View source for Quantum field theory, axioms for

The mathematical axiom systems for [[Quantum field theory|quantum field theory]] (QFT) grew out of Hilbert's sixth problem [[#References|[a6]]], that of stating the problems of quantum theory in precise mathematical terms. There have been several competing mathematical systems of axioms, and below those of A.S. Wightman [[#References|[a5]]], and of K. Osterwalder and R. Schrader [[#References|[a4]]] are given, stated in historical order. They are centred around group symmetry, relative to unitary representations of Lie groups in Hilbert space.

==Wightman axioms.==
Wightman's axioms involve:

a [[Unitary representation|unitary representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200101.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200102.png" /> as a covering of the [[Poincaré group|Poincaré group]] of relativity, and a vacuum state vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200103.png" /> fixed by the representation.

Quantum fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200104.png" />, say, as operator-valued distributions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200105.png" /> running over a specified space of test functions, and the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200106.png" /> defined on a dense and invariant domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200107.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200108.png" /> (the [[Hilbert space|Hilbert space]] of quantum states), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q1200109.png" />.

A transformation law which states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001010.png" /> is a finite-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001011.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001012.png" /> (cf. also [[Representation of a group|Representation of a group]]) acting on the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001013.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001015.png" /> acting on space-time and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001017.png" />.

The fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001018.png" /> are assumed to satisfy locality and one of the two canonical commutation relations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001019.png" />, for fermions, respectively bosons.

Finally, it is assumed that there is scattering with asymptotic completeness, in the sense <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001020.png" />.

==Osterwalder–Schrader axioms.==
The Wightman axioms were the basis for many of the spectacular developments in QFT in the 1970s, see, e.g., [[#References|[a1]]], [[#References|[a2]]], and the Osterwalder–Schrader axioms [[#References|[a3]]], [[#References|[a4]]] came in response to the dictates of path-space measures. The constructive approach involved some variant of the Feynman measure. But the latter has mathematical divergences that can be resolved with an analytic continuation, so that the mathematically well-defined [[Wiener measure|Wiener measure]] becomes instead the basis for the analysis. Two analytical continuations were suggested in this connection: in the mass-parameter, and in the time-parameter, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001021.png" />. With the latter, the Newtonian quadratic form on space-time turns into the form of relativity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001022.png" />. One gets a [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001023.png" /> that is: symmetric, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001024.png" />; stationary, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001025.png" />; and Osterwalder–Schrader positive, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001027.png" /> test functions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001029.png" /> denoting a path space measure.

Specifically: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001030.png" />, then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001032.png" /></td> </tr></table>

By Minlos' theorem, there is a [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001033.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001034.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001036.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001037.png" /> is a positive measure, one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001038.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001039.png" />, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001040.png" />. When combining (a1) and (a2), one can note that this limit-measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001041.png" /> then accounts for the time-ordered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001043.png" />-point functions which occur on the left-hand side in (a1). This observation is further used in the analysis of the stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001045.png" />. But, more importantly, it can be checked from the construction that one also has the following reflection positivity: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001048.png" />, and set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001049.png" /></td> </tr></table>

Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001050.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001051.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001052.png" />, which is one version of Osterwalder–Schrader positivity.

===Relation to unitary representations of Lie groups.===
Since the [[Killing form|Killing form]] of Lie theory may serve as a finite-dimensional metric, the Osterwalder–Schrader idea [[#References|[a4]]] turned out also to have implications for the theory of unitary representations of Lie groups. In [[#References|[a3]]], P.E.T. Jorgensen and G. Ólafsson associate to Riemannian symmetric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001053.png" /> of tube domain type (cf. also [[Symmetric space|Symmetric space]]), a duality between complementary series representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001054.png" /> on one side, and highest-weight representations of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001056.png" />-dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001057.png" /> on the other side. The duality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001058.png" /> involves [[Analytic continuation|analytic continuation]], in a sense which generalizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001059.png" />, and the reflection positivity of the Osterwalder–Schrader axiom system. What results is a new Hilbert space, where the new representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001060.png" /> is  "physical"  in the sense that there is positive energy and causality, the latter concept being defined from certain cones in the [[Lie algebra|Lie algebra]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001061.png" />.

A [[Unitary representation|unitary representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001062.png" /> acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001063.png" /> is said to be reflection symmetric if there is a unitary operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001064.png" /> such that

R1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001065.png" />;

R2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001067.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001069.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001070.png" />.

A closed convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001071.png" /> is hyperbolic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001072.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001073.png" /> is semi-simple (cf. also [[Semi-simple representation|Semi-simple representation]]) with real eigenvalues for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001074.png" />.

Assume the following, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001075.png" />:

PR1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001076.png" /> is reflection symmetric with reflection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001077.png" />.

PR2) There is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001078.png" />-invariant hyperbolic cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001079.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001080.png" /> is a closed [[Semi-group|semi-group]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001081.png" /> is diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001082.png" />.

PR3) There is a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001083.png" />, invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001084.png" />, satisfying the positivity condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001085.png" /></td> </tr></table>

Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001086.png" /> satisfies PR1)–PR3). Then the following hold:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001087.png" /> acts via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001088.png" /> by contractions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001089.png" /> (the Hilbert space obtained by completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001090.png" /> in the norm from PR3)).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001091.png" /> be the simply-connected Lie group with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001092.png" />. Then there exists a unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001093.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001094.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001095.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001097.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001098.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001099.png" />.

The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q120010100.png" /> is irreducible if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q120010101.png" /> is irreducible.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Glimm,   A. Jaffe,   "Quantum field theory and statistical mechanics (a collection of papers)" , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Glimm,   A. Jaffe,   "Quantum physics" , Springer  (1987)  (Edition: Second)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.E.T. Jorgensen,   G. Ólafsson,   "Unitary representations of Lie groups with reflection symmetry"  ''J. Funct. Anal.'' , '''158'''  (1998)  pp. 26–88</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Osterwalder,   R. Schrader,   "Axioms for Euclidean Green's functions"  ''Comm. Math. Phys.'' , '''31/42'''  (1973/75)  pp. 83–112;281–305</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R.F. Streater,   A.S. Wightman,   "PCT, spin and statistics, and all that" , Benjamin  (1964)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.S. Wightman,   "Hilbert's sixth problem: Mathematical treatment of the axioms of physics"  F.E. Browder (ed.) , ''Mathematical Developments Arising from Hilbert's Problems'' , ''Proc. Symp. Pure Math.'' , '''28:1''' , Amer. Math. Soc.  (1976)  pp. 241–268</TD></TR></table>