nLab A first idea of quantum field theory -- Geometry

Geometry

The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces $\mathbb{R}^n$ with smooth functions between them. Here we briefly review the basics of differential geometry on Cartesian spaces.

In principle the only background assumed of the reader here is

1. usual naive set theory (e.g. Lawvere-Rosebrugh 03);

2. the concept of the continuum: the real line $\mathbb{R}$, the plane $\mathbb{R}^2$, etc.

3. the concepts of differentiation and integration of functions on such Cartesian spaces;

hence essentially the content of multi-variable differential calculus.

We now discuss:

• Abstract coordinate systems

• Fiber bundles

• Synthetic differential geometry

• Differential forms

As we uncover Lagrangian field theory further below, we discover ever more general concepts of “space” in differential geometry, such as smooth manifolds, diffeological spaces, infinitesimal neighbourhoods, supermanifolds, Lie algebroids and super Lie ∞-algebroids. We introduce these incrementally as we go along:

more general spaces in differential geometry introduced further below

											higher differential geometry
differential geometry	smooth manifolds (def. )	$\hookrightarrow$	diffeological spaces (def. )	$\hookrightarrow$	smooth sets (def. )	$\hookrightarrow$	formal smooth sets (def. )	$\hookrightarrow$	super formal smooth sets (def. )	$\hookrightarrow$	super formal smooth ∞-groupoids (not needed in fully perturbative QFT)
infinitesimal geometry, Lie theory							infinitesimally thickened points (def. )		superpoints (def. )		Lie ∞-algebroids (def. )
											higher Lie theory
needed in QFT for:	spacetime (def. )		space of field histories (def. )				Cauchy surface (def. ), perturbation theory (def. )		Dirac field (expl. ), Pauli exclusion principle		infinitesimal gauge symmetry/BRST complex (expl. )

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Abstract coordinate systems

What characterizes differential geometry is that it models geometry on the continuum, namely the real line $\mathbb{R}$, together with its Cartesian products $\mathbb{R}^n$, regarded with its canonical smooth structure (def. below). We may think of these Cartesian spaces $\mathbb{R}^n$ as the “abstract coordinate systems” and of the smooth functions between them as the “abstract coordinate transformations”.

We will eventually consider below much more general “smooth spaces” $X$ than just the Cartesian spaces $\mathbb{R}^n$; but all of them are going to be understood by “laying out abstract coordinate systems” inside them, in the general sense of having smooth functions $f \colon \mathbb{R}^n \to X$ mapping a Cartesian space smoothly into them. All structure on generalized smooth spaces $X$ is thereby reduced to compatible systems of structures on just Cartesian spaces, one for each smooth “probe” $f\colon \mathbb{R}^n \to X$. This is called “functorial geometry”.

Notice that the popular concept of a smooth manifold (def./prop. below) is essentially that of a smooth space which locally looks just like a Cartesian space, in that there exist sufficiently many $f \colon \mathbb{R}^n \to X$ which are (open) isomorphisms onto their images. Historically it was a long process to arrive at the insight that it is wrong to fix such local coordinate identifications $f$, or to have any structure depend on such a choice. But it is useful to go one step further:

In functorial geometry we do not even focus attention on those $f \colon \mathbb{R}^n \to X$ that are isomorphisms onto their image, but consider all “probes” of $X$ by “abstract coordinate systems”. This makes differential geometry both simpler as well as more powerful. The analogous insight for algebraic geometry is due to Grothendieck 65; it was transported to differential geometry by Lawvere 67.

This allows to combine the best of two superficially disjoint worlds: On the one hand we may reduce all constructions and computations to coordinates, the way traditionally done in the physics literature; on the other hand we have full conceptual control over the coordinate-free generalized spaces analyzed thereby. What makes this work is that all coordinate-constructions are functorially considered over all abstract coordinate systems.

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Definition

(Cartesian spaces and smooth functions between them)

For $n \in \mathbb{N}$ we say that the set $\mathbb{R}^n$ of n-tuples of real numbers is a Cartesian space. This comes with the canonical coordinate functions

$$x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}$$

which send an n-tuple of real numbers to the $k$th element in the tuple, for $k \in \{1, \cdots, n\}$.

For

$$f \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'}$$

any function between Cartesian spaces, we may ask whether its partial derivative along the $k$th coordinate exists, denoted

$$\frac{\partial f}{\partial x^k} \;\colon\; \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n'} \,.$$

If this exists, we may in turn ask that the partial derivative of the partial derivative exists

$$\frac{\partial^2 f}{\partial x^{k_1} \partial x^{k_2}} \coloneqq \frac{\partial}{\partial x^{k_2}} \frac{\partial f}{\partial x^{k_1}}$$

and so on.

A general higher partial derivative obtained this way is, if it exists, indexed by an n-tuple of natural numbers $\alpha \in \mathbb{N}^n$ and denoted

(1)$$\partial_\alpha \;\coloneqq\; \frac{ \partial^{\vert \alpha \vert} f }{ \partial^{\alpha_1} x^1 \partial^{\alpha_2} x^2 \cdots \partial^{\alpha_n} x^n } \,,$$

where ${\vert \alpha\vert} \coloneqq \underoverset{n}{i = 1}{\sum} \alpha_i$ is the total order of the partial derivative.

If all partial derivative to all orders $\alpha \in \mathbb{N}^n$ of a function $f \colon \mathbb{R}^n \to \mathbb{R}^{n'}$ exist, then $f$ is called a smooth function.

Of course the composition $g \circ f$ of two smooth functions is again a smooth function.

$$\array{ && \mathbb{R}^{n_2} \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ \mathbb{R}^{n_1} && \underset{g \circ f}{\longrightarrow} && \mathbb{R}^{n_3} } \,.$$

The inclined reader may notice that this means that Cartesian spaces with smooth functions between them constitute a category (“CartSp”); but the reader not so inclined may ignore this.

For the following it is useful to think of each Cartesian space as an abstract coordinate system. We will be dealing with various generalized smooth spaces (see the table below), but they will all be characterized by a prescription for how to smoothly map abstract coordinate systems into them.

Example

(coordinate functions are smooth functions)

Given a Cartesian space $\mathbb{R}^n$, then all its coordinate functions (def. )

$$x^k \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}$$

are smooth functions (def. ).

For

$$f \colon \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$$

any smooth function and $a \in \{1, 2, \cdots, n_2\}$ write

$$f^a \coloneqq x^a \circ f \;\colon\; \mathbb{R}^{n_1} \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{x^a}{\longrightarrow} \mathbb{R}$$

. for its composition with this coordinate function.

Example

(algebra of smooth functions on Cartesian spaces)

For each $n \in \mathbb{N}$, the set

$$C^\infty(\mathbb{R}^n) \;\coloneqq\; Hom_{CartSp}(\mathbb{R}^n, \mathbb{R})$$

of real number-valued smooth functions $f \colon \mathbb{R}^n \to \mathbb{R}$ on the $n$-dimensional Cartesian space (def. ) becomes a commutative associative algebra over the ring of real numbers by pointwise addition and multiplication in $\mathbb{R}$: for $f,g \in C^\infty(\mathbb{R}^n)$ and $x \in \mathbb{R}^n$

1. $(f + g)(x) \coloneqq f(x) + g(x)$

2. $(f \cdot g)(x) \coloneqq f(x) \cdot g(x)$.

The inclusion

$$\mathbb{R} \overset{const}{\hookrightarrow} C^\infty(\mathbb{R}^n)$$

is given by the constant functions.

We call this the real algebra of smooth functions on $\mathbb{R}^n$:

$$C^\infty(\mathbb{R}^n) \;\in\; \mathbb{R} Alg \,.$$

If

$$f \;\colon\; \mathbb{R}^{n_1} \longrightarrow \mathbb{R}^{n_2}$$

is any smooth function (def. ) then pre-composition with $f$ (“pullback of functions”)

$$\array{ C^\infty(\mathbb{R}^{n_2}) &\overset{f^\ast}{\longrightarrow}& C^\infty(\mathbb{R}^{n_1}) \\ g &\mapsto& f^\ast g \coloneqq g \circ f }$$

is an algebra homomorphism. Moreover, this is clearly compatible with composition in that

$$f_1^\ast(f_2^\ast g) = (f_2 \circ f_1)^\ast g \,.$$

Stated more abstractly, this means that assigning algebras of smooth functions is a functor

$$C^\infty(-) \;\colon\; CartSp \longrightarrow \mathbb{R} Alg^{op}$$

from the category CartSp of Cartesian spaces and smooth functions between them (def. ), to the opposite of the category $\mathbb{R}$Alg of $\mathbb{R}$-algebras.

Definition

(local diffeomorphisms and open embeddings of Cartesian spaces)

A smooth function $f \colon \mathbb{R}^{n} \to \mathbb{R}^{n}$ from one Cartesian space to itself (def. ) is called a local diffeomorphism, denoted

$$f \;\colon\; \mathbb{R}^{n} \overset{et}{\longrightarrow} \mathbb{R}^n$$

if the determinant of the matrix of partial derivatives (the “Jacobian” of $f$) is everywhere non-vanishing

$$det \left( \array{ \frac{\partial f^1}{\partial x^1}(x) &\cdots& \frac{\partial f^n}{\partial x^1}(x) \\ \vdots && \vdots \\ \frac{\partial f^1}{\partial x^n}(x) &\cdots& \frac{\partial f^n}{\partial x^n}(x) } \right) \;\neq\; 0 \phantom{AAAA} \text{for all} \, x \in \mathbb{R}^n \,.$$

If the function $f$ is both a local diffeomorphism, as above, as well as an injective function then we call it an open embedding, denoted

$$f \;\colon\; \mathbb{R}^n \overset{\phantom{A}et\phantom{A}}{\hookrightarrow} \mathbb{R}^n \,.$$

Definition

(good open cover of Cartesian spaces)

For $\mathbb{R}^n$ a Cartesian space (def. ), a differentiably good open cover is

• an indexed set

  $$\left\{ \mathbb{R}^n \underoverset{et}{\phantom{AA}f_i\phantom{AA}}{\hookrightarrow} \mathbb{R}^n \right\}_{i \in I}$$

  of open embeddings (def. )

such that the images

$$U_i \coloneqq im(f_i) \subset \mathbb{R}^n$$

satisfy:

1. (open cover) every point of $\mathbb{R}^n$ is contained in at least one of the $U_i$;

2. (good) all finite intersections $U_{i_1} \cap \cdots \cap U_{i_k} \subset \mathbb{R}^n$ are either empty set or themselves images of open embeddings according to def. .

The inclined reader may notice that the concept of differentiably good open covers from def. is a coverage on the category CartSp of Cartesian spaces with smooth functions between them, making it a site, but the reader not so inclined may ignore this.

(Fiorenza-Schreiber-Stasheff 12, def. 6.3.9)

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fiber bundles

Given any context of objects and morphisms between them, such as the Cartesian spaces and smooth functions from def. it is of interest to fix one object $X$ and consider other objects parameterized over it. These are called bundles (def. ) below. For reference, we briefly discuss here the basic concepts related to bundles in the context of Cartesian spaces.

Of course the theory of bundles is mostly trivial over Cartesian spaces; it gains its main interest from its generalization to more general smooth manifolds (def./prop. below). It is still worthwhile for our development to first consider the relevant concepts in this simple case first.

For more exposition see at fiber bundles in physics.

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Definition

(bundles)

We say that a smooth function $E \overset{fb}{\to} X$ (def. ) is a bundle just to amplify that we think of it as exhibiting $E$ as being a “space over $X$”:

$$\array{ E \\ \downarrow\mathrlap{fb} \\ X } \,.$$

For $x \in X$ a point, we say that the fiber of this bundle over $x$ is the pre-image

(2)$$E_x \coloneqq fb^{-1}(\{x\}) \subset E$$

of the point $x$ under the smooth function. We think of $fb$ as exhibiting a “smoothly varying” set of fiber spaces over $X$.

Given two bundles $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ over $X$, a homomorphism of bundles between them is a smooth function $f \colon E_1 \to E_2$ (def. ) between their total spaces which respects the bundle projections, in that

$$fb_2 \circ f = fb_1 \phantom{AAAA} \text{i.e.} \phantom{AAA} \array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X } \,.$$

Hence a bundle homomorphism is a smooth function that sends fibers to fibers over the same point:

$$f\left( (E_1)_x \right) \;\subset\; (E_2)_x \,.$$

The inclined reader may notice that this defines a category of bundles over $X$, which is in fact just the slice category $CartSp_{/X}$; the reader not so inclined may ignore this.

Definition

(sections)

Given a bundle $E \overset{fb}{\to} X$ (def. ) a section is a smooth function $s \colon X \to E$ such that

$$fb \circ s = id_X \phantom{AAAAA} \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow\mathrlap{fb} \\ X &=& X } \,.$$

This means that $s$ sends every point $x \in X$ to an element in the fiber over that point

$$s(x) \in E_x \,.$$

We write

$$\Gamma_X(E) \coloneqq \left\{ \array{ && E \\ & {}^{\mathllap{s}}\nearrow & \downarrow^\mathrlap{fb} \\ X &=& X } \phantom{fb} \right\}$$

for the set of sections of a bundle.

For $E_1 \overset{f_1}{\to} X$ and $E_2 \overset{f_2}{\to} X$ two bundles and for

$$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X }$$

a bundle homomorphism between them (def. ), then composition with $f$ sends sections to sections and hence yields a function denoted

$$\array{ \Gamma_X(E_1) &\overset{f_\ast}{\longrightarrow}& \Gamma_X(E_2) \\ s &\mapsto& f \circ s } \,.$$

Example

(trivial bundle)

For $X$ and $F$ Cartesian spaces, then the Cartesian product $X \times F$ equipped with the projection

$$\array{ X \times F \\ \downarrow^\mathrlap{pr_1} \\ X }$$

to $X$ is a bundle (def. ), called the trivial bundle with fiber $F$. This represents the constant smoothly varying set of fibers, constant on $F$

If $F = \ast$ is the point, then this is the identity bundle

$$\array{ X \\ \downarrow\mathrlap{id} \\ X } \,.$$

Given any bundle $E \overset{fb}{\to} X$, then a bundle homomorphism (def. ) from the identity bundle to $E \overset{fb}{\to} X$ is equivalently a section of $E \overset{fb}{\to} X$ (def. )

$$\array{ X && \overset{s}{\longrightarrow} && E \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{fb}} \\ && X }$$

Definition

(fiber bundle)

A bundle $E \overset{fb}{\to} X$ (def. ) is called a fiber bundle with typical fiber $F$ if there exists a differentiably good open cover $\{U_i \hookrightarrow X\}_{i \in I}$ (def. ) such that the restriction of $fb$ to each $U_i$ is isomorphic to the trivial fiber bundle with fiber $F$ over $U_i$. Such diffeomorphisms $f_i \colon U_i \times F \overset{\simeq}{\to} E\vert_{U_i}$ are called local trivializations of the fiber bundle:

$$\array{ U_i \times F &\underoverset{\simeq}{f_i}{\longrightarrow}& E\vert_{U_i} \\ & {}_{\mathllap{pr_1}}\searrow & \downarrow\mathrlap{fb\vert_{U_i}} \\ && U_i } \,.$$

Definition

(vector bundle)

A vector bundle is a fiber bundle $E \overset{vb}{\to} X$ (def. ) with typical fiber a vector space $V$ such that there exists a local trivialization $\{U_i \times V \underoverset{\simeq}{f_i}{\to} E\vert_{U_i}\}_{i \in I}$ whose gluing functions

$$U_i \cap U_j \times V \overset{f_i\vert_{U_i \cap U_j}}{\longrightarrow} E\vert_{U_i \cap U_j} \overset{f_j^{-1}\vert_{U_i \cap U_j}}{\longrightarrow} U_i \cap U_j \times V$$

for all $i,j \in I$ are linear functions over each point $x \in U_i \cap U_j$.

A homomorphism of vector bundle is a bundle morphism $f$ (def. ) such that there exist local trivializations on both sides with respect to which $g$ is fiber-wise a linear map.

The inclined reader may notice that this makes vector bundles over $X$ a category (denoted $Vect_{/X}$); the reader not so inclined may ignore this.

Example

(module of sections of a vector bundle)

Given a vector bundle $E \overset{vb}{\to} X$ (def. ), then its set of sections $\Gamma_X(E)$ (def. ) becomes a real vector space by fiber-wise multiplication with real numbers. Moreover, it becomes a module over the algebra of smooth functions $C^\infty(X)$ (example ) by the same fiber-wise multiplication:

$$\array{ C^\infty(X) \otimes_{\mathbb{R}} \Gamma_X(E) &\longrightarrow& \Gamma_X(E) \\ (f,s) &\mapsto& (x \mapsto f(x) \cdot s(x)) } \,.$$

For $E_1 \overset{fb_1}{\to} X$ and $E_2 \overset{fb_2}{\to} X$ two vector bundles and

$$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{fb_1}}\searrow && \swarrow_{\mathrlap{fb_2}} \\ && X }$$

a vector bundle homomorphism (def. ) then the induced function on sections (def. )

$$f_\ast \;\colon\; \Gamma_X(E_1) \longrightarrow \Gamma_X(E_2)$$

is compatible with this action by smooth functions and hence constitutes a homomorphism of $C^\infty(X)$-modules.

The inclined reader may notice that this means that taking spaces of sections yields a functor

$$\Gamma_X(-) \;\colon\; Vect_{/X} \longrightarrow C^\infty(X) Mod$$

from the category of vector bundles over $X$ to that over modules over $C^\infty(X)$.

Example

(tangent vector fields and tangent bundle)

For $\mathbb{R}^n$ a Cartesian space (def. ) the trivial vector bundle (example , def. )

$$\array{ T \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times \mathbb{R}^n \\ \mathllap{tb}\downarrow && \downarrow\mathrlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$$

is called the tangent bundle of $\mathbb{R}^n$. With $(x^a)_{a = 1}^n$ the coordinate functions on $\mathbb{R}^n$ (def. ) we write $(\partial_a)_{a = 1}^n$ for the corresponding linear basis of $\mathbb{R}^n$ regarded as a vector space. Then a general section (def. )

$$\array{ && T \mathbb{R}^n \\ & {}^{\mathllap{v}}\nearrow& \downarrow\mathrlap{tb} \\ \mathbb{R}^n &=& \mathbb{R}^n }$$

of the tangent bundle has a unique expansion of the form

$$v = v^a \partial_a$$

where a sum over indices is understood (Einstein summation convention) and where the components $(v^a \in C^\infty(\mathbb{R}^n))_{a = 1}^n$ are smooth functions on $\mathbb{R}^n$ (def. ).

Such a $v$ is also called a smooth tangent vector field on $\mathbb{R}^n$.

Each tangent vector field $v$ on $\mathbb{R}^n$ determines a partial derivative on smooth functions

$$\array{ C^\infty(\mathbb{R}^n) &\overset{D_v}{\longrightarrow}& C^\infty(\mathbb{R}^n) \\ f &\mapsto& \mathrlap{ D_v f \coloneqq v^a \partial_a (f) \coloneqq \sum_a v^a \frac{\partial f}{\partial x^a} } } \,.$$

By the product law of differentiation, this is a derivation on the algebra of smooth functions (example ) in that

1. it is an $\mathbb{R}$-linear map in that

   $$D_v( c_1 f_1 + c_2 f_2 ) = c_1 D_v f_1 + c_2 D_v f_2$$

2. it satisfies the Leibniz rule

   $$D_v(f_1 \cdot f_2) = (D_v f_1) \cdot f_2 + f_1 \cdot (D_v f_2)$$

for all $c_1, c_2 \in \mathbb{R}$ and all $f_1, f_2 \in C^\infty(\mathbb{R}^n)$.

Hence regarding tangent vector fields as partial derivatives constitutes a linear function

$$D \;\colon\; \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \longrightarrow Der(C^\infty(\mathbb{R}^n))$$

from the space of sections of the tangent bundle. In fact this is a homomorphism of $C^\infty(\mathbb{R}^n)$-modules (example ), in that for $f \in C^\infty(\mathbb{R}^n)$ and $v \in \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n)$ we have

$$D_{f v}(-) = f \cdot D_v(-) \,.$$

Example

(vertical tangent bundle)

Let $E \overset{fb}{\to} \Sigma$ be a fiber bundle. Then its vertical tangent bundle $T_\Sigma E \overset{T fb}{\to} \Sigma$ is the fiber bundle (def. ) over $\Sigma$ whose fiber over a point is the tangent bundle (def. ) of the fiber of $E \overset{fb}{\to}\Sigma$ over that point:

$$(T_\Sigma E)_x \coloneqq T(E_x) \,.$$

If $E \simeq \Sigma \times F$ is a trivial fiber bundle with fiber $F$, then its vertical vector bundle is the trivial fiber bundle with fiber $T F$.

Definition

(dual vector bundle)

For $E \overset{vb}{\to} \Sigma$ a vector bundle (def. ), its dual vector bundle is the vector bundle whose fiber (2) over $x \in \Sigma$ is the dual vector space of the corresponding fiber of $E \to \Sigma$:

$$(E^\ast)_x \;\coloneqq\; (E_x)^\ast \,.$$

The defining pairing of dual vector spaces $(E_x)^\ast \otimes E_x \to \mathbb{R}$ applied pointwise induces a pairing on the modules of sections (def. ) of the original vector bundle and its dual with values in the smooth functions (def. ):

(3)$$\array{ \Gamma_\Sigma(E) \otimes_{C^\infty(X)} \Gamma_\Sigma(E^\ast) &\longrightarrow& C^\infty(\Sigma) \\ (v,\alpha) &\mapsto& (v \cdot \alpha \colon x \mapsto \alpha_x(v_x) ) }$$

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synthetic differential geometry

Below we encounter generalizations of ordinary differential geometry that include explicit “infinitesimals” in the guise of infinitesimally thickened points, as well as “super-graded infinitesimals”, in the guise of superpoints (necessary for the description of fermion fields such as the Dirac field). As we discuss below, these structures are naturally incorporated into differential geometry in just the same way as Grothendieck introduced them into algebraic geometry (in the guise of “formal schemes”), namely in terms of formally dual rings of functions with nilpotent ideals. That this also works well for differential geometry rests on the following three basic but important properties, which say that smooth functions behave “more algebraically” than their definition might superficially suggest:

Proposition

(the three magic algebraic properties of differential geometry)

1. embedding of Cartesian spaces into formal duals of R-algebras

   For $X$ and $Y$ two Cartesian spaces, the smooth functions $f \colon X \longrightarrow Y$ between them (def. ) are in natural bijection with their induced algebra homomorphisms $C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)$ (example ), so that one may equivalently handle Cartesian spaces entirely via their $\mathbb{R}$-algebras of smooth functions.

   Stated more abstractly, this means equivalently that the functor $C^\infty(-)$ that sends a smooth manifold $X$ to its $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions (example ) is a fully faithful functor:

   $$C^\infty(-) \;\colon\; SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op} \,.$$

   (Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10)

2. embedding of smooth vector bundles into formal duals of R-algebra modules

   For $E_1 \overset{vb_1}{\to} X$ and $E_2 \overset{vb_2}{\to} X$ two vector bundle (def. ) there is then a natural bijection between vector bundle homomorphisms $f \colon E_1 \to E_2$ and the homomorphisms of modules $f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)$ that these induces between the spaces of sections (example ).

   More abstractly this means that the functor $\Gamma_X(-)$ is a fully faithful functor

   $$\Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod$$

   (Nestruev 03, theorem 11.29)

   Moreover, the modules over the $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions on $X$ which arise this way as sections of smooth vector bundles over a Cartesian space $X$ are precisely the finitely generated free modules over $C^\infty(X)$.

   (Nestruev 03, theorem 11.32)

3. vector fields are derivations of smooth functions.

   For $X$ a Cartesian space (example ), then any derivation $D \colon C^\infty(X) \to C^\infty(X)$ on the $\mathbb{R}$-algebra $C^\infty(X)$ of smooth functions (example ) is given by differentiation with respect to a uniquely defined smooth tangent vector field: The function that regards tangent vector fields with derivations from example

   $$\array{ \Gamma_X(T X) &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}& Der(C^\infty(X)) \\ v &\mapsto& D_v }$$

   is in fact an isomorphism.

   (This follows directly from the Hadamard lemma.)

Actually all three statements in prop. hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. below; if only we generalize in the second statement from free modules to projective modules. However for our development here it is useful to first focus on just Cartesian spaces and then bootstrap the theory of smooth manifolds and much more from that, which we do below.

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differential forms

We introduce and discuss differential forms on Cartesian spaces.

Definition

(differential 1-forms on Cartesian spaces and the cotangent bundle)

For $n \in \mathbb{N}$ a smooth differential 1-form $\omega$ on a Cartesian space $\mathbb{R}^n$ (def. ) is an n-tuple

$$\left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n$$

of smooth functions (def. ), which we think of equivalently as the coefficients of a formal linear combination

$$\omega = \omega_i d x^i$$

on a set $\{d x^1, d x^2, \cdots, d x^n\}$ of cardinality $n$.

Here a sum over repeated indices is tacitly understood (Einstein summation convention).

Write

$$\Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set$$

for the set of smooth differential 1-forms on $\mathbb{R}^k$.

We may think of the expressions $(d x^a)_{a = 1}^n$ as a linear basis for the dual vector space $\mathbb{R}^n$. With this the differential 1-forms are equivalently the sections (def. ) of the trivial vector bundle (example , def. )

$$\array{ T^\ast \mathbb{R}^n &\coloneqq& \mathbb{R}^n \times (\mathbb{R}^n)^\ast \\ \mathllap{cb}\downarrow && \downarrow\mathrlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$$

called the cotangent bundle of $\mathbb{R}^n$ (def. ):

$$\Omega^1(\mathbb{R}^n) = \Gamma_{\mathbb{R}^n}(T^\ast \mathbb{R}^n) \,.$$

This amplifies via example that $\Omega^1(\mathbb{R}^n)$ has the structure of a module over the algebra of smooth functions $C^\infty(\mathbb{R}^n)$, by the evident multiplication of differential 1-forms with smooth functions:

1. The set $\Omega^1(\mathbb{R}^k)$ of differential 1-forms in a Cartesian space (def. ) is naturally an abelian group with addition given by componentwise addition

   $$\begin{aligned} \omega + \lambda & = \omega_i d x^i + \lambda_i d x^i \\ & = (\omega_i + \lambda_i) d x^i \end{aligned} \,,$$

2. The abelian group $\Omega^1(\mathbb{R}^k)$ is naturally equipped with the structure of a module over the algebra of smooth functions $C^\infty(\mathbb{R}^k)$ (example ), where the action $C^\infty(\mathbb{R}^k) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k)$ is given by componentwise multiplication

   $$f \cdot \omega = ( f \cdot \omega_i) d x^i \,.$$

Accordingly there is a canonical pairing between differential 1-forms and tangent vector fields (example )

(4)$$\array{ \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \otimes_{\mathbb{R}} \Gamma_{\mathbb{R}^n}(T \ast \mathbb{R}^n) &\overset{\iota_{(-)}(-) }{\longrightarrow}& C^\infty(\mathbb{R}^n) \\ (v,\omega) &\mapsto& \mathrlap{ \iota_v \omega \coloneqq v^a \omega_a } }$$

With differential 1-forms in hand, we may collect all the first-order partial derivatives of a smooth function into a single object: the exterior derivative or de Rham differential is the $\mathbb{R}$-linear function

(5)$$\array{ C^\infty(\mathbb{R}^n) &\overset{d}{\longrightarrow}& \Omega^1(\mathbb{R}^n) \\ f &\mapsto& \mathrlap{ d f \coloneqq \frac{\partial f}{ \partial x^a} d x^a } } \,.$$

Under the above pairing with tangent vector fields $v$ this yields the particular partial derivative along $v$:

$$\iota_v d f = D_v f = v^a \frac{\partial f}{\partial x^a} \,.$$

We think of $d x^i$ as a measure for infinitesimal displacements along the $x^i$-coordinate of a Cartesian space. If we have a measure of infintesimal displacement on some $\mathbb{R}^n$ and a smooth function $f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n$, then this induces a measure for infinitesimal displacement on $\mathbb{R}^{\tilde n}$ by sending whatever happens there first with $f$ to $\mathbb{R}^n$ and then applying the given measure there. This is captured by the following definition:

Definition

(pullback of differential 1-forms)

For $\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k$ a smooth function, the pullback of differential 1-forms along $\phi$ is the function

$$\phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k})$$

between sets of differential 1-forms, def. , which is defined on basis-elements by

$$\phi^* d x^i \;\coloneqq\; \frac{\partial \phi^i}{\partial \tilde x^j} d \tilde x^j$$

and then extended linearly by

$$\begin{aligned} \phi^* \omega & = \phi^* \left( \omega_i d x^i \right) \\ & \coloneqq \left(\phi^* \omega\right)_i \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j \\ & = (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j} d \tilde x^j \end{aligned} \,.$$

This is compatible with identity morphisms and composition in that

(6)$$(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)} \phantom{AAAA} (g \circ f)^\ast = f^\ast \circ g^\ast \,.$$

Stated more abstractly, this just means that pullback of differential 1-forms makes the assignment of sets of differential 1-forms to Cartesian spaces a contravariant functor

$$\Omega^1(-) \;\colon\; CartSp^{op} \longrightarrow Set \,.$$

The following definition captures the idea that if $d x^i$ is a measure for displacement along the $x^i$-coordinate, and $d x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way to get a measure, to be called $d x^i \wedge d x^j$, for infinitesimal surfaces (squares) in the $x^i$-$x^j$-plane. And this should keep track of the orientation of these squares, with

$$d x^j \wedge d x^i = - d x^i \wedge d x^j$$

being the same infinitesimal measure with orientation reversed.

Definition

(exterior algebra of differential n-forms)

For $k,n \in \mathbb{N}$, the smooth differential forms on a Cartesian space $\mathbb{R}^k$ (def. ) is the exterior algebra

$$\Omega^\bullet(\mathbb{R}^k) \coloneqq \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k)$$

over the algebra of smooth functions $C^\infty(\mathbb{R}^k)$ (example ) of the module $\Omega^1(\mathbb{R}^k)$ of smooth 1-forms.

We write $\Omega^n(\mathbb{R}^k)$ for the sub-module of degree $n$ and call its elements the differential n-forms.

Explicitly this means that a differential n-form $\omega \in \Omega^n(\mathbb{R}^k)$ on $\mathbb{R}^k$ is a formal linear combination over $C^\infty(\mathbb{R}^k)$ (example ) of basis elements of the form $d x^{i_1} \wedge \cdots \wedge d x^{i_n}$ for $i_1 \lt i_2 \lt \cdots \lt i_n$:

$$\omega = \omega_{i_1, \cdots, i_n} d x^{i_1} \wedge \cdots \wedge d x^{i_n} \,.$$

Now all the constructions for differential 1-forms above extent naturally to differential n-forms:

Definition

(exterior derivative or de Rham differential)

For $\mathbb{R}^n$ a Cartesian space (def. ) the de Rham differential $d \colon C^\infty(\mathbb{R}^n) \to \Omega^1(\mathbb{R}^n)$ (5) uniquely extended as a derivation of degree +1 to the exterior algebra of differential forms (def. )

$$d \;\colon\; \Omega^\bullet(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n)$$

meaning that for $\omega_i \in \Omega^{k_i}(\mathbb{R})$ then

$$d(\omega_1 \wedge \omega_2) \;=\; (d \omega_1) \wedge \omega_2 + \omega_1 \wedge d \omega_2 \,.$$

In components this simply means that

$$\begin{aligned} d \omega & = d \left(\omega_{i_1 \cdots i_k} d x^{i_1} \wedge \cdots \wedge d x^{i_k}\right) \\ & = \frac{\partial \omega_{i_1 \cdots i_k}}{\partial x^{a}} d x^a \wedge d x^{i_1} \wedge \cdots \wedge d x^{i_k} \end{aligned} \,.$$

Since partial derivatives commute with each other, while differential 1-form anti-commute, this implies that $d$ is nilpotent

$$d^2 = d \circ d = 0 \,.$$

We say hence that differential forms form a cochain complex, the de Rham complex $(\Omega^\bullet(\mathbb{R}^n), d)$.

Definition

(contraction of differential n-forms with tangent vector fields)

The pairing $\iota_v \omega = \omega(v)$ of tangent vector fields $v$ with differential 1-forms $\omega$ (4) uniquely extends to the exterior algebra $\Omega^\bullet(\mathbb{R}^n)$ of differential forms (def. ) as a derivation of degree -1

$$\iota_v \;\colon\; \Omega^{\bullet+1}(\mathbb{R}^n) \longrightarrow \Omega^\bullet(\mathbb{R}^n) \,.$$

In particular for $\omega_1, \omega_2 \in \Omega^1(\mathbb{R}^n)$ two differential 1-forms, then

$$\iota_{v} (\omega_1 \wedge \omega_2) \;=\; \omega_1(v) \omega_2 - \omega_2(v) \omega_1 \;\in\; \Omega^1(\mathbb{R}^n) \,.$$

Proposition

(pullback of differential n-forms)

For $f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}$ a smooth function between Cartesian spaces (def. ) the operationf of pullback of differential 1-forms of def. extends as an $C^\infty(\mathbb{R}^k)$-algebra homomorphism to the exterior algebra of differential forms (def. ),

$$f^\ast \;\colon\; \Omega^\bullet(\mathbb{R}^{n_2}) \longrightarrow \Omega^\bullet(\mathbb{R}^{n_1})$$

given on basis elements by

$$f^* \left( dx^{i_1} \wedge \cdots \wedge dx^{i_n} \right) = \left(f^* dx^{i_1} \wedge \cdots \wedge f^* dx^{i_n} \right) \,.$$

This commutes with the de Rham differential $d$ on both sides (def. ) in that

$$d \circ f^\ast = f^\ast \circ d \phantom{AAAAA} \array{ \Omega^\bullet(X) &\overset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y) \\ \mathllap{d}\downarrow && \downarrow\mathrlap{d} \\ \Omega^\bullet(X) &\underset{f^\ast}{\longleftarrow}& \Omega^\bullet(Y) }$$

hence that pullback of differential forms is a chain map of de Rham complexes.

This is still compatible with identity morphisms and composition in that

(7)$$(id_{\mathbb{R}^n})^\ast = id_{\Omega^1(\mathbb{R}^n)} \phantom{AAAA} (g \circ f)^\ast = f^\ast \circ g^\ast \,.$$

Stated more abstractly, this just means that pullback of differential n-forms makes the assignment of sets of differential n-forms to Cartesian spaces a contravariant functor

$$\Omega^n(-) \;\colon\; CartSp^{op} \longrightarrow Set \,.$$

Proposition

(Cartan's homotopy formula)

Let $X$ be a Cartesian space (def. ), and let $v \in \Gamma(T X)$ be a smooth tangent vector field (example ).

For $t \in \mathbb{R}$ write $\exp(t v) \colon X \overset{\simeq}{\to} X$ for the flow by diffeomorphisms along $v$ of parameter length $t$.

Then the derivative with respect to $t$ of the pullback of differential forms along $\exp(t v)$, hence the Lie derivative $\mathcal{L}_v \colon \Omega^\bullet(X) \to \Omega^\bullet(X)$, is given by the anticommutator of the contraction derivation $\iota_v$ (def. ) with the de Rham differential $d$ (def. ):

$$\begin{aligned} \mathcal{L}_v &\coloneqq \frac{d}{d t } \exp(t v)^\ast \omega \vert_{t = 0} \\ & = \iota_v d \omega + d \iota_v \omega \,. \end{aligned}$$

Finally we turn to the concept of integration of differential forms (def. below). First we need to say what it is that differential forms may be integrated over:

Definition

(smooth singular simplicial chains in Cartesian spaces)

For $n \in \mathbb{N}$, the standard n-simplex in the Cartesian space $\mathbb{R}^n$ (def. ) is the subset

$$\Delta^n \;\coloneqq\; \left\{ (x^i)_{i = 1}^n \;\vert\; 0 \leq x^1 \leq \cdots \leq x^n \right\} \;\subset\; \mathbb{R}^n \,.$$

More generally, a smooth singular n-simplex in a Cartesian space $\mathbb{R}^k$ is a smooth function (def. )

$$\sigma \;\colon\; \mathbb{R}^n \longrightarrow \mathbb{R}^k \,,$$

to be thought of as a smooth extension of its restriction

$$\sigma\vert_{\Delta^n} \;\colon\; \Delta^n \longrightarrow \mathbb{R}^k \,.$$

(This is called a singular simplex because there is no condition that $\Sigma$ be an embedding in any way, in particular $\sigma$ may be a constant function.)

A singular chain in $\mathbb{R}^k$ of dimension $n$ is a formal linear combination of singular $n$-simplices in $\mathbb{R}^k$.

In particular, given a singular $n+1$-simplex $\sigma$, then its boundary is a singular chain of singular $n$-simplices $\partial \sigma$.

Definition

(fiber-integration of differential forms) over smooth singular chains in Cartesian spaces)

For $n \in \mathbb{N}$ and $\omega \in \Omega^n(\mathbb{R}^n)$ a differential n-form (def. ), which may be written as

$$\omega = f d x^1 \wedge \cdots d x^n \,,$$

then its integration over the standard n-simplex $\Delta^n \subset \mathbb{R}^n$ (def. ) is the ordinary integral (e.g. Riemann integral)

$$\int_{\Delta^n} \omega \;\coloneqq\; \underset{0 \leq x^1 \leq \cdots \leq x^n \leq 1}{\int} f(x^1, \cdots, x^n) \, d x^1 \cdots d x^n \,.$$

More generally, for

1. $\omega \in \Omega^n(\mathbb{R}^k)$ a differential n-forms;

2. $C = \underset{i}{\sum} c_i \sigma_i$ a singular $n$-chain (def. )

in any Cartesian space $\mathbb{R}^k$. Then the integration of $\omega$ over $x$ is the sum of the integrations, as above, of the pullback of differential forms (def. ) along all the singular n-simplices in the chain:

$$\int_C \omega \;\coloneqq\; \underset{i}{\sum} c_i \int_{\Delta^n} (\sigma_i)^\ast \omega \,.$$

Finally, for $U$ another Cartesian space, then fiber integration of differential forms along $U \times C \to U$ is the linear map

$$\int_C \;\colon\; \Omega^{\bullet + dim(C)}(U \times C) \longrightarrow \Omega^\bullet(U)$$

which on differential forms of the form $\omega_U \wedge \omega$ with ($\omega_U$ pulled back from $U$ and $\omega$ from $C$) is given by:

$$\int_C \omega_U \wedge \omega \;\coloneqq\; (-1)^{\vert \omega_U\vert} \Big( \textstyle{\int}_C \omega \Big) \omega_U \,.$$

Proposition

(Stokes theorem for fiber-integration of differential forms)

For $\Sigma$ a smooth singular simplicial chain (def. ) the operation of fiber-integration of differential forms along $U \times \Sigma \overset{pr_1}{\longrightarrow} U$ (def. ) is compatible with the exterior derivative $d_U$ on $U$ (def. ) in that

$$\begin{aligned} d \int_\Sigma \omega & = (-1)^{dim(\Sigma)} \int_\Sigma d_U \omega \\ & = (-1)^{dim(\Sigma)} \left( \int_\Sigma d \omega - \int_{\partial \Sigma} \omega \right) \end{aligned} \,,$$

where $d = d_U + d_\Sigma$ is the de Rham differential on $U \times \Sigma$ (def. ) and where the second equality is the Stokes theorem along $\Sigma$:

$$\int_\Sigma d_\Sigma \omega = \int_{\partial \Sigma} \omega \,.$$

$\,$

This concludes our review of the basics of (synthetic) differential geometry on which the following development of quantum field theory is based. In the next chapter we consider spacetime and spin.