The Haag-Kastler Axioms for the $\mathscr{P}(\varphi)_2$ Model on the De Sitter Space

0.14 MB·

by  Christian D. Jäkel

The Haag-Kastler Axioms for the P(ϕ)
2
7
1 Model on the De Sitter Space
0
2

r Christian D. Jäkel∗and JensMund†
a
M
March 14,2017

3
1

]
h Abstract
p
- WeestablishtheHaag-Kastleraxiomsforaclassofmodelsonthetwo-
h dimensionaldeSitterspace,includingtheP(ϕ) model.
2
t
a
m
1 Introduction
[

2 Inrecentwork[2],wehaveprovidedanovelconstructionoftheP(ϕ) model
2
v
on the de Sitter space. Our work was presented in the canonical formalism,
1
3 and although finite speed of light was established, we find it worthwhile to
2 provideaproofoftheHaag-Kastleraxioms.
8
0
1. 2 One-particle space
0
7
Thetwo-dimensionaldeSitterspace
1
: .
v dS= x R1+2 |x2−x2−x2 =−r2 , r>0, (1)
i (cid:8) ∈ 0 1 2 (cid:9)
X
canbeviewedasaone-sheetedhyperboloid,embeddedinthe(1+2)-dimensional
ar MinkowskispaceR1+2. Theembedding(1)iscompatiblewiththemetricand
thecausalstructure,i.e.,thedeSitterspacedSinheritsitsmetricandthecausal
structurefromtheambientMinkowskispace.
TheisometrygroupofdSisO(1,2). Theconnectedcomponentcontaining
theidentityisSO (1,2). Thelatterisgeneratedbytherotations
0
1 0 0
.
R (α) = 0 cosα −sinα , α [0,2π),
0
  ∈
0 sinα cosα
 
∗[email protected],UniversidadedeSãoPaulo(USP),Brasil
†mund@fisica.ufjf.br,DepartamentodeFisica,UniversidadedeJuizdeFora,Brasil
1
andtheLorentzboosts
cosht 0 sinht
.
Λ (t) = 0 1 0 , t R.
1   ∈
sinht 0 cosht
 
Accordingtoourconvention, the boostsΛ (t)keepthe x -axisinvariant,and
1 1
thereforecorrespondtoboostsinthex -direction.
2
Thecircle
.
S1 ={x dS|x =0}
0
∈
forms a Cauchy surface for dS. For two points x = (0,rsinψ,rcosψ) and
y=(0,rsinψ′,rcosψ′)onthecircleS1,theWightmantwo-pointfunctionofa
scalarfreefield(analysedin[4])equals
W(2)(x,y)=cνPs+(−cos(ψ−ψ′)). (2)
HerePs+ istheLegendrefunctionfortheparameter
s± =−1 iν, (3)
2 ∓
with
i 1 −ζ2 if0<ζ<1/2,
ν= 4
 qζ2− 1 ifζ>1/2,
4
andζtheeigenvalueoftheCqasimiroperatorofSO (1,2).
0
Thetwo-pointfunction(2)givesrisetoascalarproductonC (S1):
∞
.
hh,h′iH =cνZ rdψZ rdψ′ h(ψ) (4)
S1 S1
Ps+ −cos(ψ−ψ′) h′(ψ′).
×
Thevalueofthepositivenormalisationco(cid:0)nstantis (cid:1)
1 1
c =− = .
ν 2sin(πs+) 2cos(iνπ)
Note that the singularity for ψ = ψ′ is integrable. The completion of C (S1)
w.r.t.thisscalarproductisaone-particleHilbertspace,whichwedenote∞byH.
Lemma2.1. Thescalarproduct(4)canbeexpressedas
h,h′ H = h, 1 h′ ,
h i 2ω L2(S1,rdψ)
(cid:10) (cid:11)
withωastrictlypositiveself-adjointoperatoronL2(S1,rdψ)withFouriercoefficients
Γ k+s+ Γ k+1−s+
ω(k)=r−1(k+s+) 2 2 , k Z.
Γ(cid:16)k−s+(cid:17) Γ(cid:16)k+1+s+(cid:17) ∈
2 2
e (cid:0) (cid:1) (cid:0) (cid:1)
2
Oneofthekeyresultsin[2]isthatHcarriesaunitaryirreduciblerepresenta-
tionoftheLorentzgroup:
Theorem2.2. Therotations
u(R (α))h (ψ)=h(ψ−α), α [0,2π), h H,
0
∈ ∈
(cid:0) (cid:1)
andtheboosts
u(Λ (t))=eitωrcosψ , t R,
1
∈
generatearepresentationofSO (1,2)onH.
0
Definition2.3. LetI+ betheopenhalf-circle
.
I+ ={x S1 |x2 >0}.
∈
WenowdefineR-linearsubspacesofH:
i.) Forthewedge
.
W = x dS|x >|x |
1 2 0
∈
(cid:8) (cid:9)
weset
.
H(W1)= h H|supp ℜh, ω−1ℑh I+ I+ .
∈ ⊂ ×
(cid:8) (cid:9)
(cid:0) (cid:1)
ii.) ForanarbitrarywedgeW =ΛW ,Λ SO (1,2),weset
1 0
∈
.
H(W)=u(Λ)H(W ). (5)
1
iii.) Foracausallycomplete,openandboundedregionO,weset
.
H(O)= H(W). (6)
O⊂W
\
ThenetO H(O)hasanumberofinterestingproperties,whichwillgive
7→
risetothelocalstructureoffreequantumfieldsonthedeSitterspace.
Proposition2.4. ThesubspacesintroducedinDefinition2.3havethefollowingprop-
erties:
i.) (WedgeDuality).TheR-linearsubspaceH(W′)fortheoppositewedge
.
W′ = x dS|xspace-likeseparatedfromW
∈
(cid:8) (cid:9)
equalsthesymplecticcomplement
.
H(W)′ = h H|ℑ h,g =0 g H(W)
∈ h i ∀ ∈
(cid:8) (cid:9)
ofH(W).
3
ii.) (Localisation). For I a bounded open interval in S1, let O = I′′ denote the
I
causalcompletionoftheintervalIindS. Then
H(O )= h H|supp ℜh, ω−1ℑh I I .
I
∈ ⊂ ×
(cid:8) (cid:9)
iii.) (Covariance). ForOacausallycomplete,op(cid:0)enandbound(cid:1)edregionsOandΛ
∈
SO (1,2),onefinds
0
H(ΛO)=u(Λ)H(O).
Inparticular,
u Λ (t) H(W )=H(W ) t R.
1 1 1
∀ ∈
iv.) (Microcausality).Fortw(cid:0)ospac(cid:1)e-likeseparatedcausallycomplete,openandbounded
regionsO andO ,onefinds
1 2
ℑ h1,h2 H =0 hi H(Oi), i=1,2.
h i ∀ ∈
Proof. ForthespecialcaseW = W ,propertyi.)followsfromthedefinitionof
1
H(W ),thedefinitionofthesymplecticcomplement,andthefactthat
1
L2(I,dψ)⊥ =L2(Ic,dψ).
Thegeneralcasefollowsfromdefinition(5).
Next,let
.
W(α)=R (α)W , α [0,2π),
0 1
∈
beawedgewhoseedgesliesonS1. AsO iscausallycomplete,
I
W =O =W(α) W(β)
I
∩
OI\⊂W
forsomefixedα,β [0,2π).Inspectingthedefinitions,wefindthat
∈
h H|supp ℜh, ω−1ℑh I I
∈ ⊂ ×
(cid:8) (cid:9)
isequalto (cid:0) (cid:1)
H W(α) H W(β) .
∩
AsbothW(α)andW(β)arew(cid:0)edges(cid:1)whic(cid:0)hconta(cid:1)inOI,wehave
H(O ) H W(α) H W(β) .
I
⊆ ∩
Next,weassumethatWisanarbitrar(cid:0)ywedg(cid:1)ewh(cid:0)ichcon(cid:1)tainsOI.Theopposite
wedgeW′ofW is,likeanywedge,oftheformΛβ(t)R (α)W forsuitableα,β
0 1
andt.
Asaconsequenceoffinitespeedoflight(seeProposition4.10.2in[2]),we
have
H(W′)=u Λ(β)(t) H W(α)
h(cid:0) H|s(cid:1)up(cid:0)p ℜh,(cid:1)ω−1ℑh J J ,
⊂ ∈ ⊂ ×
(cid:8) (cid:9)
withJ = Γ Λ(β)(t)R0(α)I+ S1,where(cid:0)Γ(M)isthe(cid:1)domainofdependenceof
∩
asetM,i.e.,theunionofthefutureΓ+(M)andthepastΓ−(M)ofM.
(cid:0) (cid:1)
4
Notethat
i.) Γ Λ(β)(t)R0(α)I+ =Γ(W′);
ii.) W(cid:0)′isspace-liketo(cid:1)I,sinceW containsO .
I
.
HenceΓ(W′) S1isintheinteriorIc =S1\IofthecomplementofIwithinS1.
∩
Thus
H W′ h H|supp ℜh, ω−1ℑh Ic Ic .
⊂ ∈ ⊂ ×
(cid:8) (cid:9)
Itfollowsthat (cid:0) (cid:1) (cid:0) (cid:1)
H W =H W′ ′
(cid:0) (cid:1) h(cid:0) H(cid:1) |supp ℜh, ω−1ℑh I I .
⊇ ∈ ⊂ ×
(cid:8) (cid:9)
=H(W(cid:0)(α))∩H(W(β))(cid:1)
| {z }
Thisverifiespropertyii.). Inordertoverifypropertyiii.),wecompute
H(ΛO)= H(ΛW)= H(ΛW)
ΛO⊂ΛW O⊂W
\ \
= u(Λ)H(W)
O⊂W
\
=u(Λ) H(W) .
O⊂W !
\
=H(O)
| {z }
Finally,propertyiv.)followsfromthefactthatifO andO aretwospace-like
1 2
separated causally complete, open and bounded regions, then there exists a
wedgeW =ΛW suchthat
1
O W and O W′.
1 2
⊂ ⊂
Applyingu(Λ)totheidentityH(W )′ =H(W )′,weseethatlocalityisacon-
1 1
sequenceofcovarianceandwedgeduality.
3 Second Quantization
The bosonic Fock space F = Γ(H) over H is defined as the direct sum of the
n-particlespaces:
Γ(H)=. H⊗ns , H⊗0s =. C,
⊕n=0
∞
withH⊗ns then-foldtotallysymmetrictensorproduct s ofHwithitself. The
⊗
coherentvectors
. 1
Γ(h)= h h
⊕n=0√n! ⊗s···⊗s
∞
n−times
| {z }
5
form a total set in F. The vector Ω = Γ(0) is called the Fock vacuum. One
◦
canalsodefinesecondquantizedoperators. LetAbeaclosed,denselydefined
linearoperatoronFwithdomainD(A). Then
Γ(A): F F
→
istheclosureofthelinearoperatoractingonthelinearcombinationsofcoher-
entvectorswithexponentinD(A)suchthat
Γ(A)Γ(h)=Γ(Ah).
Thisexponentiationpreservesself-adjointness,positivityandunitarity.
4 Nets of Local Algebras
Forh,g H,therelations
∈
V(h)V(g)=e−iℑhh,giV(h+g),
V(h)Ω◦ =e−12khk2Γ(ih),
defineunitaryoperators,calledtheWeyloperators. Theysatisfy
V∗(h)=V(−h) and V(0)=1.
WeusetheWeyloperatorstoassociateavonNeumannalgebrasactingonthe
Fockspace F to the wedge W : letA (W )denote the von Neumannalgebra
1 ◦ 1
generatedbytheWeyloperators
V(h)|h H(W ) .
1
∈
(cid:8) (cid:9)
Given a representation U(Λ), Λ SO (1,2), of the Lorentz group acting on
0
∈
FockspacewhichactskineticallyontheCauchysurfaceS1,wecanthandefine
von Neumann algebras associated to arbitrary bounded, causally complete,
convex regions. We proceed in steps, repeating the ideas which lie behind
Definition 2.3 and starting from the free algebra of the wedge W . Note that
1
ΛW W Λ=Λ (t)forsomet R.
1 1 1
⊂ ⇔ ∈
Definition 4.1. Given a unitary representation Λ U(Λ) of the Lorentz group
7→
SO (1,2)actingonFockspaceFandsatisfyingthecondition
0
U Λ (t) A (W )U Λ (t) −1 =A (W ), t R,
1 0 1 1 0 1
∈
(cid:0) (cid:1) (cid:0) (cid:1)
wedefinethefollowingvonNeumannalgebras:
i.) ForanarbitrarywedgeW =ΛW ,Λ SO (1,d),weset
1 0
∈
.
A(W)=U(Λ)A W U(Λ)−1 . (7)
◦ 1
(cid:0) (cid:1)
6
ii.) For an arbitrary bounded, causally complete, convex region (these are the de
Sitteranalogsofthedoublecones)O dS,weset
⊂
.
A(O)= A W . (8)
W⊃O
\ (cid:0) (cid:1)
Theinclusionpreservingmap
O A(O)
7→
is called the net of local von Neumann algebras for the bosonic field on the de Sitter
spacedStransformingunderU.
Remarks4.1.
i.) IncaseU U ,
◦
≡
.
U (Λ)=Γ(u(Λ)), Λ SO (1,2),
◦ 0
∈
wewilldenotethegeneratorofone-parameterunitarygroupt U Λ (t) by
◦ 1
7→
L andthelocalalgebrabyA (O). ItfollowsfromaresultbyAraki(Theorem1
◦ ◦ (cid:0) (cid:1)
in[1])andProposition2.4ii.)that
A (O )= V(h)|h H, supp ℜh, ω−1ℑh I I ′′;
◦ I
∈ ⊂ ×
(cid:8) (cid:9)
(cid:0) (cid:1)
justasonemighthaveexpected.
ii.) For the P(ϕ) model on the de Sitter space, the representation U is given as
2
follows.
a.) Therotationsarethefreeones,
U R (α) =Γ(u(R (α)), α [0,2π); (9)
0 0
∈
(cid:0) (cid:1)
b.) Thegenerator of the one-parameterunitary groupt U Λ (t) canbe
1
7→
expressed in terms of canonicalfields and canonical momenta (see [2] for
(cid:0) (cid:1)
details):
L=L + lim rcosψdψ:P ϕ(δ (. −ψ)) :,
◦ ǫ→0ZS1 ǫ
(cid:0) (cid:1)
wherePisapolynomial,boundedfrombelow,ϕ(h)isthegeneratorofthe
one-parameterunitarygroups V(sh),andδ approximatestheDirac
ǫ
7→
deltafunctionasǫ 0. Asusual,the :: indicatesnormalordering.
→
In the general case, we will need a criterium for the representation of the
Lorentzgroupwhichensuresthattheintersectionin(8)isnottrivial. Asuffi-
cientconditionisthefollowing: assumethevonNeumannalgebrasaredefined
asinDefinition4.1.Thenetoflocalalgebrasissaidtosatisfyfinitespeedoflight,
7
ifforanywedgeW,thealgebraA(W)iscontainedinthetime-zeroWeylalge-
bra
V(h)|h H; supp ℜh, ω−1ℑh J J ′′, (10)
∈ ⊂ ×
(cid:8) (cid:9)
whereJ=Γ(W) S1. (cid:0) (cid:1)
∩
Theorem4.2. Assumethenetoflocalalgebrassatisfiesfinitespeedoflight. Then
thelocalalgebrasassociatedtoanintervalI S1ontheCauchysurfacecoincidewith
⊂
thoseofthefreetheory,i.e.,
A(O )=A (O ), I S1 .
I ◦ I
⊂
Proof. TheprooffollowstheideasexposedintheproofofProposition2.4ii.).
ThusthekeystepistoshowthatforanywedgeWwhichcontainsO ,wehave
I
A(W′) V(h)|h H, supp ℜh, ω−1ℑh Ic Ic ′′,
⊂ ∈ ⊂ ×
(cid:8) (cid:9)
.
whereIc = S1\I. AstheedgesofW aren(cid:0)ecessarilys(cid:1)pace-orlight-liketoO ,
I
thisinclusionfollowsfromfinitespeedoflightasexpressedin(10). Byduality,
A(W) V(h)|h H, supp ℜh, ω−1ℑh I I ′′,
⊃ ∈ ⊂ ×
(cid:8) (cid:9)
=A(cid:0)◦(OI) (cid:1)
| {z }
wheneverW includesO .
I
Remarks4.2.
i.) FortheP(ϕ) modelonthedeSitterspace,property(10),whichencodesfinite
2
speedoflight,wasestablishedinTheorem10.1.1in[2].
ii.) The circle S1, which we use to identify the free field and the interacting field,
couldbereplacedbyanyspace-likegeodesicΛS1,Λ SO (1,2).
0
∈
5 The Haag-Kastler Axioms
AssumewehavedefinedanetO A(O)oflocalalgebrasaccordingtoDefi-
7→
nition4.1,withtherotationsbeingimplementedbythefreerepresentation(9).
In case this net respects finite speed of light, it will also satisfy the following
Haag-Kastleraxioms:
Theorem5.1. ThenetO A(O)sharesthefollowingproperties:
7→
i.) (Locality). Thelocalalgebrassatisfy
A(O ) A(O )′ if O O′ .
1 ⊂ 2 1 ⊂ 2
HereO′denotesthespace-likecomplementofOindSandA(O)′isthecommu-
tantofA(O)inB(F).
8
ii.) (Covariance). The representation U: Λ U(Λ) act geometrically, i.e., for
7→
Λ SO (1,2),
0
∈
U(Λ)A(O)U(Λ)−1 =A(ΛO).
iii.) (ExistenceandUniquenessoftheVacuum[3]). Thereexistsaunique(upto
aphase)unitvectorΩ F,which
∈
a.) isinvariantundertheactionofU(SO (1,2));
0
b.) satisfies the geodesic KMS condition: for every wedge W = ΛW , Λ
1
∈
SO (1,2),thepartialstate
0
.
ω↾A(W)(A)= Ω,AΩ , A A(W),
h i ∈
satisfiestheKMS-conditionatinversetemperatureβ = 2πrwithrespect
totheone-parametergroupt U(Λ (t/r)),t R.
W
7→ ∈
iv.) (Additivity). ForXadoubleconeorawedge,thereholds
A(X)= A(O). (11)
O⊂X
_
The right hand side denotes the von Neumann algebra generated by the local
algebras associated to double cones O contained in X. (It thus makes sense to
defineA(X)forarbitraryregionsXbyEqu.(11).)
iv′.) (Weakadditivity).ForeachdoubleconeO dSthereholds
⊂
A(ΛO)=A(dS) =B(F) .
Λ∈S_O0(1,2) (cid:0) (cid:1)
v.) (Time-slice axiom [5]). Let G be a causally complete region and let U be a
neighborhoodofageodesicCauchysurfaceofGsuchthatG=U′′. Then
A(U)=A(G),
wherebothalgebrasaredefinedviaEq.(11). Inparticular,thealgebraofobserv-
ables located withinan arbitrary small time–slice coincideswith thealgebra of
allobservables.
Proof. If O and O aretwo space-like separatedcausallycomplete, open and
1 2
boundedregions,thenthereexistsawedgeW =ΛW suchthat
1
O W and O W′.
1 2
⊂ ⊂
Nowtheinteractingnetinheritswedgeduality
A(W′)=A(W)′
fromtheidentityA (W′)=A (W )′ using(7). Thesefactsimplylocality.
◦ 1 ◦ 1
9
Next,letusprovecovariance. LetΛ SO (1,2)befixed. Byconstruction,
0
∈
thesetofallwedgesequals{ΛW |Λ SO (1,d)}. Thus
1 0
∈
A(ΛO)= A(ΛW)= α A(W)
Λ
ΛO⊂ΛW O⊂W
\ \ (cid:0) (cid:1)
=α A(W) =α A(O) ,
Λ Λ
(cid:16)O\⊂W (cid:17) (cid:0) (cid:1)
provingcovariance.Fortheproofofpropertyiii.)wereferthereaderto[2].
Next,wewillestablishpropertyiv.). Theinclusion
A(X) A(O).
⊃
O⊂X
_
is a consequence of isotony. Moreover, if X is a double cone, then X itself is
among the double cones on the right hand side, so the inclusion automat-
⊂
ically holds. It remains to prove the inclusion if X is a wedge. If X = W ,
1
⊂
thenA(X)coincideswithA (X),forwhichtheadditivitypropertyfortheone-
◦
particlespaceimplies
A (X)= A (RO ),
◦ ◦ I
RR∈I_S⊂OI(+2)
.
whereIisany(arbitrarilysmall)intervalcontainedinI+ =W1 S1,andOI =
∩
I′′. Thus,
A(X)= A(RO ) A(O).
I
⊂
RR∈I_S⊂OI(+2) O_⊂X
Thus,forX=W theinclusion holds. Bycovariance,italsoholdsifXisany
1
⊂
otherwedge.
Propertyiv′.)follows forma similar argument: for eachdouble cone O
⊂
dSthereexistsaLorentztransformationΛ SO (1,2)suchthatO=Λ O for
0 0 0 I
∈
someopenintervalI S1. Now
⊂
A ΛO = A ΛΛ O
0 I
Λ∈S_O0(2,1) (cid:0) (cid:1) Λ∈S_O0(2,1) (cid:0) (cid:1)
= A ΛO A RO
I I
⊃
Λ∈S_O0(2,1) (cid:0) (cid:1) R∈S_O(2) (cid:0) (cid:1)
= A RO =B(F).
◦ I
R∈S_O(2) (cid:0) (cid:1)
Again, the last equality relies on the additivity property for the one-particle
space.Hence,propertyiv′.)isestablished.
10