H-space

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## Definition

An __$H$-space__ ("H" for [[Heinz Hopf|Hopf]], as in _[[Hopf construction]]_) is a [[magma]] [[internalization|internal to]] the [[classical homotopy category]] of [[topological spaces]] [[Ho(Top)]], or in the homotopy category $Ho(Top)_*$ of [[pointed topological spaces]],  which has a [[unit]] up to [[homotopy]].

Similarly:

An **[[H-monoid|$H$-monoid]]** is a [[monoid object]] in [[Ho(Top)]], hence an $H$-space is an $H$-monoid if the product of the magma is [[associativity|associative]] up to [[homotopy]].

An __$H$-group__ is a [[group object]] in [[Ho(Top)]], so an $H$-monoid is an $H$-group if it also has [[inverses]] up to homotopy.

An **$H$-ring** is a [[ring object]] in (pointed) [[Ho(Top)]].

To continue in this pattern, one could say that an **[[H-category]]** is a category internal to [[Ho(Top)]].

Notice that here the homotopies for units, associativity etc. are only required to _exist_ for an H-space, not required to be equipped with higher [[coherence|coherent]] homotopies. An $H$-monoid equipped with such higher and coherent homotopies  is instead called a _strongly homotopy associative_ space or  _$A_\infty$-[[A-infinity space|space]]_ for short.  If it has only higher homotopies up to level $n$, it is called an $A_n$-[[A-n space|space]].

A better name for an $H$-space might be be $H$-[[unitoid]], but it is rarely used. The $H$ stands for [[Heinz Hopf]], and reflects the sad fact that the natural name 'homotopy group' was [[homotopy group|already occupied]]; Hopf and A. Borel found necessary algebraic conditions for a space to admit an $H$-space structure.

There are dual notions of $H$-counitoid (or $H'$-space, or [[co-H-space]]), $H$-comonoid (or $H'$-monoid) and $H$-[[cogroup]] (or $H'$-group) having co-operations with the usual identities up to homotopy.

## Examples

### Loop spaces

The main example of an $H$-group is the [[loop space]] $\Omega X$ of a space $X$, which however is naturally even an [[A-infinity space]].

### Suspensions

The main example of an [[co-H-space|H-cogroup]] in $Top_*$ is the [[suspension]] $S X= S^1\wedge X$ of a pointed topological space $X$.

### Spheres

The only [[n-spheres]] $S^n$ which have $H$-space structure are those for $n = 0,1,3,7$, and their H-space structure is given by identifying them as the unit spheres in one of the four [[normed division algebras]] ([[real numbers]], [[complex numbers]], [[quaternions]], [[octonions]]) and taking the product to be that induced by the algebra product.

An example of an H-space that does not lift to an [[A-infinity space]] is the [[7-sphere]] $S^7$. It can't be delooped because its [[delooping]] would have [[cohomology group]] a [[polynomial ring]] on a generator in degree 8, and this is impossible by mod $p$ [[Steenrod operations]] for any odd $p$, see [Lemma 2, Adams61](#Adams61). The 7-sphere is also not an $H$-group.

### Mapping spaces into H-groups

If $K$ is an $H$-group then for any topological space $X$, the set of [[homotopy classes]] $[X,K]$ has a natural group structure in the strict sense; analogously if $K'$ is an $H$-cogroup then $[K',X]$ has a group structure. If there is more than one $H$-group structure on a space, then the induced group structures on the set of homotopy classes coincide.

If an $H$-space is equivalent to a [[delooping|deloopable one]], then it is a [[groupoid object in an (infinity,1)-category|group object in the (∞,1)-category]] [[Top]]. In other words, in that case, the associativity and other axioms hold up to **coherent homotopy**.

### K-Theory space

The [[classifying space]] $B U \times \mathbb{Z}$ for (complex) [[topological K-theory]] is an H-ring space (p. 205 (213 of 251) in _[[A Concise Course in Algebraic Topology]]_.)

### Dwyer-Wilkerson space

See at _[[Dwyer-Wilkerson H-space]]_

## Properties

Let $A$ be a [[connected]] H-space. Then for every $a:A$, the [[maps]] $\mu(a,-),\mu(-,a):A \to A$ are [[homotopy equivalences]].

### Nilpotency

Every [[connected topological space|connected]] H-space is [[nilpotent topological space|nilpotent]] (see [there](nilpotent+topological+space#Properties)).

### Relation to $A_\infty$-spaces

Given an [[A-∞ space]] in the [[(∞,1)-category]] [[∞Grpd]]/[[Top]], its image in the corresponding [[homotopy category of an (∞,1)-category]] [[Ho(Top)]] in an $H$-space. Conversely, refining an H-space to a genuine $A_\infty$-space means lifting the structure of a [[monoid object in an (∞,1)-category]] from the [[homotopy category of an (∞,1)-category|homotopy category]] [[Ho(Top)]] to the genuine [[(∞,1)-category]] [[∞Grpd]]/[[Top]].

Further discussion of this is also at _[loop space -- Homotopy associative structure](loop+space#AInfinityStructure)_.

### Ring structure on homology
 {#RingStructureOnHomology}

The operations on an [[H-space]] $X$ equip its [[homology]] with the [[mathematical structure|structure]] of  [[ring]]. At least for [[ordinary homology]] this is known as the _[[Pontrjagin ring]]_ $H_*(X)$ of $X$.

### Rational homotopy
 {#RationalHomotopy}

Every [[connected topological space|connected]] H-space of rational [[finite type]] (i.e., whose [[rational cohomology]] in each degree is a [[finite-dimensional vector space|finite-dimensional]] [[rational vector space]]) is [[rational homotopy equivalence|rationally homotopy equivalent]] to a [[product space|product]] of [[Eilenberg-MacLane spaces]] (e.g. [May & Ponto 2011, Thm. 9.1.1](#MP11concise)). This means equivalently that its minimal [[Sullivan model]] has vanishing [[differential]], see for instance the example of [[Sullivan models of based loop spaces]].

If such an H-space in addition admits the structure of a [[finite CW-complex]] then it has the [[rational homotopy equivalence|rational homotopy type]] of a finite [[product topological space|product]] of [[odd number|odd]]-[[dimension of a manifold|dimensional]] [[rational n-sphere|rational spheres]] (Cor. 9.1.2. of [May & Ponto 2011](#MP11concise)).

For example, since [[smooth manifolds]] have CW-complex structure (see [there](CW+complex#SmoothManifoldsAdmitCWComplexStructure)) this  is the case for all [[Lie groups]] (e.g. [Lin 1995](#Lin95)).

## H-Spaces in homotopy type theory

In [[homotopy type theory]], an *H-Space* consists of

* A type $A$,
* A basepoint $e:A$
* A binary operation $\mu : A \to A \to A$
* A left unitor
$$\lambda:\prod_{(a:A)} \mu(e,a)=a$$
* A right unitor
$$\rho:\prod_{(a:A)} \mu(a,e)=a$$

A __homomorphism of H-spaces__ between two H-spaces $A$ and $B$ consists of

* A function $\phi:A \to B$ such that
  * The basepoint is preserved
$$\phi(e_A) = e_B$$
  * The binary operation is preserved
$$\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) = \mu_B(\phi(a),\phi(b))$$

* A function

$$\phi_\lambda:\left(\prod_{(a:A)} \mu(e_A,a)=a\right) \to \left(\prod_{(b:B)} \mu(e_B,b)=b\right)$$

such that the left unitor is preserved:

$$\phi_\lambda(\lambda_A) = \lambda_B$$

* A function

$$\phi_\rho:\left(\prod_{(a:A)} \mu(a, e_A)=a\right) \to \left(\prod_{(b:B)} \mu(b, e_B)=b\right)$$

such that the right unitor is preserved:

$$\phi_\rho(\rho_A) = \rho_B$$

### Examples

* There is a H-space structure on the [[circle]]. See Lemma 8.5.8 of the [[HoTT book]].

\begin{proof}
We define $\mu : S^1 \to S^1 \to S^1$ by circle induction:
$$\mu(base)\equiv id_{S^1}\qquad ap_{\mu}\equiv funext(h)$$
where $h : \prod_{x : S^1} x = x$ is defined in Lemma 6.4.2 of the [[HoTT book]]. We now need to show that $\mu(x,e)=\mu(e,x)=x$ for every $x : S^1$. Showing $\mu(e,x)=x$ is quite simple, the other way requires some more manipulation. Both of which are done in the book.
\end{proof}

* Every [[loop space]] is naturally a [[H-space]] with [[path]] concatenation as the operation. In fact every [[loop space]] is a [[group]].

* The type of [[maps]] $A \to A$ has the structure of a H-space, with basepoint $id_A$, operation function composition.

* An [[A3-space]] $A$ is an H-space with an equality $\mu(\mu(a, b),c)=\mu(a,\mu(b,c))$ for every $a:A$, $b:A$, $c:A$ representing associativity up to homotopy.

* A unital magma is a 0-truncated H-space.

## Related concepts

* [[topological monoid]]

* [[Pontrjagin ring]]

* [[group completion]]

* [[H-space ring spectrum]], [[H-group ring spectrum]]

* [[H-spaceoid]]

* [[A3-space]]

* [[H-spatial groupoid]]

* [[k-tuply associative n-groupoid]]

## References

Introduction and survey:

* {#Hatcher02} [[Allen Hatcher]], §3.C in: *Algebraic Topology*, Cambridge University Press (2002) [[ISBN:9780521795401](https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB&isbn=9780521795401), [webpage](https://pi.math.cornell.edu/~hatcher/AT/ATpage.html)]

* {#Arkowitz11} [[Martin Arkowitz]], *H-Spaces and Co-H-Spaces*, chapter 2 in: *Introduction to homotopy theory*, Springer (2011) 37-70 [[doi:10.1007/978-1-4419-7329-0_2](https://doi.org/10.1007/978-1-4419-7329-0_2)]

The terminology "$H$-space" appear sin

* [[Jean-Pierre Serre]], Chapter IV, Section 1 of: _Homologie singulière des espaces fibrés. Applications._, Ann. Math. __54__ (1951), 425--505.

Early discussion:

* [[William Browder]], *Torsion in H-Spaces*, Annals of Mathematics, Second Series **74** 1 (1961) 24-51  [[jstor:1970305](https://www.jstor.org/stable/1970305)]

More general notion of [[internalization|internal]] H-objects:

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III primitive categories_,  Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))

Original articles:

* [[Edwin Spanier]], [[Henry Whitehead]], _On fibre spaces in which the fibre is contractible_, Comment. Math. Helv. __29__, 1955, 1--8.

* Arthur H. Copeland, _On $H$-spaces with two nontrivial homotopy groups_, Proc. AMS __8__, 1957, 109--129.

* [[Masahiro Sugawara]]:
  * _$H$-spaces and spaces of loops_, Math. J. Okayama Univ, __5__, 1955, 5--11;
  * _A condition that a space is an $H$-space_, Math. J. Okayama Univ. __6__, 1957, 109--129; ([pdf](https://ousar.lib.okayama-u.ac.jp/files/public/3/33737/20160528030627675220/fulltext.pdf))
  * _A condition that a space is group-like_, Math. J. Okayama Univ. __7__, 1957, 123--149.

* F. I. Karpelevič, A. L. Oniščik, _Algebra of homologies of space of paths_, Dokl. Akad. Nauk. SSSR, (N.S.), 106 (1956), 967--969. MR0081478

The theory of $H$-spaces was widely established in the 1950s and studied by Serre, Postnikov, Spanier, Whitehead, Dold, Eckmann and Hilton and many others.

The deloopable case has more coherent structure which has been discovered few years later in J. Stasheff's thesis and published in

* [[Jim Stasheff]], _Homotopy associative $H$-spaces I, II_, Trans. Amer. Math. Soc. 108, 1963, 275-312

For a historical account see

* [[John McCleary]], _An appreciation of the work of Jim Stasheff_ ([pdf](http://www.math.unc.edu/Faculty/jds/jds.pdf))

The description in terms of [[groupoid object in an (∞,1)-category]] is due to

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

see last remark of section 6.1.2 .

Wikipedia\'s definition (at time of this writing, and phrased in the language of [[homotopy theory]]) is rather a [[unitoid]] object in the $(\infty,1)$-category [[Top]].

* MathOverflow: [homotopy-associative-h-space-and-coh-space](http://mathoverflow.net/questions/16711/homotopy-associative-h-space-and-coh-space)

For H-spaces in [[homotopy type theory]]:

* [[Univalent Foundations Project]], [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

* [[Ulrik Buchholtz]], [[J. Daniel Christensen]], [[Jarl G. Taxerås Flaten]], [[Egbert Rijke]], *Central H-spaces and banded types* [[arXiv:2301.02636](https://arxiv.org/abs/2301.02636)]

See also

* [[John Adams]], _Finite $H$-Spaces and Lie groups_, Journal of Pure and Applied Algebra 19 (1980) 1-8 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/adamse8.pdf))

* {#Adams61} [[John Adams]], _The sphere, considered as an $H$-space mod $p$_, Quart. J. Math. Oxford. Ser. (2) vol 12 52-60 ([pdf](http://qjmath.oxfordjournals.org/content/12/1/52.citation))

* {#MP11concise} [[Peter May]], [[Kate Ponto]]. *[[More Concise Algebraic Topology]]*, University of Chicago Press (2011). [[doi:10.7208/chicago/9780226511795.001.0001](https://www.doi.org/10.7208/chicago/9780226511795.001.0001), [pdf](https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf)&rbrack

* {#Lin95} James P. Lin. "H-spaces with Finiteness Conditions". In: *Handbook of Algebraic Topology*
(1995), pp. 1095, 1097-1141. ([doi](https://doi.org/10.1016/B978-044481779-2/50023-7))

[[!redirects H-spaces]]

[[!redirects homomorphism of H-spaces]]
[[!redirects homomorphisms of H-spaces]]
[[!redirects H-space homomorphism]]
[[!redirects H-space homomorphisms]]

[[!redirects H-group]]
[[!redirects H-groups]]

[[!redirects H-ring]]
[[!redirects H-rings]]

[[!redirects H-unitoid]]
[[!redirects H-cogroup]]
[[!redirects H-comonoid]]


H-spaceoid

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## Idea ##

The [[oidification]] of an [[H-space]].

## Definition ##

### Classically ###

A **H-spaceoid** is a [[unital magmoid]] [[internalization|internal to]] the [[classical homotopy category]] of [[topological spaces]] [[Ho(Top)]], or in the homotopy category $Ho(Top)_*$ of [[pointed topological spaces]],  which has a [[unit]] up to [[homotopy]].

### In homotopy type theory ###

In [[homotopy type theory]], a **H-spaceoid** is a [[type]] $A$ with

* For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called **arrows** or **morphisms**.

* For each $a:A$, a morphism $1_a:hom_A(a,a)$, called the **identity morphism**.

* For each $a,b,c:A$, a functor
$$hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)$$
called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.

* For each $a,b:A$ and $f:hom_A(a,b)$, terms $p:f = 1_b \circ f$ and $q:f=f\circ 1_a$.

## See also ##

* [[H-space]]
* [[unital magmoid]]
* [[H-magmoid]]
* [[H-category]] (category object internal to a homotopy category)
* [[homotopy category]]

[[!redirects H-spaceoids]]


h-topology

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## Idea

Let $S$ be a [[noetherian scheme]].  The **h-topology** is a [[Grothendieck topology]] on the category of schemes, used by [[Vladimir Voevodsky]] to construct a [[triangulated category]] $DM$ of [[mixed motive|mixed]] [[motives]].

## Definition

A morphism $p : X \to Y$ of [[schemes]] is called a [[topological epimorphism]] if the underlying topological space of $Y$ is a quotient space of the underlying topological space of $X$, i.e. $p$ is surjective and a subset $A \subset Y$ is open iff its preimage $p^{-1}(A) \subset X$ is open.  Such a morphism is further called a [[universal]] topological epimorphism if this property is preserved under any [[base change]].

Consider the [[pretopology]] on the category of [[schemes]] whose covering families are finite families $(p_i : U_i \to X)_i$ such that $p_i$ are of finite type and $\sqcup p_i : \sqcup U_i \to X$ is a [[topological epimorphism|universal topological epimorphism]].  The [[Grothendieck topology]] generated by this pretopology is called the **h-topology** on the category of schemes.

The h-topology is stronger than the [[etale topology|etale]] and [[proper topology|proper]] topologies.

## See also

* [[motives]]
* [[qfh-topology]]
* [[cdh-topology]]
* [[etale topology]]

## References

* [[Vladimir Voevodsky]], _Homology of schemes and covariant motives_, thesis, ProQuest LLC, Ann Arbor, MI, pp. 64, 1992, [pdf](https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/homology_of_schemes_thesis_published.pdf).

* [[Vladimir Voevodsky]], _Homology of schemes, I_.  1994, K-theory Preprint Archives, [url](http://www.math.uiuc.edu/K-theory/0031/)

* [[David Rydh]], _Submersions and effective descent of étale morphisms_,
Bull. Soc. Math. France 138(2) (2010), 181–230, [publ](http://smf4.emath.fr/Publications/Bulletin/138/html/smf_bull_138_181-230.php), [pdf](https://people.kth.se/~dary/submersion20100305.pdf).


H. Blaine Lawson

* [Wikipedia entry](http://en.wikipedia.org/wiki/H._Blaine_Lawson)

## Selected writings

On the [[Calabi-Penrose fibration]]:

* {#Lawson85} [[H. Blaine Lawson]], _Surfaces minimales et la construction de Calabi-Penrose_,  Séminaire Bourbaki : volume 1983/84, exposés 615-632, Astérisque no. 121-122  (1985), Talk no. 624, p. 197-211 ([numdam:SB_1983-1984__26__197_0](http://www.numdam.org/item/SB_1983-1984__26__197_0))

On [[spin geometry]]:

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

On [[calibration|calibrated geometry]]:

* {#HarveyLawson82} [[Reese Harvey]], [[H. Blaine Lawson]], _Calibrated geometries_, Acta Math. Volume 148 (1982), 47-157 ([Euclid:1485890157](https://projecteuclid.org/euclid.acta/1485890157))

On [[quaternionic orbifolds]] and [[quaternion-Kähler orbifolds]]:

*  K. Galicki, [[H. Blaine Lawson]], _Quaternionic Reduction and Quaternionic Orbifolds_, Mathematische Annalen (1988) Volume: 282, Issue: 1, page 1-22 ([dml:164446](https://eudml.org/doc/164446))

## Publications

* [[Spin geometry]]

* [[Sp(n).Sp(1)]]

category: people


H. Dieter Zeh

* [Wikipedia entry](https://en.wikipedia.org/wiki/H._Dieter_Zeh)

## Selected writings

Introducing the idea of dynamical [[quantum decoherence]] in the [[quantum measurement]] [[quantum measurement channel|process]]:

* {#Zeh70} [[H. Dieter Zeh]], *On the interpretation of measurement in quantum theory*, Found Phys **1** (1970) 69–76 [[doi:10.1007/BF00708656](https://doi.org/10.1007/BF00708656)]

* [[Erich Joos]], [[H. Dieter Zeh]], *The emergence of classical properties through interaction with the environment*, Z. Physik B -- Condensed Matter **59** (1985) 223–243 [[doi:10.1007/BF01725541](https://doi.org/10.1007/BF01725541)]

category: people

[[!redirects Heinz-Dieter Zeh]]


h1-meson

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## Idea

One of the [[light meson|light]] [[vector mesons]], with [[Wigner classification]] being a [[Lorentz group]] [[bivector]] ([DHR 11](#DHR11), [DHR 12](#DHR12)) and [[isospin]] [[scalar]].

## Related concepts

* [[b1-meson]]

* [[B-field]]

## References

* J. C. R. Bloch, Yu. L. Kalinovsky, C. D. Roberts, S. M. Schmidt, _Describing $a_1$ and $b_1$ decays_, Phys. Rev. D60:111502, 1999 ([arXiv:nucl-th/9906038](https://arxiv.org/abs/nucl-th/9906038))

Discussion within [[holographic QCD]]:

* {#DHR11} Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Completing the framework of AdS/QCD: $h_1$/$b_1$ mesons and excited $\omega$/$\rho$'s_, JHEP 05(2011)107 ([arXiv:1101.3315](https://arxiv.org/abs/1101.3315))

* {#DHR12}  Sophia K. Domokos, [[Jeffrey Harvey]], Andrew B. Royston, _Successes and Failures of a More Comprehensive Hard Wall AdS/QCD_, JHEP 04(2013)104 ([arXiv:1210.6351](https://arxiv.org/abs/1210.6351))

[[!redirects h1-mesons]]


Haag's theorem

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## Idea

_Haag's theorem_ [[Haag (1955)](#Haag55), [Hall & Whightman (1957)](#Hall57)] says that for a non-[[finite number]] of generators the [[canonical commutation relations]] do _not_ have a unique (up to [[isomorphism]]) [[irreducible representation|irreducible]] [[unitary representation]].

This is in contrast to the [[Stone-von Neumann theorem]] which says that for a [[finite number]] of generators the [[Schrödinger representation]] is, up to isomorphism, the unique irrducible unitary representation of the [[canonical commutation relations]].

While [[canonical commutation relations]] with a [[finite number]] of generators appear in [[quantum mechanics]], those appearing in [[quantum field theory]] (see e.g. [[Wick algebras]] for [[free fields]]) generically have infinitely many generators and hence are subject to Haag's theorem.

> More pertinently, the theorem goes on to say that, therefore, the [[interaction picture]] in interacting quantum field theory does not behave as in quantum mechanics...

## References

Haag's theorem was first stated in

* {#Haag55} [[Rudolf Haag]], _On quantum field theories_, Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser **29** 12 (1955) 1-37 [[cds:212242](http://cdsweb.cern.ch/record/212242), [pdf](http://cdsweb.cern.ch/record/212242/files/p1.pdf)]

but the proof had some gaps. It was completed in

* {#Hall57} D. Hall, [[Arthur Wightman]]  _A theorem on invariant analytic functions with applications to relativistic quantum field theory_, Det Kongelige Danske Videnskabernes Selskab, Matematisk-fysiske Meddelelser **31** 5 (1957) 1-41 [[pdf](http://gymarkiv.sdu.dk/MFM/kdvs/mfm%2030-39/mfm-31-5.pdf)]

A brief statement in context is in

* [[Rudolf Haag]], pages 54-55 of _[[Local Quantum Physics -- Fields, Particles, Algebras]]_

A thorough discussion of meaning and implications of Haag's theorem (pointing out plenty of flaws on this point in the standard literature) is in

* John Earman, Doreer Fraser, _Haag's theorem and its implications for the foundations of quantum field theory_ ([pdf](http://philsci-archive.pitt.edu/2673/1/earmanfraserfinalrevd.pdf))

Textbook account in a context of [[non-perturbative quantum field theory]]:

* {#Strocchi13} [[Franco Strocchi]], Thm. 3.1 in: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

See also

* Wikipedia, _[Haag's theorem](https://en.wikipedia.org/wiki/Haag%27s_theorem)_


Haag-Kastler axioms

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## Idea
 {#Idea}

The Haag--Kastler axioms ([Haag-Kastler 64](#HaagKastler64)) (sometimes also called Araki--Haag--Kastler axioms) try to capture in a mathematically precise way the notion of [[quantum field theory]] (QFT), by axiomatizing how its algebras of [[quantum observables]] should depend on [[spacetime]] regions, namely as _[[local nets of observables]]_.

The main point of these axioms is to say that

1. to every [[causally closed subset]] $\mathcal{O} \subset X$ of [[spacetime]] $X$ there is associated a [[C*-algebra]] $\mathcal{A}(\mathcal{O})$;

1. for every inclusion $\mathcal{O}_1 \hookrightarrow \mathcal{O}_2$ of such spacetime regions there is a corresponding inclusion $\mathcal{A}(\mathcal{O}_1) \hookrightarrow \mathcal{A}(\mathcal{O}_2)$

1. such that this respects [[composition]] of inclusions (hence such that $\mathcal{A}$ is a [[functor]] from the [[poset]] of [[causally closed subsets]] to [[C*-algebras]])

1. ([[causal locality]]) and such that whenever $\mathcal{O}_1, \mathcal{O}_2 \subset \mathcal{O} \subset X$ are [[spacelike]] separated, then the elements of the corresponding algebras of observables (graded-)commute with each other:

   $$
     [\mathcal{A}(\mathcal{O}_1), \mathcal{A}(\mathcal{O}_2)]
     = t
     0
     \phantom{AAAA}
     \in \mathcal{A}(\mathcal{O})
     \,.
   $$

Moreover one wants these assignments to behave well with spacetime symmetry.

The formulation of [[quantum field theory]] via these axioms has come to be known as _[[algebraic quantum field theory]]_ or _[[AQFT]]_, for short. There are various further axioms in the list such as the [[time slice axiom]].
The precise details of the list of axioms is in flux as the theory develops.

The Haag-Kastler axioms in their original form aim for the description of [[non-perturbative quantum field theory]] on [[Minkowski spacetime]]. They could be used to give rigorous [[proof]] of structural statements of [[QFT]] such as the [[spin-statistics theorem]] or the [[PCT theorem]], but examples of interacting field theories in dimensions $\geq 4$ are missing.

There are however variants:

* In [[perturbative AQFT]] one drops the requirement of [[C*-algebras]] and instead considers just [[formal power series algebras]] (in [[Planck's constant]] $\hbar$) in order to describe [[perturbative quantum field theory]] and connect to the traditional construction of pQFTs via the [[Feynman perturbation series]].

The observation that [[causal perturbation theory]] yields [[quantum observables]] that satisfy the Haag-Kastler axioms, except that [[C*-algebras]] are replaced by [[formal power series algebras]], is due to ([Il'in-Slavnov 78](#IlinSlavnov78), [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00), [Brunetti-Dütsch-Fredenhagen 09](#BrunettiDuetschFredenhagen09)). See at _[S-Matrix -- Causal Locality and Quantum Observables](S-matrix#CausalLocality)_.

* In [[locally covariant perturbative AQFT]] one generalizes from [[Minkowski spacetime]] to more general [[globally hyperbolic spacetimes]] in order to describe [[QFT on curved spacetimes]].

* In [[homotopical AQFT]] one consider [[homotopical algebras]] and commutativity only up to [[coherence law|coherent]] [[homotopy]] in order to discuss [[gauge theory]] with non-trivial topological ([[instanton]]) sectors.

### Terminology
 {#Terminology}

The approch to quantum field theory based on these axioms is often called _[[AQFT]]_ : either for **axiomatic quantum field theory** (since it was among the first attempts to put the edifice of QFT on solid [[axiom]]atic grounds) or **algebraic quantum field theory** (since it amplifies the algebras of local [[observable]]s over the spaces of [[state]]s). Neither of these terms is very descriptive. First there is  another, dual, axiomatization which does axiomatize the propagation of [[states]] -- see _[[FQFT]]_ -- which, second, is also "algebraic" in some sense, even though algebras of observables to not appear directly.

Another common term for these axioms is **local quantum field theory** (see the title of the standard textbook ([Haag](#Haag))) since, as becomes clear [below](#Formulation), they are focused on encoding the locality properties of QFT in terms of the algebras of observables. However, also the core aspect of _[[extended topological quantum field theory|extended]]_ [[FQFT]] is all about the notion of locality  of QFT.

Therefore neither of the traditional terms for QFT as axiomatized by Haag-Kastler is truly descriptive in that it genuinely distinguishes from the other, the Atiyah-Segal axiomatization by [[FQFT]]. What does distinguish the two approaches  may be characterized in traditional terminology of quantum theory as follows ([Schreiber](#Schreiber), [SatiSchreiber](#SatiSchreiber)):

* [[FQFT]] axiomatizes the _[[Schrödinger picture]]_ of QFT, which encodes the propagation of [[state]]s through spacetimes;

* [[AQFT]] axiomatizes the _[[Heisenberg picture]]_ of QFT, which encodes the way that [[observable]]s depend on spacetime.

### Motivation
 {#Motivation}

A central difference between the Haag-Kastler axioms and traditionally more widespread formulations of QFT (usually far from being formalized in any way) is the emphasis of the [[algebra of observables]]  of a QFT (Heisenberg picture) and the de-emphasis of the ([[Hilbert space|Hilbert]]) [[spaces of states]] ([[Schrödinger picture]]). This emphasis receives motivation from the the fact that many technical problems of QFT simply disappear when one is not trying to form its spaces of states, while at the same time no real information about the theory is lost.

Examples of technical problems that formulation in terms of spaces of states bring with them are the following:

* In [[quantum field theory]] as opposed to [[quantum mechanics]], the [[Stone-von Neumann theorem]] fails, making the [[unitary representation]] of the [[Heisenberg group]] on the spaces of states non-unique, hence requiring an explicit choice of representation. There is no generally good theory available for how to make this choice.

* More seriously, [[Haag's theorem]] says that at a crucial step in [[perturbation theory]] where one wants to pass from the representation "free fields" to that of "interacting fields", the two representations are necessarily inequivalent, contrary to what is (silently or explicitly) assumed in much traditional QFT literature (see [EarmanFraser](#EarmanFraser)).

* In the [[renormalization]] or [[perturbation theory]] the formulation in terms of states brings with it _infrared problems_ that are simply absent when formulating renormalization just in terms of observables ([DuetschFredenhagen](#DuetschFredenhagen)).

## Formulation
 {#Formulation}

We formulate the ideas of the core axioms of Haag-Kastler, and their intended physical meaning. For more details see [[local net of observables]].

1. **spacetime locality**

   Since the _fields_ in [[quantum field theory]] (such as the [[electromagnetic field]]) exhibit and are characterized by
their _local excitations_ (for instance the value of the electric/magnetic [[field strength]] at any point) having effects only locally (the field excitations at two points a finite distance apart do not directly influence each other) the fields over any region of [[spacetime]] form a [[subsystem]] of the fields of any larger region and
in particular of the total system.

   If "[[quantum mechanical system]]" is formalized as "[[C-star algebra]]" (of [[observables]]) then "[[subsystem]]" translates to "sub-$C^*$-algebra". Therefore the above sentence translates into: quantum fields form a [[copresheaf]] of [[C-star algebras]] on [[spacetimes]] whose co-restriction morphisms are [[monomorphism]]s.

   In [[AQFT]] such is called an **[[local net|isotonic net of algebras]]** .

   There are different approaches to define what kind of spacetime regions the algebras of observables are assigned to, hence different approaches as to what exactly the [[site]] is on which the co-presheaf is defined. A common approach is to take all [[bounded space|bounded]] [[open subspace|open]] subsets of [[Minkowski space|Minkowski]] [[spacetime]]. For more general setups see [[AQFT on curved spacetimes]].

1. **causal locality**

   If two regions of [[spacetime]] are [[spacelike]] separated, then there can be no influence between them whatsoever. Not only do the field excitations in one of the two regions not _directly_ influence those in the other region (as per item 1), but they do not even influence _indirectly_ : no [[waves]] of excitations (for instance [[electromagnetic waves]]: [[light]]) can run from one region to a spacelike separated region. Therefore the two [[subsystems]] constituted by these two regions accordording to the first point are even _[[independent subsystems]]_ . This is called _[[causal locality]]_.

   The formalization of "two [[independent subsystems]]" in [[quantum mechanics]] is: two subalgebras that _commute_ with each other inside the larger [[C-star algebra]]. (And usually one adds: and such that the algebra they generate in the larger algebra is [[isomorphic]] to their [[tensor product]].)

   Therefore this translates into the axiom: quantum fields on a
spacetime form an isotonic copresheaf of algebras such that the
algebras assigned to any two spacelike separated regions commute with
each other inside the algebra assigned to any larger region containing
these two regions.

1. **spacetime covariance**

   The geometric symmetry operations map the algebra of a region onto the algebra of the transformed region.

   (this is not an extra axiom if one defines the [[site]] of spacetime regions general enough...)

   In Minkowski spacetime the geometric symmetry group is usually taken to be the [[Poincaré group]], but note that some authors consider subgroups of the full Poincaré group, like the subgroup of translations (Borchers: "Translation group and particle representations in quantum field theory").

1. **positivity of energy**

   An axiom is needed to ensure that only nonnegative energies occur -- one possibility is the "spectrum condition", which says that the spectrum (to be more precise: the support of the [[spectral measure]]) of the operator associated with a translation is contained in the closed forward light cone, for all translations.

## Properties

### Structural theorems

It is possible to prove both a [[spin-statistics theorem]] and a [[PCT theorem]] in the Haag-Kastler approach. The mathematically precise, model independent statements and their proofs are considered to be a major breakthrough of the theory.

### Relation to Wightman axioms

Unlike the [[Wightman axioms]], the Haag--Kastler axioms do not need the notion of "[[physical field|field]]": the fields in the Wightman axioms are -- from the Haag--Kastler point of view -- only necessary to describe how the algebras of observables are constructed; any way to consistently construct the net of algebras would suffice.

## Related concepts

* [[quantum field theory]]

  * [[AQFT]]

    * **Haag-Kastler axioms**

    * [[local net of observables]], [[conformal net]]

    * [[causal perturbation theory]]

    * [[locally covariant perturbative AQFT]]

    * [[scattering amplitude]]

    * [[modular theory]]

  * [[FQFT]]

## References

The original article that introduced these axioms is

* {#Haag64} [[Rudolf Haag]], [[Daniel Kastler]], _An algebraic approach to quantum field theory_, Journal of Mathematical Physics, Bd.5, 1964, S.848-861 ([doi:10.1063/1.1704187](https://doi.org/10.1063/1.1704187), [spire:9124](https://inspirehep.net/literature/9124))

following

* {#Haag59} [[Rudolf Haag]], _Discussion des "axiomes" et des propriétés asymptotiques d’une théorie des champs locales avec particules composées, Les Problèmes Mathématiques de la Théorie Quantique des Champs_, Colloque Internationaux du CNRS LXXV (Lille 1957), CNRS Paris (1959), 151.

translated to English as:

* [[Rudolf Haag]], *Discussion of the ‘axioms’ and the asymptotic properties of a local field theory with composite particles*, EPJ H 35, 243–253 (2010) ([doi:10.1140/epjh/e2010-10041-3](https://doi.org/10.1140/epjh/e2010-10041-3))

Textbook accounts include

* {#Haag} [[Rudolf Haag]], _[[Local Quantum Physics -- Fields, Particles, Algebras]]_ Springer (1992) 2nd., rev. and enlarged ed. Springer (1996) ([ISBN 978-3-642-61458-3] (https://www.springer.com/gp/book/9783540610496))

* [[Huzihiro Araki]]: _[[Mathematical Theory of Quantum Fields]]_ Oxford University Press 1999 [ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0998.81501&format=complete).

precursors include

* Bogolyubov, Logunov, Oksak, Todorov: _General principles of quantum field theory._ ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0732.46040&format=complete))

An online reference page is here:

* Stephen J. Summers: [AQFT online] (http://www.math.ufl.edu/~sjs/aqft.html)

See also the references at _[[AQFT]]_ and at _[[perturbative AQFT]]_.

One of the founding fathers of [[perturbative quantum field theory]] wrote:

* {#Dyson72} [[Freeman Dyson]] _Missed opportunities_, Bulletin of the AMS, Volume 78, Number 5, September 1972 ([pdf](https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf))

  > These  axioms,  taken  together  with  the  axioms  defining  a  C*-algebra  are  a  distillation  into  abstract  mathematical  language  of  all  the   general  truths  that  we  have  learned  about  the  physics  of  microscopic   systems  during  the  last  50 years. They  describe a  mathematical  structure of  great  elegance  whose  properties  correspond  in  many  respects  to  the facts  of  experimental  physics.  In  some  sense,  the  axioms  represent  the most  serious  attempt  that  has  yet  been  made  to  define  precisely  what physicists mean by the words "observability, causality, locality, relativistic invariance," which  they are constantly  using  or  abusing  in their  everyday speech. $[$...$]$ I  therefore  propose  as  an  outstanding  opportunity  still  open  to  the  pure mathematicians, to  create a mathematical structure preserving the  main   features  of  the  Haag-Kastler axioms but possessing E-invariance instead of P-invariance.

  (on the latter see _[[AQFT on curved spacetimes]]_)

The observation that [[causal perturbation theory]] yields [[quantum observables]] that satisfy the Haag-Kastler axioms, except that [[C*-algebras]] are replaced by [[formal power series algebras]] 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))

* {#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))

* {#BrunettiDuetschFredenhagen09} [[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))

An introduction into [[Tomita-Takesaki modular theory]]:

* Stephen J. Summers: "Tomita-Takesaki Modular Theory" ([arXiv] (http://xxx.uni-augsburg.de/abs/math-ph/0511034))

and on its role in AQFT:

* [[Hans-Jürgen Borchers]], *On Revolutionizing of Quantum Field Theory with Tomita's Modular Theory*, ESI Preprint 773 (1999) [[pdf](https://www.mat.univie.ac.at/~esiprpr/esi773.pdf)]

A discussion of how the Haag-Kastler axioms (those concerning locality) follow from an extended [[FQFT]] with Lorentzian structure is in

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

A discussion putting the state of the art of the [[AQFT]]-axiomatization in context with that of the [[FQFT]]-axiomatization is in

* {#SatiSchreiber} [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Mathematical Foundations of Quantum Field and String Theory]]_

A discussion of [[perturbation theory]] and [[renormalization]] in terms of the Haag-Kastler axioms is in

* {#DuetschFredenhagen} [[Michael Dütsch]], [[Klaus Fredenhagen]], _A local (perturbative)_ construction of observables in gauge theores: the example of qed_ , Commun. Math. Phys. 203 (1999), no.1, 71-105,  ([arXiv:hep-th/9807078](http://xxx.uni-augsburg.de/ps/hep-th/9807078)).

[[Haag's theorem]] and its meaning and implication is discussed thoroughly in

* {#EarmanFraser} John Earman, Doreer Fraser, _Haag's theorem and its implications for the foundations of quantum field theory_ ([pdf](http://philsci-archive.pitt.edu/2673/1/earmanfraserfinalrevd.pdf))

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Haag-Kastler vacuum representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### AQFT
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
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=--

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

## Idea
This page lays down a specific set of axioms of the [[Haag-Kastler axioms]] to
[[AQFT]], knowing that this set is not _the set_ of Haag-Kastler axioms, but one specific choice. The purpose is to state and prove some of the classical results of the theory in a mathematically precise and model independent way. In fact some experts will find the chosen axioms to be stronger than necessary, this is deliberately so to ease the exposition.

It is possible to construct examples that fulfill the axioms, to show that they are not empty, but we will not engage in this task here, at least not now. Note however that up to now there was no success in the task to construct systems in 4 dimensions with interactions, which has led to some doubts about the usefulness of this approach in the physics community: It has yet to be shown if the approach does or does not capture the essential features that makes possible the tremendous success of the standard model of particle physics.

## Abstract
We first collect some terms that are needed to then formulate the axioms of the vacuum representation. A few simple consequences are cited and some links to further concepts are provided.

###Notation

####The Poincaré Group
The universal covering $SL(2, \C) \ltimes \R^4$ of the restricted [[Poincaré group]] $\mathcal{P}^{\dagger}_+$ will be denoted by $\mathcal{P}$, the [[abelian group|abelian]] [[subgroup]] of all translations by $\mathcal{T}$.

####The Minkowski Spacetime

We talk about 4-dimensional [[Minkowski spacetime]] $\mathcal{Min}$ only, i.e. $\mathcal{Min}$ is the [[vector space]] $\R^{4} = \R \times \R^{3}$ equipped with the scalar product $\lt x, y \gt := x_0 y_0 - (\vec x, \vec y)$ with $( \cdot , \cdot  )$ being the Euclidean scalar product on $\R^{3}$.
Open bounded subsets of $\mathcal{Min}$ will be denoted by $\mathcal{O}$. The union of these $\mathcal{O}$ form an index set $\mathcal{J}$, that is [[poset|partially ordered]] by inclusion.

If two sets are spacelike separated, this will be denoted by $\mathcal{O}_1 \perp \mathcal{O}_2$, the interior of the spacelike complement of a given set $\mathcal{O}$ will be denoted by $\mathcal{O}'$ or $\mathcal{O}^{\perp}$. The bounded open sets thus form a [[causal index set]].

We denote the open forward (light)cone at x by $V_+(x)$, similar $V_-(x)$ is the open backward cone at $x$, if $x=0$ we simply write $V_+$ and $V_-$.
A **double cone** or **diamond** is an intersection of an open forward cone and an open backward cone that is nonempty and will be denoted by $\mathcal{K}$.

An important class of unbounded regions are the **wedges**: Choose an inertial frame and define the right wedge as
$$
       W_R := \{ x = (t, x_1, x_2, x_3) \in \mathbb{R}^4 \mid x_1 \gt \shortmid t \shortmid \}
$$
The set of wedges is then defined to be
$$
      \mathcal{W} := \{\lambda W_R \mid \lambda \in \mathcal{P}^{\dagger}_+ \}
$$
While the definition of the right wedge depends on the chosen initial frame, the definition of the set of wedges does not.

####Operator Algebras

See [[operator algebra]] and [[von Neumann algebra]] here on the nLab.

[[von Neumann algebra|Von Neumann algebras]] $\mathcal{M}$ will always be concrete [[operator algebra]]s acting on a given [[Hilbert space]] $\mathcal{H}$, as is the rule in the literature (see also [[von Neumann algebra]]). The [[commutant]] of $\mathcal{M}$ will be denoted by $\mathcal{M}'$, the positive cone by $\mathcal{M}^+$. The minimal von Neumann algebra that contains two given ones $\mathcal{M}_1$ and $\mathcal{M}_2$ will be denoted by:
$$
      \mathcal{M}_1 \vee \mathcal{M}_2 :=  {(\mathcal{M}_1 \cup \mathcal{M}_2)}''
$$
As in this formula we will make frequently use of the [[bicommutant theorem]] to denote by $\mathcal{M}''$ the weak resp. strong closure resp. the generated von Neumann algebra of a give set of bounded operators.

An [[automorphism]] of an algebra $\alpha \mathcal{M} \rightarrow \mathcal{M}$ is called an **inner** automorphism if there is an invertible element $u \in \mathcal{M}$ such that $\alpha$ is given by conjugation with $u: \alpha(m) = u \quad m \quad u^{-1} \qquad \forall m\in \mathcal{M}$ (note that our convention here differs from that used by [Wikipedia] (http://en.wikipedia.org/wiki/Inner_automorphism)).

####Group Representations

In this paragraph we will collect some links and remarks about unitary [[representation]]s of [[topological group]]s on Hilbert spaces that are relevant to our topic and less commonly used in the literature.
In the following $\mathcal{G}$ will be a topological group, $\mathcal{H}$ a (complex) Hilbert space and $\mathcal{U}$ an unitary representation of $\mathcal{G}$ in the algebra of bounded operators of $\mathcal{H}$.

Definition (**analytical vector**): Let $\mathcal{G}$ be a n-dimensional real Lie group. Fix a $f \in \mathcal{H}$, a neighbourhood B of 0 in $\R^n$ and a parametrization $\phi: B \to \mathcal{G}$ of a neighbourhood of 1 in $\mathcal{G}$. Then we can define a function $\mathcal{U}_f$ by
$$
      \mathcal{U}_f: B \to \mathcal{H}
$$
$$
                     x \mapsto \mathcal{U}(\phi(x)) f
$$
If $\mathcal{U}_f$ has an extension to an analytic function on a neighbourhood of 0 in $\C^n$, the vector $f$ is called an analytic vector (for $\mathcal{U}$).

See [planetmath] (http://planetmath.org/encyclopedia/BanachSpaceValuedAnalyticFunctions.html) for the definition of Banach space valued analytic functions.

###Definition of Vacuum Representations
A net of von Neumann algebras $\mathcal{M}(\mathcal{O})$ on a common Hilbert space $(\mathcal{H})$, indexed by $\mathcal{O} \in \mathcal{J}$, is called a **vacuum respresentation** (on the 4-dimensional Minkowski spacetime) if it satisfies the following axioms:

+-- {: .num_theorem #isotony}
* (V1) **isotony**: $\mathcal{O}_1 \subset \mathcal{O}_2$ implies $\mathcal{M}(\mathcal{O}_1) \subseteq \mathcal{M}(\mathcal{O}_2)$
=--

+-- {: .num_theorem #additivity}
* (V2) **additivity**: $\mathcal{O} = \bigcup_j \mathcal{O}_j$  implies   $\mathcal{M}(\mathcal{O}) = ( \bigcup_j \mathcal{M}(\mathcal{O}_j) )'' $
=--

+-- {: .num_theorem #locality}
* (V3) **locality**, see [[local net]]: $\mathcal{O}_1 \perp \mathcal{O}_2 implies \mathcal{M}(\mathcal{O}_1) \subseteq  (\mathcal{M}(\mathcal{O}_2))' $
=--

+-- {: .num_theorem #covariance}
* (V4) **covariance**: There is a strongly continuous unitary representation $\mathcal{U}(\mathcal{P})$ of $\mathcal{P}$ such that for each $g \in \mathcal{P}$ the following holds: $\mathcal{M}(g\mathcal{O}) = \mathcal{U}(g) \mathcal{M}(\mathcal{O}) \mathcal{U}(g)^{-1}$
=--

+-- {: .num_theorem #spectrum_condition}
* (V5) **spectrum condition**: $spec\mathcal{U}(\mathcal{T}) \subseteq clo \mathcal{V}_+$
=--

_Remark_ (mathematical viewpoint): $\mathcal{T}$ is the abelian subgroup of translations, and $\mathcal{V}_+$ is the (open) forward cone at 0, see above. For the definiton of the spectrum of the representation $\mathcal{U}(\mathcal{T})$ see [[spectral measure]].

+-- {: .num_theorem #existence_of_vacuum_vector}
* (V6) **existence of a vacuum vector**: There exists a vector $\Omega \in \mathcal{H}, \|\Omega\| = 1$, such that $\bigl( \bigcup_{\mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr) \Omega$ is dense in $\mathcal{H}$ and $\mathcal{U}(g)\Omega = \Omega$ for all $g \in \mathcal{P}$
=--

+-- {: .un_remark}
###### Remark
**(choice of axioms)** The _uniqueness_ is sometimes part of the axioms, but not here. Instead we will cite theorems that will specify necessary and sufficient conditions to ensure that there is a _unique_ vacuum vector.

=--

###Additional Notations and Notions of Vacuum Representations
A short hand notation for vacuum representations will be $\mathcal{M}(\mathcal{J})$ in the following.

The algebras $\mathcal{M}(\mathcal{O})$ are sometimes called **local algebras**.

The $C^*-$algebra $\mathcal{A} := clo_{\| \cdot \|} \bigl( \bigcup_{\mathcal{O}\in\mathcal{J}}\mathcal{M}(\mathcal{O}) \bigr) $ is called **quasilocal algebra**, the smallest von Neuman algebra that contains $\mathcal{A}$ is called the **global algebra** and denoted by $\mathcal{R}$.

A vacuum representation is called **irreducible** if $\mathcal{R} = \mathcal{L}(\mathcal{H})$ (the global algebra is the whole algebra of all bounded linear operators on the given Hilbert space), it is called **factorial** if $\mathcal{R}$ is a factor.

* Wikipedia on [factors] (http://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors) of von Neumann algebras.

The subspace of $\mathcal{H}$ that is invariant under the action of the translation group $\mathcal{T}$ is not trivial due to the axiom \ref{existence_of_vacuum_vector}. If it is one-dimensional, we will say that the vacuum representation has a **unique vacuum vector** (the space is then necessarily the subspace $\C\Omega$).

### first consequences and the Reeh-Schlieder theorem

* **Definition (algebras of unbounded open set)**
To any unbounded open set $\mathcal{U}$ we associate a von Neumann algebra by
$$
           \mathcal{M}(U) := \bigl( \bigcup_{\mathcal{O} \subset \mathcal{U}} \mathcal{M}(\mathcal{O}) \bigr)''
$$

* **Theorem (weak additivity)**
Let $\mathcal{O} \in \mathcal{J}$ be arbitrary, then we have
$$
         \bigl( \bigcup_{a \in \mathcal{T}} \mathcal{M}(a \mathcal{O}) \bigr)'' = \mathcal{R}
$$

_Remark_ (choice of axioms): Weak additivity of our nets are a direct consequence of our axioms, but this is often enough not so in the literature: Depending on the choice of axioms, it is either stated as an axiom or an auxiliary property of a net that is called for when needed.

_Proof_: Let $\mathcal{O}_0$ be given (arbitrary but fixed).
Let us first note that thanks to isotony and additivity we can restrict the index set of the union in the definition of $\mathcal{R}$ from open bounded sets to diamonds, since every bounded set is contained in a diamond:
$$
\bigl( \bigcup_{\mathcal{O} \in \mathcal{J}} \mathcal{M}(\mathcal{O}) \bigr)'' = \bigl( \bigcup_{\mathcal{K}} \mathcal{M}(\mathcal{K}) \bigr)'' = \mathcal{R}
$$
Given an arbitrary diamond $\mathcal{K}$ we have $\mathcal{K} \subset \bigl( \bigcup_{a \in \mathcal{K}}(\mathcal{O}_0 + a) \bigr)$ while the latter is a bounded open subset $\in \mathcal{J}$. This means that we arrive at
$$
\bigcup_{\mathcal{K}} \mathcal{M}(\mathcal{K}) = \bigcup_{a\mathcal{O}_0, a \in {\mathcal{T}}} \mathcal{M}(a\mathcal{O}_0)
$$
which implies that the generated von Neumann algebras coincide, too, of course.

* **Theorem (Borchers, translations are inner)**: The representatives of the translations are elements of the global algebra $\mathcal{R}$, i.e. they are inner automorphisms of $\mathcal{R}$: $\mathcal{U}(\mathcal{T}) \subseteq \mathcal{R}$.

* **Theorem (uniqueness of vacuum)**: Every factorial vacuum representation is irreducible. A vacuum representation is irreducible iff it has an unique vacuum vector.

* **[[Reeh-Schlieder Theorem]]**: The vacuum vector is cyclic and separating for all local algebras.

Let $x, y \in \R^4$ and define $[x, y] := \{\lambda x + \mu y | \lambda, \mu \geqq 0, \lambda + \mu = 1\} $.
Define

$$
           \mathcal{M}[x, y] := \cap_{[x, y] \subset \mathcal{O}} \mathcal{M}(\mathcal{O})
$$

+-- {: .un_theorem}
###### Theorem
**triviality of algebras of spacelike segments**

If the segment $[x, x + b]$ is spacelike, i.e. $\langle b, b \rangle \lt 0$, and the vacuum respresentation has a unique vacuum vector, then $\mathcal{M}[x, x + b] = \mathbb{C} \mathbb{1}$, i.e. the algebra associated with the segment is trivial.
=--

+-- {: .un_theorem}
###### Theorem
**triviality of algebras of points**
The conclusion of the preceding statement holds if we put $b = 0$, i.e. if we consider the algebra associated with one point.
=--

_Remark_ (physical viewpoint): The preceding theorem is sometimes summarized as _there are no non-trivial observables at the point_. There are two possible ways to interpret this result: The pragmatic approach says that, since no detector can be built that measures observables precisley at one point of spacetime, there is no need of a theory to support the concept of observables localized at a point. The philosophical approach takes this one step further and states that our relativistic quantum theory tells us that the concepts of points and observables localized at points are an idealization with no relevance to nature.

The local algebras fulfill the [[Borchers property]].

There are several directions one can pursue next, for example

* the [[classification of local algebras]] or

* [[superselection theory]]

## References
This page is based upon:

* Hellmut Baumgärtel: Operator algebraic Methods in Quantum Field Theory. A series of lectures. Akademie Verlag 1995 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0839.46063&format=complete))

A secondary reference is this:

* Hellmut Baumgärtel, Manfred Wollenberg: Causal nets of operator algebras, Berlin: Akademie Verlag 1992 ([ZMATH entry] (http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0749.46038&format=complete))


Haag-Ruelle scattering theory

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]], _Haag-Ruelle scattering theory_ refers to formalization of [[scattering]] processes  in the framework of [[algebraic quantum field theory]].

## References

* [[Rudolf Haag]]

* [[Nikolay Bogolyubov]], A. A. Logunov, A. I. Oksak, I. T. Todorov, G. G. Gould, *Haag-Ruelle Scattering Theory* [[doi:10.1007/978-94-009-0491-0_12](https://doi.org/10.1007/978-94-009-0491-0_12)],  Chapter in: *General principles of quantum field theory*, Mathematical Physics and Applied Mathematics **10**, Kluwer (1990) [[doi:10.1007/978-94-009-0491-0](https://doi.org/10.1007/978-94-009-0491-0)]

[[!redirects Hag-Ruelle theory]]


Haag–Łopuszański–Sohnius theorem

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###Context###
#### Algebraic Quantum Field Theory
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[[!include AQFT and operator algebra contents]]
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#### Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Super-Algebra and Super-Geometry
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[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Coleman-Mandula theorem]] says that the only possible [[symmetry]] [[Lie algebra]] of the [[S-matrix]] of a 4-dimensional [[quantum field theory]] with [[mass gap]] is a [[direct sum]] of the [[Poincare Lie algebra]] and some other Lie algebra of internal [[gauge equivalence|gauge]] symmetries, hence that there cannot be a non-trivial mixing between the [[spacetime]] symmetry and the internal symmetry at the level of [[Lie algebras]] and in the presence of a [[mass gap]].

(This applies to fundamental [[local symmetry]] of the [[S-matrix]], not to [[global symmetry]] after [[spontaneous symmetry breaking]].)

The theorem of [Haag–Łopuszański–Sohnius 75](#HaagLopuszanskiSohnius75) makes the analogous statement for [[super Lie algebras]]: the only possible [[super Lie algebra]] symmetry of an [[S-matrix]] is a [[direct sum]] of the [[super Poincare Lie algebra]] ("[[supersymmetry]]") and another super Lie algebra.

Notice that the [[super Poincare Lie algebra]] itself does mix the plain bosonic [[Poincare Lie algebra]] symmetry with a kind of "internal" symmetry, see also at [[extended supersymmetry]].

## References

The original article:

* {#HaagLopuszanskiSohnius75} [[Rudolf Haag]], [[Jan Łopuszański]], [[Martin Sohnius]], _All possible generators of supersymmetries of the S-matrix_, Nuclear Physics B, 88: 257–274 (1975) (doi:10.1016/0550-3213(75)90279-5, [spire:90850](https://inspirehep.net/literature/90850))

On the history of the result:

* [[Jan Łopuszański]], *The story of central charges*, p. 16-17 in: [[Steven Duplij]], [[Warren Siegel]], [[Jonathan Bagger]] (eds.), *Concise encyclopedia of supersymmetry -- and noncommutative structures in mathematics and physics*, Springer 2005 ([doi:10.1007/1-4020-4522-0](https://doi.org/10.1007/1-4020-4522-0) [pdf](https://ivv5hpp.uni-muenster.de/u/douplii/susy/SUSYEnc_Story.pdf))

See also:

* Wikipedia,  _[Haag-Łopuszański-Sohnius theorem](https://en.wikipedia.org/wiki/Haag%E2%80%93%C5%81opusza%C5%84ski%E2%80%93Sohnius_theorem)_

[[!redirects Haag–Lopuszanski–Sohnius theorem]]

[[!redirects Haag-Łopuszański-Sohnius theorem]]


Haar integral

[[!redirects Haar measure]]

#Contents#
* table of contents
{: toc}

## Idea

If $G$ is a locally compact Hausdorff [[topological group]], with $C_c(G)$ the algebra of compactly supported continuous functions from $C_c(G)$ to $\mathbb{R}$,  a _Haar integral_ $\int_G $ is a continuous linear map $C_c(G) \rightarrow \mathbb{R}$ which is invariant under the aparent action of $G$ on $C_c(G)$. It turns out that there exists a unique such integral, up to a scalar multiple. The Riesz representation theorem then allows one to conclude the existence of a unique _Haar measure_, which is a $G$-invariant Borel measure  on $G$.

The archetypal example of Haar measure is the [[Lebesgue measure]] on the (additive group underlying) [[cartesian space]] $\mathbb{R}^n$.

## Definition

Let $G$ be a locally compact Hausdorff group. Let $C_c(G)$ denote the vector space of continuous real-valued functionals with compact support on $G$. This is a [[locally convex topological vector space]] where the locally convex structure is specified by the family of seminorms

$$\rho_K(f) = \sup_{x \in K} |f(x)|$$

$K$ ranging over compact subsets of $G$. Recall that a [[Radon measure]] on $G$ may be described as a continuous linear functional

$$\int_G : C_c(G) \to \mathbb{R}$$

Such a Radon measure defines a measure $\mu$ on the $\sigma$-algebra of Borel sets in the usual sense of [[measure theory]], where

$$\mu(B) = sup \left\{\int_G f : supp(f) = K \subseteq B, \rho_K(f) = 1 \right\}$$

In this context, a (left) **Haar integral** on $G$ is a nonzero such linear functional $\int_G$ such that
$$ \int_G f \geq 0 \text{ when } f \geq 0$$
$$ \int_G f^g = \int_G f$$
for each $f \in C_c(G)$ and each $g \in G$, where $f^g : G \rightarrow \mathbb{R}$ sends $x$ to $f(gx)$.

Correspondingly, a (left) **Haar measure** on $G$ is a nonzero Radon measure $\mu$ such that

$$\mu(g B) = \mu(B)$$

for all $g \in G$ and all Borel sets $B$.

## An Analogy with the Finite Case

There is an analogy between Haar measure and scaled-cardinality on a [[finite group]]. In fact, the latter is a special case of the former, as we may view a finite group as a discrete topological group. While measure on a (discrete) finite group is subsumed by the notion of Haar measure, it may be of interest to build intuition.

Let $G_{fin}$ be a finite group.  Let $G_{fin} \text{-Rep}$ be the category of $G$-[[representations]] of $R$-modules, where $R$ is a ring in which $|G_{fin}|$ is invertible. This category is equivalent to $R[G]$-mod. We can view $R$ as a trivial $G_{fin}$-representation, where $ga = a$ for each $a \in R$ and each $g \in G$.

Let $G$ be a [[compactum]]. Let $G \text{-Ban}$ be the category of Banach representations of $G$. Objects in $G \text{-Ban}$ are banach spaces $X$ over $\mathbb{R}$ with a continuous norm preserving action $G \times X \rightarrow X$, i.e. $||gx|| = ||x|| \forall g \in G \forall x \in X$. Maps in $G \text{-Ban}$ are short maps which are $G$-equivariant. (Alternatively, $G \text{-Ban}$ can be viewed as a category of certain $\mathbb{R}[G]$-modules.) We can view $\mathbb{R}$ as a trivial $G$-representation, where $ga = a$ for each $a \in \mathbb{R}$ and each $g \in G$.

Let $C(G_{fin})$ be the ring of set-maps from $G_{fin}$ to $R$. $G$ acts on $C(G_{fin})$ where $f^g : G_{fin} \rightarrow R$ sends $x$ to $f(gx)$. There is a map of abelian groups $\int_{G_{fin}} : C(G_{fin}) \rightarrow R$ sending $f$ to $\frac{1}{|G|} \sum_{g \in G} f(g)$, analogous to the Haar integral. Indeed,
$$\int_{G_{fin}} f + g = \int_{G_{fin}} f + \int_{G_{fin}} g \forall f, g \in C(G_{fin}) $$
$$\int_{G_{fin}} a f = a \int_{G_{fin}} f \forall f \in C(G_{fin}) \forall a \in R$$
$$\int_{G_{fin}} f^g = \int_{G_{fin}} f \forall f \in C(G_{fin}) \forall g \in G$$
$$\int_{G_{fin}} f \geq 0 \text{ when } f \geq 0 \forall f \in C(G_{fin}) $$

What corresponds to the Haar measure on $G$ is simply cardinality (though we must appropriately divide by the cardinality of $G$, to get a function $\mu_{fin} : P(G) \rightarrow R$ from the power set of $G$ to $R$ sending $S$ to $\frac{|S|}{|G|}$).

Using the Haar integral, we may define [[convolution product]]: $* : C(G \times G) \rightarrow C(G)$ sending $f : G \times G \rightarrow \mathbb{R}$ to the map $G \rightarrow \mathbb{R}$ sending $\int_{h \in G} f(gh^{-1}, h) = \int_{hk = g} f(h, k)$. This is analogous to the map $*_{fin} : C(G_{fin} \times G_{fin}) \rightarrow C(G_{fin})$ sending $f$ to the map $C(G_{fin} )$ sending $g$ to $\frac{1}{|G_{fin}|} \sum_{h k = g} f(h, k)$.

In both cases, we get a "[[bar construction]]". For the compact Hausdorff case, we get:
$$
    \cdots
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
   C(G  \times G)
    \stackrel{\stackrel{* }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto \int_{h \in G} f( g, h) \right)  }{\longrightarrow}}
    C(G) \stackrel{\int_G}{\longrightarrow} \mathbb{R}
$$
and for the finite case, we get:
$$
    \cdots
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
   C(G_{fin}  \times G_{fin})
    \stackrel{\stackrel{*_{fin} }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto  \frac{1}{|G_{fin}|} \sum_{h \in G} f( g, h) \right)  }{\longrightarrow}}
    C(G_{fin}) \stackrel{f \mapsto \frac{1}{|G_{fin}|} \sum_{g \in G} f(g)}{\longrightarrow} \mathbb{R}
$$
Calling this a bar resolution is a slight abuse of terminology; the "monad" involved is actually has no unit, as $C(G)$ has no unit for the convolution product. However, convolution makes $C(G)$ an assocciative nonunitial algebra, so that the resolution is still a _unitless_ monad. Hence there are evident face maps without degeneracies.

While $C(G)$ has no unit, there is a canonical way of adding units to it: take products with $\mathbb{R}[G]$. Write $\sum_{g \in G} a_g \delta_g$ for the elements of $\mathbb{R}[G]$; since $\delta_1$ is a formal unit for convolution, we may think of it as a Dirac delta function. Now the bar resolution
$$
    \cdots
    \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}
   C(G  \times G) \prod \mathbb{R}[G]
    \stackrel{\stackrel{* }{\longrightarrow}}{\stackrel{ f \mapsto \left( g \mapsto \int_{h \in G} f( g, h) \right)  }{\longrightarrow}}
    C(G) \prod \mathbb{R}[G] \stackrel{\int_G}{\longrightarrow} \mathbb{R}
$$
has degeneracies as well.

## Properties

### Existence and Uniqueness

Any locally compact Hausdorff topological group $G$ admits a Haar integral (and therefore Haar measure) that is unique up to scalar multiple. This result was first proven by Weil. A proof can be found in these online [notes](http://simonrs.com/HaarMeasure.pdf) by Rubinstein-Salzedo. A different, constructive proof, due to E.M. Alfsen, can be found in the article "[A simplified and constructive proof of the existence and uniqueness of haar measure](https://www.mscand.dk/article/view/10675)".

We here give a lesser known proof of the existence of the Haar integral, specifically on compact Hausdorff groups $G$, which uses convex sets and the Krein Milman theorem instead of measure theory. Let $G \text{-Ban}$ be the category of Banach representations of $G$ (see "Analogy with the Finite Case").

$C_c(G) = C(G)$ is such a Banach representation. We may view $\mathbb{R}$ as a Banach representation of $G$ as well, where $gz = z$ for each $z \in \mathbb{R}$ and each $g \in G$. $\mathbb{R}$ embeds into $C(G)$ as constant functions. We may then consider the exact sequence
$$0 \rightarrow \mathbb{R} \rightarrow C(G) \rightarrow C(G)/ \mathbb{R} \rightarrow 0$$

A Haar integral on the $G$-representation $C(G)$ is equivalently a retract $\int_G : C(G) \rightarrow \mathbb{R}$ for the injection $\mathbb{R} \rightarrow C(G)$. In other words, it is a function $\int_G : C(G) \rightarrow \mathbb{R}$ such that

$$ \int_G (f_1 + f_2) = \int_G f_1 + \int_G f_2  \forall f_1, f_2 \in C(G)$$
$$ \int_G a f = a \int_G f   \forall f \in C(G), a \in \mathbb{R}$$
$$ \int_G f^g = \int_G f \forall f \in C(G), g \in G$$
$$ \exists C \in  \mathbb{R}_{\geq 0 } :  \left| \left| \int_G f \right| \right| \leq C \int_G ||f||\forall  [G, \mathbb{C}]_{\text{Top}}$$
The last of these requirements, given the others, is equivalent to continuity of $\int_G$.

In some sense, we might wish to show that $\text{Ext}^1_{G \text{-Ban}}(C(G), \mathbb{R})$ vanishes; this would show that the sequence
$$0 \rightarrow \mathbb{R} \rightarrow C(G) \rightarrow C(G)/ \mathbb{R} \rightarrow 0$$
splits by the usual characterization of extensions via $\text{Ext}^1$. On further contemplation, it is sufficient to show that the trivial $G$-representation $\mathbb{R}$ is an injective object in $G \text{-Ban}$. This could be seen as an equivariant Hahn-Banach theorem.

**Proof:** We show that $\mathbb{R}$ is an injective object in $G \text{-Ban}$. Take an injection of Banach representations of $G$, $X \rightarrow Y$. Let $f : X \rightarrow \mathbb{R}$ be a map of Banach representations of $G$. By the (usual) Hahn-Banach theorem, there exists a map $g : Y \rightarrow \mathbb{R}$ in the category of Banach spaces and short maps extending $f$, though it may lack $G$-invariance.

Consider the subset of all extensions of $f$ to $Y$. Let $S$ be the collection of $G$-invariant compact convex subsets of this set. $S$ contains the convex hull of $G g$, where $g$ is some chosen extension of $f$ to $Y$, so $S$ is nonempty. Using compactness and [[Zorn's Lemma]], we may find a minimal element of $S$ in this collection, where $S$ is ordered where $A \leq B$ when $A \subset B$. Call this element $H$. $H$ must be a singleton. If $H$ contains a point which is not extremal then it contains the convex hull of the orbit of that point, which would be a proper $G$-invariant compact convex subset of $H$ (see Krein Milman theorem). Therefore $H$ is a singleton, and its unique element is a $G$-invariant functional extending $f$.

In particular, since $\mathbb{R}$ has been shown to be injective, the map $\text{Id}_{\mathbb{R}}  : \mathbb{R} \rightarrow \mathbb{R}$ lifts along the inclusion
$$0 \rightarrow \mathbb{R} \rightarrow C(G)$$
giving a retract $\int_G : C(G) \rightarrow \mathbb{R}$ for $0 \rightarrow \mathbb{R} \rightarrow C(G)$ in $G \text{-Ban}$.

It follows that $\int_G$ has norm $1$, and from this positivity follows immediately.

### Extensive and Intensive Properties

In Lawvere's thinking about [extensive and intensive quantities](https://ncatlab.org/nlab/show/intensive+or+extensive+quantity),

* $C(G)$ is a space of _intensive quantities_ on $G$.

* $[C(G), \mathbb{R}]_{\text{Ban}}$ is the space of _extensive quantities_ on $X$, where $\text{Ban}$ is the category of Banach spaces with bounded maps as maps. The Haar integral is the unique $G$-invariant element of this space of norm $1$.

* the [[integration]] map is the canonical [[evaluation]] pairing

  $$
    \int_G \;\colon\; C(G) \times [C(G), \mathbb{R}]_{\text{Ban}}  \longrightarrow \mathbb{R}
    \,.
  $$

If we suggestively write $\int_G f d \phi$ for $\int_G (f, \phi) = \phi(f)$, then $\int_G - d \phi$ becomes a way of writing $\phi$. In particular, choosing $\phi$ to be the Haar measure, we can write $\phi$ as $\int_G - d \phi$.

## Left and Right Haar Measures that Differ

The left and the right Haar measure may or may not coincide, groups for which they coincide are called **unimodular**.
Consider the matrix subgroup
$$
G := \left\{ \left.\, \begin{pmatrix} y & x \\ 0 & 1 \end{pmatrix}\,\right|\, x, y \in \mathbb{R}, y \gt 0  \right\}
$$
The left and right invariant measures are, respectively,
$$
  \mu_L = y^{-2} \,\mathrm{d}x \,\mathrm{d}y,\quad   \mu_R = y^{-1} \,\mathrm{d}x \,\mathrm{d}y
$$
and so G is not unimodular.

[[Abelian groups]] are obviously unimodular; so are [[compactum|compact]] groups and [[discrete topology|discrete]] groups.

## References

* {#Bredon72} [[Glen Bredon]], Section 0.3 of: _[[Introduction to compact transformation groups]]_, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1), [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

[1]: https://arxiv.org/abs/math/0606794

[[!redirects Haar measure]]
[[!redirects Haar measures]]
[[!redirects haar measure]]
[[!redirects haar measures]]