# Learn the Geometry of Mathematical Quantum Field Theory

Common Topics: def, smooth, differential, cartesian, bundle

This is the first chapter in a series on Mathematical Quantum Field Theory.

The next chapter is 2. Spacetime.

## 1. 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:

As we uncover the Lagrangian field theory further below, we discover ever more general concepts of “space” in differential geometry, such as smooth manifolds, diffeological spaces, infinitesimal neighborhoods, 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. 3.32) ##\hookrightarrow## diffeological spaces (def. 3.8) ##\hookrightarrow## smooth sets (def. 3.12) ##\hookrightarrow## formal smooth sets (def. 3.22) ##\hookrightarrow## super formal smooth sets (def. 3.38) ##\hookrightarrow## super formal smooth ∞-groupoids (not needed in fully perturbative QFT) infinitesimal geometry, Lie theory infinitesimally thickened points (def. 3.18) superpoints (def. 3.35) Lie ∞-algebroids (def. 10.10) higher Lie theory needed in QFT for: spacetime (def. 2.17) space of field histories (def. 3.10) Cauchy surface (def. 8.1), perturbation theory (def. 7.37) Dirac field (expl. 3.48), Pauli exclusion principle infinitesimal gauge symmetry/BRST complex (expl. 10.15)

## 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. 1.1 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. 3.32 below) is essential that o 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 us 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 overall abstract coordinate systems.

### Definition 1.1. (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

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

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

 $$\label{PartialDerivativeWithManyIndices} \partial^\alpha \;:=\; \frac{ \partial^{\vert \alpha \vert} f }{ \partial^{\alpha_1} x^1 \partial^{\alpha_2} x^2 \cdots \partial^{\alpha_n} x^n } \,,$$ (1)

where ##{\vert \alpha\vert} := \underset{n}{\overset{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.

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

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 1.2. (coordinate functions are smooth functions)

Given a Cartesian space ##\mathbb{R}^n##, then all its coordinate functions (def. 1.1)

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

are smooth functions (def. 1.1).

For

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

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

$$f^a := x^k \circ f \;\colon\; \mathbb{R}^{n_1} \overset{f}{\longrightarrow} \mathbb{R}^{n_2} \overset{x^a}{\longrightarrow} \mathbb{R}$$ .
for its composition with this coordinate function.

### Example 1.3. (algebra of smooth functions on Cartesian spaces)

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

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

of real number-valued smooth functions ##f \colon \mathbb{R}^n \to \mathbb{R}## on the ##n##-dimensional Cartesian space (def. 1.1) 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) := f(x) + g(x)##
2. ##(f \cdot g)(x) := 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. 1.1) then pre-composition with ##f## (“pullback of functions”)

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

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. 1.1), to the opposite of the category ##\mathbb{R}##Alg of ##\mathbb{R}##-algebras.

### Definition 1.4. (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. 1.1) 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 1.5. (good open cover of Cartesian spaces)

For ##\mathbb{R}^n## a Cartesian space (def. 1.1), a differentiably good open cover is

• an indexed set$$\left\{ \mathbb{R}^n \underset{et}{\overset{\phantom{AA}f_i\phantom{AA}}{\hookrightarrow}} \mathbb{R}^n \right\}_{i \in I}$$of open embeddings (def. 1.4)

such that the images

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

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. 1.4.

The inclined reader may notice that the concept of differentiably good open covers from def. 1.5
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)

### fiber bundles

Given any context of objects and morphisms between them, such as the Cartesian spaces and smooth functions from def. 1.1
it is of interest to fix one object ##X## and consider other objects parameterized over it. These are called bundles (def. 1.6) 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. 3.32 below). It is still worthwhile for our development to first consider the relevant concepts in this simple case first.

For more exposition see fiber bundles in physics.

### Definition 1.6. (bundles)

We say that a smooth function ##E \overset{fb}{\to} X## (def. 1.1) is a bundle just to amplify that we think of it as exhibiting ##E## as being a “space over ##X##”:

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

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

 $$\label{FiberOfAFiberBundle} E_x := fb^{-1}(\{x\}) \subset E$$ (2)

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. 1.1) between their total spaces which respects the bundle projections, in that

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

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 1.7. (sections)

Given a bundle ##E \overset{fb}{\to} X## (def. 1.6) a section is a smooth function ##s \colon X \to E## such that

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

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

$$\Gamma_X(E) := \left\{ \array{ && E \\ & {}^{\llap{s}}\nearrow & \downarrow^\rlap{fb} \\ X &=& X } \phantom{fb} \right\}$$

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

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

a bundle homomorphism between them (def. 1.6), 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 1.8. (trivial bundle)

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

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

to ##X## is a bundle (def. 1.6), 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

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

Given any bundle ##E \overset{fb}{\to} X##, then a bundle homomorphism (def. 1.6) from the identity bundle to ##E \overset{fb}{\to} X## is equivalently a section of ##E \overset{fb}{\to} X## (def. 1.7)

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

### Definition 1.9. (fiber bundle)

A bundle ##E \overset{fb}{\to} X## (def. 1.6) 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. 1.5) 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:

$$\array{ U_i \times F &\underset{\simeq}{\overset{f_i}{\longrightarrow}}& E\vert_{U_i} \\ & {}_{\llap{pr_1}}\searrow & \downarrow\rlap{fb\vert_{U_i}} \\ && U_i } \,.$$

### Definition 1.10. (vector bundle)

A vector bundle is a fiber bundle ##E \overset{vb}{\to} X## (def. 1.9) with typical fiber a vector space ##V## such that there exists a local trivialization ##\{U_i \times V \underset{\simeq}{\overset{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. 1.6) 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 1.11. (module of sections of a vector bundle)

Given a vector bundle ##E \overset{vb}{\to} X## (def. 1.10), then its set of sections ##\Gamma_X(E)## (def. 1.6) 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 1.3) 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

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

a vector bundle homomorphism (def. 1.10) then the induced function on sections (def. 1.7)

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

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 1.12. (tangent vector fields and tangent bundle)

For ##\mathbb{R}^n ## a Cartesian space (def. 1.1) the trivial vector bundle (example 1.8, def. 1.10)

$$\array{ T \mathbb{R}^n &:=& \mathbb{R}^n \times \mathbb{R}^n \\ \llap{tb}\downarrow && \downarrow\rlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$$

is called the tangent bundle of ##\mathbb{R}^n##. With ##(x^a)_{a = 1}^n## the coordinate functions on ##\mathbb{R}^n## (def. 1.2) 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. 1.7)

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

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. 1.1).

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

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

By the product law of differentiation, this is a derivation on the algebra of smooth functions (example 1.3) 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 1.11), 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 1.13. (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. 1.9) over ##\Sigma## whose fiber over a point is the tangent bundle (def. 1.12) of the fiber of ##E \overset{fb}{\to}\Sigma## over that point:

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

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 1.14. (dual vector bundle)

For ##E \overset{vb}{\to} \Sigma## a vector bundle (def. 1.10), 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##:

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

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. 1.11) of the original vector bundle and its dual with values in the smooth functions (def. 1.1):

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

### 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 1.15. (the three magic algebraic properties of differential geometry)

1. embedding of Cartesian spaces into formal duals of R-algebrasFor ##X## and ##Y## two Cartesian spaces, the smooth functions ##f \colon X \longrightarrow Y## between them (def. 1.1) are in natural bijection with their induced algebra homomorphisms ##C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)## (example 1.3), 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 1.3) 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 modulesFor ##E_1 \overset{vb_1}{\to} X## and ##E_2 \overset{vb_2}{\to} X## two vector bundle (def. 1.10) 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 1.11).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 1.1), then any derivation ##D \colon C^\infty(X) \to C^\infty(X)## on the ##\mathbb{R}##-algebra ##C^\infty(X)## of smooth functions (example 1.3) 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 1.12$$\array{ \Gamma_X(T X) &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}& Der(C^\infty(X)) \\ v &\mapsto& D_v }$$is in fact an isomorphism.(This follows directly from the Hadamard lemma.)

Actually all three statements in prop. 1.15
hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. 3.32 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.

## differential forms

We introduce and discuss differential forms on Cartesian spaces.

### Definition 1.16. (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. 1.1) is an n-tuple

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

of smooth functions (def. 1.1), 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. 1.7) of the trivial vector bundle (example 1.8, def. 1.10)

$$\array{ T^\ast \mathbb{R}^n &:=& \mathbb{R}^n \times (\mathbb{R}^n)^\ast \\ \llap{cb}\downarrow && \downarrow\rlap{pr_1} \\ \mathbb{R}^n &=& \mathbb{R}^n }$$

called the cotangent bundle of ##\mathbb{R}^n## (def. 1.16):

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

This amplifies via example 1.11 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. 1.16) 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 1.3), 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 1.12)

 $$\label{PairingVectorFieldsWithDifferential1Forms} \array{ \Gamma_{\mathbb{R}^n}(T \mathbb{R}^n) \otimes_{\mathbb{R}} \Gamma_{\mathbb{R}^n}(T \ast \mathbb{R}^n) &\overset{\iota_{(-)}(-) }{\longrightarrow}& C^\infty(\mathbb{R}^n) \\ (v,\omega) &\mapsto& \rlap{ \iota_v \omega := v^a \omega_a } }$$ (4)

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

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

Under the above pairing with tangent vector fields ##v## this yields the particular partial derivative along with ##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 with the ##x^i##-coordinate of Cartesian space. If we have a measure of infinitesimal 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 1.17. (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. 1.16, which is defined on basis-elements by

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

and then extended linearly by

\begin{aligned} \phi^* \omega & = \phi^* \left( \omega_i d x^i \right) \\ & := \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

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

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 with 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 1.18. (exterior algebra of differential n-forms)

For ##k,n \in \mathbb{N}##, the smooth differential forms on a Cartesian space ##\mathbb{R}^k## (def. 1.1) is the exterior algebra

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

over the algebra of smooth functions ##C^\infty(\mathbb{R}^k)## (example 1.3) 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 1.3) 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 1.19. (exterior derivative or de Rham differential)

For ##\mathbb{R}^n## a Cartesian space (def. 1.1) 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. 1.18)

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

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 1.20. (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. 1.18) 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 1.21. (pullback of differential n-forms)

For ##f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}## a smooth function between Cartesian spaces (def. 1.1) the operationf of pullback of differential 1-forms of def. 1.16 extends as an ##C^\infty(\mathbb{R}^k)##-algebra homomorphism to the exterior algebra of differential forms (def. 1.18),

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

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. 1.19) in that

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

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

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

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 1.22. (Cartan’s homotopy formula)

Let ##X## be a Cartesian space (def. 1.1), and let ##v \in \Gamma(T X)## be a smooth tangent vector field (example 1.12).

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. 1.20) with the de Rham differential ##d## (def. 1.19):

\begin{aligned} \mathcal{L}_v &:= \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. 1.24 below). First, we need to say what it is that differential forms may be integrated over:

### Definition 1.23. (smooth singular simplicial chains in Cartesian spaces)

For ##n \in \mathbb{N}##, the standard n-simplex in the Cartesian space ##\mathbb{R}^n## (def. 1.1) is the subset

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

More generally, a smooth singular n-simplex in a Cartesian space ##\mathbb{R}^k## is a smooth function (def. 1.1)

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

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 1.24. (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. 1.18), 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. 1.23) is the ordinary integral (e.g. Riemann integral)

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

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. 1.23)

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. 1.21) along all the singular n-simplices in the chain:

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

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## is given by

$$\int_C \omega_U \wedge \omega \;:=\; (-1)^{\vert \omega_U\vert} \int_C \omega \,.$$

### Proposition 1.25. (Stokes theorem for fiber-integration of differential forms)

For ##\Sigma## a smooth singular simplicial chain (def. 1.24) the operation of fiber-integration of differential forms along ##U \times \Sigma \overset{pr_1}{\longrightarrow} U## (def. 1.24) is compatible with the exterior derivative ##d_U## on ##U## (def. 1.19) 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. 1.19) 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.

Tags:
33 replies
1. strangerep says:
mattt

By the way Urs, I am just starting to review this fantastic series of yours on mathematically rigorous Relativistic Perturbative Quantum Field Theory, and I think it is one of the best things that has happened on physicsforums in a long time, Heh, great to hear. Maybe one day you can explain it in simpler terms to the rest of us rubes. :confused:

2. mattt says:

By the way Urs, I am just starting to review this fantastic series of yours on mathematically rigorous Relativistic Perturbative Quantum Field Theory, and I think it is one of the best things that has happened on physicsforums in a long time, so thank you dearly for that! ( and keep them coming! )

3. Urs Schreiber says:
mattt

Below definition 1.6Thanks! Fixed now.

4. mattt says:

Below definition 1.6 (bundles) it shows: $$fb_2 circ f = fb_2$$ and it should be $$fb_2 circ f = fb_1$$

Also in definition 1.7 (sections) it shows: $$fb circ f = id_X$$ and it should be $$fb circ s = id_X$$

5. Greg Bernhardt says:
Urs Schreiber

That might be an idea. But if the goal is to make publicity for PF, maybe that would better be served if PF is the exclusive host of this material?

dextercioby

I agree. Your work is to the benefit of this heterogenous community and we should keep it here.Obviously we cherish our Insight authors and their unique contributions to our community. But just to be clear, our authorship guidelines do grant authors the freedom to publish elsewhere after publishing on PF. :smile:

6. dextercioby says:
Urs Schreiber

That might be an idea. But if the goal is to make publicity for PF, maybe that would better be served if PF is the exclusive host of this material?I agree. Your work is to the benefit of this heterogenous community and we should keep it here. I see arxiv as a repository for research paper drafts. Your notes are singular (in the sense that the material in them is nowhere else to be seen), so this part of research work is retained, but, unless you plan to sell this material to Springer Verlag, we would like to have it here. Indeed, a pdf with active links to your encyclopedic website would be ideal, as it would offer immediate, unlimited offline access to the contents. I told you, I plan to print them, study them and store them in my physical library alongside other teasures.

7. Urs Schreiber says:
thephystudent

wouldn't it be a good idea to put these articles on the arXiv and link to them?

I don't think that means downplaying PhysicsForums; on the contrary if the article's headers clearly mention the link to PF I think it could be great publicity for the forum.That might be an idea. But if the goal is to make publicity for PF, maybe that would better be served if PF is the exclusive host of this material?

8. thephystudent says:

I am far from an experienced senior like you guys, but wouldn't it be a good idea to put these articles on the arXiv and link to them?

I don't think that means downplaying PhysicsForums; on the contrary if the article's headers clearly mention the link to PF I think it could be great publicity for the forum.

9. Urs Schreiber says:

At the very least, I know how to produce a readable single-file pdf from my source, formatted as web-display, but readable. I'll provide that when the series is finished.

But I am also in contact with people who think about looking into proper LaTeX conversion of my source. With a little luck, this will work out.

10. Greg Bernhardt says:
eys_physics

I fully agree with the suggestion by vanhees71. It would be very nice if at least some of the Insight articles could be read as pdf, at least the ones containing many equations.This is not easily possible as the latex is rendered by the web browser

11. eys_physics says:

I fully agree with the suggestion by vanhees71. It would be very nice if at least some of the Insight articles could be read as pdf, at least the ones containing many equations.

12. vanhees71 says:

This brings up again my older suggestion to allow also to reach in Insight articles as pdf. Then the author can choose his or her preferred way of writing (LaTeX or even, horribile dictu, Word & Co) and provide a well readable and printable pdf. In the best of all worlds one should have both the pdf and the html version.

I think a way to provide the latter, using LaTeX should be the following: One types LaTeX and at the end translates via query replace to the format (I still don't know what it is, mathjax, WordPress, or something else?) of the Insights editor. The only quibble is that one has to provide also a list of private macro definitions used in the LaTeX. So it's perhaps a bit of cumbersome work to get the final Insights html version to look right.

13. Greg Bernhardt says:
A. Neumaier

Also, the pdf for the above Insight article has the leading table truncated.
Please place the print/pdf button at the top!PDF conversion won't work because it won't render the latex. I'll have to find a friendly print option.

14. A. Neumaier says:
Greg Bernhardt

Working on that now. First try looked good but now the latex doesn't render. Also, the pdf for the above Insight article has the leading table truncated.
Please place the print/pdf button at the top!

15. Greg Bernhardt says:
A. Neumaier

maybe you can make a button that generates a print version. In this case it would be useful to have a choice whether one wants to have all links simply suppressed, or listed in the way suggested above.Working on that now. First try looked good but now the latex doesn't render.

16. A. Neumaier says:
Greg Bernhardt

There are some things I can trymaybe you can make a button that generates a print version. In this case it would be useful to have a choice whether one wants to have all links simply suppressed, or listed in the way suggested above.

17. Greg Bernhardt says:

Ok there is now a Print/PDF/Email link at the bottom of the article. Looks good!

18. Greg Bernhardt says:
A. Neumaier

This makes it almost unreadable as there happen to be lots of links!

To be readable, each link should be replaced in the print version by a reference such as [3], and at the end of the article the reference numbers should be listed together with their link:
[3] http://xxxx.yy….Right, that is what I meant. There are some things I can try. Be back later to report.

19. A. Neumaier says:
Greg Bernhardt

To be readable, each link should be replaced in the print version by a reference such as [3], and at the end of the article the reference numbers should be listed together with their link:
[3] http://xxxx.yy….

20. Greg Bernhardt says:
A. Neumaier

I tried to print some articles from your sequence using the browser's print facility (from chrome), and the result was very poor….One of the main issues is that all the keyword text links turn into URL address links. Make it almost readable.

21. A. Neumaier says:
Urs Schreiber

There is two different things you could be asking for here:

2. A reformatting in standard LaTeX style compiled to a pdf.

I'll provide the former when the series is finished. For the latter we would need somebody offering to write a conversion tool that would convert my source code to LaTeX.I tried to print some articles from your sequence using the browser's print facility (from chrome), and the result was very poor….

22. vanhees71 says:

As long as the pdf is as readable as the web site and formatted in a way that it can be printed out nicely, I don't care with which tool it is created. Thanks anyway for all this work. I have to struggle to keep up reading and have negative available time at the moment :-(.

23. Urs Schreiber says:

There is two different things you could be asking for here:

2. A reformatting in standard LaTeX style compiled to a pdf.

I'll provide the former when the series is finished. For the latter we would need somebody offering to write a conversion tool that would convert my source code to LaTeX.

24. vanhees71 says:
Urs Schreiber

I know what you mean. On the other hand, with the script that some kind soul has written for me (who prefers to remain anonymous), 99% of the technical problems have now gone away (for conversion from Instiki to WordPress that is, but something analogous could be done for conversion from LaTeX to Wordress.) I am confident that the last remaining little issues as above will be sorted out.Still I prefer pdf's, particularly since I'm still preferring to print out manuscripts like this on paper. I'm a digital immigrant!

25. thephystudent says:

Thanks for sharing this very interesting guide which you clearly took very seriously!

I actually think pdf-notes would be more comfortable to read (and more printer-friendly as well) though, especially on complex subjects like this.

26. Urs Schreiber says:
Urs Schreiber

But an issue remains. The use of "substack" here is a hack anyway. I replaces what in my source is "mathrlap" or "mathllap", neither of which seem to be supported here. If anyone has a suggestion, please let me know.Found the solution! Turns out that the functionality of "mathrlap" and "mathllap" is supported here after all, but it needs to be called just as "rlap" and "llap".

Now everything works! (Or so it seems, please let me know if you spot further issues.)

27. Urs Schreiber says:
vanhees71

It's offtopic a bit, because it's about technical things, but I'd suggest Greg should also admit collections of pdf's as Insights. Then it would be much easier to write Insights of such typographically challenging texts simply in LaTeX. The greatest obstacle writing Insights articles for me are these technical problems!I know what you mean. On the other hand, with the script that some kind soul has written for me (who prefers to remain anonymous), 99% of the technical problems have now gone away (for conversion from Instiki to WordPress that is, but something analogous could be done for conversion from LaTeX to Wordress.) I am confident that the last remaining little issues as above will be sorted out.

28. vanhees71 says:

It's offtopic a bit, because it's about technical things, but I'd suggest Greg should also admit collections of pdf's as Insights. Then it would be much easier to write Insights of such typographically challenging texts simply in LaTeX. The greatest obstacle writing Insights articles for me are these technical problems!

29. Urs Schreiber says:
strangerep

Some diagrams which work ok contain:

downarrowsubstack{fb}

But both the faulty ones have: downarrow^substack{fb}Thanks again. Yes, that's what triggers the problem. I have removed the "substack", and now it displays.

But an issue remains. The use of "substack" here is a hack anyway. I replaces what in my source is "mathrlap" or "mathllap", neither of which seem to be supported here. If anyone has a suggestion, please let me know.

30. strangerep says:
Urs Schreiber

I am not sure yet what is causing the two rendering issues. Will investigate… Some diagrams which work ok contain:

downarrowsubstack{fb}

But both the faulty ones have: downarrow^substack{fb}

31. Urs Schreiber says:
strangerep

I encounter some non-rendered latex in Definition 1.7 (sections) and Example 1.8 (trivial bundle).
Anyone else seeing this?

[Edit:] In Definition 1.14. sub-para "1": Typo? ##dx^j## in 2nd line should be ##dx^i## ?Thanks. I fixed the indexing in Def. 1.14 now. I am not sure yet what is causing the two rendering issues. Will investigate…

32. strangerep says:

I encounter some non-rendered latex in Definition 1.7 (sections) and Example 1.8 (trivial bundle).
Anyone else seeing this?