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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.
 
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.
  
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Wightman's axioms involve:
 
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.
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a [[Unitary representation|unitary representation]] $U$ of $G = \operatorname{SL} ( 2 , {\bf C} ) \rtimes {\bf R} ^ { 4 }$ as a covering of the [[Poincaré group|Poincaré group]] of relativity, and a vacuum state vector $\psi_0$ 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" />.
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Quantum fields $\varphi _ { 1 } ( f ) , \dots , \varphi _ { n } ( f )$, say, as operator-valued distributions, $f$ running over a specified space of test functions, and the operators $\varphi _ { i } ( f )$ defined on a dense and invariant domain $D$ in $\mathcal{H}$ (the [[Hilbert space|Hilbert space]] of quantum states), and $\psi _ { 0 } \in D$.
  
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" />.
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A transformation law which states that $U ( g ) \varphi_j ( f ) U ( g ^ { - 1 } )$ is a finite-dimensional representation $R$ of the group $G$ (cf. also [[Representation of a group|Representation of a group]]) acting on the fields $\varphi _ { i } ( f )$, i.e., $\sum _ { i } R _ { j i } ( g ^ { - 1 } ) \varphi _ { i } ( g [ f ] )$, $g$ acting on space-time and $g [ f ] ( x ) = f ( g ^ { - 1 } x )$, $x \in \mathbf{R} ^ { 4 }$.
  
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.
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The fields $\varphi_j ( f )$ are assumed to satisfy locality and one of the two canonical commutation relations of $[ A , B ] _ { \pm } = A B \pm B A$, 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" />.
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Finally, it is assumed that there is scattering with asymptotic completeness, in the sense $\mathcal{H} = \mathcal{H} ^ { \text{in} } = \mathcal{H} ^ { \text{out} }$.
  
 
==Osterwalder–Schrader axioms.==
 
==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.
+
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., $t \mapsto \sqrt { - 1 }t$. With the latter, the Newtonian quadratic form on space-time turns into the form of relativity, $x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } - t ^ { 2 }$. One gets a [[Stochastic process|stochastic process]] $\mathcal{X} _ { t }$ that is: symmetric, i.e., ${\cal X} _ { t } \sim {\cal X}_{ - t }$; stationary, i.e., $\mathcal{X} _ { t + s } \sim \mathcal{X} _ { s }$; and Osterwalder–Schrader positive, i.e., $\int _ { \Omega } f _ { 1 } \circ \mathcal{X} _ { t _ { 1 } } \ldots f _ { n } \circ \mathcal{X} _ { t _ { n } } d P \geq 0$, $f _ { 1 } , \ldots , f _ { n }$ test functions, $- \infty &lt; t _ { 1 } \leq \ldots \leq t _ { n } &lt; \infty$, and $P$ denoting a path space measure.
  
Specifically: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001030.png" />, then
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Specifically: If $- t / 2 &lt; t _ { 1 } \leq \ldots \leq t _ { n } &lt; t / 2$, 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>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001031.png"/></td> <td style="width:5%;text-align:right;" valign="top">(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>
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\begin{equation*} = \operatorname { lim } _ { t \rightarrow \infty } \int \prod _ { k = 1 } ^ { n } A _ { k } ( q ( t _ { k } ) ) d \mu _ { t } ( q ( \cdot ) ). \end{equation*}
  
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
+
By Minlos' theorem, there is a [[Measure|measure]] $\mu$ on ${\cal D} ^ { \prime }$ 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>
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\begin{equation} \tag{a2} \operatorname { lim } _ { t \rightarrow \infty } \int e ^ { i q ( f ) } d \mu _ { t } ( q ) = \int e ^ { i q ( f ) } d \mu ( q ) = : S ( f ) \end{equation}
  
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
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for all $f \in \mathcal{D}$. Since $\mu$ 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>
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\begin{equation*} \sum _ { k } \sum _ { l } \overline { c } _ { k } c _ { l } S ( f _ { k } - \overline { f } _ { l } ) \geq 0 \end{equation*}
  
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
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for all $c_1 , \ldots , c_n \in \mathbf{C}$, and all $f _ { 1 } , \dots , f _ { n } \in \mathcal{D}$. When combining (a1) and (a2), one can note that this limit-measure $\mu$ then accounts for the time-ordered $n$-point functions which occur on the left-hand side in (a1). This observation is further used in the analysis of the stochastic process $\mathcal{X} _ { t }$, $\mathcal{X} _ { t } ( q ) = q ( t )$. But, more importantly, it can be checked from the construction that one also has the following reflection positivity: Let $( \theta f ) ( s ) : = f ( - s )$, $f \in \mathcal{D}$, $s \in \mathbf{R}$, 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>
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\begin{equation*} \mathcal{D} _ { + } = \{ f \in \mathcal{D} : f \ \text { real valued, } f ( s ) = 0 \text { for } s &lt; 0 \}. \end{equation*}
  
 
Then
 
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>
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\begin{equation*} \sum _ { k } \sum _ { l } \overline { c } _ { k } c_{ l} S ( \theta (\, f _ { k } ) - f _ { l } ) \geq 0 \end{equation*}
  
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.
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for all $c_1 , \ldots , c_n \in \mathbf{C}$ and all $f _ { 1 } , \dots , f _ { n } \in \mathcal{D} _ { + }$, which is one version of Osterwalder–Schrader positivity.
  
 
===Relation to unitary representations of Lie groups.===
 
===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" />.
+
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 $G / K$ of tube domain type (cf. also [[Symmetric space|Symmetric space]]), a duality between complementary series representations of $G$ on one side, and highest-weight representations of a $c$-dual $G ^ { c }$ on the other side. The duality $G \leftrightarrow G ^ { c }$ involves [[Analytic continuation|analytic continuation]], in a sense which generalizes $t \mapsto \sqrt { - 1 }t$, and the reflection positivity of the Osterwalder–Schrader axiom system. What results is a new Hilbert space, where the new representation of $G ^ { c }$ 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 $G$.
  
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
+
A [[Unitary representation|unitary representation]] $\pi$ acting on a Hilbert space $\mathcal{H} ( \pi )$ is said to be reflection symmetric if there is a unitary operator $J : \mathcal{H} ( \pi ) \rightarrow \mathcal{H} ( \pi )$ such that
  
R1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001065.png" />;
+
R1) $J ^ { 2 } = \operatorname{id}$;
  
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" />.
+
R2) $J \pi ( g ) = \pi ( \tau ( g ) ) J$, $g \in G$. Here, $\tau \in \operatorname { Aut } ( G )$, $\tau ^ { 2 } = \operatorname{id}$, and $H = \{ g \in G : \tau ( g ) = g \}$.
  
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" />.
+
A closed convex cone $C \subset \text{q}$ is hyperbolic if $C ^ { o } \neq \emptyset$, and if $\operatorname { ad } X$ is semi-simple (cf. also [[Semi-simple representation|Semi-simple representation]]) with real eigenvalues for every $X \in C ^ { o }$.
  
Assume the following, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001075.png" />:
+
Assume the following, for $( G , \pi , \tau , J )$:
  
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" />.
+
PR1) $\pi$ is reflection symmetric with reflection $J$.
  
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" />.
+
PR2) There is an $H$-invariant hyperbolic cone $C \subset \text{q}$ such that $S ( C ) = H \operatorname { exp } C$ is a closed [[Semi-group|semi-group]] and $S ( C ) ^ { o } = H \operatorname { exp } C ^ { o }$ is diffeomorphic to $H \times C ^ { o }$.
  
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
+
PR3) There is a subspace $0 \neq \mathcal{K} _ { 0 } \subset \mathcal{H} ( \pi )$, invariant under $S ( C )$, 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>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><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:
+
Assume that $( \pi , C , \mathcal{H} , J )$ 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)).
+
$S ( C )$ acts via $s \mapsto \widetilde{\pi} ( s )$ by contractions on $\mathcal{K}$ (the Hilbert space obtained by completion of $\mathcal{K} _ { 0 }$ 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" />.
+
Let $G ^ { c }$ be the simply-connected Lie group with Lie algebra $\mathfrak { g } ^ { c }$. Then there exists a unitary representation $\tilde{\pi} ^ { c }$ of $G ^ { c }$ such that $d \tilde { \pi } ^ { c } ( X ) = d \tilde { \pi } ( X )$ for $X \in \mathfrak { h }$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q120/q120010/q12001097.png"/> for $Y \in C$, where $\mathfrak { h } = \{ X \in \mathfrak { g } : \tau ( X ) = X \}$.
  
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.
+
The representation $\tilde{\pi} ^ { c }$ is irreducible if and only if $\tilde{\pi}$ is irreducible.
  
 
====References====
 
====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>
+
<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>

Revision as of 16:59, 1 July 2020

The mathematical axiom systems for quantum field theory (QFT) grew out of Hilbert's sixth problem [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 [a5], and of K. Osterwalder and R. Schrader [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 $U$ of $G = \operatorname{SL} ( 2 , {\bf C} ) \rtimes {\bf R} ^ { 4 }$ as a covering of the Poincaré group of relativity, and a vacuum state vector $\psi_0$ fixed by the representation.

Quantum fields $\varphi _ { 1 } ( f ) , \dots , \varphi _ { n } ( f )$, say, as operator-valued distributions, $f$ running over a specified space of test functions, and the operators $\varphi _ { i } ( f )$ defined on a dense and invariant domain $D$ in $\mathcal{H}$ (the Hilbert space of quantum states), and $\psi _ { 0 } \in D$.

A transformation law which states that $U ( g ) \varphi_j ( f ) U ( g ^ { - 1 } )$ is a finite-dimensional representation $R$ of the group $G$ (cf. also Representation of a group) acting on the fields $\varphi _ { i } ( f )$, i.e., $\sum _ { i } R _ { j i } ( g ^ { - 1 } ) \varphi _ { i } ( g [ f ] )$, $g$ acting on space-time and $g [ f ] ( x ) = f ( g ^ { - 1 } x )$, $x \in \mathbf{R} ^ { 4 }$.

The fields $\varphi_j ( f )$ are assumed to satisfy locality and one of the two canonical commutation relations of $[ A , B ] _ { \pm } = A B \pm B A$, for fermions, respectively bosons.

Finally, it is assumed that there is scattering with asymptotic completeness, in the sense $\mathcal{H} = \mathcal{H} ^ { \text{in} } = \mathcal{H} ^ { \text{out} }$.

Osterwalder–Schrader axioms.

The Wightman axioms were the basis for many of the spectacular developments in QFT in the 1970s, see, e.g., [a1], [a2], and the Osterwalder–Schrader axioms [a3], [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 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., $t \mapsto \sqrt { - 1 }t$. With the latter, the Newtonian quadratic form on space-time turns into the form of relativity, $x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } - t ^ { 2 }$. One gets a stochastic process $\mathcal{X} _ { t }$ that is: symmetric, i.e., ${\cal X} _ { t } \sim {\cal X}_{ - t }$; stationary, i.e., $\mathcal{X} _ { t + s } \sim \mathcal{X} _ { s }$; and Osterwalder–Schrader positive, i.e., $\int _ { \Omega } f _ { 1 } \circ \mathcal{X} _ { t _ { 1 } } \ldots f _ { n } \circ \mathcal{X} _ { t _ { n } } d P \geq 0$, $f _ { 1 } , \ldots , f _ { n }$ test functions, $- \infty < t _ { 1 } \leq \ldots \leq t _ { n } < \infty$, and $P$ denoting a path space measure.

Specifically: If $- t / 2 < t _ { 1 } \leq \ldots \leq t _ { n } < t / 2$, then

(a1)

\begin{equation*} = \operatorname { lim } _ { t \rightarrow \infty } \int \prod _ { k = 1 } ^ { n } A _ { k } ( q ( t _ { k } ) ) d \mu _ { t } ( q ( \cdot ) ). \end{equation*}

By Minlos' theorem, there is a measure $\mu$ on ${\cal D} ^ { \prime }$ such that

\begin{equation} \tag{a2} \operatorname { lim } _ { t \rightarrow \infty } \int e ^ { i q ( f ) } d \mu _ { t } ( q ) = \int e ^ { i q ( f ) } d \mu ( q ) = : S ( f ) \end{equation}

for all $f \in \mathcal{D}$. Since $\mu$ is a positive measure, one has

\begin{equation*} \sum _ { k } \sum _ { l } \overline { c } _ { k } c _ { l } S ( f _ { k } - \overline { f } _ { l } ) \geq 0 \end{equation*}

for all $c_1 , \ldots , c_n \in \mathbf{C}$, and all $f _ { 1 } , \dots , f _ { n } \in \mathcal{D}$. When combining (a1) and (a2), one can note that this limit-measure $\mu$ then accounts for the time-ordered $n$-point functions which occur on the left-hand side in (a1). This observation is further used in the analysis of the stochastic process $\mathcal{X} _ { t }$, $\mathcal{X} _ { t } ( q ) = q ( t )$. But, more importantly, it can be checked from the construction that one also has the following reflection positivity: Let $( \theta f ) ( s ) : = f ( - s )$, $f \in \mathcal{D}$, $s \in \mathbf{R}$, and set

\begin{equation*} \mathcal{D} _ { + } = \{ f \in \mathcal{D} : f \ \text { real valued, } f ( s ) = 0 \text { for } s < 0 \}. \end{equation*}

Then

\begin{equation*} \sum _ { k } \sum _ { l } \overline { c } _ { k } c_{ l} S ( \theta (\, f _ { k } ) - f _ { l } ) \geq 0 \end{equation*}

for all $c_1 , \ldots , c_n \in \mathbf{C}$ and all $f _ { 1 } , \dots , f _ { n } \in \mathcal{D} _ { + }$, which is one version of Osterwalder–Schrader positivity.

Relation to unitary representations of Lie groups.

Since the Killing form of Lie theory may serve as a finite-dimensional metric, the Osterwalder–Schrader idea [a4] turned out also to have implications for the theory of unitary representations of Lie groups. In [a3], P.E.T. Jorgensen and G. Ólafsson associate to Riemannian symmetric spaces $G / K$ of tube domain type (cf. also Symmetric space), a duality between complementary series representations of $G$ on one side, and highest-weight representations of a $c$-dual $G ^ { c }$ on the other side. The duality $G \leftrightarrow G ^ { c }$ involves analytic continuation, in a sense which generalizes $t \mapsto \sqrt { - 1 }t$, and the reflection positivity of the Osterwalder–Schrader axiom system. What results is a new Hilbert space, where the new representation of $G ^ { c }$ is "physical" in the sense that there is positive energy and causality, the latter concept being defined from certain cones in the Lie algebra of $G$.

A unitary representation $\pi$ acting on a Hilbert space $\mathcal{H} ( \pi )$ is said to be reflection symmetric if there is a unitary operator $J : \mathcal{H} ( \pi ) \rightarrow \mathcal{H} ( \pi )$ such that

R1) $J ^ { 2 } = \operatorname{id}$;

R2) $J \pi ( g ) = \pi ( \tau ( g ) ) J$, $g \in G$. Here, $\tau \in \operatorname { Aut } ( G )$, $\tau ^ { 2 } = \operatorname{id}$, and $H = \{ g \in G : \tau ( g ) = g \}$.

A closed convex cone $C \subset \text{q}$ is hyperbolic if $C ^ { o } \neq \emptyset$, and if $\operatorname { ad } X$ is semi-simple (cf. also Semi-simple representation) with real eigenvalues for every $X \in C ^ { o }$.

Assume the following, for $( G , \pi , \tau , J )$:

PR1) $\pi$ is reflection symmetric with reflection $J$.

PR2) There is an $H$-invariant hyperbolic cone $C \subset \text{q}$ such that $S ( C ) = H \operatorname { exp } C$ is a closed semi-group and $S ( C ) ^ { o } = H \operatorname { exp } C ^ { o }$ is diffeomorphic to $H \times C ^ { o }$.

PR3) There is a subspace $0 \neq \mathcal{K} _ { 0 } \subset \mathcal{H} ( \pi )$, invariant under $S ( C )$, satisfying the positivity condition

Assume that $( \pi , C , \mathcal{H} , J )$ satisfies PR1)–PR3). Then the following hold:

$S ( C )$ acts via $s \mapsto \widetilde{\pi} ( s )$ by contractions on $\mathcal{K}$ (the Hilbert space obtained by completion of $\mathcal{K} _ { 0 }$ in the norm from PR3)).

Let $G ^ { c }$ be the simply-connected Lie group with Lie algebra $\mathfrak { g } ^ { c }$. Then there exists a unitary representation $\tilde{\pi} ^ { c }$ of $G ^ { c }$ such that $d \tilde { \pi } ^ { c } ( X ) = d \tilde { \pi } ( X )$ for $X \in \mathfrak { h }$ and for $Y \in C$, where $\mathfrak { h } = \{ X \in \mathfrak { g } : \tau ( X ) = X \}$.

The representation $\tilde{\pi} ^ { c }$ is irreducible if and only if $\tilde{\pi}$ is irreducible.

References

[a1] J. Glimm, A. Jaffe, "Quantum field theory and statistical mechanics (a collection of papers)" , Birkhäuser (1985)
[a2] J. Glimm, A. Jaffe, "Quantum physics" , Springer (1987) (Edition: Second)
[a3] P.E.T. Jorgensen, G. Ólafsson, "Unitary representations of Lie groups with reflection symmetry" J. Funct. Anal. , 158 (1998) pp. 26–88
[a4] K. Osterwalder, R. Schrader, "Axioms for Euclidean Green's functions" Comm. Math. Phys. , 31/42 (1973/75) pp. 83–112;281–305
[a5] R.F. Streater, A.S. Wightman, "PCT, spin and statistics, and all that" , Benjamin (1964)
[a6] 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
How to Cite This Entry:
Quantum field theory, axioms for. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quantum_field_theory,_axioms_for&oldid=50316
This article was adapted from an original article by Palle E.T. JorgensenGestur Ólafsson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article