swarrow
\\
Spec(o)
}$$

commutes. $\tau:=\Spec(\sigma)$ is an example of a *cover automorphism* (also called *cover transformation* or *Deck transformation*). Since $Spec:Rings\to Schemes$ is an equivalence of categories we have an isomorphism

$$Gal(L/K)\simeq Aut_{Spec(o)}(Spec(O))$$

where the object to the right we have the group of cover automorphism.

One can show that there is a *maximal unramified extension* ($e_i=1$ and the extensions $O/P_i:o/p$ being separable) $\tilde K$ of $K$ and the scheme $\tilde Y:=Spec(\tilde o)$ where $\tilde o$ denotes the integral closure of $o$ in $\tilde K$. Then $f:\tilde Y\to Y:=Spec(O)$ satisfies the axioms of a *[[nLab:universal cover|universal covering]]* and consequently we define on the side of schemes the *[[nLab:fundamental group]]*

$$\pi_1(Y):=Aut_Y(\tilde Y)\simeq Gal(\tilde K/ K)$$

## References

* Jürgen Neukirch, algebraic number theory, I.§13.6{#NeuNum}


(...)

An instance is the following:

Let $k$ be a field, let $k_s$ denote the separable closure of $k$.

+-- {: .num_theorem}
###### Theorem
([[the fundamental theorem of Galois theory|fundamental theorem of Galois theory]])

The functor

$$\begin{cases}
Sch_{et}\to Gal(k_s / k)- Mod
\\
X\mapsto X(k_s)
\end{cases}$$

from [[etale scheme|étale schemes]] to the category of [[Galois module|Galois modules]] $Gal(k_s/s)-Mod$ is an equivalence of categories.
=--


The Geometry of Physics - An Introduction

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

This page provides a hyperlinked index for the book

* [[Theodore Frankel]]:

  \linebreak

  **The Geometry of Physics -- An introduction**

  \linebreak

  Cambridge University Press (1997, 2004, 2012)

  [doi:10.1017/CBO9781139061377](https://doi.org/10.1017/CBO9781139061377)

  [website](http://www.math.ucsd.edu/~tfrankel/) with errata and preface for 3rd edition)

on methods of [[differential geometry]] and their meaning and use in [[physics]], especially [[gravity]] and [[gauge theory]].

Among the nice aspects of the book are

* it discusses [[pseudoforms]] on top of ordinary [[differential forms]], instead of just assuming that all [[manifolds]]  are [[orientation|oriented]] as often done — and what\'s more, it explains the physical meaning of this!  (But it still uses [[Riemannian metric|metrics]] more than necessary.)

Related books:

* [[Mikio Nakahara]], _[[Geometry, Topology and Physics]]_, Graduate Student Series in Physics (2003).

* [[Raoul Bott]], [[Loring Tu]], _[[Differential Forms in Algebraic Topology]]_

  (more advanced [[algebraic topology]])

* [[Werner Greub]], [[Stephen Halperin]], [[Ray Vanstone]], _[[Connections, Curvature, and Cohomology]]_

* [[Chris Isham]], _[[Modern Differential Geometry for Physicists]]_

  ([[differential geometry]] in [[mathematical physics]])

* [[David Hestenes]],  _[[geometric algebra|Space-time Algebra]]_, New York (1966)

  ([[Clifford algebra]] and [[spinors]])

#Contents#
* table of contents
{:toc}

## I Manifolds, Tensors and Exterior Forms

### 1 Manifolds and Vector Fields

#### 1.1 Submanifolds of Euclidean space

#### 1.1a Submanifolds of $\mathbb{R}^N$

* [[Cartesian space]]

* [[submanifold]]

#### 1.1b The Geometry of Jacobian Matrices: The "Differential"

#### 1.1c The main theorem on submanifolds of $\mathbb{R}^N$

* [[Whitney embedding theorem]]

#### 1.1d A Nontrivial Example: The Configuration Space of a Rigid Body

* [[rigid body]]

#### 1.2 Manifolds

* [[manifold]]

* [[smooth manifold]]

#### 1.3 Tangent Vectors and Mappings

* [[tangent bundle]]

* [[coordinates]]

#### 1.4 Vector Fields and Flows

* [[vector field]]

* [[flow]]

### 2 Tensors and Exterior Forms

#### 2.1 Covectors and Riemannian Metrics

* [[differential form]]

* [[Riemannian metric]]

#### 2.2 The Tangent Bundle

* [[tangent bundle]]

#### 2.3 The Cotangent Bundle and Phase Space

* [[cotangent bundle]]

* [[phase space]]

#### 2.4 Tensors

* [[tensor]]

#### 2.5 The Grassmann or Exterior Algebra

* [[Grassmann algebra]]

* [[exterior algebra]]

#### 2.6 Exterior Differentiation

* [[de Rham differential]]

#### 2.7 Pull-Backs

* [[pullback of a differential form]]

#### 2.8 Orientation and Pseudoforms

* [[orientation]]

* [[pseudoform]]

#### 2.9 Interior Products and Vector Analysis

#### 2.10 Dictionary

### 3 Integration of Differential Forms

* [[integration]]

* [[integration of differential forms]]

#### 3.1 Integration over a Parameterized Subset

#### 3.2 Integration over Manifolds with Boundary

#### 3.3 Stokes' Theorem

* [[Stokes' theorem]]

#### 3.4 Integration of Pseudoforms

#### 3.5 Maxwell's Equations

* [[Maxwell's equations]]

### 4 The Lie Derivative

* [[Lie derivative]]

#### 4.1 The Lie Derivative of a Vector Field

#### 4.2 The Lie Derivative of a Form

#### 4.3 Differentiation of Integrals

#### 4.4 A Problem Set on Hamiltonian Mechanics

* [[Hamiltonian mechanics]]

### 5 The Poincaré Lemma and Potentials

* [[electromagnetic potential]]

### 6 Holonomic and Nonholonomic Constraints

## II Geometry and Topology

### 7 $\mathbb{R}^3$ and Minkowski Space

#### 7.1 Curvature and Special Relativity

* [[curvature]]

* [[special relativity]]

#### 7.2 Electromagnetism in Minkowski Space

* [[electromagnetism]]

### 8 The Geometry of Surfaces in $\mathbb{R}^3$

#### 8.1 The First and Second Fundamental Form

#### 8.2 Gaussian and Mean Curvature

#### 8.3 The Brouwer Degree of a Map: A Problem Set

#### 8.4 Area, Mean Curvature, and Soap Bubbles

* [[area]]

* [[mean curvature]]

#### 8.5 Gauss' _Theorema Egregium_

* [[theorema egregium]]

#### 8.6 Geodesics

* [[geodesic]]

#### 8.7 The Parallel Displacement of Levi-Civita

* [[parallel transport]]

* [[Levi-Civita connection]]

### 9 Covariant Differentiation and Curvature

#### 9.1 Covariant Differentiation

* [[covariant derivative]]

#### 9.2 The Riemannian Connection

* [[Levi-Civita connection]]

#### 9.3 Cartan's Exterior Covariant Differential

#### 9.4 Change of Basis anf Gauge Transformations

* [[basis]]

* [[gauge transformation]]

#### 9.5 The Curvature Forms in a Riemannian Manifold

#### 9.6 Parallel Displacement and Curvature on a Surface

#### 9.7 Riemann's Theorem and Horizontal Distribution

### 10 Geodesics

#### 10.1 Geodesics and Jacobian Fields

* [[geodesic]]

#### 10.2 Variational Principles in Mechanics

* [[variational calculus]]

* [[Euler-Lagrange equations]]

#### 10.3 Geodesics, Spiders, and the Universe

* [[observable universe]]

### 11 Relativity, Tensors, and Curvature

#### 11.1 Heuristic's of Einstein's Theory

* [[general covariance]]

#### 11.2 Tensor analysis

#### 11.3 Hilbert's Action Principle

* [[Einstein-Hilbert action]]

#### 11.4 The Second Fundamental Form in the Riemannian Case

#### 11.5 The Geometry of Einstein's Equations

* [[Einstein's equations]]

### 12 Curvature and Topology: Synge's Theorem

### 13 Betti Numbers and De Rham's Theorem

#### 13.1 Singular Chains and Their Boundaries

* [[singular chain complex]]

#### 13.2 The Singular Homology Groups

* [[singular homology]]

#### 13.3 Homology Groups of Familiar Manifolds

* [[Betti number]]

#### 13.4 De Rham's Theorem

* [[de Rham theorem]]

### 14 Harmonic Forms

#### 14.1 The Hodge Operators

* [[Hodge star operator]]

#### 14.2 Harmonic Forms

* [[harmonic differential form]]

#### 14.3 Boundary Values, Relative Homology, and Morse Theory

* [[relative homology]]

* [[Morse theory]]

## II Lie Groups, Bundles and Chern Forms

### 15 Lie Groups

#### 15.1 Lie Groups, Invariant Vector Fields and Forms

* [[Lie group]]

* [[Maurer-Cartan form]]

#### 15.2 One Parameter Subgroups

#### 15.3 The Lie Algebra of a Lie Group

* [[Lie algebra]]

* [[Lie theory]]

#### 15.4 Subgroups and Subalgebras

* [[subgroup]]

### 16 Vector Bundles in Geometry and Physics

#### 16.1 Vector Bundles

* [[vector bundle]]

#### 16.2 Poincaré's Theorem and the Euler Characteristic

* [[Euler characteristic]]

#### 16.3 Connections in a Vector Bundle

* [[connection on a vector bundle]]

#### 16.4 The Electromagnetic Connection

* [[line bundle with connection]]

### Fiber Bundles, Gauss-Bonnet, and Topological Quantization

#### 17.1 Fiber Bundles and Principal Bundles

* [[fiber bundle]]

* [[principal bundle]]

#### 17.2 Coset Spaces

* [[coset]]

#### 17.3 Chern's Proof of the Gauss-Bonnet-Poincaré Theorem

* [[Gauss-Bonnet theorem]]

#### 17.4 Line Bundles, Topological Quantization, and Berry Phase

* [[line bundle]]

* [[Berry phase]]

### 18 Connections and Associated Bundles

#### 18.1 Forms with Values in a Lie Algebra

* [[Lie algebra valued 1-form]]

#### 18.2 Associated Bundles and Connections

* [[associated bundle]]

* [[connection on a bundle]]

#### 18.3 $r$-Form Sections of a Vector Bundle: Curvature

* [[section]]

* [[curvature]]

### 19 The Dirac Equation

#### 19.1 The Groups $SO(3)$ and $SU(2)$

* [[special orthogonal group]]

* [[special unitary group]]

#### 19.2 Hamilton, Clifford, and Dirac

* [[Clifford algebra]]

* [[spinor]]

#### 19.3 The Dirac Algebra

* [[Lorentz group]]

#### 19.4 The Dirac Operator in Minkowski Space

* [[Dirac operator]]

#### 19.5 The Dirac Operator in Curved Space-Time

### 20 Yang-Mills Fields

#### 20.1 Noether's Theorem for Internal Symmetries

* [[Noether's theorem]]

#### 20.2 Weyl's Gauge Invariance Revisited

* [[gauge invariance]]

#### 20.3 The Yang-Mills Nucleon

* [[nucleon]]

#### 20.4 Compact Groups and Yang-Mills Action

* [[compact Lie group]]

* [[Yang-Mills theory]]

#### 20.5 The Yang-Mills Equation

* [[Yang-Mills equation]]

#### 20.6 The Yang-Mills Instanton

* [[Yang-Mills instanton]]

### 21 Betti Numbers and Covering Spaces

#### 21.1 Bi-invariant Forms on Compact Groups

#### 21.2 The Fundamental Group and Covering Spaces

* [[fundamental group]]

* [[covering space]]

* [[universal covering space]]

#### 21.3 The Theorem of S.B. Myers: A Problem Set

#### 21.4 The Geometry of a Lie Group

* [[Lie group]]

### 22 Chern Forms and Homotopy Groups

#### 22.1 Chern-Forms and Winding Numbers

* [[invariant polynomial]]

* [[Chern form]]

#### 22.2 Homotopies and Extensions

* [[homotopy]]

* [[extension]]

#### 22.3 The Higher Homotopy Groups $\pi_k(M)$

* [[homotopy group]]

#### 22.4 Some Computations of Homotopy Groups

#### 22.5 Chern Forms as Obstructions

* [[obstruction]]

### Appendix A. Forms in Continuum Mechanics

### Appendix B. Harmonic Chains and Kirchhoff's Circuit Law

### Appendix C. Symmetries, Quarks, and Meson Masses

### Appendix D. Representations and Hyperelastic Bodies

### Appendix E. Orbits and Morse-Bott Theory in Compact Lie Groups

category: reference


the ideal (I:a)

if R is a ring and I an ideal in R and a an element in R we define (I:a)={x in R such that xa belong ot I


The Impact of D-Branes on Mathematics

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

This page collects links related to

* [[Gregory Moore]]:

  \linebreak

  **The Impact of D-Branes on Mathematics**

  \linebreak

  talk at [PolchinskiFest 2014](http://online.kitp.ucsb.edu/online/joefest-c14/)

  [pdf](http://www.physics.rutgers.edu/~gmoore/JOEFEST-THOUGHTS.pdf), [[Moore-ImpactOfDBranesOnMath.pdf:file]]

on the concept of [[D-branes]] in  [[mathematical physics]] and [[mathematics]].

Related:

* Greg Moore, _[[Physical Mathematics and the Future]]_, 2014

See also the _[[string theory FAQ]]_

category: reference


The Interpretation of Quantum Mechanics

This page collects links related to the book

* [[Roland Omnès]]:

  \linebreak

  **The Interpretation of Quantum Mechanics**

  \linebreak

  Princeton University Press (1994)

  ([ISBN:9780691036694](http://press.princeton.edu/titles/5525.html))

on the [[interpretation of quantum mechanics]] ([[philosophy of physics|philosophy of]] [[quantum mechanics]]).

#Index#
* table of contents
{:toc}

## Elementary Quantum Mechanics

* [[quantum mechanics]]

## The Problems of Measurement Theory

* [[measurement]]

* [[Stern-Gerlach experiment]]

## Foundations and Properties

* [[quantum state]]

* [[quantum observable]]

* [[Gleason's theorem]]

## Histories

## The Logical Framework of Quantum Mechanics

* [[quantum logic]]

## Recovering Classical Physics

* [[classical limit]]

## Decoherence

* [[decoherence]]

## Measurement theory

* [[quantum measurement]]

## Questioning Quantum Mechanics

* [[Einstein-Podolsky-Rosen paradox]]

## Nonclassical Macroscopic Systems

## Experiments

## Summary and Outlook

\linebreak

***

category: reference


The join construction

* [[Egbert Rijke]],

  **The join construction**

  preprint (2017)

  [arXiv:1701.07538](https://arxiv.org/abs/1701.07538)

on the [[join of maps]] in [[homotopy type theory]].

> **Abstract.** In [[homotopy type theory]] we can define the [[join of maps]] as a [[binary operation]] on [[maps]] with a common [[co-domain]]. This operation is [[commutative]], [[associative]], and the unique map from the [[empty type]] into the common codomain is a [[neutral element]]. Moreover, we show that the [[idempotents]] of the join of maps are precisely the [[embeddings]], and we prove the 'join connectivity theorem', which states that the [[connectivity]] of the join of maps equals the join of the connectivities of the individual maps.
We define the [[image]] of a [[map]] $f:A \to X$ in $U$ via the join construction, as the [[colimit]] of the finite join powers of $f$. The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity.
A modified version of the join construction can be used to show that for any map $f:A\to X$ in which $X$ is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of [[hset|set]]-[[quotients]], the [[Rezk completion]] of a [[precategory]], and we define the $n$-[[truncation]] for any $n: \mathbb{N}$. Thus we see that each of these are definable operations on a [[univalence|univalent]] [[type universe|universe]] for Martin-Löf type theory with a natural numbers object, that is moreover closed under homotopy [[coequalizers]].

category: reference


the little n-disk operad is formal

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Statement

The [[little n-disk operad]] is [[formal dg-algebra|formal]] in the sense that for each of its component [[topological spaces]] there is a [[zig-zag]] of [[quasi-isomorphisms]] between their [[de Rham cohomology]] and their [[de Rham complex]], and such that these morphisms are compatible with the induced [[cooperad]]-[[structure]] on both sides.

Concretely, the [[zig-zags]] may be taken to consist of one [[span]] of [[quasi-isomorphisms]] out of a suitable [[graph complex]] to the [[de Rham cohomology]]/[[de Rham complex]] of the [[Fulton-MacPherson operad]], which in turn is [[weak equivalence|weakly equivalent]] to the [[little n-disk operad]] ([this Prop.](Fulton-MacPherson+operad#WeakEquivalenceToLittleNDiskOperad)).

Here the morphism from the [[graph complex]] to the [[de Rham complex]] of the [[Fulton-MacPherson operad]] regards the latter as the [[compactification]] of a [[configuration space of points]], regards [[functions]]/[[differential forms]] on [[configuration spaces of points]] as [[n-point functions]] of a [[topological quantum field theory|topological]] [[quantum field theory]], regards suitable [[graphs]] as [[Feynman diagrams]] and proceeds by sending each such graph/Feynman diagram to a corresponding [[Feynman amplitude]].

This idea of a proof was sketched in [Kontsevich 99](#Kontsevich99), a full account is due to [Lambrechts-Volic 14](#LambrechtsVolic14).

## Related concepts

* [[Kontsevich formality]]

## References

The special case of formality of $E_2$ (see also _[[Kontsevich formality]]_) was finally proven in

* [[Dmitry Tamarkin]], _Another proof of M. Kontsevich formality theorem_ ([arXiv:math/9803025](https://arxiv.org/abs/math/9803025))

* [[Dmitry Tamarkin]], _Formality of Chain Operad of Small Squares_

See [Kontsevich 99, p. 15](#Kontsevich99) for the history of this result.

A general proof of formality of $E_n$ for all $n$ was sketched in

* {#Kontsevich99} [[Maxim Kontsevich]], section 3, around Def. 15 and Lemma 3 in _Operads and Motives in Deformation Quantization_, Lett.Math.Phys.48:35-72,1999 ([arXiv:math/9904055](https://arxiv.org/abs/math/9904055))

and fully spelled out in

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society ; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

Further discussion of the graph complex as a model for the [[de Rham cohomology]] of  [[configuration spaces of points]] is in

* {#CamposWillwacher16} [[Ricardo Campos]], [[Thomas Willwacher]], _A model for configuration spaces of points_ ([arXiv:1604.02043](https://arxiv.org/abs/1604.02043))

* [[Ricardo Campos]], Najib Idrissi, [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, Najib Idrissi, [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

The following paper constructs a canonical chain of formality quasiisomorphisms for the operad of chains on framed little disks and the operad of chains on little disks. The construction is done in terms of logarithmic algebraic geometry and is
remarkable for being rational (and indeed definable integrally) in de Rham
cohomology:

* [[Dmitry Vaintrob]], _Formality of little disks and algebraic geometry_, [arXiv:2103.15054](https://arxiv.org/abs/2103.15054).

[[!redirects formality of the little n-disk operad]]

[[!redirects the Fulton-MacPherson operad is formal]]
[[!redirects formality of the Fulton-MacPherson operad]]


the logic T(m)

# The epistemic logics $T$ and $T_{(m)}$
* table of contents
{:toc}

## Idea

The flavor of _[[modal logic]]_ called $T$ is [[propositional logic]] equipped with a single [[modality]] usually written "$\Box$" subject to the rules that for all [[propositions]] $p, q \colon Prop$ we have

1. $\Box K \colon \Box(p \to q) \to (\Box p \to \Box q)$ ([[K modal logic]])

1. $\Box T \colon \Box p \to p$ (T modal logic)

Often one considers adding one more axiom:

* $\Box 4 \colon \Box p \to \Box \Box p$. ([[S4 modal logic]]).

In some applications especially in Artificial Intelligence,  $\Box p$ is interpreted as representing a statement that some (fixed) 'agent' knows the proposition $p$. It is then a short step to handling the idea of knowledge in a [[knowledge in a multiagent system|multiagent system]] where there may be $m$ different 'agents'.  In that setting, $\Box_i p$ interprets as 'agent $i$ knows that $p$'.  This leads to modelling of the passage of the interchange of information between neighbours, e.g. we might have three agents and the proposition that 'agent 2 knows that agent 3 does not know $p$'.

The [[the logic K(m)|basic epistemic logics]], $K$ and $K_{(m)}$, do not reflect much of our intuition of 'knowledge'.  The $K$-axiom merely says that, if an agent knows $\phi$ and also that agent knows $\phi\to \psi$, then the agent knows $\psi$.  There are a series of additional axioms proposed as being appropriate for knowledge, (although, it seems, each has their supporters and detractors!)  These are called $T$, $(4)$, and $B$ (and please don't ask why, ... each has its own history).

## The axioms $T_i$

This is found in two equivalent forms

* $p\to M_i p$

and

* $K_i p\to p$.

The first interprets as _if $p$ is true, then agent $i$ considers it possible_ and the second as _atomic statements known by agent $i$ are  true_ .

## The logics $T$ and $T_{(m)}$

These logics are generated by $K$ (resp. $K_{(m)}$) and the axiom $T$, (resp. axioms $T_i$ for each $i = 1,\ldots m$.

## Semantics

First looking in the monomodal case, suppose that we have a frame $\mathfrak{F} = (W,R)$ then

+-- {: .un_prop}
###### Proposition

$\mathfrak{F}\models T$ if and only if $\mathfrak{F} \models  \forall w\in W,\; R w w$.
=--

So the frames that support models for the logic $T$ are exactly the reflexive frames.
+-- {: .un_proof}
###### Proof

Suppose $\mathfrak{F}$ is a reflexive frame and take an arbitrary valuation $V$ on $\mathfrak{F}$ and a state $w$ in $\mathfrak{F}$ so that $(\mathfrak{F},V),w\models p$. We use the first form of $T$ above, and have to prove that $M p$ holds at $w$, i.e., that $p$ holds at some state 'accessible' from $w$, but as $R$ is reflexive, $w$ is accessible from itself, ... .

For the converse, we will suppose $R$ is not reflexive, so there is some state, $w \in W$ which is not $R$-related to itself. We will falsify the formula $T$ if we can find a valuation $V$ and a state $v$ such that $p$ holds at $v$ but $M p$ does not. (Recall the semantics of $M$: $\mathfrak{M},w \models M \phi$ if and only if, for some $v \in W$ such that $ R w v$, $\mathfrak{M},v \models \phi$.)

We need a state with this property and we only know about one namely $w$, so that is the obvious to try! We need a valuation such that 1) $w\in V(p)$ and 2) $\{x\in W\mid R w x\}\cap V(p)= \empty$. If we set $V(p) = \{w\}$ this works since $w$ is not related to itself. (Other values of $V$ are irrelevant.) If $w$ has no $R$-successors, then we are finished since clearly in that case, $\neg(w \models M \phi)$, so suppose that $v$ is any $R$-successor of $w$, i.e., $R w v$, then $w\neq v$, so $\neg(v\models p)$, hence $\neg(w\models M p)$ as required.$\blacksquare$
=--

## Related entries

[[!include flavors of modal logics and relations -- table]]

* [[epistemic modal logic]]

* [[necessity and possibility]]

[[!redirects the logic T(m)]]
[[!redirects The logic T(m)]]
[[!redirects T(m)]]

[[!redirects T modal logic]]


The many faces of the superworld

This page collect pointers related to the references

* [[Mikhail Shifman]] (ed.)

  _The Many Faces of the Superworld_

  [[Yuri Golfand]] memorial volume

  World Scientific, 2000

  [doi:10.1142/4332](https://doi.org/10.1142/4332)

on [[superalgebra]], [[supergeometry]], [[supersymmetry]], [[supergravity]] and  [[superstrings]].

**Contents**

Introducing the [[super Poincaré Lie algebra]] ("[[supersymmetry]]"):

* {#GolfandLikhtman72} [[Yuri Golfand]], [[Evgeny Likhtman]],_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: [[Victor Ginzburg]] et al. (eds.) _I. E. Tamm Memorial Volume Problems of Theoretical Physics_, (Nauka, Moscow 1972), page 37,

  translated and reprinted in: [[Mikhail Shifman]] (ed.) _[[The Many Faces of the Superworld]]_ pp. 44-53,  World Scientific (2000) ([doi:10.1142/9789812793850_0006](https://doi.org/10.1142/9789812793850_0006))

On the [[DBI action]] for [[D-branes]]:

* [[Arkady Tseytlin]], _Born-Infeld action, supersymmetry and string theory_, in: [[Mikhail Shifman]] (ed.)  _[[The many faces of the superworld]]_, pp. 417-452, World Scientific (2000) ([arXiv:hep-th/9908105](https://arxiv.org/abs/hep-th/9908105), [doi:10.1142/9789812793850_0025](https://doi.org/10.1142/9789812793850_0025))

On [[type I' string theory]]:

* {#Schwarz00} [[John Schwarz]], _Some Properties of Type I' String Theory_, in: [[Mikhail Shifman]] (ed.), _[[The Many Faces of the Superworld]]_, pp. 388-397 (2000) ([arXiv:hep-th/9907061](https://arxiv.org/abs/hep-th/9907061), [doi:10.1142/9789812793850_0023](https://doi.org/10.1142/9789812793850_0023))

On [[super Yang-Mills theory]]:

* {#KacSmilga00} [[Victor Kac]], A.V. Smilga, _Vacuum structure in supersymmetric Yang-Mills theories with any gauge group_, in _[[The Many Faces of the Superworld]], pp. 185-234 World Scientific (2000)_ ([arXiv:hep-th/9902029](https://arxiv.org/abs/hep-th/9902029), [doi:10.1142/9789812793850_0014](https://doi.org/10.1142/9789812793850_0014))

On the [[Witten index]] of [[D=3 N=1 super Yang-Mills theory]]:

* [[Edward Witten]], _Supersymmetric Index Of Three-Dimensional Gauge Theory_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9903005](https://arxiv.org/abs/hep-th/9903005))

On [[multi-trace operators]] in the [[AdS/CFT correspondence]]:

* [[Eric D'Hoker]], [[Daniel Freedman]], [[Samir Mathur]], A. Matusis, [[Leonardo Rastelli]], _Extremal Correlators in the AdS/CFT Correspondence_, in: _[[The many faces of the superworld]]_ ([arXiv:hep-th/9908160](https://arxiv.org/abs/hep-th/9908160))

* ...

category: reference

[[!redirects The Many Faces of the Superworld]]


The Multifaceted Skyrmion

This page collectes material related to the book

* [[Mannque Rho]], [[Ismail Zahed]] (eds.)

  _The Multifaceted Skyrmion_,

  World Scientific

  Second edition, 2016

  [doi:10.1142/9710](https://doi.org/10.1142/9710)

on [[Skyrmions]], particularly as a model for [[confinement]] in [[QCD]] of [[quarks]] within [[hadrons]] (but also in [[solid state physics]]) and their origin as [[instantons]] of a [[AdS/CFT|holographic]] 5d [[Yang-Mills theory]] ([[holographic QCD]], [[AdS/QCD correspondence]]).

#Contents#
* table of contents
{:toc}

### Introduction

* {#RhoEtAl16} [[Mannque Rho]] et al., _Introduction_

> One can make $[$[[chiral perturbation theory]]$]$ consistent with [[QCD]] by suitably matching the [[correlators]] of the [[effective field theory|effective theory]] to those of [[QCD]] at a [[scale]] near $\Lambda$. Clearly this procedure is not limited to only one set of [[vector mesons]]; in fact, one can readily generalize it to an infinite number of hidden [[gauge fields]] in an [[effective field theory|effective]] [[Lagrangian density|Lagrangian]]. In so doing, it turns out that a fifth dimension is "deconstructed" in a (4+1)-dimensional (or 5D) [[Yang-Mills theory|Yang–Mills]] type form. We will see in Part III that such a structure arises, [[top-down model building|top-down]], in [[string theory]].

> $[...]$

> $[$this [[holographic QCD]]$]$ model comes out to describe — unexpectedly well — low-energy properties of both [[mesons]] and [[baryons]], in particular those properties reliably described in quenched [[lattice QCD]] simulations.

> $[...]$

> One of the most noticeable results of this [[AdS/QCD|holographic model]] is the first derivation of [[vector meson dominance|vector dominance]] (VD) that holds both for [[mesons]] and for [[baryons]]. It has been somewhat of an oddity and a puzzle that [[Jun John Sakurai|Sakurai's]] [[vector meson dominance|vector dominance]] — with the lowest [[vector mesons]] [[rho meson|ρ]] and ω — which held very well for [[pion|pionic]] [[form factors]] at low momentum transfers famously failed for [[nucleon]] [[form factors]]. In this [[AdS/QCD|holographic model]], the [[vector meson dominance|VD]] comes out automatically for both the [[pion]] and the [[nucleon]] provided that the infinite $[$[[Kaluza-Klein mechanism|KK-]]$]$tower is included. While the [[vector meson dominance|VD]] for the [[pion]] with the infinite tower is not surprising given the successful Sakurai VD, that the [[vector meson dominance|VD]] holds also for the [[nucleons]] is highly nontrivial. $[...]$ It turns out to be a consequence of a [[AdS/QCD|holographic]] [[Cheshire cat principle|Cheshire Cat phenomenon]]

### Hadrons and Nuclear Matter

**Chapter 1**

On [[Skyrmions]] as [[model (in theoretical physics)|models]] for [[atomic nuclei]] all the way up to [[carbon]]:

* {#BattyeMantonSutcliffe10} [[Richard A. Battye]], [[Nicholas Manton]], [[Paul Sutcliffe]], _Skyrmions and Nuclei_, pp. 3-39 (2010) ([doi:10.1142/9789814280709_0001](https://doi.org/10.1142/9789814280709_0001))

\linebreak

**Chapter 2**

* {#Holzwarth05} Gottfried Holzwarth, _Electromagnetic Form Factors of the Nucleon in Chiral Soliton Models_ ([arXiv:hep-ph/0511194](https://arxiv.org/abs/hep-ph/0511194)), Chapter 2 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

\linebreak

**Chapter 3**

* [[Dmitri Diakonov]], [[Victor Petrov]], _Exotic baryon resonances in the Skyrme model_ ([arXiv:0812.1212](https://arxiv.org/abs/0812.1212), [doi:10.1142/9789814704410_0004](https://doi.org/10.1142/9789814704410_0004)), Chapter 3 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> It is astounding that [[Tony Skyrme|Skyrme]] had suggested [[Skyrmion|his model]] as early as in 1961 before it has been generally accepted that [[pions]] are (pseudo) [[Goldstone bosons]] associated with the [[spontaneous symmetry breaking|spontaneous breaking]] of [[chiral symmetry]], and of course long before [[QCD|Quantum Chromodynamics (QCD)]] has been put forward as the microscopic theory of [[strong nuclear force|strong interactions]].

> The revival of the Skyrme idea in 1983 is [due to Witten](Skyrmion#Witten83) who explained the _raison d'ˆetre_ of the [[Skyrmion|Skyrme model]] from the viewpoint of [[QCD]]. In the [[chiral perturbation theory|chiral limit]] when the light [[quark]] masses $m_u$, $m_d$, $m_s$ tend to zero, such that the octet of the pseudoscalar mesons [[pion|π]], [[kaon|K]] , η become nearly massless (pseudo) [[Goldstone bosons]], they are the lightest degrees of freedom of [[QCD]]. The [[effective field theory|effective]] chiral Lagrangian (EχL) for pseudoscalar mesons, understood as an infinite expansion in the derivatives of the pseudoscalar (or chiral) fields, encodes, in principle, full information about QCD. The famous two-term Skyrme Lagrangian can be understood as a low-energy truncation of this infinite series. Witten has added an important four-derivative [[WZW-term|Wess–Zumino term]] to the original Skyrme Lagrangian and pointed out that the overall coefficient in front of the EχL is proportional to the number of quark [[color charge|colours]] $N_c$.

> $[...]$

> Soon after Witten's work it has been realized that it is possible to bring the Skyrme model and the Skyrmion even closer to QCD and to the more customary language of constituent quarks. It has been first noticed $[$[6](#DiakonovEides83), [7a](#DharWadia84), [7b](#DarShankarWadia85), [8](#DiakonovPetrov86)$]$ that a simple chiral invariant Lagrangian for massive (constituent) quarks $Q$ interacting with the octet chiral field $\pi_A$ $(A = 1, ..., 8)$,

> $$\mathcal{L} = \overline{Q} \left( \partial\!\!\!\!/ - M e^{ \tfrac{i \pi^A \lambda^A \gamma_5}{F_\pi}  }  \right) Q$$

> induces, via a quark loop in the external pseudoscalar fields (see Fig. 3.1), the EχL whose lowest-derivative terms coincide with the Skyrme Lagrangian, including
automatically the Wess–Zumino term, with the correct coefficient!

> $[...]$

> The condition that the winding number of the trial field is unity needs to be imposed to get a deeply bound state, that is to guarantee that the baryon number is unity. $[$[9](#DiakonovPetrovPobylitsa88)$]$ The Skyrmion is, thus, nothing but the **mean chiral field binding quarks in a baryon**.

\linebreak

* {#DiakonovEides83} [[Dmitri Diakonov]], Michael I. Eides, _Chiral Lagrangian from a functional integral over quarks_, JETP Letters 38.7 (1983): 433-436 ([pdf](http://jetpletters.ac.ru/ps/1483/article_22635.pdf), [[DiakonovEides83.pdf:file]])

* {#DharWadia84} A. Dhar, Spenta R. Wadia, _Nambu—Jona-Lasinio Model: An Effective Lagrangian for Quantum Chromodynamics at Intermediate Length Scales_, Phys. Rev. Lett. 52, 959 (1984) ([doi:10.1103/PhysRevLett.52.959](https://doi.org/10.1103/PhysRevLett.52.959))

* {#DarShankarWadia85} Avinash Dhar, R. Shankar, Spenta R. Wadia, _Nambu–Jona-Lasinio–type effective Lagrangian: Anomalies and nonlinear Lagrangian of low-energy, large-N QCD_, Phys. Rev. D 31, 3256 (1985) ([doi:10.1103/PhysRevD.31.3256](https://doi.org/10.1103/PhysRevD.31.3256))

* {#DiakonovPetrov86} [[Dmitri Diakonov]], [[Victor Petrov]], _A theory of light quarks in the instanton vacuum_, Nuclear Physics B Volume 272, Issue 2, 21 July 1986, Pages 457-489 (doi:10.1016/0550-3213(86)90011-8)

* {#DiakonovPetrovPobylitsa88} [[Dmitri Diakonov]], [[Victor Petrov]], P.V. Pobylitsa, _A Chiral Theory of Nucleons_, Nucl. Phys. B306 (1988) 809 ([spire:247700](http://inspirehep.net/record/247700), doi:10.1016/0550-3213(88)90443-9)

\linebreak

**Chapter 4**

On [[Skyrmions]] with [[heavy mesons]]:

* [[Norberto Scoccola]], _Heavy quark skyrmions_, ([arXiv:0905.2722](https://arxiv.org/abs/0905.2722), [doi:10.1142/9789814280709_0004](https://doi.org/10.1142/9789814280709_0004)), Chapter 4 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

\linebreak

**Chapter 5**

On the [[Skyrmion model]] in [[thermal QFT]], with emphasis on the [[omega-meson]]/[[baryon current]]-coupling:

* {#ParkVento09} Byung-Yoon Park, Vicente Vento, _Skyrmion approach to finite density and temperature_ ([arXiv:0906.3263](https://arxiv.org/abs/0906.3263), [doi:10.1142/9789814280709_0005](https://doi.org/10.1142/9789814280709_0005)), Chapter 5 in:  _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

\linebreak

**Chapter 6**

On [[mesons]] as [[gauge bosons]] of a [[hidden local symmetry]] and then as [[Kaluza-Klein mechanism|KK-modes]] of a 5-dimensional holographic [[Yang-Mills theory]]:

* Hyun Kyu Lee, [[Mannque Rho]], _Half-Skyrmion Hadronic Matter at High Density_ ([doi:10.1142/9789814280709_0006](https://doi.org/10.1142/9789814280709_0006)), Chapter 6 in: [[Mannque Rho]] et al., _The Multifaceted Skyrmion_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

### Condensed matter

On [[Skyrmions]] in [[condensed matter physics]]:

(...)

### String theory

On the [[skyrmion]] [[model (in theoretical physics)|model]] for [[baryons]] appearing within the [[Sakai-Sugimoto model]] of [[holographic quantum chromodynamics]] in [[string theory]], as the [[boundary field theory]] of [[D=5 super Yang-Mills theory]]:

**Chapter 15**

* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmions and String theory_

**Chapter 16**

* {#Yi09} [[Piljin Yi]], _Holographic Baryons_ ([arXiv:0902.4515](https://arxiv.org/abs/0902.4515), [doi:10.1142/9789814280709_0016](https://www.worldscientific.com/doi/abs/10.1142/9789814280709_0016)), Chapter 16 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

> [[QCD]] is a challenging theory. Its most interesting aspects, namely the [[confinement]] of [[color charge|color]] and the [[chiral symmetry breaking]], have defied all analytical approaches. While there are now many data accumulated from the [[lattice gauge theory]], the methodology falls well short of giving us insights on how one may understand these phenomena analytically, nor does it give us a systematic way of obtaining a low energy theory of [[QCD]] below the [[confinement]] [[scale]].

> $[...]$

> it has been proposed early on that [[baryons]] are topological [[solitons]], namely [[Skyrmions]]  $[$but$]$ the usual [[Skyrmion]] picture of the [[baryon]] has to be modified significantly in the context of full [[QCD]]. $[...]$ the holographic picture naturally brings a gauge principle in the bulk description of the [[flavor physics|flavor]] dynamics in such a way that all spin one [[mesons]] as well as [[pions]] would enter the $[$ [[Skyrmion|skyrmionic]]-$]$construction of [[baryons]] on the equal footing.

> $[...]$

> [[holographic QCD]] is similar to the [[chiral perturbation theory]] in the sense that we deal with exclusively [[gauge invariance|gauge-invariant]] [[observables|operators]] of the theory. The huge difference is, however, that this new approach tends to treat all [[gauge invariance|gauge-invariant]] objects together. Not only the light [[meson]] [[field (physics)|fields]] like [[pions]] but also heavy [[vector mesons]] and [[baryons]] appear together, at least in principle. In other words, a [[holographic QCD]] deals with all [[color charge|color]]-[[trivial representation|singlets]] simultaneously, giving us a lot more predictive power.

> $[...]$

> The expectation that there exists a more intelligent theory consisting only of [[gauge invariance|gauge-invariant]] objects in the [[large N limit|large Nc limit]] is thus realized via [[string theory]] in a somewhat surprising manner that the master fields, those truly physical degrees of freedom, actually live not in four dimensional Minkowskian world but in five or
higher dimensional curved geometry. This is not however completely unanticipated, and was heralded in the celebrated work by [Eguchi and Kawai in early 1980's](#EguchiKawai82) which is all the more remarkable in retrospect.

> $[...]$

> To compare against actual [[QCD]], we must fix $[$the [['t Hooft coupling]] and the [[Kaluza-Klein compactification|KK]]-[[scale]]$]$ to fit
both the [[pion]] decay constant $f_\pi$ and the [[mass]] of the first [[vector meson]]. After this fitting, all other infinite number of [[masses]] and [[coupling constants]] are fixed. This version $[$the holographic [[WSS model]]$]$ of the [[holographic QCD]]  is extremely predictive.

> $[...]$

> $[$this$]$ elevates the classic [[skyrmion|Skyrme picture]] based on [[pions]] to a unified model involving all [[vector meson|spin one mesons]] in addition to [[pions]]. This is why the picture is extremely predictive.

> As we saw in this note, for low momentum processes, such as soft [[pion]] [[scattering amplitude|processes]], soft [[rho meson]] exchanges, and soft elastic scattering of [[photons]], the $[$holographic [[WSS model|WSS]]-$]$model's predictions compare extremely well with experimental data. It is somewhat mysterious that the [[baryon]] sector works out almost as well as the [[meson]] sector

\linebreak

**Chapter 17**

On deriving the [[Cheshire cat principle]] of the [[quark bag model]] from [[AdS/QCD]]:

* [[Holger Bech Nielsen]], [[Ismail Zahed]], _The Cheshire Cat Principle from Holography_ ([doi:10.1142/9789814280709_0017](https://doi.org/10.1142/9789814280709_0017), [arXiv:0904.0429](https://arxiv.org/abs/0904.0429)), Chapter 17 in: _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

\linebreak

**Chapter 18**

On deriving [[baryon]]-[[masses]] from [[holographic QCD]]:

* {#PomarolWulzer09} Alex Pomarol, Andrea Wulzer, _Baryon physics in a five-dimensional model of hadrons_ ([arXiv:0904.2272](https://arxiv.org/abs/0904.2272)), Chapter 18 in: [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

(...)

category: reference


* Stewart Shapiro, _[[The Oxford handbook of philosophy of mathematics and logic]]_, Oxford University Press 2005
[doi](http://dx.doi.org/10.1093/oxfordhb/9780195325928.001.0001)

Contents:

> After the Introduction (chapter 1), the book begins with a historical section, consisting of three chapters. Chapter 2 deals with the modern period—Kant and his intellectual predecessors; chapter 3 concerns later empiricism, including John Stuart Mill and logical positivism; and chapter 4 focuses on Ludwig Wittgenstein.

> The volume then turns to the "big three" views that dominated the philosophy and foundations of mathematics in the early decades of the twentieth century: logicism, formalism, and intuitionism. There are three chapters on logicism, one dealing with the emergence of the program in the work of Frege, Russell, and Dedekind (chapter 5); one on neologicism, the contemporary legacy of Fregean logicism (chapter 6); and one called "Logicism Reconsidered," which provides a technical assessment of the program in its first century (chapter 7). This is followed by a lengthy chapter on formalism, covering its historical and philosophical aspects (chapter 8). Two of the three chapters on intuitionism overlap considerably. The first (chapter 9) provides the philosophical background to intuitionism, through the work of L. E. J. Brouwer, Arend Heyting, and others. The second (chapter 10) takes a more explicitly mathematical perspective. Chapter 11, "Intuitionism Reconsidered," focuses largely on technical issues concerning the logic.

> The next section of the volume deals with views that dominated in the later twentieth century and beyond. Chapter 12 provides a sympathetic reconstruction of Quinean holism and indispensability. This is followed by two chapters that focus directly on naturalism. Chapter 13 lays out the principles of some prominent naturalists, and chapter 14 is critical of the main themes of naturalism. Next up are nominalism and structuralism, which get two chapters each. One of these is sympathetic to at least one variation on the view in question, and the other "reconsiders."

> Chapter 19 is a detailed and sympathetic treatment of a predicative approach to both the philosophy and the foundations of mathematics. This is followed by an extensive treatment of the application of mathematics to the sciences; chapter 20 lays out different senses in which mathematics is to be applied, and draws some surprising philosophical conclusions.

> The last six chapters of the volume focus more directly on logical matters, in three pairs. There are two chapters devoted to the central notion of logical consequence. Chapter 21 presents and defends the role of semantic notions and model theory, and chapter 22 takes a more "constructive" approach, leading to proof theory. The next two chapters deal with the so‐called paradoxes of relevance, chapter 23 arguing that the proper notion of logical consequence carries a notion of relevance, and chapter 24 arguing against this. The final two chapters concern higher‐order logic. Chapter 25 presents higher‐order logic and provides an overview of its various uses in foundational studies. Of course, chapter 26 reconsiders.


The Principles of Quantum Mechanics

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### Context
#### Physics
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#### Quantum field theory
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[[!include functorial quantum field theory - contents]]
=--
=--
=--

This entry collects hyperlinks related to the classic textbook

* [[Paul Dirac]]:

  \linebreak

  **The Principles of Quantum Mechanics**

  \linebreak

  International series of monographs on physics,

  Oxford University Press (1930, 1935, 1947)

  [ISBN:9780198520115](https://global.oup.com/academic/product/the-principles-of-quantum-mechanics-9780198520115)

  [GoogleBooks entry](https://books.google.de/books/about/The_Principles_of_Quantum_Mechanics.html?id=XehUpGiM6FIC&redir_esc=y)

  [Wikipedia entry](http://en.wikipedia.org/wiki/The_Principles_of_Quantum_Mechanics)

on [[quantum mechanics]] (introducing the "$\hbar$"-notation for the reduced [[Planck's constant]]) and (from the third edition on) a bit of [[quantum field theory]] ([[quantum electrodynamics]]).

category: reference


The QED Project

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Foundations
+-- {: .hide}
[[!include foundations - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _QED project_ was a project -- or at least a manifesto for a project, see [below](TheQED) -- for the formalization of large parts of [[mathematics]] ([[formal proof]]) in formal language supported by [[proof assistants]].

* [The QED Project Webpage](http://mizar.org/qed/)

While as a project this seems to have died out around 1996, the idea of the manifesto itself is still pertinent and waiting for its implementation (see also [Wiedijk 07](#Wiedijk07)).

Notice that since then the issues discussed in ([Wiedijk 07](#Wiedijk07)) may appear in a different light in view of developments in [[homotopy type theory]] ("univalent foundations for mathematics").

## The QED manifesto
 {#TheQED}

The following is taken literally from _[The QED Manifesto](ftp://mizar.uwb.edu.pl/pub/qed/manifesto)_.

> The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. -- [[Kurt Gödel]]

> If civilization continues to advance, in the next two thousand years the overwhelming novelty in human thought will be the dominance of mathematical understanding.  -- [[Alfred North Whitehead]]

### 1 What Is the QED Project and Why Is It Important?

QED is the very tentative title of a project to build a computer
system that effectively represents all important mathematical
knowledge and techniques.  The QED system will conform to the highest
standards of mathematical rigor, including the use of strict formality
in the internal representation of knowledge and the use of mechanical
methods to check proofs of the correctness of all entries in the
system.

The QED project will be a major scientific undertaking requiring the
cooperation and effort of hundreds of deep mathematical minds,
considerable ingenuity by many computer scientists, and broad support
and leadership from research agencies. In the interest of enlisting a
wide community of collaborators and supporters, we now offer reasons
that the QED project should be undertaken.

First, the increase of mathematical knowledge during the last two
hundred years has made the knowledge, let alone understanding, of all,
or even of the most important, mathematical results something beyond the
capacity of any human.  For example, few mathematicians, if any, will
ever understand the entirety of the recently settled structure of
simple finite groups or the proof of the four color theorem.
Remarkably, however, the creation of mathematical logic and the
advance of computing technology have also provided the means for
building a computing system that represents all important mathematical
knowledge in an entirely rigorous and mechanically usable fashion.
The QED system we imagine will provide a means by which mathematicians
and scientists can scan the entirety of mathematical knowledge for
relevant results and, using tools of the QED system, build upon such
results with reliability and confidence but without the need for
minute comprehension of the details or even the ultimate foundations
of the parts of the system upon which they build.  Note that the
approach will almost surely be an incremental one: the most important
and applicable results will likely become available before the more
obscure and purely theoretical ones are tackled, thus leading to a
useful system in the relatively near term.

Second, the development of high technology is an endeavor of fabulously
increasing mathematical complexity.  The internal documentation of the next
generation of microprocessor chips may run, we have heard, to thousands of
pages.  The specification of a major new industrial system, such as a
fly-by-wire airliner or an autonomous undersea mining operation, is likely to
be even an order of magnitude greater in complexity, not the least reason being
that such a system would perhaps include dozens of microprocessors.  We believe
that an industrial designer will be able to take parts of the QED system and
use them to build reliable formal mathematical models of not only a new
industrial system but even the interaction of that system with a formalization
of the external world.  We believe that such large mathematical models will
provide a key principle for the construction of systems substantially more
complex than those of today, with no loss but rather an increase in
reliability.  As such models become increasingly complex, it will be a major
benefit to have them available in stable, rigorous, public form for use by
many.  The QED system will be a key component of systems for verifying and
even synthesizing computing systems, both hardware and software.

The third motivation for the QED project is education.  Nothing is more important than mathematics education to the creation of infrastructure for technology-based economic growth.  The development of mathematical ability is notoriously dependent upon 'doing' rather than upon 'being told' or 'remembering'.  The QED system will provide, via such techniques as interactive proof checking algorithms and an endless variety of mathematical results at all levels, an opportunity for the one-on-one presenting, checking, and debugging of mathematical technique, which it is so expensive to provide by the method of one trained mathematician in dialogue with one student.  QED can provide an engaging and non-threatening framework for the carrying out of proofs by students, in the same spirit as a long-standing program of Suppes at Stanford for example.  Students will be able to get a deeper understanding of mathematics by seeing better the role that lemmas play in proofs and by seeing which kinds of manipulations are valid in which kinds of structures.  Today few students get a grasp of mathematics at a detailed level, but via experimentation with a computerized laboratory, that number will increase.  In fact, students can be used (eagerly, we think) to contribute to the development of the body of definitions and proved theorems in QED.  Let also us make the observation that the relationship of QED to education may be seen in the following broad context: with increasing technology available, governments will look not only to cut costs of education but will increasingly turn to make education and its delivery more cost-effective and beneficial for the state and the individual.

Fourth, although it is not a practical motivation, nevertheless
perhaps the foremost motivation for the QED project is cultural.
Mathematics is arguably the foremost creation of the human mind.  The
QED system will be an object of significant cultural character,
demonstrably and physically expressing the staggering depth and power
of mathematics.  Like the great pyramids, the effort required
(especially early on) may be great; but the rewards can be even more
staggering than this effort.  Mathematics is one of the most basic
things that unites all people, and helps illuminate some of the most
fundamental truths of nature, even of being itself.  In the last one
hundred years, many traditional cultural values of our civilization
have taken a severe beating, and the advance of science has received
no small blame for this beating.  The QED system will provide a
beautiful and compelling monument to the fundamental reality of truth.
It will thus provide some antidote to the degenerative effects of
cultural relativism and nihilism.  In providing motivations for
things, one runs the danger of an infinite regression.  In the end, we
take some things as inherently valuable in themselves.  We believe
that the construction, use, and even contemplation of the QED system
will be one of these, over and above the practical values of such a
system.  In support of this line of thought, let us cite Aristotle,
the Philosopher, the Father of Logic, 'That which is proper to each
thing is by nature best and most pleasant for each thing; for man,
therefore, the life according to reason is best and pleasantest, since
reason more than anything else is man.'  We speculate that this
cultural motivation may be the foremost motivation for the QED
project.  Sheer aesthetic beauty is a major, perhaps the major, force
in the motivation of mathematicians, so it may be that such a
cultural, aesthetic motivation will be the key motivation inciting
mathematicians to participate.

Fifth, the QED system may help preserve mathematics from corruption.
We must remember that mathematics essentially disappeared from Western
civilization once, during the dark ages.  Could it happen again?  We
must also remember how unprecedented in the history of mathematics is
the clarity, even perfection, that developed in this century in regard
to the idea of formal proof, and the foundation of essentially the
entirety of known mathematics upon set theory.  One can easily imagine
corrupting forces that could undermine these achievements.  For
example, one might suspect that there is already a trend towards
believing some recent "theorems" in physics because they offer some
predictive power rather than that they have any meaning, much less
rigorous proof, with a possible erosion in established standards of
rigor.  The QED system could offer an antidote to any such tendency.
The standard, impartial answer to the question 'Has it been proved?'
could become 'Has it been checked by the QED system?'.  Such a
mechanical proof checker could provide answers immune to pressures of
emotion, fashion, and politics.

Sixth, the 'noise level' of published mathematics is too high.  It
has been estimated that something between 50 and 100 thousand
mathematical papers are published per year.  Nobody knows for sure
how many contain errors or how many are repetitions, but some
pessimists claim the number of both is high.  QED can help to reduce
the level of noise, both by helping to find errors and by helping to
support computer searches for duplication.

Seventh, QED can help to make mathematics more coherent.  There are
similar techniques used in various fields of mathematics, a fact that
category theory has exploited very well.  It is quite natural for
formalizers to generalize definitions and propositions because it can
make their work much easier.

Eighth, by its insistence upon formalization, the QED project will
add to the body of explicitly formulated mathematics.  There is
mathematical knowledge that is neither taught in classes nor
published in monographs.  It is below what mathematicians call
"folklore," which is explicitly formulated.  Let us call this lower
level of unformulated knowledge "mathlore".  In formalization
efforts, we must formalize everything, and that includes mathlore
lemmas.

Ninth, the QED project will help improve the low level of
self-consciousness in mathematics.  Good mathematicians understand
trends and connections in their field.  The QED project will enable
mathematicians to analyze, perhaps statistically, the whole structure
of the mathematics, to discover new trends, to forecast developments
and so on.

## Related projects

[[!include proof assistants and formalization projects -- list]]

## Related entries

* [[formal proof]]

## References

* _The QED Manifesto_ in _Automated Deduction - CADE 12_, Springer-Verlag, Lecture Notes in Artificial Intelligence, Vol. 814, pp. 238-251, 1994. ([web](ftp://mizar.uwb.edu.pl/pub/qed/manifesto))

* [Wikipedia article](http://en.wikipedia.org/wiki/QED_manifesto)

* {#Wiedijk07} [[Freek Wiedijk]], _The QED Manifesto Revisited_, 2007 ([pdf](http://mizar.org/trybulec65/8.pdf))

[[!redirects The QED project]]


* [[Ulrik Buchholtz]], [[Egbert Rijke]]:

  **The real projective spaces in homotopy type theory**

  Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science

  LICS 2017 **86** (2017) 1–8

  [arxiv:1704.05770](https://arxiv.org/abs/1704.05770)

  [doi:10.5555/3329995.3330081](https://dl.acm.org/doi/10.5555/3329995.3330081)

on the [[real projective space]] in [[homotopy type theory]].

> **Abstract.** [[homotopy type theory|Homotopy type theory]] is a version of [[Martin-Löf dependent type theory|Martin-Löf type theory]] taking advantage of its [[relation between category theory and type theory|homotopical models]]. In particular, we can use and construct [[objects]] of [[homotopy theory]] and reason about them using [[higher inductive types]]. In this article, we construct the [[real projective spaces]], key players in [[homotopy theory]], as certain [[higher inductive types]] in [[homotopy type theory]]. The classical definition of [[real projective space|$\mathbb{R}P^n$]], as the [[quotient space]] identifying [[antipode|antipodal points]] of the [[n-sphere|$n$-sphere]], does not translate directly to homotopy type theory. Instead, we define $\mathbb{R}P^n$ by [[induction]] on $n$ simultaneously with its [[tautological line bundle|tautological bundle]] of $2$-element sets. As the base case, we take $\mathbb{R}P^{-1}$ to be the [[empty type]]. In the [[induction|inductive step]], we take $\mathbb{R}P^{n+1}$ to be the [[mapping cone]] of the projection map of the [[tautological line bundle|tautological bundle]] of $\mathbb{R}P^{n}$, and we use its universal property and the [[univalence axiom]] to define the [[tautological line bundle|tautological bundle]] on $\mathbb{R}P^{n+1}$.
By showing that the total space of the [[tautological line bundle|tautological bundle]] of $\mathbb{R}P^n$ is the [[n-sphere]], we retrieve the classical description of $\mathbb{R}P^{n+1}$ as $\mathbb{R}P(n)$ with an $(n+1)$-cell [[space attachment|attached]] to it. The infinite dimensional [[real projective space]], defined as the [[sequential colimit]] of the $\mathbb{R}P^n$ with the canonical inclusion maps, is equivalent to the [[Eilenberg-MacLane space]] $K(\mathbb{Z}/2\mathbb{Z},1)$, which here arises as the subtype of the [[universe]] consisting of $2$-element types. Indeed, the infinite dimensional [[projective space]] classifies the $0$-sphere bundles, which one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of homotopy type theory, including the interplay of [[type theory|type theoretic]] and [[homotopy theory|homotopy theoretic]] ideas.

category: reference


The Relation of Cobordism to K-Theories

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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
=--
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

This page compiles material related to the book

* [[Pierre Conner]], [[Edwin Floyd]],

  **The Relation of Cobordism to K-Theories**,

  Lecture Notes in Mathematics __28__ Springer 1966

  [doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091),

  [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511)

on [[cobordism theory]] and [[topological K-theory]], meeting in the notion of the [[e-invariant]].

#Contents#
* table of contents
{:toc}

## The Thom Isomorphism in K-theory

* [[Thom isomorphism]]

* [[topological K-theory]]

### 1. Exterior algebra

* [[exterior algebra]]

### 2. Tensor products of exterior algebras

### 3. Application to bundles

### 4. Thom classes of line bundles

* [[projective spaces]], in particular [[quaternionic projective space]]

* [[tautological line bundle]]

* [[Thom class]]

* [[Whitehead-generalized cohomology theory]]

* [[multiplicative cohomology theory]]

### 5. Cobordism and homomorphism into K-theory

* [[cobordism cohomology theory]]

* [[MFr]]$\to$[[MSp]]$\to$[[MSU]]$\to$[[MU]]

* [[Conner-Floyd orientation]]

### 6. The homomorphism $\mu_c$

* [[Thom isomorphism]]

* [[rational Todd class is Chern character of Thom class]]

## Cobordism Characteristic Classes

### 7. A theorem of Dold

* [[Leray-Hirsch theorem]]

* [[quaternionic oriented cohomology]], [[complex oriented cohomology]]

* [[Conner-Floyd Chern classes]]

### 8. Characteristic classes on cobordism

* [[MSp]]

* [[Conner-Floyd Chern class]]

### 9. Characteristic classes in K-theory

* [[Chern classes]]

* [[Chern character]]

* [[Pontrjagin character]]

### 10. A cobordism interpretation for $K^\ast(X)$

* [[Conner-Floyd isomorphism]]

  ([[Landweber exact functor theorem]] for [[KU]] and [[KSp]], see also _[[cobordism theory determining homology theory]]_)

### 11. Mappings into spheres

## $U$-Manifolds with Framed Boundaries

### 12. The $U$-bordism groups $\Omega^U_\bullet$

* [[MU]]

### 13. Characteristic numbers from K-theory

* [[Todd class]]

* [[Todd genus]]

### 14. The theorem of Stong and Hattori

### 15. $U$-Manifolds with stably framed boundaries

### 16. The bordism groups $\Omega^{U,fr}_\bullet$

* [[manifold with boundary]]

* [[MUFr]], [[(U,fr)-manifolds]]

* [[rational Todd class is Chern character of Thom class]]

* [[e-invariant]]

* [[e-invariant is Todd class of cobounding (U,fr)-manifold]]

### 17. The groups $\Omega^{\mathrm{U}, SU}_\bullet$

### 18. The image of $\Omega^{fr}_{\bullet}$ in $\Omega^{SU}_\bullet$

category: reference


The Rising Sea

# Contents
* table of contents
{: toc}

The "rising sea" is a metaphor due to [[Alexander Grothendieck]] (see the quote [below](#Idea)), meaning to illuminate how the development of [[general abstract]] theory eventually brings with it effortless solutions to [[concrete particular]] problems, much like a hard nut may be cracked not immediately by sheer punctual force, but eventually by gently immersing it into a whole body of water.

Following this metaphor, "The Rising Sea" was chosen as the name of a [website](http://therisingsea.org/) of [[Daniel Murfet]] with notes on [[algebraic geometry]] as developed in the school of Grothendieck.

Additionally, "The Rising Sea" is the name of a [textbook on algebraic geometry](http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf)
by [[Ravi Vakil]].

## Idea
 {#Idea}

As a metaphor for his approach of problem solving via theory building, [[Alexander Grothendieck]] once referred to the how the rising sea slowly but eventually surrounds solid land masses. This metaphor became more or less proverbial as "The Rising Sea" analogy.

From [Grothendieck 85, pp. 552-3-1](#Grothendieck85):

> I can illustrate the second approach with the same image of a nut to be opened.

> The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better,and otherwise you let time pass. The shell becomes more flexible through weeks and months -- when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado!

> A different image came to me a few weeks ago.

> The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it.. yet it finally surrounds the resistant substance.

For more see ([McLarty 03](#McLarty03)).

## Website

[[Daniel Murfet]] has written a body of useful notes on fundamentals of [[algebraic geometry]], [[commutative algebra]], and [[category theory]], which can be found at his website [The Rising Sea](http://therisingsea.org/).

## Related entries

Grothendieck's metaphor is also discussed at

* [[nPOV]], [[applications of (higher) category theory]]

Other writings by Grothendieck:

* [[EGA]], [[FGA]], [[SGA]]

## References

* {#Grothendieck85} [[Alexander Grothendieck]], _[[Récoltes et semailles]]_, 1985-1987,

* {#McLarty03} [[Colin McLarty]], _The Rising Sea: Grothendieck on simplicity and generality I_, 2003 ([pdf](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Mathbiographies/mclarty1.pdf), [[McLartyRisingSea.pdf:file]])

category: reference

[[!redirects The Rising Sea]]
[[!redirects the rising sea]]
[[!redirects la mer qui monte]]