Osterwalder-Schrader axioms - Wightman Axioms

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by  Palle E. T. Jorgensen

Osterwalder–Schrader axioms—Wightman Axioms—The mathematical
axiomsystemsforquantumfieldtheory(QFT)grewoutofHilbert’s sixthproblem
[6], that of stating the problems ofquantum theory in precisemathematical terms.
There have been several competing mathematical systems of axioms, and here we
shall deal with those of A.S. Wightman [5] and of K. Osterwalder and R. Schrader
[4], stated in historical order. They are centered around group symmetry, relative
tounitaryrepresentationsofLiegroupsinHilbertspace. We alsomentionhowthe
Osterwalder–Schraderaxiomshaveinfluencedthetheoryofunitaryrepresentations
0
of groups, making connection with [3]. Wightman’s axioms involve: (1) a unitary
0 representationU of G:=SL(2,C)⋊R4 as a coverofthe Poincar´egroupof relativ-
0
2 ity, and a vacuum state vector ψ0 fixed by the representation, (2) quantum fields
ϕ (f),...,ϕ (f), say, as operator-valueddistributions, f running over a specified
n 1 n
space of test functions, and the operators ϕ (f) defined on a dense and invariant
a i
J domain D in H (the Hilbert space of quantum states), and ψ D, (3) a trans-
6 formation law which states that U(g)ϕj(f)U g−1 is a finite0-d∈imensional repre-
sentation R of the group G acting on the fields(cid:0)ϕ (f(cid:1)), i.e., R g−1 ϕ (g[f]),
i i ji i
1 g acting on space-time and g[f](x) = f g−1x , x R4. (4P) The fi(cid:0)elds(cid:1)ϕ (f) are
v ∈ j
assumed to satisfy locality and one of th(cid:0)e two(cid:1)canonical commutation relations of
0
1 [A,B]± = AB BA, for fermions, resp., bosons; and (5) it is assumed that there
±
0 is scattering with asymptotic completeness, in the sense H=Hin =Hout.
1
The Wightmanaxiomswere the basisfor many ofthe spectaculardevelopments
0
inQFTintheseventies,see,e.g.,[1,2],andtheOsterwalder–Schraderaxioms[3,4]
0
0 cameinresponsetothedictatesofpathspacemeasures. Theconstructiveapproach
/ involvedsomevariantoftheFeynmanmeasure. Butthelatterhasmathematicaldi-
h
vergencesthatcanberesolvedwithananalyticcontinuationsothatthemathemat-
p
- icallywell-definedWiener measurebecomesinsteadthe basisforthe analysis. Two
h
analyticalcontinuations were suggestedin this connection: in the mass-parameter,
t
a and in the time-parameter, i.e., t √ 1t. With the latter, the Newtonian qua-
m draticformonspace-timeturnsint7→othe−formofrelativity,x2+x2+x2 t2. Weget
1 2 3−
v: a stochastic process Xt: symmetric, i.e., Xt ∼ X−t; stationary, i.e., Xt+s ∼ Xs;
and Osterwalder–Schrader positive, i.e., f X f X f X dP 0,
i Ω 1 ◦ t1 2 ◦ t2 ··· n ◦ tn ≥
X f ,...,f test functions, < t t R t < , and P denoting a path
1 n 1 2 n
−∞ ≤ ≤ ··· ≤ ∞
r space measure.
a
Specifically: If t/2<t t t <t/2, then
1 2 n
− ≤ ≤···≤
(1) Ω A e−(t2−t1)HˆA e−(t3−t2)HˆA A Ω
1 2 3 n
D ··· E
n
= lim A (q(t )) dµ (q( )).
t→∞Z Y k k t ·
k=1
By Minlos’ theorem, there is a measure µ on ′ such that
D
(2) lim eiq(f)dµ (q)= eiq(f)dµ(q)=:S(f)
t→∞Z t Z
for all f . Since µ is a positive measure, we have
∈D
c¯ c S f f¯ 0
k l k l
− ≥
Xk Xl (cid:0) (cid:1)
for all c ,...,c C, and all f ,...,f . When combining (1) and (2), we
1 n 1 n
∈ ∈ D
notethatthislimit-measureµthenaccountsforthetime-orderedn-pointfunctions
1
2
which occur on the left-hand side in formula (1). This observation is further used
in the analysis of the stochastic process X , X (q)=q(t). But, more importantly,
t t
it can be checked from the construction that we also have the following reflection
positivity: Let (θf)(s):=f( s), f , s R, and set
− ∈D ∈
= f f real valued, f(s)=0 for s<0 .
+
D { ∈D| }
Then
c¯ c S(θ(f ) f ) 0
k l k l
− ≥
XX
k l
forallc ,...,c C,andallf ,...,f , whichisone versionofOsterwalder–
1 n 1 n +
∈ ∈D
Schrader positivity.
SincetheKillingformofLietheorymayserveasafinite-dimensionalmetric,the
Osterwalder–Schrader idea [4] turned out also to have implications for the theory
of unitary representations of Lie groups. In [3], the authors associate to Riemann-
iansymmetricspacesG/K oftube domaintype,adualitybetweencomplementary
series representations of G on one side, and highest weight representations of a
c-dual Gc on the other side. The duality G Gc involves analytic continua-
↔
tion, in a sense which generalizes t √ 1t, and the reflection positivity of the
7→ −
Osterwalder–Schrader axiom system. What results is a new Hilbert space where
thenewrepresentationofGc is“physical”inthe sensethatthereispositiveenergy
andcausality,thelatterconceptbeingdefinedfromcertainconesintheLiealgebra
of G.
Aunitaryrepresentationπ actingonaHilbertspaceH(π)issaidtobereflection
symmetric if there is a unitary operator J :H(π) H(π) such that
→
R1) J2 =id.
R2) Jπ(g)=π(τ(g))J, g G,
∈
where τ Aut(G), τ2 =id, and H := g G τ(g)=g .
∈ { ∈ | }
A closed convex cone C q is hyperbolic if Co = and if adX is semisimple
⊂ 6 ∅
with real eigenvalues for every X Co.
∈
Assume the following for (G,π,τ,J):
PR1) π is reflection symmetric with reflection J.
PR2) Thereis anH-invarianthyperbolicconeC q suchthat S(C)=HexpC
⊂
is a closed semigroup and S(C)o =HexpCo is diffeomorphic to H Co.
×
PR3) There is a subspace 0 = K H(π) invariant under S(C) satisfying the
0
6 ⊂
positivity condition
v v := v J(v) 0, v K .
h iJ h i≥ ∀ ∈ 0
Assume that (π,C,H,J) satisfies (PR1)–(PR3). Then the following hold:
1) S(C)actsvias π˜(s)bycontractionsonK(=theHilbertspaceobtained
7→
by completion of K in the norm from (PR3)).
0
2) LetGc bethesimplyconnectedLiegroupwithLiealgebragc. Thenthere
exists a unitary representation π˜c of Gc such that dπ˜c(X) = dπ˜(X) for
X handidπ˜c(Y)=dπ˜(iY)forY C,whereh:= X g τ(X)=X .
∈ ∈ { ∈ | }
3) The representationπ˜c is irreducible if and only if π˜ is irreducible.
3
References
[1] Glimm, J., and Jaffe, A.: Quantum field theory and statistical mechanics (a collection of
papers),Birkha¨user,Boston,1985.
[2] Glimm,J.,andJaffe,A.: Quantum physics (2nded.),Springer-Verlag,NewYork,1987.
[3] Jorgensen,P.E.T.,andO´lafsson,G.: ‘UnitaryrepresentationsofLiegroupswithreflection
symmetry’,J. Funct. Anal. 158(1998), 26–88.
[4] Osterwalder, K., and Schrader, R.: ‘Axioms for Euclidean Green’s functions’, Comm.
Math. Phys.31(1973), 83–112; 42(1975), 281–305.
[5] Streater, R.F., and Wightman A.S.: PCT, spin and statistics, and all that, W.A. Ben-
jamin,Inc.,NewYork,1964.
[6] Wightman,A.S.: ‘Hilbert’ssixthproblem: Mathematicaltreatmentoftheaxiomsofphysics’,
inF.E.Browder(ed.): Vol.28(part1)ofProc.Symp.PureMath.,Amer.Math.Soc.,1976,
pp.241–268.
Palle E.T. Jorgensen: [email protected]
Gestur O´lafsson: [email protected]