The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

The Relation b etween KMSStates

for Dierent Temp eratures

Christian D Jakel

Vienna Preprint ESI March

Supp orted by Federal Ministry of Science and Transp ort Austria

Available via httpwwwesiacat

Preprint March

The Relation b etween KMSStates

for Dierent Temp eratures

CHRISTIAN D JAKEL

Inst f Theoretische Physik Universitat Wien Boltzmanng A Wien Austria

email cjaekelesiacat

Abstract Some years ago Buchholz and Junglas BJu established the existence of thermal equilibrium states

KMSstates for a large class of quantum eld theories The KMSstates were constructed as limit p oints of

states which represent strictly lo calized excitations of the vacuum We adjust their metho d such that it resp ects

the general structure of thermal quantum eld theory Ja b In a rst step we construct states which lo ok like

4

KMSstates for a new temp erature in a lo cal region O IR but coincide with the given KMSstate in the spatial



complement of a slightly larger region O The existence of limit p oints of these nets of states which are global

KMSstates for the new temp erature dep ends crucially on the surface energy contained in the layer in b etween

the b oundaries of O and O Intro ducing an auxiliary structure and applying a generalized cluster theorem



we can controll the surface eects in all thermal theories with a physically sensible numb er of lo cal degrees of

freedom Ie given a thermal eld theory that corresp onds to a certain temp erature our metho d shows that it

is p ossible to construct the theory at any p ositive nite temp erature provided the numb er of lo cal degrees of

freedom is restricted in a physically sensible manner

Mathematics Sub ject Classications T

Intro duction

Algebraic quantum statistical mechanics starts from a C algebra A and a strongly con

tinuous oneparameter group of automorphisms f g see eg BRERSeTh is

I

t t R

called the timeevolution and the pair A is called a C dynamical system Quantum

ergo dic theory can b e set up in this framework and it is fascinating to study the mixing

prop erties of a physical system sp ecied by the pair A see eg Ja Equilibrium

states are characterized by their timeevolution invariance and stability against small p er

turbations or adiabatic p erturbations NT of the timeevolution HKTP Adding a

few technical assumptions this characterisation of an equilibrium state leads to a sharp

mathematical criterion HHW named for Kub o K Martin and Schwinger MS A state

I

over A is called a KMS state for some R fg if

a b ba

i

for all a b in a norm dense invariant subalgebra of A A Here A denotes the set

of analytic elements for We note that there are C dynamical systems A where a

I

KMS state exists at one and only one value R BR But if a mo del reects

the basic elements of physical reality one exp ects for high temp eratures and low densities

that the set of KMSstates contains a unique element but at lower temp erature it should

For nonrelativistic fermions with pairinteraction see Ja

C D Jakel

contain many disjoint extremal elements and their convex combinations corresp onding to

various thermo dynamic phases and their p ossible mixtures As is well known an arbitrary

KMSstate can b e represented in a unique manner as a convex sup erp osition of extremal

KMSstates and therefore the symmetry or lack of symmetry of the extremal KMSstates

is automatically determined One can say that by this mechanism sp ontaneous symmetry

breaking o ccures naturally if we start from a unique KMSstate for some high temp erature

and then co ol down the physical system

Nonrelativistic quantum eld theories and spin systems t nicely into this framework

The latter are quite p opular and in low dimensions they have b een worked out in great

detail see eg BR But only recently the b enets of formulating thermal quantum

eld theory TQFT in the algebraic framework were emphasized in a series of recent

pap ers BJu BB NJa abc Let us briey outline how a thermal quantum

eld theory ts into the framework of algebraic quantum statistical mechanics Assume

a QFT is formulated in terms of quantum elds ie op erator valued distributions x

I

over Minkowski space R In a rst step prop osed by Haag and Kastler way back in the

sixties HHK the given QFT is casted into a net

I

O AO O R

of abstract C algebras The algebra of lo cal observables AO asso ciated with a space

I

time region O R may b e thought of as b eing generated by b ounded functions of the

underlying quantum elds currents etc BoY If x is any such eld and if f x is any

real test function with supp ort in a b ounded region O of spacetime then the corresp onding

unitary op erator

Z

a exp i dx f xx

would b e a typical element of AO In general the Hermitian elements of the algebra AO

are interpreted as observables which can b e measured at times and lo cations in O The

C inductive limit of the lo cal algebras AO is denoted by A The assumption that the

timeevolution is given by a strongly continuous automorphism A A do es not imp ose

any essential restrictions of generality By restricting the algebra A see eg S Prop

to a lo cal net which complies with the continuity condition one can always pro ceed

Once we have asso ciated a C dynamical system A with a given QFT we know how to

pro ceed Assume for a moment that the C dynamical system A has a unique vacuum

state and that it shows decent phase space prop erties in the vacuum representation Then

we can apply the metho d of Buchholz and Junglas BJu and construct KMSstates

But no matter how we get it given a KMSstate the GNSrepresentation H

y

provides us with a lo cal thermal eld theory

I

O R O AO O R

y

For the Lagrangian formulation of a TQFT we refer the reader to the b o ok by Umezawa U and the

excellent review article by Landsman and van Weert LvW Recent work in the Wightman framework can b e found in BB St

The relation b etween KMSstates for dierent temp eratures

Under fairly general circumstances dierent values of the inverse temp erature lead to

unitary inequivalent GNSrepresentations TBR Thus thermal eld theories for

dierent temp eratures are frequently treated as completely disjoint ob jects even if they

refer to the same vacuum theory ie even if they show identical interactions on the mi

croscopic level Note that phase transitions may o ccure while we change the temp erature

Consequently there is no simple prescription for connecting the KMS states

for dierent s cf BR p One exception is well known Assume that can b e

approximated by a net of inner automorphisms such that for a A xed

iz h iz h

 

lim k a e ae k h h A

z

I

uniformly in z on compact subsets of C If A has a KMS state at some then

the net of states

0 0

1 1

h h

 

2 2

ae e

a A a

0

h



e

a a A are has convergent subsequences and the limit p oints a lim

i

i

KMSstates Pe In general the situation is more involved

For instance the class of mo dels of a countable numb er of free scalar particles prop osed by

Hagedorn Ha provides us with quantum eld theories which ob ey all the Wightman and

HaagKastler axioms but in which no equilibrium states exist ab ove a certain temp erature

BJu Nevertheless if the quantum eld theory shows decent phasespace prop erties

in the vacuum representation then KMSstates exist for all BJu and it seems

highly desirable to understand the relations b etween the corresp onding disjoint thermal

eld theories

Let us sp ell out our result Starting form a given thermal eld theory O R which

corresp onds to a certain temp erature we show that it is p ossible to construct the

thermal eld theory

0

I

O R R

at any p ositive nite temp erature provided the numb er of lo cal degrees of freedom

of the given thermal eld theory is restricted in a physically sensible manner The metho d

we use is essentially due to Buchholz and Junglas BJu Although we almost rep eat

their line of arguments there are some nontrivial deviations due to the mathematical

structure one encounters in thermal eld theory In a rst step we construct pro duct

states O O which up to b oundary eects lo ok like KMSstates for the

I

new temp erature in a lo cal region O R but coincide with the given KMSstate

in the spatial complement of a slightly larger region O ie

ab a b a AO b AO

At this p oint our metho d is semiconstructive It do es not uniquely x the pro duct

states We exp ect that dierent choices will manifest themself in dierent

C D Jakel

exp ectation values for observables lo calized in b etween the two regions O and O ie we

exp ect

a AO O such that a a

Intuitively the choice of a pro duct state corresp onds to a choice of the b oundary

conditions which decouple the lo cal region O where the state already lo oks like an equi

librium state for the new temp erature from the outside O ie the spatial complement

of O Phase transitions are not excluded by our metho d by cho osing dierent b oundary

conditions we may encounter disjoint KMSstates for the new temp erature in the ther

mo dynamic limit Whether or not the thermo dynamic limit exists at all ie whether or

not the net of states has convergent subsequences dep ends on the surface energy

contained in b etween the two regions O and O At this p oint it is favourable to intro duce

an auxiliary structure which can b e understo o d as a lo cal purication Ja b Applying a

generalized cluster theorem Ja c we can than controll the surface eects in all thermal

y

theories with a physically sensible numb er of lo cal degrees of freedom Ie there exists a

i

i

sequence O O such that the limit p oints

i

0

a a A a lim

i

i

are KMS states for the new inverse temp erature Lo osely sp eaking we

have found a metho d to heat up or co ol down a quantum eld theory Before we pro ceed let

us mention a p ossible application Recall that there are quantum eld theories in which

two disjoint vacuum states exist while the thermal state is unique ab ove some critical

temp erature To start from one vacuum and construct the other by simply combining the

metho d provided by Buchholz and Junglas to heat up the system with our metho d to co ol

down the system sounds like a challenging task for the future

We conclude the intro duction with a list of our assumptions

i Thermal QFT A thermal QFT is sp ecied by a von Neumann algebra R acting on

a Hilb ert space H together with a net

I

Net structure O R O O R

of subalgebras asso ciated with op en b ounded spacetime regions O in Minkowski space

More precisely we assume that the net O R O satises

Isotony R O R O O O

y

Such theories have decent phasespace prop erties which can b e expressed by certain nuclearity conditions

BWBDAL abBY

We have simplied the notation here a little bit In fact we will have to adjust the relative sizes of a triple

(i)

(i) (i)

O O O of spacetime regions

i 

The relation b etween KMSstates for dierent temp eratures

and

Lo cality R O R O O O

As b efore O denotes the spacelike complement of O

ii Dynamical law The timeevolution is given by a strongly continuous oneparameter

group of unitary op erators

iH t

I

fe t Rg

The timeevolution acts geometrically ie

iH t iH t

I

e R O e R O te t R

Here e is a unitvector in the timedirection in a xed Lorentzframe

iii Unique KMSvector We assume that there exists a distinguished vector some

times called the thermal vacuum vector cyclic and separating for R such that

the asso ciated vector state satises the KMScondition Restricting at

tention to pure phases we assume that is the unique up to a phase normalized

eigenvector with eigenvalue fg of H

iv ReehSchlieder prop erty We assume that is cyclic and separating for the lo cal

I

algebras R O whenever the spacelike complement of O R is not empty

Remark The ReehSchlieder prop erty is a consequence Ja b of additivity and the rel

ativistic KMScondition prop osed by Bros and Buchholz BB If the KMSstate is

lo cally normal wrt the vacuum representation then the standard KMScondition to

gether with additivity of the net in the vacuum representation is also sucient to ensure

the ReehSchlieder prop erty of the KMSvector J

v Nuclearity condition We assume that the lo cal algebras R O satisfy the following

nuclearity condition The maps R O H given by

O

A A

O

are nuclear for In addition we assume that for large diameters r of O the

following b ound holds for the nuclear norm for and

d m m

cr

k k e

O

where c m d are p ositive constants We exp ect that the constant d in this b ound can b e

put equal to the dimension of space in realistic theories But we do not make such an

assumption here The constant m will dep end on the TQFT under consideration

The net O R O is called additive if O O R O R O

i i i i

C D Jakel

In order to controll the thermo dynamic limit we have to sharp en the nuclearity con

R O H given by dition We assume that the map

O

jH j

A e A A

O

m d

is nuclear and satises for large in comparison with r the following b ound on the

nuclear norm

d m

k const r k

O

m

k comes from taking the limit large in comparison The b ound on the nuclear norm k

O

d

with r in the expression

d m

cr

e

where the one comes from the subtraction of the thermal exp ectation value

vi Regularity from the outside We assume that the net of lo cal algebras of observables

R O is regular from the outside ie

I

R O R O C l O O

Remark If the KMSstate is lo cally normal wrt vacuum representation then it is su

cient to assume that

I

RO RO C l O O

holds in the vacuum representation For the free eld this prop erty was shown by Araki

As a consequence of the assumption i one and ii we can asso ciate the following

mo dular structure with a given thermal QFT The p olar decomp osition S J of the

closeable op erator

S A A A R

provides us with a conjugatelinear isometric mapping J from H onto H and a p ositive

selfadjoint in general unb ounded but densely dened and invertible op erator acting

on H satisfying the conditions J l and

J A A A R

J and are called the mo dular ob jects asso ciated to the pair R The mo dular

conjugation J induces an antiisomorphism j A JA J b etween the algebra of quasi

lo cal observables R and its commutant Tomitas theorem The opp osite net

I

O j R O O R

The relation b etween KMSstates for dierent temp eratures

is

I

provides a p erfect mirror image of the net of observables The unitary op erator s R

implements a automorphism R R

s

is is

I

A A s R

s

is called the mo dular automorphism Takesaki has shown that is a KMS

s

state Moreover is uniquely determined by this condition We conclude that in a TQFT

the mo dular automorphism coincides up to a scaling factor with the timeevolution

Consequently the mo dular automorphism resp ects the net structure ie

I

R O R O s e s R

s

I

The real parameter R app earing until now was just a dummy index in

distinguishes a length scale called the thermal wavelength Let us turn the argument

up side down given a thermal eld theory R sp ecied by the net O R O it

is not necessary to provide an explicit expression for the eective Hamiltonian H It is

already uniquely sp ecied by the pair R By Stones theorem there exists a unique

selfadjoint generator H called the eective Hamiltonian such that

H

e

Mo dular theory implies that for the op erator H is not semib ounded from

b elow but its sp ectrum is symmetric and consists typically of the whole real line A

tBW

In general the mo dular automorphism t will not b e strongly continuous eval

t

uated p ointwise on the elements of R Thus R will not form a C dynamical sys

tem Nevertheless by a suitable smo othening pro ceedure we can asso ciate an abstract

C dynamical system A together with a net

I

O AO O R

of subalgebras of A to the given thermal eld theory O R O More precisely once

again we refer the reader to S there exists

i a C algebra A and a representation A B H such that A is a

weakly dense C subalgebra of R

AO is a weakly dense ii a net O AO of C subalgebras of A such that

I

C subalgebra of R O O R

iii a strongly continuous automorphism A A of A ie

lim k a ak a A

t

t

such that a a for all a A

t t

C D Jakel

Moreover the net O AO satises isotony and lo cality and resp ects the lo cal structure

of the net O AO ie

I

AO AO te t R

t

Note that AO will always denote the algebra generated by the set fa A a b

AO in B H b AO g and not the commutant of

Remark From the viewp oint of algebraic quantum statistical mechanics it is more naturall

to start from a dynamical system A and then characterize equilibrium states and

thermal representations with resp ect to the dynamics But here we want to assume

that we are given a thermal eld theory O R O Thus we have to reconstruct the

C dynamical system A from the W dynamical system R

Doubling the Degrees of Freedom

Our rst step can b e understo o d as a lo cal purication Assume there exists a such

that

O te O jtj

Recently it has b een shown that as a consequence of the nuclearity condition the

split prop erty holds for the net of von Neumann algebras O R O Thus ensures

that there exists a typ e I factor N such that

R O N R O

Moreover there exists a pro duct vector H cyclic and separating for R O R O

p

such that

AB A B

p p

for all A R O and B R O Ja b The pro duct vector can b e utilized to dene

p

a linear op erator V H H H by linear extension from

V AB A B

p

where A R O and B R O The unitary op erator V maps the typ e I factor N

onto B H l ie

V N V B H l

Moreover R O N and R O N implies

V R O V R O l and V R O V l R O

The relation b etween KMSstates for dierent temp eratures

and the inclusion j R O R O implies that the von Neumann algebra

M O R O j R O

is naturally isomorphic to the tensor pro duct of R O and j R O as long as the

complement of O is not empty The denition of V implies that as a consequence

of the ReehSchlieder prop erty of the pro duct vector is cyclic and separating

p

for M O

The algebras R O and j R O are weakly statistically indep endent ie

A R O and B R O implies AB Schlieder prop erty Ja b In this

sense one can sp eak of a doubling of degrees of freedom The elements of M O will in

general neither b elong to R nor R In fact they will not show analyticity prop erties

with resp ect to Recall that the analyticity prop erties enco ded in the KMScondition

simply reect the basic stability and passivity prop erties of an equilibrium state Thus it

seems that the essence of a thermal eld theory gets lost when we double the degrees of

freedom and consider the net

I

O M O O R

instead of the net of observables quantities O R O But due to the tensor pro duct

structure of M O we can recover certain analyticity prop erties with resp ect to the

pro duct vector as long as O is b ounded To do so we simply have to intro duce

p

an auxiliary oneparameter group of unitary op erators which resp ects the natural tensor

pro duct structure of M O

Denition Given a pro duct vector H cyclic and separating for R O R O

p

is

we dene a oneparameter group of unitary op erators H H and an antiunitary

op erator J H H by linear extension from

p

is is is

I

s R AB V A B

p

p

and resp ectively

JA JB J AB V

p p

for all A R O and B R O

Assume there exists a spacetime region O and some such that

O te O jtj

is

For a AO and js j the group of unitary op erators coincides up to

p

rescaling with the timeevolution

is is

a a js j

s

p p

C D Jakel

is

Moreover for jsj suciently small resp ects the lo cal structure of M O ie

p

is is

M O M O s e js j

p p

Stones theorem implies that there exists a unique selfadjoint op erator K such that

K

e

p

For p we have

p max p p

and the sp ectral resolution of the p ositive op erator implies

p

M O D

p

p

By denition J l J K and

p p p p

p

A B J AB V

p p

p

A B AB

p p

for all A R O and B j R O Nevertheless J and are not the mo dular

p p

ob jects asso ciated to the pair R O j R O

p

Lemma If the nuclearity conditions and hold for the algebra R O then

the maps M O H

O

M M

p

p

M O H and

O

jK j

M e M M

p p p

are nuclear The nuclear norm of can b e estimated by

O

d m m

cr

c m d k k e

O

where r denotes the diameter of O and c m d are the constants app earing in the nuclearity

condition for R O

Proof Let A R O and B j R O By denition

A B AB V

O

and the maps A A and B B are nuclear for The tensor

pro duct of two nuclear maps is again a nuclear map and the norm is b ounded by the

pro duct of the nuclear norms P tu

The relation b etween KMSstates for dierent temp eratures

Recently it was recognized that the nuclearity condition imp oses uniform b ounds

on the correlations b etween elements in M O and M O in the KMSstate

Note Ja c Quantitative information can b e extracted from the nuclear norm of

O



that in once again we have subtracted the unique exp ectation values for the discrete

O



eigenvalue fg of K

Theorem Let O O and O denote three spacetime regions as sp ecied in

and Consider two families of op erators M M O and N M O j

j j

f j g Assume is large enough such that there exists some n IN with

m+1 m+1

m m

n n

If there exist p ositive constants c d and m such that the nuclear norm of is b ounded

O



by

d m

k cr k

O



then

j j

 

X X

d m m

M M N M N cr e n

j j j j j

j j

2

m

m+1

Remark For large compared to the correlations decrease like Note that

the thermal wavelength sets a natural length scale in our units

Lo calized Excitations of a KMSstate

Taking the auxiliary structure develop ed in the previous section into account we can now

adapt the metho d of Buchholz and Junglas to a thermal representation Let O and O

denote two spacetime regions as sp ecied in From the split inclusions

R O N R O and j R O j N j R O

such that we infer that there exists a typ e I factor N j N

M O M O N j N

All innite typ e I factors with innite commutant on the separable Hilb ert space H are

unitarily equivalent to B H l KR Chapter Thus there exists a unitary op erator

W H H H such that

N j N W B H l W

C D Jakel

From M O N j N and M O N j N we infer

W M O W M O l and W M O W l M O

is cyclic and separating for M O and is cyclic and separating for M O Thus

p

is cyclic and separating for M O M O and the split inclusion is

p

standard DL We conclude that there exists a unique choice for W which maps the p ositive

natural cone P M O M O onto the p ositive natural cone P M O

M O DL

p

Now consider and as two normal states over

p p p

M O and M O resp ectively Set

C WCW C M O M O

p p

is normal and satises

p

MN M N

p p

for all M M O and N M O In the presence of a separating vector each normal

state is a vector state KR and there exists a unique vector in the natural p ositive

cone P M O M O such that

M N MN M N

p p p

for all M M O and N M O BR Insp ecting the relevant natural

p ositive cones we infer

W M N M N

p

for all M M O and N M O

Denition Lo calized excitations of the KMSstate Let O O and O denote three

spacetime regions as sp ecied in and ie there exist some such that

O te O jtj and O te O jtj

We dene the Hilb ert space H H O O O of lo calized excitations of the

KMSstate

M O H

The pro jection onto H is denoted by E Recall that M O denotes the von Neumann

y

algebra generated by R O and j R O and denotes the unique pro duct vector

in the natural p ositive cone P M O M O satisfying

M N M N

p p

for all M M O and N M O Finally denotes the the unique pro duct vector

p

in the natural p ositive cone P R O R O satisfying

AB A B

p p

for all A R O and B R O

y

Fixing the pro duct vector with resp ect to some natural cone is mathematically convenient but not all

necessary In fact we exp ect that dierent b oundary conditions are realized by dierent choices of In the

thermo dynamic limit the dierent choices of b oundaries might lead to dierent phases

The relation b etween KMSstates for dierent temp eratures

Recall that W is unitary and fullls

WMNW M N

for M M O and N M O Using the isometry W we can write

M O W H H W

p p

and E W l P W

p

Propsition Given a triple O O O of spacetime regions as sp ecied in

we nd

i H is invariant under the application of elements from M O ie

M O H H

ii The vectors of H induce pro duct states on M O M O Let H

then

MN M N

p p

for all M M O and N M O

iii The vectors states asso ciated with H represent strictly lo calized excitations of

the KMSstate ie they coincide with in the spacelike complement of O

Let H then

a a a AO

Note that AO denotes the commutant of AO in A

iv H is complete in the following sense to every normal state on M O there

exists a H such that

A A

for all A M O

Proof We simply adapt the pro of of the corresp onding result by Buchholz and Junglas to

our situation

i follows from the denition

ii follows from and

iii follows from and

iv Since M O has a cyclic and separating vector there exists a vector H which

H induces the given normal state on M O By denition W

p

satises iv

tu

C D Jakel

Now we show that the restriction of the op erator to the subspace H is of trace

p

class for

Propsition Let O O O b e a triple of spacetime regions as sp ecied in

The op erator E acting on the Hilb ert space H is of traceclass and

p

k E k k k

O

p

where is the nuclear map dened in

O

Proof The pro of of this prop osition is more or less identical to the one given by Buchholz

and Junglas BJu in the vacuum case

i The rst step is to construct a convenient orthonormal basis of H Let f g b e

i iIN

an orthonormal basis of H with Set

U W M l W

ij ij

where M B H are matrix units given by

ij

M H

ij j i

M O we infer from that U N j N Since W B H l W N j N

ij

Furthermore we nd

N

X

U l U U U U s lim U

ii ji ij k l jk il

ij

N

i

Set

U W

i i p

fU g is the desired orthonormal basis of H Intro ducing an isometry S N

i iIN

M O by j N

M O N SN N N j N

p

we can represent this orthonormal basis by vectors

U U S

i i i p

where U S N j N M O By construction

i

D i IN

i

p

Esp ecially D for all

p

The relation b etween KMSstates for dierent temp eratures

ii By p olar decomp osition of the closeable op erator E we get

p

E F j E j

p p

where F is a partial isometry with range in H Intro ducing a set of linear functionals

i

which can b e chosen to b e continuous with resp ect to the ultraweakly top ology induced

by M O BDAF b and vectors H corresp onding to the nuclear map we

i O

obtain

X

U S F U S Tr j E j

i p i p

p p

i

X

U S F U S

i p O i

i

X X

U S U S F

n i i p n

n

i

X X

j U S j kU F k

n i i n

n

i

Buchholz and Junglas have shown the following inequality BJu

X

j U j kU k k k kk

i i

i

which holds for any vector H and any ultraweakly continuous linear functional

on M O Consequently

X

E j k k k k Tr j

n n

p

n

Taking the inmum with resp ect to all decomp ositions of the resp ective nuclear maps we

nd

Tr j E j k k

O

p

tu

We will now give a precise meaning to the statement that a normal state with resp ect

to the given thermal representation satises a lo cal KMScondition for a new temp era

ture in some b ounded region O Note that we can reach any p ositive temp erature

T simply by decomp osing into some and some minimal n IN

such that n

C D Jakel

Denition Let and let n IN b e the smallest natural numb er such that

n for some

A normal state satises the lo cal KMScondition at inverse temp erature in some

I

b ounded spacetime region U R i for any subregion U U whose closure is contained

in the interior of U there exists a function F for every pair of op erators a b AU

ab

such that

i F is dened on

ab

I

G fz C z n gn

n

I

n fz C jz j t z k k n g

ii F is b ounded and analytic in its domain of denition

ab

iii F is continuous for z k and z k k n

ab

iv F is continuous at the b oundary for z and z n

ab

I

v For t R the resp ective b oundary values are

F t a b

jtj t

ab t

for

jtj t

ba F t in

t ab

Remark To heat up the system lo cally is quite simple For we nd n ie

I

no cuts app ear in G fz C z g The real problem is to co ol down the

system lo cally One needs at least n cuts where n is the minimal natural numb er such

that n Whether or not it is useful to op erate with more cuts then

necessary is unknown to us

Prop osition allows us to dene lo cal quasiGibbs states which are lo cal

KMS states for the temp erature T in the interior of O and KMS states for

the temp erature outside of O

Propsition Let O O O b e a triple of spacetime regions as sp ecied in

Let n IN b e the minimal natural numb er such that n Set for

n IN and xed

n

E E

p

and a Tr a a A

n

Tr E

p

Then is a density matrix ie and Tr and the following statements hold

true

The relation b etween KMSstates for dierent temp eratures

i The states are pro duct states which coincide with the given KMSstate outside

of O ie

ab a b

for all a AO b AO

ii The states are lo cal n KMSstates for the spacetime region O

AO thus Proof i Let a AO and b AO Then E M O

E a and E W l P W implies

p

E b E b E b AO

Using the cyclicity of the trace we nd

n

aE b E Tr E E

p

ab

n

E Tr

p

a b

ii Consider the case n Let and let O b e an op en spacetime region such

that O te O for all jT j Let a b AO By assumption a b AO

t

for all jtj We set

iz iz

Tr aE b E E

p p

p

F z

ab

E Tr

p

for z The function F z is analytic in its domain and continuous at the

ab

it it

b oundary We recall that b b AO for jtj Using

p p

t

we concude that AO once again the cyclicity of the trace and E

E E b Tr aE

z

p p

z lim F

ab

z

E Tr

p

a b E E Tr

z

p

Tr E

p

a b jz j

z

On the other hand for jz j

E b E Tr aE

z

p p

lim F z

ab

z

E Tr

p

Tr aE E b E

z

p p

Tr E

p

C D Jakel

For z we set

iz iz

E b E Tr aE

p p

p

F z

ab

E Tr

p

The function F z is analytic in its domain and continuous at the b oundary By deni

ab

tion

Tr aE E b E

z

p p

lim F z

ab

z

Tr E

p

lim F z jz j

ab

z

Furthermore F satises the b oundary condition

ab

E Tr aE b E

z

p p

z lim F

ab

z

Tr E

p

E Tr a E b

z

p

Tr E

p

ba jzj

z

Using the EdgeoftheWedge theorem SW one concludes that F and F are the

ab ab

restrictions to the upp er resp lower half of the double cut strip

I I

G fz C z g n fz C jz j z g

of a function

F z

z

ab

F z for

ab

z

F z

ab

dened and analytic for z G From and we infer

ba jtj a b F t i F t

t t ab ab

Analogous results for arbitrary n IN can b e established by the same line of arguments

but with considerable more eort tu

The relation b etween KMSstates for dierent temp eratures

The Thermo dynamic Limit

The unit ball in A is weak compact thus for every net of states

O O O n xed

there exists a subsequence f g converging to some state Whether or not this state

iIN

i

is a n KMS state dep ends on the energy contained in the b oundary ie the choice

of the relative size of O O and O The necessary quantitative information restricting the

surface energy comes from the b ounds on the nuclear norm of the maps and

O

O

Before we analyse the surface energy in detail we intro duced subalgebras A of almost

p

lo cal elelements in A which are analytic with resp ect to the timetranslations BJu

For the existence of the these subsalgebras it is crucial that the timeevolution automor

phism A A is strongly continuous

Lemma Buchholz and Junglas Let p IN b e xed and let A A b e the algebra

p

generated by all nite sums and pro ducts of op erators of the form

Z

af dt f t a

t

where f is any one of the functions

2p

tw

f t const e

I

with w C and a AO any lo cal op erator Then

O

i Each b A is an analytic element with resp ect to ie the op eratorvalued

p

I

function t b can b e extended to a holomorphic function on C

t

ii Each b A is almost lo cal in the sense that for any r there is a lo cal

p i

i

op erator b AO such that

i

2p

r

i

kb bk const e

i

where the constant do es not dep end on r

i

I

iii The algebra A is invariant under z C and norm dense in A

p z

i

i i

Let O O O b e a sequence of triples of double cones with diame

i

ters r r r We can now exploit the fact that the elements of A p IN have go o d

i i i p

lo calization prop erties More precisely we will show that there exists some p IN such

that

a b A a b ba

i p in

i i

C D Jakel

with as i Note that we will have to controll the surface energy by adjusting

i

the relative size of r r and r Insp ecting the denition of we recognize that in

i i i

i

order to prove it is sucient to controll

k n b E Tr a

ik

i i

Let a b A p IN xed Then also

p

b A k n

ik p

Let us consider the case n Since a b c b and d b are almost

i i

i

lo calized in O they almost commute with E Thus for example

i

a b E E Tr

i

i i

pi

Tr a b E

i

i

E Tr

i

pi

k b d E k Tr j E j kak

i i

i i

pi

E Tr

i

pi

kak

k b d k Tr j E j

i i

pi i

E Tr

i

pi

2p

d

C

C r C r

2 3

i

i

e e

E Tr

i

pi

i

for certain p ositive constants c c and c Here d AO denotes the lo cal approxi

i

mation of d b A such that E b In the last inequality we made use of

i p i

i

Prop osition the b ound on the nuclear norm of the map and the second part

O

of Lemma We conclude that the numerator in vanishes in the thermo dynamic

limit if we set p d and

i IN r r

i

i

We will now try to controll the denominator In order to ensure that for and

n IN xed the quasipartition function

n

i

i i

O O O Z n Tr E E

i

i i i

pi

i

is b ounded from b elow in the thermo dynamic limit it is necessary that O grows rapidely

i

with O Otherwise the energy contained in the b oundary which is necessary to decouple

the lo cal region from the outside lessens the eigenvalues of E E so drastically that

i i

pi

it might outrun the increase in the numb er of states contributing to the trace by enlarg

i

ing O Note that this can not happ en if H tends to the whole Hilb ert space H ie

i

s lim E l

i i

The relation b etween KMSstates for dierent temp eratures

Lemma Let j N b e any set of mutually orthogonal and normalized

j

vectors Then s lim E l implies

i

i

N

X

E E E lim inf Tr E E lim inf

j j

i pi i pi i i pi i

i i

j

N

X

C lim inf

j j

pi

i

j

Proof Note that is by denition see a p ositive op erator and is the unique

pi

pi

eigenvector of K for the discrete eigenvalue fO g tu

Thus following corollary tells us that it is sucient to controll the distance b etween

k k as i go es to innity

i

Corollary Buchholz and Junglas For an increasing sequence such that

i

k k i

i

l tends to the whole Hilb ert space H ie s lim E H

i

i i

Proof Since W converges to the unitary op erators W see

i pi i

i

fulll

H W

pi

i

I

is the pro jection onto C W where P W l P as i Recall that E

pi i

pi pi i

i

Therefore

W W l P W E

pi

i pi i i

H W

pi

i

We conclude that s lim E l tu

i

i

i

i

If we cho ose the vectors in the p ositive natural cone P M O M O

i

then we know from a result of Araki A that

C C k k sup

i i i i i

kC k

i

where the supremum has to b e evaluated over all

i

i

C M O M O

i

C D Jakel

By denition see and

M N M N

i i pi pi

i

i i

for all M M O and N M O Note that for N M O we nd

N N

pi pi

i

i

Let denote the pro duct vector in the p ositive natural cone P M O M O

i

such that

M N M N

i i

i

i

for all M M O and N M O Theorem provides us with the following

i

estimate

j j

 

2

X X

m

m+1

M N const r r M N M N

j j i i i j j i j j

j j

for r r large compared to Thus by the result of Araki cited ab ove

i i

k k r r

i i i

i i

But sofar there was no restriction on the relative size of the regions O and O We

can exploit this freedom if the net lo cal algebras of observables R O is regular from the

outside One easily shows that this implies

I

M O M O C l O O

Thus we can once again use the result of Araki A concerning the distance of vectors

i

i

in the p ostive natrual cone P M O M O to prove

i i

k k O O

i i

The details are as follows

Theorem Assume that the net of algebras of observables O R O is regular

from the outside Furthermore assume there exist p ositve constants m c d such

that the nuclear norm of is b ounded by

O

d m

k cr m c d k

O

with Then there exists a sequence of triples of spacetime regions

i

iIN

r r m m

i

i

and

r r i

i i

suciently fast such that

lim k k

i

i

and consequently

n

Z n Tr E E C i IN

i i pi i

The relation b etween KMSstates for dierent temp eratures

i

i

Proof Consider two sequences of double cones fO g and fO g eventually ex

iIN iIN

ik

I

hausting all of R For each i N xed we consider a sequence of double cones fO g

k IN

such that

ik i

O O k

In order to ease the notation we set

i

i ik

A M O B M O C M O

i i ik

and

i i

i ik

C M O M O D M O M O

i ik

i

Let denotes the unique pro duct vector in the natural p ositive cone P R O

pik

ik

R O satisfying

AB A B

pik pik

i ik

for all A R O and B R O If we cho ose pro duct vectors and such

ik i

that

M N M N

ik ik pik pik

and

M N M N

i i

for all M A N B in the natural cone P C then we know from the result of

i i

i

Araki A that

k k sup C C

ik i ik i ik i i i

C C kC k

i i i

Let i b e xed and consider a sequence fB g of algebras B B such that

k k IN k k

I

B A C l

k k

i

Then there exists some k IN such that note that k k as i

i i

sup C C k k k k

ik ik i i i i

C C kC k

i

Otherwise for each i IN xed there would exist a sequence fD D kD k g

ik ik ik k IN

such that

D D lim k k

ik ik ik i ik i i

k

The linear functional is ultraweakly continuous on the von Neu

ik ik i i

mann algebra C We conclude that the sequence fD D kD k g has a

i ik ik ik k IN

weak limit p oint D

i

w lim D D D C D

ik i i i k ik

k

C D Jakel

such that

k k k k D D

i i ik i ik i i i

and some k IN in contradiction to

i

D D k IN D D

ik k ik ik ik ik i ik i

i ik

i

We conclude that if we set O O then

k k k k k k k k i

i i i i

tu

Making use of the restrictions imp osed on the relative size ofr r andr in the previ

i i i

g uos theorem we can now establish the KMSprop erty for all weak limit p oints of f

iIN

i

i

i i

provided the regions O O O tend to the whole spacetime in agreement with

i

our assumptions

i

i i

Theorem Let n IN and b e xed and let O O O b e a

i

sequence of triples of double cones with diameters r r and r

i i i

m m r r

i

i

and

r r i

i i

suciently fast such that

lim k k

i

i

Then every weak limit p oint of the sequence f g is a KMSstate at inverse tem

iIN

i

p erature n

Proof Let us consider the case n The pro of pro ceeds in two steps

i Let denote the limes of a convergent subsequence f g Then for every

iIN

i

we can nd an index i IN such that

a b ba a b ba

i i

i

ii We can now approximate b b and b by lo cal elements in AO and apply

i i

the commutator estimate several times We get for and suitable large i IN

The relation b etween KMSstates for dierent temp eratures

a b ba a b ba

i i

i

a b E E Tr

i

i i

pi

ba

i

E Tr

i

pi

b E Tr aE

i

i i

pi

ba

i

E Tr

i

pi

Tr aE b E E

i

i i i

pi pi

ba

i

E Tr

i

pi

Tr a E bE

i i

pi

ba

i

E Tr

i

pi

b Tr a E E

i i

pi

ba

i

E Tr

i

pi

Thus we have found a norm dense invariant subalgebra A of the set A of analytic

p

elements for where the state satises

a b ba a b A

i p

Consequently is a KMS state Similiar results for arbitrary n IN can b e

established by the same line of arguments but with considerable more eort tu

0 0

Once we have constructed a KMSstate the GNSrepresentation provides

us with a lo cal thermal eld theory

0 0

I

O R O AO O R

0 0

Note that there might b e several with GNSvector acting on a Hilb ert space H

extremal n KMSstates which might lead to unitary inequivalent representations

ie disjoint thermal eld theories

Acknow ledgments For critical reading of the manuscript many thanks are due to M Muger

and D Schlingemann Kind hospitality of the I I Institute for theoretical physics Univer

sity of Hamburg the Institute for theoretical physics University of Vienna and the Erwin

Schrodinger Institute ESI Vienna are greatly acknowleged The present work started

in collab oration with D Buchholz The present formulation is strongly infuenced by his

constructive criticism and by several substantial hints from him

C D Jakel

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