Cauchy sequence

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# Cauchy sequences
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## Idea

A _Cauchy sequence_ is an infinite [[sequence]] which ought to [[convergence|converge]] in the sense that successive terms get arbitrarily close together, as they would if they were getting arbitrarily close to a limit.  Among sequences, only Cauchy sequences will converge; in a [[complete space]], all Cauchy sequence converge.

## Definitions

The precise definition varies with the context.

A sequence $(x_i)_i$ of [[real numbers]] is __Cauchy__ if, for every positive number $\epsilon$, almost all terms are within $\epsilon$ of one another.  Explicitly:
$$ \forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; |x_i - x_j| \lt \epsilon .$$

In a [[metric space]], a sequence $(x_i)_i$ is __Cauchy__ under the same condition, now relative to the metric $d$ on that space.  Explicitly:
$$ \forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; d(x_i,x_j) \lt \epsilon .$$

The same definition immediately applies to an extended quasipseudometric space (aka a Lawvere metric space), or anything in between.

In a [[gauge space]], a sequence $(x_i)_i$ is __Cauchy__ if this condition is satisfied for each gauging distance separately.  Explicitly:
$$ \forall d,\; \forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; d(x_i,x_j) \lt \epsilon .$$

In a [[Booij premetric space]], a sequence $(x_i)_i$ is __Cauchy__ if this condition is satisfied for the premetric for all positive rational numbers $\epsilon$.  Explicitly:
$$ \forall \epsilon,\; \exists N,\; \forall i, j \geq N,\; x_i \sim_\epsilon x_j .$$

In a [[uniform space]], a sequence $(x_i)_i$ is __Cauchy__ if an analogous condition is satisfied for each [[entourage]] $U$.  Explicitly:
$$ \forall U,\; \exists N,\; \forall i, j \geq N,; x_i \approx_U x_j .$$

In a [[preuniform space]], a sequence $(x_i)_i$ is __Cauchy__ if this condition is satisfied for each [[preuniformity]] $U$.  Explicitly:
$$ \forall U,\; \exists N,\; \forall i, j \geq N,\; x_i \approx_U x_j .$$

Generalizing both [[Booij premetric spaces]] and [[preuniform spaces]], in a $T$-premetric space with set $T$, a sequence $(x_i)_i$ is __Cauchy__ if this condition is satisfied for each element $U \in T$. Explicitly:
$$ \forall U,\; \exists N,\; \forall i, j \geq N,\; x_i \sim_U x_j .$$

In a [[Cauchy space]], a sequence $(x_i)_i$ is __Cauchy__ if it generates a [[Cauchy filter]].  Explicitly:
$$ \{ A \;|\; \exists N,\; \forall i, j \geq N,\; x_i \in A \} \in \mathcal{C} ,$$
where $\mathcal{C}$ is the collection of [[Cauchy filters]] that defines the structure of the Cauchy space.

All of the above are in fact special cases of this.

## Generalizations

### Multivalued sequences

A [[multivalued function|multivalued]] [[sequence]] on a set $A$ is a function $f:\mathbb{N} \to \mathcal{P}(T)$ such that for every natural number $n \in \mathbb{N}$, $f(n)$ is an [[inhabited]] [[subset]] of $A$.

A multivalued sequence $x:\mathbb{N} \to \mathcal{P}(\mathbb{R})$ of [[real numbers]] is __Cauchy__ if, for every positive number $\epsilon$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist real numbers $a, b$ such that $x(i)(a)$ and $x(j)(b)$ holds and $\vert a - b \vert \lt \epsilon$.

In a [[metric space]] $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is __Cauchy__ if, for every positive number $\epsilon$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $d(a, b) \lt \epsilon$.

In a [[gauge space]] $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is __Cauchy__ if, for every positive number $\epsilon$ and every gauge $d:S \times S \to \mathbb{R}_{\geq 0}$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $d(a, b) \lt \epsilon$.

In a [[uniform space]] $S$, a multivalued sequence $x:\mathbb{N} \to \mathcal{P}(S)$ is __Cauchy__ if, for every [[entourage]] $U$, there exist a natural number $N$ such that for all $i, j \geq N$, there exist elements $a, b \in S$ such that $x(i)(a)$ and $x(j)(b)$ holds and $a \approx_U b$.

### Nets and filters

A [[net]] is a generalization of a sequence; the definitions above serve to define a __Cauchy net__ without any change, other than allowing $(x_i)_i$ to be a net.  This is precisely the structure of a [[Cauchy space]]; instead of defining Cauchy nets in terms of Cauchy filters as above, we may equally well define a Cauchy filter to be a [[proper filter]] whose [[canonical net]] is Cauchy.

Note that in a [[complete space]], every Cauchy net has a limit, not just every Cauchy sequence.  Rather, a space in which every Cauchy sequence converges is called __sequentially complete__ or __[[sequentially Cauchy complete]]__.  Note that a metric space (or even a Lawvere metric space) is in fact complete if it is sequentially complete (although this result is not valid in some weak [[foundations]]); in particular, the [[real line]] $\mathbf{R}$ is complete.

When [[Bill Lawvere]] idenitified Lawvere metric spaces with [[enriched categories]] over the [[closed monoidal category|closed monoidal]] [[poset]] $(\mathbf{R}^+,+)$, he identified Cauchy sequences in such spaces with certain [[adjunctions]] of [[bimodules]], enough so that a metric space would be a Cauchy-[[complete space]] if and only if every adjunction of bimodules is induced by an [[enriched functor]].  Generalising this condition from $(\mathbf{R}^+,+)$ to an arbitrary closed monoidal category, we have the concept of [[Cauchy-complete category]].

## References

* Wikipedia, _[Cauchy sequence](https://en.wikipedia.org/wiki/Cauchy_sequence)_

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Cauchy space

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# Cauchy spaces
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## Idea

A _Cauchy space_ is a generalisation of a [[metric space]] with a bare minimum of structure for the concepts of _[[Cauchy sequence]]_, _Cauchy-continuous map_, and _[[Cauchy completion of a Cauchy space|Cauchy completion]]_ to make sense.  Topologically (that is, up to [[continuous maps]]), any Cauchy space is a [[convergence space]], but not much more than that.  Besides Cauchy sequences, we can also speak more generally of [[Cauchy nets]] and [[Cauchy filters]] in a Cauchy space; in fact, the usual definition is in terms of Cauchy filters.

## Definitions

A __Cauchy space__ is a [[set]] $S$ together with a collection of [[proper filters]] declared to be __[[Cauchy filters]]__.  These must satisfy [[axioms]]:

1. Centred: The [[principal ultrafilter]] $F_x = \{ A \;|\; x \in A \}$ at $x$ is Cauchy;
2. Isotone: If $F \subseteq G$ and $F$ is a Cauchy filter (and $G$ is at least a proper filter), then $G$ is Cauchy;
3. Locally filtered: If $F$ and $G$ are Cauchy filters and $F \vee G$ (the filter generated by $F \cup G$) is proper, then $F \cap G$ is Cauchy.

That is, the set of Cauchy filters is a local filter of proper filters that contains all principal ultrafilters (sort of a tongue twister).

The definition can also be phrased in terms of [[nets]]; a __[[Cauchy net]]__ is a net whose [[eventuality filter]] is Cauchy.  In particular, a __[[Cauchy sequence]]__ is a [[sequence]] whose eventuality filter is Cauchy.

The [[morphisms]] of Cauchy spaces are the Cauchy-continuous functions; a [[function]] $f$ between Cauchy spaces is __Cauchy-continuous__ if $f(F)$ is a ([[filterbase|base]] of a) [[Cauchy filter]] whenever $F$ is.  In this way, Cauchy spaces form a [[concrete category]] $Cau$.

## Examples

Any [[metric space]] is a Cauchy space: $F$ is a Cauchy filter iff it has elements of arbitrarily small diameter.  This reconstructs the usual definitions of Cauchy sequence and Cauchy-continuous map for metric spaces.  (In particular, a map between metric spaces is Cauchy-continuous iff it maps every Cauchy sequence to a Cauchy sequence; the result for general nets follows since a metric space is [[sequential space|sequential]].)  The [[forgetful functor]] from $Met$ (metric spaces and [[short maps]]) to $Cau$ is [[faithful functor|faithful]] but not [[full functor|full]].

More generally, any [[uniform space]] is a Cauchy space: $F$ is a Cauchy filter if, given any [[entourage]] $U$, $A \times A \subseteq U$ for some $A \in F$.  This reconstructs the usual definitions of Cauchy net and Cauchy-continuous map for uniform spaces.  (In general, we need nets rather than just sequences here.)  A map between uniform spaces is Cauchy continuous iff its [[restrictions]] to [[precompact space|precompact subspaces]] are always uniformly continuous.  The [[forgetful functor]] from $Unif$ (uniform spaces and [[uniformly continuous maps]]) to $Cau$ is [[faithful functor|faithful]] but still not [[full functor|full]].

Assuming the [[ultrafilter principle]] and [[excluded middle]], we may take any collection $U$ of [[free ultrafilters]] and define a proper filter $F$ to be Cauchy if (hence iff) every free ultrafilter that refines $F$ belongs to $U$.

## Properties

Every Cauchy space is a [[convergence space]]; $F \to x$ if the intersection of $F$ with the principal ultrafilter $F_x$ is Cauchy.  Note that any convergent proper filter must be Cauchy.  Conversely, if every Cauchy filter is convergent, then the Cauchy space is called __[[complete space|complete]]__.

The set of [[Cauchy filters]] on a Cauchy space has a natural Cauchy structure which is complete and (as a convergence space) [[preregular space|preregular]]; we identify the indistinguishable Cauchy filters to get a [[Hausdorff space]], the Hausdorff __[[Cauchy completion of a Cauchy space|completion]]__ of the original Cauchy space.  The complete Hausdorff Cauchy spaces thus form a [[reflective subcategory]] of $Cau$.  This completion agrees with the completion of a metric or uniform space; that is, Cauchy completion, even of a metric space, is an operation on its Cauchy structure only.

Conversely, every Hausdorff convergence space becomes a complete Hausdorff Cauchy space upon declaring that the Cauchy filters are precisely the convergent proper filters.  The convergence structure on this Cauchy space matches the original convergence structure.

A Cauchy space $S$ is __[[totally bounded space|precompact]]__ (or __totally bounded__) if every proper filter is contained in a Cauchy filter.  Equivalently (assuming the [[ultrafilter principle]]), $S$ is precompact iff every [[ultrafilter]] is Cauchy.  A Cauchy space is [[compact space|compact]] (as a convergence space) iff it is both complete and precompact.  Conversely, it is precompact iff its completion is compact.

## See also

* [[Cauchy structure]]
* [[sequentially Cauchy complete space]]

## References

* Eva Lowen-Colebunder, _Function Classes of Cauchy Continuous Maps_ Dekker, New York, 1989

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Cauchy structure

[[!redirects Cauchy structures]]
[[!redirects Cauchy structure homomorphism]]
[[!redirects Cauchy structure homomorphisms]]
[[!redirects category of Cauchy structures]]

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## Definition ##

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}$$

be the set of positive rational numbers.

A __Cauchy structure__ is a [[Booij premetric space]] $(S, \sim)$ with a function $rat:\mathbb{Q}_{+} \to S$ and a function $lim:C(S) \to S$, where $C(S)$ is the set of [[Cauchy approximation]]s in $S$, such that

* for all elements $a \in S$ and $b \in S$ and all positive rational numbers $\epsilon$, $a \sim_\epsilon b$ implies $a = b$.

* for all elements $a \in S$ and $b \in S$ and all positive rational numbers $\epsilon$, $\vert a - b \vert \lt \epsilon$ implies that $rat(a) \sim_\epsilon rat(b)$

* for all elements $a \in S$ and Cauchy approximations $b \in C(S)$ and all positive rational numbers $\epsilon$ and $\eta$, $rat(a) \sim_\epsilon b(\eta)$ implies that $rat(a) \sim_{\epsilon + \eta} lim(b)$

* for all Cauchy approximations $a \in C(S)$ and elements $b \in S$ and all positive rational numbers $\delta$ and $\epsilon$, $a(\delta) \sim_\epsilon rat(b)$ implies that $lim(a) \sim_{\delta + \epsilon} rat(b)$

* for all Cauchy approximations $a \in C(S)$ and $b \in C(S)$ and all positive rational numbers $\delta$, $\epsilon$ and $\eta$, $a(\delta) \sim_\epsilon b(\eta)$ implies that $lim(a) \sim_{\delta + \epsilon + \eta} lim(b)$

For two Cauchy structures $S$ and $T$, a __Cauchy structure homomorphism__ is a function $f:S \to T$ such that

* for all elements $a \in S$ and $b \in S$ and all positive rational numbers $\epsilon$, $a \sim_{\epsilon} b$ implies $f(a) \sim_{\epsilon} f(b)$.

* for all rational numbers $r \in \mathbb{Q}$, $f(rat(r)) = rat(r)$

* for all Cauchy approximations $r \in C(S)$, $f(lim(r)) = lim(f \circ r)$

* for all elements $a \in S$ and $a \in S$, suppose $P(a, b)$ is the predicate that if for all positive rational numbers $\epsilon$ $a \sim_\epsilon b$ is true, then $a = b$, and $Q(a, b)$ is the predicate that for all positive rational numbers $\epsilon$ $f(a) \sim_\epsilon f(b)$ is true, then $f(a) = f(b)$. Then for all elements $a \in S$ and $a \in S$, $P(a, b)$ implies $Q(a, b)$

The __category of Cauchy structures__ is a [[category]] $CauchyStr$ whose objects $Ob(CauchyStr)$ are Cauchy structures and whose morphisms $Mor(A, B)$, for objects $A \in Ob(CauchyStr)$ and $B \in Ob(CauchyStr)$, are Cauchy structure homomorphisms. The initial object in the category of Cauchy structures is the [[HoTT book real numbers]].

### In homotopy type theory ###

Let $\mathbb{Q}$ be the [[rational numbers]] and let

$$\mathbb{Q}_{+} \coloneqq \sum_{x:\mathbb{Q}} 0 \lt x$$

be the positive rational numbers.

A __Cauchy structure__ is a [[Booij premetric space]] $S$ with

* a function $inj: \mathbb{Q} \to S$

* a function $lim: C(S) \to S$, where $C(S)$ is the type of $R_{+}$-[[Cauchy approximation]]s in $S$

* dependent families of terms

$$a:S, b:S \vdash eq(a, b): \Vert \prod_{\epsilon:\mathbb{Q}_{+}} (a \sim_{\epsilon} b) \Vert \to (a =_S b)$$
$$a:\mathbb{Q}, b:\mathbb{Q}, \epsilon:\mathbb{Q}_{+} \vdash \mu_{\mathbb{Q}, \mathbb{Q}}(a, b, \epsilon): (\vert a - b \vert \lt \epsilon) \to (inj(a) \sim_{\epsilon} inj(b))$$
$$a:\mathbb{Q}, b:C(S), \epsilon:\mathbb{Q}_{+}, \eta:\mathbb{Q}_{+} \vdash \mu_{\mathbb{Q}, C(S)}(a, b, \epsilon, \eta): (inj(a) \sim_{\epsilon} b_{\eta}) \to (inj(a) \sim_{\epsilon + \eta} lim(b))$$
$$a:C(S), b:\mathbb{Q}, \delta:\mathbb{Q}_{+}, \epsilon:\mathbb{Q}_{+} \vdash \mu_{C(S), \mathbb{Q}}(a, b, \delta, \epsilon): (a_{\delta} \sim_{\epsilon} inj(b) ) \to (lim(a) \sim_{\delta + \epsilon} inj(b))$$
$$a:C(S), b:C(S), \delta:\mathbb{Q}_{+}, \epsilon:\mathbb{Q}_{+}, \eta:\mathbb{Q}_{+} \vdash \mu_{C(S), C(S)}(a, b, \delta, \epsilon, \eta): (a_{\delta} \sim_{\epsilon} b_{\eta} ) \to (lim(a) \sim_{\delta + \epsilon + \eta} lim(b))$$

For two Cauchy structures $S$ and $T$, a __Cauchy structure homomorphism__ consists of

* a function $f:S \to T$

* a dependent family of functions
$$a:S, b:S, \epsilon:\mathbb{Q}_{+} \vdash \pi(a, b, \epsilon): (a \sim_{S\epsilon} b) \to (f(a) \sim_{T\epsilon} f(b))$$

* a dependent family of types
$$r:\mathbb{Q} \vdash \iota(r): f(inj_S(r)) = inj_T(r)$$

* a dependent family of types
$$r:C(S) \vdash \lambda(r): f(lim_S(r)) = lim_T(f \circ r)$$

* a dependent family of types
$$a:S, b:S, c:\Vert \prod_{\epsilon:\mathbb{Q}_{+}} (a \sim_{\epsilon} b) \Vert \vdash \eta(a, b, c): f(eq_S(a, b, c)) = eq_T(f(a), f(b), f(c))$$

## See also ##

* [[Booij premetric space]]
* [[Cauchy approximation]]
* [[HoTT book real numbers]]

## References ##

* Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf))


Cauchy sum theorem

# The Cauchy sum theorem
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## Overview

In his influential 1821 textbook _[[Cours d'Analyse]]_, [[Augustin Cauchy]] states a [[theorem]] that is now widely regarded as [[false]], attributed to a confusion between [[pointwise convergence]] and [[uniform convergence]].  This mistake --- if indeed it was a mistake --- is of both pedagogical and philosophical-historical interest.

[[counterexample|Counterexamples]] (specific [[Fourier series]]) were known already to [[Joseph Fourier]], and [[Niels Abel]] specifically pointed them out as counterexamples to Cauchy\'s claim.  However, Cauchy denied that these were [[counterexamples]], on the grounds that the series did not converge everywhere (which is a hypothesis in the theorem).  [[Imre Lakatos]] has argued that the confusion rests on different conceptions of the [[continuum]], so that Cauchy\'s notion of convergence everywhere is really more like Weierstrass\'s notion of uniform convergence than pointwise convergence, and the theorem as he intended it is true.

Although Cauchy\'s original formulation was about the sum of an [[infinite series]], we will consider it in the slightly more elementary context of the limit of an [[infinite sequence]]; the modern isomorphism between these was already well established and used explicitly by Cauchy.

## Statements

+-- {: .num_theorem #mistake}
###### Non-theorem (attributed to Cauchy, 1821)

Let $f = (f_1, f_2, \ldots)$ be an [[infinite sequence]] of [[continuous functions]] from the [[real line]] to itself.  Suppose that, for every [[real number]] $x$, the sequence $(f_1(x), f_2(x), \ldots)$ converges to some (necessarily unique) real number $f_\infty(x)$, defining a [[function]] $f_\infty$; in other words, the sequence $f$ [[pointwise convergence|converges pointwise]] to $f_\infty$.  Then $f_\infty$ is also continuous.
=--

+-- {: .num_theorem #uniform}
###### Theorem (attributed to Seidel, 1847)

Let $f = (f_1, f_2, \ldots)$ be an [[infinite sequence]] of [[continuous functions]] from the [[real line]] to itself.  Suppose that the sequence $f$ [[uniform convergence|converges uniformly]] to a function $f_\infty$.  Then $f_\infty$ is also continuous.
=--

+-- {: .num_theorem #equicontinuous}
###### Theorem (attributed to ???)

Let $f = (f_1, f_2, \ldots)$ be an [[equicontinuous family|equicontinuous]] [[infinite sequence|sequence]] of (necessarily [[continuous functions|continuous]]) functions from the [[real line]] to itself.  Suppose that, for every [[real number]] $x$, the sequence $(f_1(x), f_2(x), \ldots)$ converges to some (necessarily unique) real number $f_\infty(x)$, defining a [[function]] $f_\infty$; in other words, the sequence $f$ [[pointwise convergence|converges pointwise]] to $f_\infty$.  Then $f_\infty$ is also continuous.
=--

## Counterexamples

The first counterexamples to Non-Theorem \ref{mistake} arose as [[Fourier series]].  The sawtooth wave
$$ \sum_{k = 1}^\infty \frac{\sin(k x)}{k} = \frac{\pi}{2} - \frac{x \mathbin{mod} 2 \pi}{2} $$
may be the simplest.  Each partial sum of this [[infinite series]] is continuous; the sum converges pointwise as indicated for $x$ not a multiple of $2 \pi$ and to $0$ (which is the average of the limits $\pm\pi/2$ on either side) for $x$ a multiple of $2 \pi$.  However, the sequence of partial sums is not equicontinuous, nor does it converge uniformly.  And indeed, the sum is not continuous at multiples of $2 \pi$.

A simple counterexample from outside of this context is
$$ f_n(x) \coloneqq x^n $$
on $[0,1]$.  (We can make this a function on all of $\mathbb{R}$ by letting $f_n(x)$ be $f_n(0) = 0$ for $x \lt 0$ and letting $f_n(x)$ be $f_n(1) = 1$ for $x \gt 1$.)  Each $f_n$ is continuous; the pointwise limit is $f_\infty(x) = 0$ for $x \lt 1$ but $f_\infty(x) = 1$ for $x \geq 1$, which is not continuous.  Again, the sequence is not equicontinuous, and its convergence is not uniform.

Another counterexample is
$$ f_n(x) \coloneqq \exp ({-n {|x|}}) .$$
(The [[absolute value]] is here only to handle negative values of $x$; the example is [[analytic function|analytic]] on $]0,\infty[$.)  Now $f_\infty(0) = 1$, while $f_\infty(x) = 0$ for $x \ne 0$.

## Proofs

Here is Cauchy\'s argument:
+-- {: .proof}
###### Proof?

At any $x$, we have
$$
      {|f_\infty(x + h) - f_\infty(x)|}
    = {|f_\infty(x + h) - f_n(x + h) + f_n(x + h) - f_n(x) + f_n(x) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
.$$
As $n$ becomes infinite and $h$ becomes infinitesimal, all of the terms on the right become infinitesimal (respectively, because $f$ converges to $f_\infty$ at $x + h$, because $f_n$ is continuous at $x$, and because $f$ converges to $f_\infty$ at $x$).  Thus, the expression on the left also becomes infinitesimal, proving that $f_\infty$ is continuous at $x$.
=--

Nowadays, we\'d say that a variable approaches zero rather than that it becomes infinitesimal.  If we interpret this argument in modern analysis, then the proof is flawed, because (the lower bound for) $n$ depends on $h$ in the first term, while (the upper bound for the absolute value of) $h$ depends on $n$ in the second term; they cannot be chosen simultaneously.

Analysing this argument, Philipp von Seidel (first, and others afterwards) realised that it could be fixed if $f$ converges to $f_\infty$ _[[uniform convergence|uniformly]]_, so that $n$ is independent of $h$.  Another fix is to make $h$ independent of $n$, by requiring the family $f$ to be [[equicontinuous family|equicontinuous]], although this idea came later.

### In epsilontics

Writing Cauchy\'s argument in the [[epsilontic analysis|epsilontic]] language developed by [[Karl Weierstrass]], we have:
+-- {: .proof}
###### Non-proof
(Non-theorem \ref{mistake}).

Let $\epsilon$ be a [[positive number]], and consider $\epsilon/3$.  Because $f$ converges to $f_\infty$ at $x$, there is some [[natural number]] $N$ such that ${|f_n(x) - f_\infty(x)|} \lt \epsilon/3$ whenever $n \geq N$.  Because $f_n$ is continuous at $x$ for each $n$, there is some positive number $\delta$ such that ${|f_n(x + h) - f_n(x)|} \lt \epsilon/3$ whenever $h \lt \delta$.  Because $f$ converges to $f_\infty$ at $x + h$ for each $h$, there is some natural number $N$ such that ${|f_\infty(x + h) - f_n(x + h)|} \lt \epsilon/3$ whenever $n \geq N$.  Therefore, there are $N$ and $\delta$ such that
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
  \lt \epsilon/3 + \epsilon/3 + \epsilon/3
    = \epsilon
$$
whenever $n \geq N$ and $h \lt \delta$.  Fixing any such $n$, $f_\infty$ is continuous.
=--

We have used the variable names $n$ and $h$ in two different [[contexts]] each, then pretended that they arose in a single context for the final inequality.  The error can be made more explicit by using different variable names:
+-- {: .proof}
###### Non-proof
(Non-theorem \ref{mistake}).

Let $\epsilon$ be a [[positive number]], and consider $\epsilon/3$.  Because $f$ converges to $f_\infty$ at $x$, there is some [[natural number]] $N$ such that ${|f_n(x) - f_\infty(x)|} \lt \epsilon/3$ whenever $n \geq N$.  Because $f_{n'}$ is continuous at $x$ for each $n'$, there is some positive number $\delta$ such that ${|f_{n'}(x + h) - f_{n'}(x)|} \lt \epsilon/3$ whenever $h \lt \delta$.  Because $f$ converges to $f_\infty$ at $x + h'$ for each $h'$, there is some natural number $N'$ such that ${|f_\infty(x + h') - f_{n''}(x + h')|} \lt \epsilon/3$ whenever $n'' \geq N'$.  Therefore, there are $N, N'$ and $\delta$ such that
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h') - f_{n''}(x + h')|} + {|f_{n'}(x + h) - f_{n'}(x)|} + {|f_n(x) - f_\infty(x)|}
  \lt \epsilon/3 + \epsilon/3 + \epsilon/3
    = \epsilon
$$
whenever $n, n', n'' \geq max(N, N')$ and $h, h' \lt \delta$.  Fixing any such $n, n', n''$, $f_\infty$ is continuous.
=--
The final inequality is now clearly spurious, since $n, n', n''$ need not be equal, nor $h, h'$.

This can be fixed up to a point.  We may let $n'$ be $n$ and let $h'$ be $h$, but then we have no control over $n''$.  Conversely, we may let $n, n', n''$ all be (anything bounded below by) $max(N,N')$, but then we have no control over $h'$.  Indeed, Non-theorem \ref{mistake} is false, as the counterexamples show.

However, the weaker theorems with strengthened hypotheses are valid, by similar proofs:
+-- {: .proof}
###### Proof
(Theorem \ref{uniform}).

Let $\epsilon$ be a [[positive number]], and consider $\epsilon/3$.  Because $f$ converges uniformly to $f_\infty$, there is some [[natural number]] $N$ such that ${|f_n(x + h) - f_\infty(x + h)|} \lt \epsilon/3$ for every $h$ (including $h \coloneqq 0$) whenever $n \geq N$.  Because $f_n$ is continuous at $x$ for each such $n$, there is some positive number $\delta$ such that ${|f_n(x + h) - f_n(x)|} \lt \epsilon/3$ whenever $h \lt \delta$.  Therefore, there is $N$ such that, whenever $n \geq N$, there is $\delta$ such that
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
  \lt \epsilon/3 + \epsilon/3 + \epsilon/3
    = \epsilon
$$
whenever $h \lt \delta$.  Fixing any such $n$, $f_\infty$ is continuous.
=--
+-- {: .proof}
###### Proof
(Theorem \ref{equicontinuous}).

Let $\epsilon$ be a [[positive number]], and consider $\epsilon/3$.  Because $f$ converges to $f_\infty$ at $x$, there is some [[natural number]] $N$ such that ${|f_n(x) - f_\infty(x)|} \lt \epsilon/3$ whenever $n \geq N$.  Because $f$ is equicontinuous at $x$, there is some positive number $\delta$ such that ${|f_n(x + h) - f_n(x)|} \lt \epsilon/3$ for every $n$ whenever $h \lt \delta$.  Because $f$ converges to $f_\infty$ at $x + h$ for each such $h$, there is some natural number $N'$ such that ${|f_\infty(x + h) - f_n(x + h)|} \lt \epsilon/3$ whenever $n \geq N'$.  Therefore, there are $N$ and $\delta$ such that, whenever $h \lt \delta$, there is $N'$ such that
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
  \lt \epsilon/3 + \epsilon/3 + \epsilon/3
    = \epsilon
$$
whenever $n \geq max(N,N')$.  Fixing any such $n$, $f_\infty$ is continuous.
=--

### In nonstandard analysis

Writing Cauchy\'s argument in the language of [[nonstandard analysis]] developed by [[Abraham Robinson]], we have:
+-- {: .proof}
###### Non-proof
(Non-theorem \ref{mistake}).

Because $f$ converges to $f_\infty$ at $x$, ${|f_n(x) - f_\infty(x)|}$ is [[infinitesimal number|infinitesimal]] whenever $n$ is [[infinite number|infinite]].  Because $f_n$ is continuous at $x$ for each finite $n$, it is continuous at $x$ for each infinite $n$ (by the [[transfer principle]]), so ${|f_n(x + h) - f_n(x)|}$ is infinitesimal whenever $h$ is infinitesimal.  Because $f$ converges to $f_\infty$ at each [[standard point]], $f$ converges to $f_\infty$ at the nonstandard point $x + h$ (by the transfer principle), so ${|f_\infty(x + h) - f_n(x + h)|}$ is infinitesimal whenever $n$ is infinite.  Therefore,
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
$$
is also infinitesimal whenever $n$ is infinite and $h$ is infinitesimal.  Fixing any such $n$, $f_\infty$ is continuous.
=--

Both uses of the transfer principle are illegitimate, since the statements to which they\'re applied are not [[first-order logic|first-order]] in the language of the [[real numbers]].  Indeed, consider the counterexample $f_n(x) = \exp ({-n {|x|}})$ near $x = 1$.  Then ${|f_n(0 + h) - f_n(0)|} = \exp ({-n {|h|}}) - 1$ is infinitesimal as claimed iff $n h$ is infinitesimal, which may fail.  Conversely, ${|f_\infty(0 + h) - f_n(0 + h)|} = \exp ({-n {|h|}})$ (for $h \ne 0$) is infinitesimal iff $n h$ is infinite, which may also fail.  In between, when $n h$ is finite and finitesimal, both fail!

We can rewrite this proof more carefully, putting down what really follows in place of the misuse of the transfer principle:
+-- {: .proof}
###### Non-proof
(Non-theorem \ref{mistake}).

Because $f$ converges to $f_\infty$ at $x$, ${|f_n(x) - f_\infty(x)|}$ is [[infinitesimal number|infinitesimal]] whenever $n$ is [[infinite number|infinite]].  Because $f_n$ is continuous at $x$ for each finite $n$, there is some infinite $N$ such that $f_n$ is continuous at $x$ for all $n \leq N$, so then ${|f_n(x + h) - f_n(x)|}$ is infinitesimal whenever $h$ is infinitesimal.  Because $f$ converges to $f_\infty$ at each [[standard point]] and $x + h$ is [[nearstandard point|nearstandard]], there is an infinite number $N'$ such that ${|f_\infty(x + h) - f_n(x + h)|}$ is infinitesimal whenever $n \geq N'$.  Therefore,
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
$$
is also infinitesimal whenever $n \leq N$, $n \geq N'$, and $h$ is infinitesimal.  Fixing any such $n$, $f_\infty$ is continuous.
=--
This is more clearly wrong, since there may be no such $n$ in the last step (as $N'$ may be larger than $N$).

However, we can modify this to produce correct proofs of the weaker theorems:
+-- {: .proof}
###### Proof
(Theorem \ref{uniform}).

Because $f$ converges to $f_\infty$ at $x$, ${|f_n(x) - f_\infty(x)|}$ is [[infinitesimal number|infinitesimal]] whenever $n$ is [[infinite number|infinite]].  Because $f_n$ is continuous at $x$ for each finite $n$, there is some infinite $N$ such that $f_n$ is continuous at $x$ for all $n \leq N$, so then ${|f_n(x + h) - f_n(x)|}$ is infinitesimal whenever $h$ is infinitesimal.  Because $f$ converges to $f_\infty$ uniformly, $f$ converges to $f_\infty$ at the nonstandard point $x + h$, so ${|f_\infty(x + h) - f_n(x + h)|}$ is infinitesimal whenever $n$ is infinite.  Therefore,
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
$$
is also infinitesimal whenever $n \leq N$ is infinite and $h$ is infinitesimal.  Fixing any such $n$, $f_\infty$ is continuous.
=--
+-- {: .proof}
###### Proof
(Theorem \ref{equicontinuous}).

Because $f$ converges to $f_\infty$ at $x$, ${|f_n(x) - f_\infty(x)|}$ is [[infinitesimal number|infinitesimal]] whenever $n$ is [[infinite number|infinite]].  Because $f$ is equicontinuous at $x$, $f_n$ is continuous at $x$ even for infinite $n$, so ${|f_n(x + h) - f_n(x)|}$ is infinitesimal whenever $h$ is infinitesimal.  Because $f$ converges to $f_\infty$ at each [[standard point]] and $x + h$ is [[nearstandard point|nearstandard]], there is an infinite number $N$ such that ${|f_\infty(x + h) - f_n(x + h)|}$ is infinitesimal whenever $n \geq N$.  Therefore,
$$    {|f_\infty(x + h) - f_\infty(x)|}
 \leq {|f_\infty(x + h) - f_n(x + h)|} + {|f_n(x + h) - f_n(x)|} + {|f_n(x) - f_\infty(x)|}
$$
is also infinitesimal whenever $n \geq N$ and $h$ is infinitesimal.  Fixing any such $n$, $f_\infty$ is continuous.
=--
(To check that these are valid, one must know the nonstandard formulations of uniform convergence and equicontinuity that are used; but they are correct.  Specifically, a sequence $f$ of functions converges uniformly to a function $f_\infty$ iff, for every hyperreal number $x$, the sequence $f(x) = (f_n(x))_n$ of hyperreal numbers converges to the hyperreal number $f_\infty(x)$; and $f$ is equicontinuous at a real number $x$ iff, for every hyperinteger $n$, the hyperfunction $f_n$ is continuous at $x$.)

On the other hand, in the light of nonstandard analysis, we can also reevaluate Cauchy\'s original claim.  Cauchy himself, when confronted with counterexamples such as the Fourier series, denied that the sequence of functions converged everywhere.  One interpretation of this is that it fails to converge at some nonstandard points.  By this analysis, Cauchy\'s notion of convergence everywhere is our modern notion of uniform convergence, not pointwise convergence, and his theorem is true (and his proof correct, even if less detailed than one might like).

## Lakatos\'s view

In his experimental textbook _[[Proofs and Refutations]]_, [[Imre Lakatos]] used Cauchy\'s sum theorem to motivate the concept of uniform convergence.  Two years later, he partially reevaluated his discussion in light of the question of what Cauchy\'s conception of the continuum was.

According to Lakatos, it is ahistorical to interpret Cauchy\'s 1821 result using either Weierstrass\'s epsilontics or Robinson\'s nonstandard analysis.  Cauchy did not mean that $f_n(x)$ converges (in $n$) for each fixed standard real number $x$, nor that it converges for each fixed hyperreal number $x$; rather, he said that it converges for each *variable* real number.  In particular, when discussing the Fourier series
$$ \sum_{k = 1}^\infty \frac{\sin(k x)}{k} ,$$
Cauchy states (in 1853) that the sequence
\[ \sum_{k = 1}^n \frac{\sin(k x)}{k} \label{FourierPS} \]
of partial sums fails to converge when $x = 1/n$; that is, $x$ is a variable whose value depends on the position in the sequence!  Such a claim is hard to interpret in an epsilontic framework; but in a nonstandard framework, it simply means that for some particular infinite values of $n$ and infinitesimal values of $x$ (namely those for which $x = 1/n$), this partial sum is not infinitely close to $0$ (which is the value of the partial sum at $x = \lim_n (1/n) = 0$), demonstrating that the sequence is not equicontinuous.

If we (still arguably ahistorically) interpret convergence to mean to that $f_n(x_n)$ converges (in $n$) for every convergent sequence $(x_n)_n$ of real numbers, then this is equivalent (given that each $f_n$ is continuous) to uniform convergence, and Cauchy\'s result would again be true and his argument would be (when formalised with either epsilontics or nonstandard analysis) valid.

On the other hand (and this goes beyond what Lakatos wrote), one way to interpret Cauchy\'s claim that (eq:FourierPS) fails to converge when $x = 1/n$ is, using nonstandard analysis, simply that for any particular infinite value of $n$ and using the infinitesimal value $x = 1/n$, this partial sum is not infinitely close to $0$ (which is the value of the infinite sum at $x = \lim_n (1/n) = 0$).  This suggests interpreting convergence to mean that $f_n(c+h)$ is infinitely close to $f_\infty(c)$ whenever $n$ is infinite and $h$ is infinitesimal (for $c$ real), which again is equivalent (given continuity of $f_n$) to uniform convergence, again justifying Cauchy\'s proof.  However, proving that uniform convergence is equivalent to this is basically the same concept as the nonstandard proof of Theorem \ref{uniform} anyway.

## Related concepts

* [[Cauchy's theorems]]

## References

The original sum theorem is in

*  [[Augustin Cauchy]] (1821). _[[Cours d'Analyse]]_.

Cauchy\'s most thorough defence of the theorem against critics is in

*  [[Augustin Cauchy]] (1853). Note sur les séries convergentes dont les divers termes sont des fonctions continues d\'une variable réelle ou imaginaire, entre des limites données.

The reconstruction in terms of nonstandard analysis is in the historical appendix to

*  [[Abraham Robinson]] (1966). _Nonstandard Analysis_.

Lakatos\'s discussion forms Chapter 3 of

*  [[Imre Lakatos]] (1978). _Mathematics, Science, and Epistemology_.

[[!redirects Cauchy's mistake]]
[[!redirects Cauchy's mistake]]
[[!redirects Cauchy\'s mistake]]
[[!redirects Cauchy/'s mistake]]

[[!redirects Cauchy sum theorem]]
[[!redirects Cauchy's sum theorem]]
[[!redirects Cauchy\'s sum theorem]]
[[!redirects Cauchy's sum theorem]]


Cauchy surface

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### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Riemannian geometry
+-- {: .hide}
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#### Physics
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[[!include physicscontents]]
=--
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=--

# Cauchy surfaces
* table of contents
{: toc}

## Idea

A Cauchy surface is a [[hypersurface]] in [[spacetime]] (so actually a $3$-dimensional region in our $4$-dimensional spacetime) that can profitably be seen as constituting 'all of space at a given time'.

## Definition

For $(X,g)$ a [[Lorentzian manifold]], a **Cauchy surface** is an [[embedding|embedded]] [[submanifold]] $\Sigma \hookrightarrow X$ such that every [[timelike]] [[curve]] in $X$ may be extended to a timelike curve that intersects $\Sigma$ precisely in one point.

A Lorentzian manifold that does admit a Cauchy surface is called _[[globally hyperbolic]]_.

## Properties

\begin{proposition}\label{FoliationOfGloballyHyperbolicSpacetimes}
  For $X^D$ a [[smooth manifold|smooth]] [[globally hyperbolic spacetime]], it admits a smooth [[foliation]]  by smooth spacelike Cauchy surfaces $X^d$, exhibited by a [[diffeomorphism]]
$$
  X^D \,\simeq\, \mathbb{R}^{0,1} \times X^d
  \,,
$$
which is [[isometry|isometric]] with respect to a possibly non-product [[pseudo-Riemannian metric]] on the right
\end{proposition}
([Bernal & Sánchez 2005, Thm. 1.1](#BernalSánchez03), following [Geroch 1970](#Geroch70), reviewed in [Sánchez 2022 (3)](#Sánchez22)).

## Applications

One way to formulate [[causality]] in [[physics]] is that the values of all [[observables]] at all points on a single Cauchy surface in [[spacetime]] is enough information (in the sense of a [[boundary condition]] to apply to a [[differential equation]] constituting a relevant physical theory) to determine the values of all observables at all points of spacetime.  (This is not always an actual theorem of differential equations.)  Stated more intuitively, the state of the universe at any given time is enough information to determine the state of the universe at all times.

If [[spacetime]] can be equipped with a [[foliation]] of Cauchy surfaces, then we may assign a [[real number]] $t$ to each surface $\Sigma$, so that we think of $\Sigma$ as 'space at time $t$'.  Of course, there are typically many ways to do this (if any), in accordance with the principle of [[relativity of simultaneity]].  On the other hand, for some spacetimes, this may not be possible at all (because they are not globally hyperbolic).

## Related concepts

* [[Cauchy horizon]]

## References

The existence of a splitting of globally hyperbolic spacetimes into Cauchy surfaces is

in the topological category due to

* {#Geroch70} [[Robert Geroch]], §5, Thm. 11 in: *Domain of Dependence*, J. Math. Phys. **11** (1970) 437–449 [[doi:10.1063/1.1665157](https://doi.org/10.1063/1.1665157)]

and in the smooth category (needed in practice) due to

* {#BernalSánchez03} Antonio N. Bernal, [[Miguel Sánchez]], _On smooth Cauchy hypersurfaces and Geroch's splitting theorem_, Commun. Math. Phys. 243 (2003) 461-470 [[arXiv:gr-qc/0306108](http://arxiv.org/abs/gr-qc/0306108), [doi:10.1007/s00220-003-0982-6](https://doi.org/10.1007/s00220-003-0982-6)]

* {#BernalSánchez05} [[Antonio N. Bernal]], [[Miguel Sánchez]], *Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes*, Commun. Math. Phys. **257** (2005) 43-50 [[arXiv:gr-qc/0401112](https://arxiv.org/abs/gr-qc/0401112), [doi:10.1007/s00220-005-1346-1](https://doi.org/10.1007/s00220-005-1346-1)]

* {#Sánchez22} [[Miguel Sánchez]], *Globally hyperbolic spacetimes: slicings, boundaries and counterexamples*, Gen Relativ Gravit **54** 124 (2022) [[arXiv:2110.13672](https://arxiv.org/abs/2110.13672), [doi:10.1007/s10714-022-03002-6](https://doi.org/10.1007/s10714-022-03002-6)]

[[!redirects Cauchy surface]]
[[!redirects Cauchy surfaces]]


Cauchy theorem

Here are some of the many [[theorems]] named after [[Augustin Cauchy]]:

* [[Cauchy's integral theorem]] (see also [[Cauchy's integral formula]])

* the [[Cauchy sum theorem]]

* the [[Cauchy group theorem]]

* the Cauchy [[mean-value theorem]]

See also:

* [[Cauchy sequences]] (also [[Cauchy nets]], [[Cauchy filters]], and [[Cauchy spaces]])

* the [[Cauchy–Riemann equations]]

* [[Cauchy surfaces]]

* [[Cauchy real numbers]]

* [[Cauchy principal values]]

category: disambiguation

[[!redirects Cauchy theorem]]
[[!redirects Cauchy's theorem]]
[[!redirects Cauchy’s theorem]]
[[!redirects Cauchy\'s theorem]]

[[!redirects Cauchy theorems]]
[[!redirects Cauchy's theorems]]
[[!redirects Cauchy’s theorems]]
[[!redirects Cauchy\'s theorems]]


Cauchy's integral formula

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* table of contents
{:toc}

## Statement

For $U \subset \mathbb{C}$ an [[open subset]] of the [[complex plane]] $\mathbb{C}$ and for $C$ a [[Jordan curve]] in $U$, a [[holomorphic function]] $f$ of $U$ sends a point $\zeta$ enclosed by $C$ to the [[contour integral]]

$$
  f(\zeta) =  \frac{1}{2\pi\mathrm{i}}\oint_C \frac{f(z)}{z - \zeta} \,\mathrm{d}z.
$$

Hence the [[contour integral]] picks out the enclosed _[[residues]]_.

More generally, this implies, by [[Taylor series]] expansion of $f$, that for $n \in \mathbb{N}$ the $n$th complex [[derivative]] $f^{(n)}$ is

$$
  f^{(n)}(\zeta)
    =
  \frac{n!}{2\pi i}  \oint_C \frac{f(z)}{(z - \zeta )^{n+1}} d z
  \,.
$$

This is also known as _Cauchy's differentiation formula_.

## Proof in synthetic differential geometry

Here is a proof written in terms of [[synthetic mathematics|synthetic]] [[infinitesimals]] as in [[synthetic differential geometry]]:

Let $\epsilon$ be a nilpotent. Let $S_\epsilon$ denote the [[circle]] of [[radius]] $\epsilon$ centered at $\zeta$. By the holomorphicity of $f$, the [[differential 1-form]] $f(z)/(z-\zeta) \,\mathrm{d}z$ is [[closed differential form|closed]] in the region bounded by $C$ and $S_\epsilon$. By the [[Stokes theorem]],
$$\frac{1}{2\pi\mathrm{i}}\oint_C \frac{f(z)}{z - \zeta} \,\mathrm{d}z = \frac{1}{2\pi\mathrm{i}}\oint_{S_\epsilon} \frac{f(z)}{z - \zeta} \,\mathrm{d}z.$$
Parametrize $S_\epsilon$ by
$(t\mapsto \zeta + \epsilon \,\exp(2\pi\mathrm{i}t))$
to transform the above integral to
$$\frac{1}{2\pi\mathrm{i}}\int_0^1 \frac{f(\zeta + \epsilon\, \exp(2\pi\mathrm{i}t))}{\epsilon \,\exp(2\pi\mathrm{i}t))} \,\mathrm{d}(\zeta + \epsilon\, \exp(2\pi\mathrm{i}t)) = \int_0^1 f(\zeta + \epsilon\, \exp(2\pi\mathrm{i}t)) \, \mathrm{d}t$$
By the infinitesimal [[Taylor series|Taylor formula]] and the [[holomorphic function|holomorphicity]] of $f$,
$$f(\zeta + \epsilon\exp(2\pi\mathrm{i}t)) = f(\zeta) + \epsilon\,\exp(2\pi\mathrm{i}t) \frac{\partial f(\zeta)}{\partial z}.$$
Hence the above integral is equal to
$$\int_0^1 \left[f(\zeta) + \epsilon\,\exp(2\pi\mathrm{i}t) \frac{\partial f(\zeta)}{\partial z}\right] \, \mathrm{d}t = \left[f(\zeta) + \epsilon\frac{\partial f(\zeta)}{\partial z} \int_0^1 \exp(2\pi\mathrm{i}t)  \, \mathrm{d}t\right] = f(\zeta).$$

## Related concepts

* [[Cauchy's integral theorem]]

* [[Cauchy's theorems]]

* [[Goursat theorem]]

## References

* Wikipedia, _[Cauchy's integral formula](https://en.wikipedia.org/wiki/Cauchy's_integral_formula)_

[[!redirects Cauchy integral formula]]

[[!redirects Cauchy differentiation formula]]
[[!redirects Cauchy's differentiation formula]]


Cauchy-Riemann equation

## Related concepts

* [[Cauchy's theorems]]

## References

Named after [[Augustin Cauchy]] and [[Bernhard Riemann]].

* Wikipedia, _[Cauchy-Riemann equations](http://en.wikipedia.org/wiki/Cauchy–Riemann_equations)_

[[!redirects Cauchy-Riemann equations]]
[[!redirects Cauchy–Riemann equations]]
[[!redirects Cauchy-Riemann]]


Cauchy–Schwarz inequality

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* table of contents
{:toc}

## Statement

Consider an *[[inner product space]]* in the sense of a [[vector space]] $\mathscr{V}$ equipped with a [[Hermitian form]] $\langle -,- \rangle$ which is positive *semi*-definite:

$$
  v \in \mathscr{V}
  \;\;\;
  \vdash
  \;\;\;
  \langle
    v , v
  \rangle
  \;\geq 0\;
  \,.
$$

(we do *not* need to assume positive definiteness, cf. [MO:a/2548691](https://math.stackexchange.com/a/2548691/58526), [Ćurgus](#Ćurgus))

then for all [[pairs]] of [[vectors]] $v,w \,\in\, V$ the following Cauchy-Schwarz [[inequality]] holds:

$$
  \left\vert\langle u,v\rangle\right\vert^2
  \;\leq\;
  \langle u,u \rangle
  \cdot
  \langle v,v \rangle
  \,.
$$

In terms of the [[norm]] $\Vert-\Vert \,\coloneqq\, \sqrt{\langle -,-\rangle}$ and the [[absolute value]] this means equivalently:

$$
  {\big\vert \langle u,v\rangle \big\vert}
  \;\leq\;
  \left\Vert u \right\Vert
  \cdot
  \left\Vert v \right\Vert
  \,.
$$

## Related concepts

* [[GNS-construction]]

## References

Original proofs are due to [[Cauchy]] in 1821, [[Bouniakowsky]] in 1859, [[Hermann Schwarz]] in 1888.

Review:

* {#Ćurgus} Branko Ćurgus, *[Cauchy-Bunyakovsky-Schwarz inequality](https://faculty.curgus.wwu.edu/Courses/Math_pages/Math_504/Cauchy-Schwarz-Bunyakovsky.html)*

* Wikipedia, _[Cauchy-Schwarz inequality](https://en.wikipedia.org/wiki/Cauchy-Schwarz_inequality)_

[[!redirects Cauchy-Schwarz inequality]]
[[!redirects Cauchy inequality]]


causal additivity

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## Idea

The formalization of [[perturbative quantum field theory]] via [[causal perturbation theory]] states the key properties that the  [[scattering matrix]] of the theory is supposed to have as [[axioms]] (instead of imagining that the scattering matrix is defined by a [[path integral]]). The key such axiom is the statement that the [[S-matrix]] be _causally additive_ (which gives _[[causal perturbation theory]]_ its name). This essentially encodes the idea that effects in the quantum theory propagate causally, hence within the [[future cone]] and [[past cone]] of the region which causes the effect.

## Causal additivity

The [[S-matrix]] (as discussed there) is a [[functional]] of the form

$$
  S \;\colon\; \mathcal{F}_{loc} \longrightarrow \mathcal{W}[ [ \tfrac{g}{\hbar} ] ]
$$

sending [local observables](#Wick+algebra#CompactlySupportedPolynomialLocalDensities) with [[compact support]] in [[spacetime]] (i.e. [[adiabatic switching|adiabatically switched]] [[interaction]] [[Lagrangian densities]] $L_{int}$ and [[source fields]] $J$) to [[formal power series]] in the ratio of a [[coupling constant]] $g$ over [[Planck's constant]] $\hbar$ with [[coefficients]] in the [[Wick algebra]] $\mathcal{W}$ of the underlying [[free field theory]].

Causal additivity ([Epstein-Glaser 73](#EpsteinGlaser73)) is the statement/requirement that

$$
  \left( supp(J_1) \geq supp(J_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    \underset{L \in \mathcal{F}_{loc}}{\forall}
    \left(
      S(L + J_1 + J_2)
      =
      S(L + J_1) S(L)^{-1} S(L + J_2)
    \right)
  \right)
  \,.
$$

Here $supp(-)$ denotes the [[spacetime]] [[support]] of the given [local observables](#Wick+algebra#CompactlySupportedPolynomialLocalDensities) and $supp(J_1) \geq supp(J_2)$ means that $supp(J_1)$ does not intersect the [[causal past]] of $supp(J_2)$. The product on the right is the product in the [[Wick algebra]] (hence the [[normal ordered product]]).

## Causal factorization

As a special case, causal additivity immediately implies **causal factorization**:

$$
  \left( supp(L_1) \geq supp(L_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    S(L_1 + L_2)
    =
    S(L_1) S(L_2)
  \right)
  \,.
$$

This in turn directly implies that the perturbative expansion of the [[S-matrix]] is given by [[time-ordered products]].

## Causal locality of quantum observables

But the reason why the full condition is called _causal additivity_ is that it is equivalent to a simpler-looking condition on the [[generating function]]

$$
  Z_{L}(J)
  \;\colon\;
  S(L)^{-1} S(L + J)
$$

that is induced by the [[S-matrix]]. In terms of these, causal additivity is equivalently (by [this lemma](S-matrix#CausalLocalityOfThePerturbativeSMatrix)) the condition

$$
  \left( supp(J_1) \geq supp(J_2) \right)
  \;\; \Rightarrow \;\;
  \left(
    \underset{L \in \mathcal{F}_{loc}}{\forall}
    \left(
      Z_{L}(J_1 + J_2)
      =
      Z_L(J_1) Z_L(J_2)
    \right)
  \right)
  \,.
$$

Notice that what these generating functions generate is the perturbative [[quantum observables]]

$$
  \hat A
  \coloneqq
  \frac{d}{d \epsilon}
  T_{\tfrac{g}{\hbar}L_{int}}( \epsilon A )
  \vert|_{\epsilon = 0}
$$

and causal additivity implies that as long as $L_{int}$ is [[adiabatic switching|adiabatically switched]] only outside the [[causal closure]] of $supp(A)$, then the system of quantum observables satisfies **[[causal locality]]**
and hence forms a [[local net of observables]] ([this prop.](#S-matrix#PerturbativeQuantumObservablesIsLocalnet)).

This is how [[causal perturbation theory]] gives rise to _[[perturbative AQFT]]_.

## References

The concept goes back to

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

Lecture notes include

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory]]_ (around [this equation](geometry+of+physics+--+perturbative+quantum+field+theory#eq:CausalAdditivity))

See also the references at _[[causal perturbation theory]]_, _[[perturbative AQFT]]_ and _[[S-matrix]]_.

[[!redirects causally additive]]

[[!redirects causal factorization]]
[[!redirects causal factorizations]]


causal complement

+-- {: .rightHandSide}
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###Context
#### Riemannian geometry
+-- {: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The concept of a _causal complement_ is suitable to establish a [[causal disjointness relation]] on index sets where the indices are subsets of a given set. This generalizes the concept of causal complement of subsets of the [[Minkowski spacetime]], see for example [[Haag-Kastler vacuum representation]].

## Definition

The concept of causal complement, as the name indicates, a concept related to [[causality]] on [[spacetimes]], hence a concept associated with [[Lorentzian manifolds]]:

* _[On Lorentzian manifolds](#OnLorentzianManifolds)_.

One may abstract away some of the key properties of causal complements on Lorentzian manifolds to obtain a more general concept:

* _[Abstract definition](#AbstractDefinition)_

### On Lorentzian manifolds
 {#OnLorentzianManifolds}

Let $X$ be [[Lorentzian manifold]] (a [[spacetime]]), hence a [[pseudo-Riemannian manifold]] with one [[time-like]] dimension in each tangent space.

Then for $S \subset X$ any [[subset]], its _[[causal cone]]_ $J(S)$ is the subset of all points in $X$ which may be joined to a point of $S$ by a [[smooth function|smooth]] [[curve]] $\gamma \colon [0,1] \to X$ which is everywere [[timelike]] or [[lightlike]].

+-- {: .num_example #CausalComplementOfSubsetOfLorentzianManifold}
###### Definition
**(causal complement of subset of Lorentzian manifold)**

For $S \subset X$ a [[subset]] of a [[Lorentzian manifold]], its _causal complement_ $S^\perp$ is the [[complement]] of the [[causal cone]]:

$$
  S^\perp \;\coloneqq\; X \setminus J(S)
  \,.
$$

The causal complement $S^{\perp \perp}$ of the causal complement $S^\perp$ is called the _[[causal closure]]_. If

$$
  S = S^{\perp \perp}
$$

then the subset $S$ is called a _[[causally closed subset]]_.

=--

### Abstract definition
 {#AbstractDefinition}

Let $X$ be an arbitrary set and $M \subset X$. An assignment

$$
  M \mapsto M^{\perp} \subset X
$$

is called a **causal complement** if the following conditions hold:

(i) $M \subseteq M^{\perp\perp}$

(ii) $(\bigcup_j M_j)^{\perp} = \bigcap_j (M_j)^{\perp}$

(iii) $M \bigcap M^{\perp} = \emptyset$

A set $M$ is **[[causally closed subset|causally closed]]** iff $M = M^{\perp\perp}$.

The set $M^{\perp\perp}$ is the **[[causal closure]]** of $M$.

## Properties ##
Causal complements are always causally closed. The intersection of two causally closed sets is again a causally closed set. The causal complement of a set may be empty.

A [[causal disjointness relation]] on an index set of subsets of a given set $X$ can be defined via
$$
M_1 \perp M_2 \; \text{iff} \; M_1 \subseteq (M_2)^{\perp}
$$
if all sets $M$ have a causal complement and if

(iv) there is a sequence $(Y_n)_{n=1}^{\infty}$ of mutually different subsets with $Y_n^{\perp} \neq \emptyset$ and $\bigcup Y_n = X$.

The latter condition is needed to get a _$\sigma$-bounded_ [[poset]]; the $\sigma$-boundedness is part of the definition of a  [[causal index set]].

## Related concepts

* [[causal structure]]

* [[causal order]]

* [[causality]], [[microcausality]]

* [[causal locality]], [[causal additivity]]

* [[causal complement]], [[causal closure]]

* [[causal perturbation theory]]

* [[time-ordered product]]

* [[S-matrix]]

[[!redirects causal complement]]
[[!redirects causal complements]]


causal cone

+-- {: .rightHandSide}
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###Context###
#### Riemannian geometry
+--{: .hide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[time orientation|time-oriented]] [[Lorentzian manifold]] $\Sigma$, then for a pont $x \in \Sigma$

1. its _open [[future cone]]_ is the set of all points $y$ distinct from $x$ such that there is a [[future]]-directed [[time-like]] [[curve]] from $x$ to $y$;

1. its _closed future cone_ is the set of all points $y$ such that there is a [[future]]-directed [[time-like]] or [[light-like]] [[curve]] from $x$ to $y$;

1. its _open [[past cone]]_ is the set of all points $y$ distinct from $x$ such that there is a [[past]]-directed [[time-like]] [[curve]] from $x$ to $y$;

1. its _closed past cone_ is the set of all points $y$ such that there is a [[past]]-directed [[time-like]] or [[light-like]] [[curve]] from $x$ to $y$.

The [[boundary]] of the union of the past and future closed cone is the _[[light cone]]_ of the point.

Given a [[subset]] $S \subset X$, then its future/past open/closed cone is the union of that of all its points. The open cones above are _[[conical spaces]]_.

The [[complement]] of the (closed) causal cone is the _[[causal complement]]_.

## Related concepts

* [[causal locality]]

* [[causal additivity]]

* [[causal complement]], [[causal closure]]

* [[causal perturbation theory]]

* [[conal manifold]]

## References

* {#Baer14} [[Christian Bär]], section 1 of _Green-hyperbolic operators on globally hyperbolic spacetimes_, Communications in Mathematical Physics 333, 1585-1615 (2014) ([doi](http://dx.doi.org/10.1007/s00220-014-2097-7), [arXiv:1310.0738](https://arxiv.org/abs/1310.0738))

[[!redirects causal cones]]

[[!redirects closed causal cone]]
[[!redirects closed causal cones]]

[[!redirects future causal cone]]
[[!redirects future causal cones]]
[[!redirects past causal cone]]
[[!redirects past causal cones]]

[[!redirects closed future cone]]
[[!redirects closed future cones]]
[[!redirects closed past cone]]
[[!redirects closed past cones]]

[[!redirects open future cone]]
[[!redirects open future cones]]
[[!redirects open past cone]]
[[!redirects open past cones]]


causal index set

#Contents#
* the following line creates the automatic table of contents
{:toc}

## Idea ##

In the [[Haag-Kastler approach]] to [[quantum field theory]] one deals with [[local nets]] indexed by bounded open regions of [[Minkowski spacetime]]. An important axiom of this approach is that of causality, which says that observables localized in spacelike separated regions of spacetime commute.
A causal index set is an abstraction of the index set of bounded open regions that retains the relation induced by the concept of spacelike separation. It can be used to generalize the axiom of causality to nets with different or more general index sets as the one mentioned above.
Causal index sets are needed to define the notion of a [[causal net of algebras]].

## Definition ##

A relation $\perp$ on a $\sigma$-bounded [[poset]] $I$ is called a **causal disjointness relation** (and $a, b \in I$ are called causally disjoint if $a \perp b$) if the following properties are satisfied:

(i) $\perp$ is symmetric

(ii) $a \perp b$ and $c \lt b$ implies $a \perp c$

(iii) if $M \subset I$ is bounded from above, then $a \perp b$ for all $a \in M$ implies $sup M \perp b$.

(iv) for every $a \in I$ there is a $b \in I$ with $a \perp b$

A poset with such a relation is called a **causal index set**.

## Examples ##

One example is explained in the Idea section.

Let $I$ be the poset of finite subspaces of a [[separable space|separable]] [[Hilbert space]]. Define $\perp$ by orthogonality of subspaces, then $(I, \perp)$ is a causal index set.

Let $X$ be a [[countable set]] and $I$ be its [[finite power set]], that is the collection of [[finite set|finite]] [[subsets]] of $X$. Set $M \perp N$ iff $M \cap N = \emptyset$, then this defines a causal disjointness relation. This is an example of a [[causal complement]].

[[!redirects causal index set]]
[[!redirects causal index sets]]
[[!redirects causal disjointness relation]]
[[!redirects causal disjointness relations]]


causal locality

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### AQFT
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A basic characteristic of [[physics]] in the context of [[special relativity]] and [[general relativity]] is that causal influences on a [[Lorentzian manifold]] [[spacetime]] propagate in [[timelike]] or [[lightlike]] directions but not [[spacelike]].

The fact that any two spacelike-separated regions of spacetime thus behave like [independent subsystems](http://ncatlab.org/nlab/show/quantum+mechanics#Subsystems) is called **causal locality** or, with a slightly stronger technical definition, **Einstein causality**.

(One sometimes sees a further criterion to causality, that the causal influences in timelike and lightlike directions only propagate into the [[future]], but this is not so simply dealt with; it probably only makes sense as a statement about coarse-grained [[entropy]] in [[statistical physics]].)

From ([Grigor'ev 197x](#Grigoriev)):

> Microcausality condition

> a requirement that the causality condition (which states that cause must precede effect) be satisfied down to an arbitrarily small distance and time interval. The microcausality condition usually refers to distances ≲ $10^{-16}$ [[meter|cm]] and to times ≲ $10^{-24}$ [[second|sec]].

> It is shown in the theory of [[special relativity|relativity]] that the assumption of the existence of physical signals that propagate with a velocity greater than the velocity of [[light]] leads to violation of the causality requirement. Thus, the microcausality condition prohibits the propagation of signals at a velocity greater than the velocity of [[light]] "in the small".

> In [[quantum physics|quantum theory]], where [[quantum operator|operators]] correspond to physical quantities, the microcausality condition requires the interchangeability of any operators that pertain to two points of [[space-time]] if these points cannot be linked by a light signal. This interchangeability means that the physical quantities to which these operators correspond can be precisely determined independently and simultaneously. The microcausality condition is important in [[quantum field theory]], especially in the dispersion and [[AQFT|axiomatic approaches]]; these approaches are not based on specific [[model (physics)|model]] concepts of interaction and therefore can be used for direct verification of the microcausality condition. In the most highly developed branch of [[quantum field theory]] — [[quantum electrodynamics]] — the microcausality condition has been experimentally verified for distances ≲ $10^{-15}$ cm (and, correspondingly, for times ≲ $10^{-25}$ sec).

> The violation of the microcausality condition would make it necessary to radically alter the method of describing physical processes and to reject the dynamic description used in modern theories, in which the state of a physical system at a given moment of time (the effect) is determined by the states of the system at preceding times (the cause).

Notice that $10^{-15}$[[cm]] $ = 10^{-17}m  = 10^{-2}$[[fm]] and that the (charge) [[radius]] of the [[proton]] is about 0.8 [[fm]]. So the bound cited by ([Grigor'ev 197x](#Grigoriev)) in the above quote is about 1/100 the diameter of a proton.

It seems that [Grigor'ev 197x](#Grigoriev) just cited the length scale resolution of particle accelerators at that time. More recently, the [[LHC]] (see there) probes scales $\simeq 10^{-20}m$.

## In algebraic quantum field theory
  {#InAlgebraicQuantumFieldTheory}

In [[algebraic quantum field theory]] causal locality is formalized as follows. This is a key statement in the [[Haag-Kastler axioms]] on [[causally local nets of quantum observables]]:

+-- {: .num_defn #CausalLocalityOfAlgebrasOfObservables}
###### Definition
**(causal locality of algebras of observables)**

A [[co-presheaf]] of [[algebras of quantum observables]] $\mathcal{A}$ on some [[spacetime]] is _causally local_ if the algebras  $\mathcal{A}(\mathcal{O})$ localized in [[spacelike]] separated [[spacetime]] regions $\mathcal{O}$ commute with each other (inside any of the algebras of observables localized in the causal closure $\mathcal{O}$ of the union of the two spacetime regions).

$$
  \left(
    \mathcal{O}_1
     \;\text{spacelike separated from}
     \;
    \mathcal{O}2
  \right)
  \;\Rightarrow\;
  \left(
    [\mathcal{A}(\mathcal{O}_1), \mathcal{A}(\mathcal{O}_2)]
    \;=\;
    0
    \;\;\;\;
    \in \mathcal{A}( \mathcal{O} )
  \right)
  \,.
$$

=--

Under [[Wick rotation]], this causal locality becomes "statistical locality"  (see at _[[Osterwalder-Schrader theorem]]_).

In [[perturbative algebraic quantum field theory]] this condition follows from the [[causal additivity]] of the [[S-matrix]] (see there the section _[Causal locality and Quantum obsrvables](S-matrix#CausalLocality)_).

There are variants that one may consider:

+-- {: .num_defn #StrongLocality}
###### Definition
**(strong causal locality of algebras of observables)**

A [[local net of quantum observables]] is **strongly causally local** if it is causally local in that algebras $A_1 = A(O_1)$ and $A_2 = A(O_2)$ associated with  spacelike separated regions commute with each other, and in addition for all commutative subalgebras $C_1 \subset A_1$ and $C_2 \subset A_2$ the algebra $C_1 \vee C_2 \subset A(O_1 \vee O_2)$ satisfies

1. $(C_1 \vee C_2) \cap A_1 = C_1$

1. $(C_1 \vee C_2) \cap A_2 = C_2$.

=--

This is ([Nuiten 11, def. 14](#Nuiten11)).

There have been various proposals to understand these conditions from other principles:

1. In ([Schreiber 09](#Schreiber09)) the condition \ref{CausalLocalityOfAlgebrasOfObservables} is related to [[n-functor|n-functoriality]] of a corresponding ([[Schrödinger picture]]) [[functorial quantum field theory]].

1. In ([Nuiten 11, theorem 4.2](#Nuiten11)) the condition \ref{StrongLocality} us shown to be implied by the associated pre-sheaf of [[Bohr toposes]] satisfying spatial [[descent]] by [[local geometric morphisms]].

1. In ([Brunetti-Fredenhagen-Imani-Rejzner 12](#BrunettiFredenhagenImaniRejzner12)) condition def. \ref{CausalLocalityOfAlgebrasOfObservables} is shown to be equivalent to the [[co-presheaf]] of observables being a [[monoidal functor]] is a suitable way.

## In S-matrix theories and string theory

In [[S-matrix]] theories [[causal additivity]] is meant to also be incarnated in terms of [[analytic function|analyticity]] properties of the [[scattering amplitudes]] (for this reason one often speaks of "the analytic S-matrix").

One S-matrix theory is [[perturbative string theory]]. Discussion of causality in string theory includes [Martinec 95](#Martinec95) and ([Erler-Gross 04](#ErlerGross04)). The latter write in their introduction:

> Perhaps then it comes as a surprise that critical string theory produces an analytic [[S-matrix]] consistent with macroscopic causality. In absence of any other known theoretical mechanism which might explain this, despite appearances one is lead to believe that string interactions must be, in some sense, local.

and

> We find that string theory avoids problems with nonlocality in a surprising way. In particular, we find that the Witten vertex is "local enough" to allow for a nonsingular description of the theory which is completely local along a single null direction.

and

> unlike lightcone string field theory, it is clear that cubic string field theory at least has a local limit where all spacetime coordinates are taken to the midpoint. We investigate this limit with a careful choice of regulator and show that at any stage the theory is nonsingular but arbitrarily close to being local and manifestly causal. We believe that the existence of this limit, though singular, must account for the macroscopic causality of the string S-matrix. Thus, string theory is local enough to avoid the inconsistencies of a theory which is acausal and nonlocal in time, but is nonlocal enough to make string theory different from quantum field theory

Then they comment on Martinec's account above, and other's, by saying:

> To motivate our particular perspective, it seems appropriate to discuss earlier attempts to understand the role of locality, causality and time in string theory, and explain why we feel these approaches do not adequately address the problems just raised.

## Related concepts

* [[causal order]]

* [[causal structure]]

* [[causal additivity]]

* [[causal closure]]

* [[causal perturbation theory]]

* [[local prequantum field theory]]

* [[local quantum field theory]]

* in [[AQFT]]:

  [[local net of observables]], [[Haag-Kastler axioms]]

* in [[FQFT]]:

  [[extended field theory]]

* [[string theory FAQ]]: _[Is string theory causal, given that it is not local on the string scale?](string+theory+FAQ#IsStringTheoryCausal)_

## References

### In local quantum field theory

* {#Grigoriev} V. I. Grigor'ev, _Microcausality condition_, The Great Soviet Encyclopedia, 3rd Edition (1970-1979) ([web](http://encyclopedia2.thefreedictionary.com/Microcausality+Condition))

* Jessey Wright, _Quantum field theory: Motivating  the Axiom of Microcausality_, PhD thesis 2012 ([pdf](http://uwspace.uwaterloo.ca/bitstream/10012/6998/1/Wright_Jessey.pdf))

* Anthony Duncan, _The Conceptual Framework of Quantum Field Theory -- Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity_, Oxford Scholarship Online ([web](http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199573264.001.0001/acprof-9780199573264-chapter-6))

* {#Bell75} [[John Bell]], _The theory of local beables_ (1975) ([pdf](http://cds.cern.ch/record/980036/files/197508125.pdf))

* [pdf](http://ptp.oxfordjournals.org/content/52/1/335.full.pdf)

### In algebraic quantum field theory

For references on the traditional discussion in [[AQFT]] see at _[[Haag-Kastler axioms]]_ and at _[[causally local net of observables]]_. Proposals to understand the causal locality axiom from other principles include

* {#Schreiber09} [[Urs Schreiber]], _AQFT from $n$-functorial QFT_ , Comm. Math. Phys., Volume 291, Issue 2, pp.357-401 2009 ([pdf](http://ncatlab.org/schreiber/files/AQFTfromFQFT.pdf))

* {#Nuiten11} [[Joost Nuiten]], _[[schreiber:bachelor thesis Nuiten|Bohrification of local nets]]_, Proceedings of [QPL 2011](http://qpl.science.ru.nl/), [EPTCS 95, 2012](http://rvg.web.cse.unsw.edu.au/eptcs/content.cgi?QPL2011), pp. 211-218
([arXiv:1109.1397](http://arxiv.org/abs/1109.1397))

* {#BrunettiFredenhagenImaniRejzner12} [[Romeo Brunetti]], [[Klaus Fredenhagen]], Paniz Imani, [[Katarzyna Rejzner]], _The Locality Axiom in Quantum Field Theory and Tensor Products of $C^*$-algebras_ ([arXiv:1206.5484](https://arxiv.org/abs/1206.5484))

### In S-matrix theories and string theory

Discussion of causality in [[string theory]] includes the following:

A brief look at the causality property of the string 2-point function is in

* {#Martinec95} [[Emil Martinec]], _Strings and Causality_ ([pdf](http://cds.cern.ch/record/569030/files/9311129.pdf)) in L. Baulieu, V.  Dotsenko, V. Kazakov, P. Windey (eds.) Quantum Field Theory and String Theory , NATO ASI Series B: Physics Vol. 328 (1995)

An in-depth discussion of causality of the [[string scattering amplitude|string scattering]] [[S-matrix]] via [[open string field theory]] is in

* {#ErlerGross04} [[Theodore Erler]], [[David Gross]], _Locality, Causality, and an Initial Value Formulation for Open String Field Theory_ ([arXiv:hep-th/0406199](https://arxiv.org/abs/hep-th/0406199))

This rediscovered some facts that had earlier been noticed in

* M. Maeno, _Canonical quantization of Witten's string field theory using midpoint light-cone time_, Phys. Rev. D43 no. 12 (1991).

See also

* Gianluca Calcagni, Leonardo Modesto, _Nonlocality in string theory_, J. Phys. A: Math. Theor. 47 (2014) 355402 ([arXiv:1310.4957](https://arxiv.org/abs/1310.4957))

Discussion in theories with [[higher curvature corrections]] of the gravitational background and in [[string theory]] includes

* Xian O. Camanho, Jose D. Edelstein, [[Juan Maldacena]], Alexander Zhiboedov, _Causality Constraints on Corrections to the Graviton Three-Point Coupling_ ([arXiv:](http://arxiv.org/abs/1407.5597))

* Giuseppe D'Appollonio, [[Paolo Di Vecchia]], [[Rodolfo Russo]], [[Gabriele Veneziano]], _Regge behavior saves String Theory from causality violations_ ([arXiv:1502.01254](http://arxiv.org/abs/1502.01254))

[[!redirects causal locality]]

[[!redirects Einstein causality]]
[[!redirects Einstein-causality]]

[[!redirects Einstein locality]]
[[!redirects Einstein-locality]]

[[!redirects microlocality]]
[[!redirects microcausality]]

[[!redirects locality]]

[[!redirects causality]]


causal net of algebras > history

< [[causal net of algebras]]


causal order

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Relations
+-- {: .hide}
[[!include relations - contents]]
=--
#### Riemannian geometry
+--{: .hide}
[[!include Riemannian geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #CausalOrdering}
###### Definition
**(causal order)**

Let $\Sigma$ be a [[spacetime]] (a [[Lorentzian manifold]] equipped with [[time orientation]]).

Consider the [[relation]] on the set $P(\Sigma)$ of [[subsets]] of spacetime
which says a [[subset]] $S_1 \subset \Sigma$ is _not prior_ to a subset $S_2 \subset \Sigma$,
denoted $S_1 {\vee\!\!\!\wedge} S_2$, if $S_1$ does not [[intersection|intersect]] the [[causal past]] of $S_2$,
or equivalently that $S_2$ does not intersect the [[causal future]] of $S_1$:

$$
  \begin{aligned}
    S_1 {\vee\!\!\!\wedge} S_2
    & \;\coloneqq\;
    S_1 \cap \overline{V}^-(S_2) = \emptyset
    \\
    & \Leftrightarrow S_2 \cap \overline{V}^+(S_1) = \emptyset
   \end{aligned}
   \,.
$$

If $S_1 {\vee\!\!\!\wedge} S_2$ and $S_2 {\vee\!\!\!\wedge} S_1$ we say that the two subsets are _[[spacelike]] separated_ and
write

$$
   S_1 {\gt\!\!\!\!\lt} S_2
   \;\;\;\coloneqq\;\;\;
   S_1 {\vee\!\!\!\wedge} S_2
   \;\text{and}\;
   S_2 {\vee\!\!\!\wedge} S_1
   \,.
$$

=--

(e.g. [Epstein-Glaser 73, p. 5 (273)](#EpsteinGlaser73))

## Related concepts

* [[causal structure]]

* [[causality]], [[microcausality]]

* [[causal locality]], [[causal additivity]]

* [[causal complement]], [[causal closure]]

* [[causal perturbation theory]]

* [[time-ordered product]]

* [[S-matrix]]

## References

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

[[!redirects causal ordering]]


causal perturbation theory

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#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _causal perturbation theory_ is a mathematically rigorous construction of _[[perturbative quantum field theory|perturbative]]_ ([[gauge theory|gauge]]-) [[quantum field theory]], such as [[quantum electrodynamics]], based on a mathematical formulation of [[renormalization]] by Stückelberg-Bogoliubov-Epstein-Glaser ([Epstein-Glaser 73](#EpsteinGlaser73)).

Causal perturbation theory may be regarded as providing a well-defined consistent generalization from [[quantum mechanics]] to [[quantum field theory]] on [[Lorentzian manifold|Lorentzian]] [[spacetimes]] of the construction of the [[S-matrix]] via the [[Dyson formula]] ("[[time-ordered products]]") in the [[interaction picture]] .

A key idea of causal perturbation theory is that the [[interaction]] term $V$ is considered multiplied with some [[smooth function]] $g$ which has [[compact support]] on [[spacetime]]. This hence serves as a spacetime-dependent "[[coupling constant]]" which "switches off" the interaction outside a compact region, but not discontinuously as in many other schemes, but smoothly, hence "adiabatically" in terminology borrowed from [[thermodynamics]]. Therefore this is often called the "[[adiabatic switching]]" function, or similar.

The corresponding [[S-matrix]] would naively be given by the [[Dyson formula]]

$$
  S_g \coloneqq T \exp\left( \int_X g(x) V(x) dvol(x) \right)
$$

for $V$ the [[interaction]] term, and "$T$" indicating the [[time-ordered product]]. Causal perturbation theory proceeds by [[axiom|axiomatizing]] the key structural properties of this "[[adiabatic switching|adiabatically switched]]" [[S-matrix]], in particular its _[[causal additivity]]_, making sense of the "[[time-ordered product]]" by appropriate [[causal locality|causal ordering]] (whence the name of the approach) and then proving by [[induction]] that solutions to these axioms exist.

It turns out that at each step in the [[induction]] (corresponding to each loop order) a [[distribution]] has to be [extended to a point](extension%20of%20distributions#SolutionSpaceOfPointExtensions), namely to the point at which the [[interaction]] takes place. A key theorem ([Epstein-Glaser 73, section 5](#EpsteinGlaser73), [this prop.](extension+of+distributions#SpaceOfPointExtensions)) states that this extension is parameterized by a finite-dimensional space of [[point-supported distributions]] at that point. One may organize the distributions that need to be extended into [[Feynman diagrams]] ([Dütsch-Fredenhagen-Keller-Rejzner 14](#DuetschFredenhagenKellerRejzner14)). This way the freedom in [[extension of distributions]] is identified with the traditional [[renormalization]] freedom in [[perturbative quantum field theory]], see at _[[main theorem of perturbative renormalization]]_ for more on this point.

(It may be argued, vividly so in [Scharf 95, p. 181-182](#Scharf95), that the notorious "infinities" that "plague" quantum field theory in other approaches are nothing but the result of incorrectly dealing with the issue of [[extension of distributions]].)

In fact the [[interacting field algebra]] induced by the [[S-matrix]] constructed via causal perturbation theory this way is a model for [[Fedosov's formal deformation quantization]] of the given [[local Lagrangian density]] ([Collini 16](#Collini16), [Hawkins-Rejzner 16](#HawkinsRejzner16)), thus justifying the method from first principles of (perturbative) [[quantization]].

The key axiom imposed on the [[S-matrix]] in causal perturbation theory is [[causal locality]], whence the name of the approach. This axiom asks that if adiabatic switching functions $g$ and $h$ have [[spacelike]] separated [[supports]] then the S-matrix _factors_

$$
  S_{g + h} = S_g S_h
  \,.
$$

The [[time-ordered products]] $T(..)$ of [[field (physics)|fields]] are handled by splitting of [[distributions]] using tools from [[microlocal analysis]]. This is the mathematically rigorous step that takes care of what in other approaches are the "[[ultraviolet divergences]]".

Originally the idea was that in the end the [[limit of a sequence|limit]] $g \to 1$ had to be taken, removing the adiabatic switching of the the compact support of the interaction. In general this limit does not exist ("[[infrared divergences]]", e.g. [AAS 10, section 6](#AAS10)).

But in ([Il'in-Slavnov 78](#IlinSlavnov78)) it was observed that in fact the [[algebra of observables]] on any [[bounded subsets|bounded]] region may be computed with a $g$ whose [[compact support]] contains the [[causal closure]] of that region. Later in ([Brunetti-Fredenhagen 00](#BrunettiFredenhagen00)) it was pointed out that this means that causal perturbation theory in fact serves to construct the [[causally local net of observables]] of the [[perturbative quantum field theory]] (see [this prop](S-matrix#PerturbativeQuantumObservablesIsLocalnet) for details), as in the [[axioms]] for [[AQFT]] and in fact as in the axioms of [[AQFT on curved spacetimes]], but with values in [[formal power series algebras]] (as befits a [[perturbation theory]]) instead of [[C*-algebras]] (suitable for [[non-perturbative quantum field theory]]).

(This takes care of the "[[algebraic adiabatic limit]]", which defines the [[quantum observables]]. It does not yet in itself define the "[[weak adiabatic limit]]" for the would-be [[vacuum]] [[quantum state]].)

This way causal perturbation theory leads to a unification of [[AQFT]] [[AQFT on curved spacetimes]] with [[perturbative quantum field theory]]. This unification is now known as **[[locally covariant perturbative quantum field theory]]**, see there for more.

## Properties

### Main theorem of renormalization
 {#MainTheoremOfRenormalization}

A central result in causal perturbation theory is called the _[[main theorem of perturbative renormalization theory]]_. This says that any two _renormalization schemes_, hence any two solutions to the inductive construction of the [[S-matrix]] $V \mapsto S(V)$,  as indicated above, for [[interaction]] terms $V$, are related by a unique natural transformation $Z \colon V \to V'$

$$
  S' = S \circ Z
  \,.
$$

The collection of these operations $Z$ forms a [[group]] called the _[[Stückelberg-Petermann renormalization group]]_.

This is a mathematical reflection of the idea that renormalization is about regarding a [[perturbative quantum field theory]] with [[interaction]] $V$ as a [[effective field theory]] at some energy scale and then discovering that at higher energy there is a more fundamental interaction $Z(V)$ which effectively looks like $V$ at lower energy.

## Related concepts

* [[locally covariant perturbative quantum field theory]]

* [[Schwinger-Tomonaga-Feynman-Dyson]]

## References
 {#References}

Lecture notes in

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory]]_

The method is due to

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

with precursors in

* {#Stueckelberg49} [[Ernst Stückelberg]], D. Rivier, Helv. Phys. Acta, 22 (1949) 215.

* {#Stueckelberg51} [[Ernst Stückelberg]], J. Green, Helv. Phys. Acta, 24 (1951) 153.

* [[Ernst Stückelberg]], A. Peterman, , _La normalisation des constants dans la theorie des quanta_, Helv. Phys. Acta 26, 499 (1953);

* {#BogoliubovShirkov59} [[Nikolay Bogoliubov]], [[Dmitry Shirkov]], _Introduction to the Theory of Quantized Fields_, New York (1959)

whence sometimes called the _Stückelberg-Bogoliubov-Epstein-Glaser method_.

The expression of causal perturbation theory in terms of [[Feynman diagram]] techniques is due to

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], Kai Keller, [[Katarzyna Rejzner]], _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy.
55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

That causal perturbation theory (in the generality of curved spacetimes, see [below](#ReferencesOnCurvedSpacetimes)) is equivalently the ([[Fedosov deformation quantization|Fedosov]]-)[[formal deformation quantization]] of the interacting Lagrangian density was shown for the  [[scalar field]] [[phi^4 theory]] in

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

and for the interacting scalar field in the toy example of regular non-local interactions in

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

### On Minkowski spacetime

Causal perturbation theory has been worked out in detail for the example of [[quantum electrodynamics]] on [[Minkowski spacetime]] in

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

and for  [[quantum chromodynamics]] and [[perturbative quantum gravity]] on [[Minkowski spacetime]] in

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

Non-technical exposition of this includes

* {#Neumaier15} [[Arnold Neumaier]], _[Causal Perturbation Theory](https://www.physicsforums.com/insights/causal-perturbation-theory/)_ (2015)

* [[Arnold Neumaier]], _[Action-based quantum field theory and causal perturbation theory](https://www.mat.univie.ac.at/~neum/physfaq/topics/causalQFT.html)_ (one section in _[A theoretical physics FAQ](http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html)_)

Technical review includes

* {#AAS10} Andreas Aste, Cyrill von Arx, [[Günter Scharf]], _Regularization in quantum field theory from the causal point of view_, Prog. Part. Nucl. Phys.64:61-119, 2010 ([arXiv:0906.1952](https://arxiv.org/abs/0906.1952))

* Christian Pöselt, _The method of Epstein and Glaser I_, 2002 ([pdf](http://wwwthep.physik.uni-mainz.de/~scheck/Poeselt.pdf))

### On curved spacetimes
 {#ReferencesOnCurvedSpacetimes}

The generalization of causal perturbation theory to [[quantum field theory on curved spacetimes]] is developed in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661, 2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

* [[Stefan Hollands]], [[Robert Wald]], _Local Wick polynomials and time ordered products of quantum fields in curved spacetime_, Communications in Mathematical Physics, 223(2):289–326, 2001 ([arXiv:gr-qc/0103074](https://arxiv.org/abs/gr-qc/0103074))

* [[Stefan Hollands]], [[Robert Wald]], _Existence of local covariant time ordered products of quantum fields in curved spacetime_, Communications in Mathematical Physics, 231(2):309–345, 2002 ([arXiv:gr-qc/0111108](https://arxiv.org/abs/gr-qc/0111108))

* [[Stefan Hollands]], [[Robert Wald]], _On the renormalization group in curved spacetime_, Communications in Mathematical Physics, 237(1-2):123–160, 2003 ([arXiv:gr-qc/0209029](https://arxiv.org/abs/gr-qc/0209029))

* [[Stefan Hollands]], [[Robert Wald]], _Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes_, Reviews in Mathematical Physics, 17(03):227–311, 2005 ([arXi:gr-qc/0404074](https://arxiv.org/abs/gr-qc/0404074))

* [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](http://arxiv.org/abs/0901.2038))

Review is in

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))

### Local covariance

The observation that the method of causal perturbation theory naturally leads to [[locally covariant perturbative quantum field theory]] is due to

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32 ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

and was re-discovered and then popularized in

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], section 8 of _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_, Commun. Math. Phys. 208 : 623-661,2000 ([math-ph/9903028](https://arxiv.org/abs/math-ph/9903028))

Review includes

* {#FredenhagenRejzner15} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], section 4.1 of _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](https://arxiv.org/abs/1503.07814))

For more on this see the references at _[[locally covariant perturbative quantum field theory]]_.

[[!redirects causal perturbation theories]]