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Passive States and Essential Observers in Algebraic Quantum Field Theory

Permanent Link: http://ufdc.ufl.edu/UFE0021653/00001

Material Information

Title: Passive States and Essential Observers in Algebraic Quantum Field Theory
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Strich, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We show that if a real Lie group is represented unitarily and faithfully on a Hilbert space H and if a vector phi is fixed by the action of an 'essential' one-parameter subgroup then this vector is stabilized by the whole Lie group action. We exhibit Lie groups having essential one-parameter subgroups and show that compact, semisimple Lie algebras do not have essential elements. Then we show that the 'temperature' of a KMS-state with respect to the dynamics generated by a suitable Lie algebra element is fixed by the commutation relations of this Lie algebra element. Furthermore we investigate a selection criterion for physically significant, fundamental states on quantum systems on 'a priori' general space-times, namely the requirement of a state to be passive with respect to the dynamics of an essential observer. We show that this requirement has strong consequences as for instance the invariance of the state under space-time symmetries and the Reeh-Schlieder property for wedge algebras.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Robert Strich.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Summers, Stephen J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021653:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021653/00001

Material Information

Title: Passive States and Essential Observers in Algebraic Quantum Field Theory
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Strich, Robert
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We show that if a real Lie group is represented unitarily and faithfully on a Hilbert space H and if a vector phi is fixed by the action of an 'essential' one-parameter subgroup then this vector is stabilized by the whole Lie group action. We exhibit Lie groups having essential one-parameter subgroups and show that compact, semisimple Lie algebras do not have essential elements. Then we show that the 'temperature' of a KMS-state with respect to the dynamics generated by a suitable Lie algebra element is fixed by the commutation relations of this Lie algebra element. Furthermore we investigate a selection criterion for physically significant, fundamental states on quantum systems on 'a priori' general space-times, namely the requirement of a state to be passive with respect to the dynamics of an essential observer. We show that this requirement has strong consequences as for instance the invariance of the state under space-time symmetries and the Reeh-Schlieder property for wedge algebras.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Robert Strich.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Summers, Stephen J.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021653:00001


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1104ae1e5147740c3be6955a50cbcef4
efc2fa9de563f4cba90a1487ecf0270d45ecf780







PASSIVE STATES AND ESSENTIAL OBSERVERS IN ALGEBRAIC QUANTUM
FIELD THEORY



















By

ROBERT STRICH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007
































2007 Robert Strich




































To my parents









ACKNOWLEDGMENTS

I would like to thank my adviser Prof. Stephen J. Summers for -.. -I .ii-; the topic

of this dissertation for his patience and for numerous helpful discussions throughout my

work. I would also like to thank Prof. John Klauder, Prof. Scott McCollough, Prof. Paul

Robinson and Prof. Bernard Whiting for agreeing to serve on my supervisory committee.

Finally I thank Kristin Stroth and Karsten Roeseler for proof reading this work at various

stages of its becoming.









TABLE OF CONTENTS


page

ACKNOW LEDGMENTS ................................. 4

LIST OF FIGURES .................................... 6

A B ST R A CT . . . . . . . . .. . 7

CHAPTER

1 INTRODUCTION .................................. 8

2 REPRESENTATIONS OF LIE GROUPS ................. .... 12

2.1 Lie Group Representations and Invariant Vectors .... ..... .... 12
2.2 Real Lie Algebras and Essential Elements ........ ......... 19
2.2.1 Compact Real Lie Algebras and Essential Elements . ... 19
2.2.2 Noncompact Real Lie Algebras and Essential Generators ..... 21

3 ON /-KMS-STATES .................. .............. .. 24

4 APPLICATION TO ALGEBRAIC QUANTUM FIELD THEORY ...... .. .29

4.1 Basic Setup .................. ................ .. 29
4.2 Observers and Wedges .................. .......... .. 30
4.3 States . . . . . . . ..... 31
4.4 Invariance and Reeh-Schlieder Property ............. .. .. .. 33
4.5 Modular Objects and Unruh-Temperature ................. .. 36
4.6 W eak Locality .................. ............... .. 37
4.7 Concrete Examples . . . .............. 39
4.8 Further Examples Using Conformally Covariant Theories . ... 41
4.9 Einstein Static Universe ............... ......... .. 42
4.9.1 Wedges in Einstein Static Universe ................. .. 42
4.9.2 Essential Elements in Einstein Static Universe . . ... 47

REFERENCES ...... ........... .................. .. 48

BIOGRAPHICAL SKETCH ................... . .... 50









LIST OF FIGURES


Figure page

3-1 de Sitter and Anti-de-Sitter space-time .................. .. 25

4-1 Wedges in Minkowski-, de Sitter and Anti-de-Sitter space-time . ... 30

4-2 Spherical caps corresponding to a pair of wedges in S3 ............. .46









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

PASSIVE STATES AND ESSENTIAL OBSERVERS IN ALGEBRAIC QUANTUM
FIELD THEORY

By

Robert Strich

December 2007

('!C iu: S. J. Summers
Major: Mathematics

We show that if a real Lie group is represented unitarily and faithfully on a Hilbert

space -1 and if a vector Q E -1 is fixed by the action of an essential one-parameter

subgroup then this vector is stabilized by the whole Lie group action. We exhibit Lie

groups having essential one-parameter subgroups and show that compact, semisimple Lie

algebras do not have essential elements.

Then we show that the "temp. i ,I ni of a KMS-state with respect to the dynamics

generated by a suitable Lie algebra element is fixed by the commutation relations of this

Lie algebra element.

Furthermore we investigate a selection criterion for physically significant, fundamental

states on quantum systems on a prior general space-times, namely the requirement of a

state to be passive with respect to the dynamics of an essential observer. We show that

this requirement has strong consequences as for instance the invariance of the state under

space-time symmetries and the Reeh-Schlieder property for wedge algebras.









CHAPTER 1
INTRODUCTION

The algebraic formulation of quantum field theory as introduced by Haag and Kastler

(see [12]) provides a model independent mathematically rigorous approach to conceptual

questions in quantum physics. It relies on the assumption that all physical information of

a quantum system is encoded in an assignment


0 A(O) (1-1)


of open space-time regions O to C*-algebras A(O) whose -, lI iioint elements are

regarded as the observables corresponding to measurements that can be done in the

given space-time region O. Within this framework in recent years much work has been

done on one of the key problems one encounters when dealing with quantum field theory

on general space-times, namely how to choose physically relevant, fundamental states for a

quantum system. In Minkowski space-time a vacuum state can be characterized as a state

that admits a unitary representation of the space-time translations fulfilling the spectrum

condition, i.e. the joint spectrum of the four momentum operators lies in the forward

lightcone, and leaving the vacuum state invariant. The isometry group of a generic general

space-time however is trivial or in any case not big enough to contain a counterpart to the

space-time translations in Minkowski space.

One recent approach to this conceptual problem ([4]) uses an observation that was

made by Bisognano and Wichmann in their classical paper [1], namely the fact that the

modular objects associated to the algebras of certain wedge shaped regions in Minkowski

space-time and the vacuum vector act geometrically upon the net of local algebras, if this

net is generated by a (finite component) quantum field fulfilling the Wightman axioms

([12]). The authors of [4] propose to take this geometric action as a selection criterion for

physically relevant states. Their Condition of Geometric Modular Action (CGMA) can be

formulated in any space-time and does not require the presence of space-time symmetries.









One obstacle in their approach is on the other hand that the CGMA is formulated with

respect to a suitable set of space-time regions which has to be chosen carefully. Although

nontrivial examples of space-times with appropriate sets of wedges have been found

([4],[6]), it is not clear as to how applicable their approach is in general.
Another proposal of how to tackle the problem of choosing a fundamental reference

state has been made in [2], [5] and [8]. There the authors show that if one imposes certain

stability conditions on a quantum state for a quantum system on de-Sitter space-time (dS)

or Anti-de-Sitter space-time (AdS), then this has strong consequences for the quantum

system.

One such stability condition is the KMS-condition (see [2] for details) which

characterizes a state that is in thermodynamic equilibrium at a certain "temp, I I .

Another related but more fundamental condition is the condition of pIr- .:,';; which was

first discussed in [23]. This condition is a mathematical formulation of the second law of

thermodynamics which in principle -ivz that there cannot be a perpetuum mobile.

So for instance in [8] the authors show that if a state is passive ([23]) with respect

to the dynamics of a uniformly accelerated observer in AdS, then this observer sees the

given state as an equilibrium state (KMS-state) at a certain fixed temperature, the state

is invariant under the isometry group of AdS and, what is more, one can deduce weak

locality relations among the measurements that the observer in question can perform

in his maximal labor il..,; and the measurements that can be performed in an opposite

laboratory region.

Similar results were shown to hold in de-Sitter space-time ([2]) under slightly different

assumptions, where of course the precise notions of what is meant by maximal labor tl..',;

and opposite have to be adapted to the respective geometries. Also, related work has been

done for the case of Minkowski space-time in [18], but there the author imposes a different

set of assumptions.









In this dissertation we are going to generalize these results to quantum systems on

a prior general space-times. One obstacle in this attempt is the absence of concrete

geometric information such as the Lie algebraic structure of the symmetry group, which is

heavily used in computations in the aforementioned papers.

In chapter 2 we try to overcome this difficulty by introducing a somewhat ad hoc

but useful replacement for the uniformly accelerated observers used in [2] and [8] called

essential observers. These are observers that move along worldlines in our space-time

which are generated by certain essential elements in the Lie algebra of the isometry

group of the space-time, where an essential element is one that generates enough of the

Lie algebra in a certain sense (see Definition 1 for details). One of the main results of

this chapter is then the following (see Corollary 2): If a Lie group is unitarily, faithfully

represented on a Hilbert space -1R and if a vector Q E R- is invariant under a one parameter

subgroup of the Lie group that is generated by an essential Lie algebra element, then the

vector Q is stabilized by the whole group.

This property is later used in physical applications but due to this result it is also

of purely mathematical interest to investigate which Lie groups respectively which Lie

algebras have essential elements. This is done to some extent in chapter 2 as well. We

exhibit a list of (noncompact) real Lie algebras having essential elements and we also

prove that compact semisimple real Lie algebras cannot have essential elements (Theorem

1).

C'! lpter 3 deals with KMS-states. As pointed out before, the stability assumptions

on a state imply as one of rn 'viv consequences a fixed Hawking-Unruh temperature (see

[27],[11] for details) that a uniformly accelerated observer (in dS or AdS) finds that state

in. We show that in our general space-time approach under rather general assumptions

this temperature is directly related to certain structure constants of the Lie algebra of the

symmetry group in question.









In chapter 4 we then finally provide the basic setup of algebraic quantum field theory

and show how the results of the previous two chapters can be used to obtain among other

things the following result: If a state is passive for an essential observer, then under mild

assumptions

* the state is invariant under the action of the whole isometry group;

* the state has (a version of) the Reeh-Schlieder property, i.e. it is cyclic for the
algebras of a particular set of space-time regions (wedges);

* the state satisfies (a version of) the modular covariance condition, i.e. the modular
automorphism groups for the corresponding wedge algebra and the given state acts
geometrically upon the net of local algebras;

* certain weak locality results can be deduced, i.e. one can exhibit pairs of space-time
regions W1 and W2 such that if A E A(Wi) and B E A(W2) then

w(AB) = w(BA)

for our given state w.

Furthermore we extend the results to conformally invariant theories on certain

Robertson-Walker space-times. These are Lorentzian warped products that are topologically

equivalent to R x S3.

Finally we give a selection of results concerning the case of the Einstein cylindrical

space-time. We show that in at least three space-time dimensions Einstein space-time

contains nontrivial wedges (according to our definition of a wedge). Similar to the

Minkowski and de Sitter space-time case these wedges come in pairs in the case of three

dimensional Einstein space-time but unlike in the previous two examples they are not

causally disconnected.

We also show that Einstein space-time does not admit essential generators.









CHAPTER 2
REPRESENTATIONS OF LIE GROUPS

This chapter deals with questions in the representation theory of real Lie groups

which on the one hand are of mathematical interest in their own right and which on the

other hand will be used in chapter four to answer questions in physical applications.

First in section 2.1 we investigate what and how much information on a unitary

representation of a real Lie group can be gained from the fact that there is an invariant

vector for a one parameter subgroup of our Lie group. We introduce the notion of an

essential Lie group element and show its usefulness in the present setting. We also exhibit

conditions on the (non)existence of essential elements in real Lie algebras and determine

which of the classical real Lie algebras have essential elements.

In section 3 we then turn to a question concerning the "temp. i I ,I, 3 of a /-KMS

state with respect to a dynamical system that is given by a generator in our Lie group

representation. We show under very general assumptions that this 3 is uniquely fixed and

its value is tightly bound to the structure constants of our Lie algebra.

2.1 Lie Group Representations and Invariant Vectors

Let G be a finite dimensional, connected real Lie group i.e. G is a smooth real

manifold which is topologically connected and for which the operation


G x GD (g,h) g-'h G (2-1)


defines a smooth map.

Let furthermore

U: G C U-(H) (2-2)

be a strongly continuous, unitary, faithful representation of G on some Hilbert space 1.

As usual LU(I) denotes the space of unitary operators on R.









Now let's consider the following situation: There is a one-parameter subgroup

{A(t)}teR C G and a vector C E T-H which is invariant under the action of this subgroup, i.e.

U(A(t)) Q = 0 for all t c R. (2-3)

In general, this certainly has no implication for the action of the rest of the group G on Q.

As an example, consider the standard representation of the special orthogonal

group in three dimensions SO(3), i.e. the group of real, orthogonal 3 x 3 matrices with

determinant 1, on L2(3) given by


gb(x) = (g-'x) (2-4)

for every g E SO(3) and every E L2(R3).

For every one parameter subgroup of rotations there is an abundance of states Q that

are invariant under that particular subgroup but not invariant under any other rotation.

One can for example consider the one parameter group

cos(t) sin(t) 0
A(t) sin(t) cos(t) 0 (2-5)

0 0 1

of rotations in the 1-2-coordinate plane. Under the above defined action this one-parameter

group will fix the function

(xi, x2, 3) exp(-(x + x1 + 2x )). (2-6)

And certainly no other one parameter rotation group will keep this Q fixed.

But, as for instance shown in [2] and [8] using direct calculations in the respective Lie

groups, for any strongly continuous unitary representation of SO(1, n 1) or SO(2, n 2)









on some Hilbert space, any vector that is invariant under a boost subgroup

cosh(t) sinh(t) 0 ... 0

sinh(t) cosh(t) 0 ... 0
t 0 0 1 ... 0 (27)



0 0 0 01

must automatically be invariant under the whole group. We want to give a generalization

of the arguments presented there.

Let g be the Lie algebra corresponding to G, i.e. g is the tangent space at the identity

of G equipped with the natural Lie bracket operation.

To any coordinate system in a neighborhood of the identity on our Lie group G we

have a set of generators of translations in the coordinate directions mi, m2,..., nE 0 and

we can (via Stone's theorem) find a set of -1: ..-- lioint generators M1, .V_,..., VI of U(G)

such that U(exp(tmi)) = exp(i' V) for all real t and all i.

Let g be the real Lie algebra generated by the set {3I }
-1: .-- v.ioint operators acting on some common dense invariant domain of analytic vectors

in R. Since U is faithful G is isomorphic to g.

Now the following is true.

Lemma 1. Given a one parameter -;,1'.',',1 t v- A(t) of G let M cE be its generator i.e.

U(A(t)) = exp(tM) for all real t. Let furthermore E -c be such that U(A(t))O = 0 for all

t R. Then

(a) The set {N g I exp(tN) = Vt e R} is a Lie -l.,/rl I/,,, of g containing M.

(b) If Ad(M)(N) = AN for some N e g and A / 0 then N e go.

Proof. (a) Using one of the Trotter product formulas, given N1, N2 E G we have


exp (t (aNi + 3N2)) = s-lim (exp(taNi/n) exp(tlN2/n))" =0 (2-8)
n- oo









for all real t, a, f. Thus go is a linear space. Also another Trotter formula


exp(t[NI, N21])

Ss-lim (exp(-tN/n) exp(-N2/n) exp(tNi/n) exp(tN2/n))"2 (2-9)
n-*oo

guarantees that 96 is a closed under the bracket operation and hence is indeed a Lie

subalgebra.

(b) Excluding the obvious case N = 0 we first observe that A E R as -AN [M, N]*

[N*, M*] -[M, N] -AN. Now by the Baker-Campbell-Hausdorff formula we

have for all real s, t


exp(tM) exp(sN) exp(-tM) = exp(setAd(M)(N)) = exp(setxN). (2-10)


Set s re-t for some fixed but arbitrary r IR to get


exp(tM) exp(re-tN) exp(-tM) = exp(rN). (2-11)


Hence if A > 0 we get, using the fact that M E Q and that exp(tM) is unitary for

all real t:


||exp(rN)A | lim exp(rN)O ||
t-oo

Slim exp(tM) exp(re-t'N) exp(-tM) -
t--oo

= lim exp(re-tAN) -
t--oo

=0.


The latter follows because of the strong continuity of the representation. For A < 0

the same result follows taking the limit t -- -o. Thus we conclude exp(rN)o = 0

for all r IR, thereby completing the proof.



Hence it is useful to observe the following.









Lemma 2. The following two statements are equivalent for an element M E :

(a) M together with the set of eigenvectors of Ad(M) in g for nonzero .: ,.. ;.,

generate g as a Lie i,. 1,ra;

(b) Ad(M) is diagonalizable over R as a linear map from g to g and the following

equation holds:1

EI + [M, I] + [[M, g], [M, ]] = (2-12)

Proof. If Ad(M) is diagonalizable then 9, [M, Q] is just the span of the eigenvectors

belonging to nonzero eigenvalues of Ad(M). Thus (b) implies (a).

On the other hand, if (a) is fulfilled then g is generated as Lie algebra by a set of

eigenvectors of Ad(M) belonging to nonzero eigenvalues together with M, which itself is

an eigenvector for Ad(M) with eigenvalue 0. But if N1 and N2 are eigenvectors for Ad(M)

with real eigenvalues A1 and A2 respectively, one has


Ad(M)([AN, N2]) = [M, [TNI, N12 = -[2, [NM, [ N]] [NI, [N2, M]] = (Ai + A2)[N1, N21.
(2-13)

Thus the commutator [NI, N2] is either zero or an eigenvector for the action of Ad(M)

with real eigenvalue A1 + A2 which entails that actually g is already spanned by the

eigenvectors of Ad(M) as a vector space. Hence Ad(M) is diagonalizable over R.

To prove equation (2-12) let N1 = M, N2,..., Nk be a basis of g consisting of

eigenvectors of Ad(M) belonging to real eigenvalues A1, 2, A 3, ..., Ak where Ai = 0 for

i 1 ,...,r 1 and A f 0 for i > r.

Then 9, = [M, 9] is just the linear span of the set {NA}l>r.

As M together with the Ad(M)-eigenvectors for nonzero real eigenvalues generate

G as a Lie algebra, every element in g is a finite linear combination of basic nested



1 as usual we write [A, B] for the linear span of all elements of the form [a, b] with
a A, b B









commutators of the form X = [X1, [X2, [... [X,_, X,]...]]], where each Xi is either M or
one of the Nj for j > r.

We have to show that each such commutator is in R M + [M, 9] + [[M, G], [M, ]] =
RR M + ,. + [g,,g,].
This will be done by induction on the length n of the commutator. For n = 1 we have

X = X1, which equals either M or one of the Ni and thus is in R M + g,.

Now let those commutators lie in R M + Q, + [9,, Q,] for all n < no and
consider an X = [XI, [X2, [... [Xno,Xno+1]...]]] of length no + 1. If X, = M,

then X cE g and we are done. Otherwise X1 = Ni for some i > r. By inductive
hypothesis [X2, [... [Xn,, Xn+1] ...]] is a linear combination of elements of the form

aM + flNj + 7[Nk, N1] with j, k, 1 > r and real a, 3, 7. Thus X is a linear combination of
elements of the form

a[N1,, M] + p[N,, Nj] + 7[Ni, [Nk, Ni]].

The first summand is in G,, the second lies in [G,, G,]. Hence we only need to show that

[Ni, [Nk, NI]] E R M + 0, + [0,, ,]
So consider N = [Ni, [Nk, Nl]] with i, k, 1 > r. According to the computation above, N

is either zero or an eigenvalue for Ad(M) with eigenvalue A = ,Ai + k .. If A / 0 then
N GE by definition of 9,. Otherwise A = 0 but then as Ai / 0 we have Aj + A, / 0 and

thus [Nk, N1] E 9, implying N E [*, *,]. This proves the statement. E

Definition 1. If M cE fulfills either of the equivalent conditions in the lemma, we call

it an essential element in g. A,,. l.',i..~;-I we call m cE essential, if it fulfills u,,;, of the

above conditions with g replaced by g and M replaced by m.
Due to the isomorphism between g and g it follows then in particular, that if m cE is

essential and U(exp(tm)) = exp(tM), then M E g is essential and vice versa.
Corollary 1. If M cE Q is essential, then 96 = g and hence U(A)o = 0 for all A c G.









Proof. According to Lemma 1 and Lemma 2, M together with all the eigenvectors

of Ad(M) for nonzero real eigenvalues belong to the Lie subalgebra g0 of g, but also

generate g as a Lie algebra. Hence we must have g6 = Q.

Now ([28], [22]) for every element A e G we find n1, n2 E g with A = exp(ni) exp(n2).

Hence we find N1, N2 E g with U(A) = exp(Ni) exp(N2). Since N1, N2E we

conclude U(A)Q -= E

In the last part of the proof we used the fact that each Lie group is equal to the

square of the exponential image of its Lie algebra ([28]) i.e.

G = exp(g)2. (2-14)

If a Lie group G fulfills the more restrictive equation

G = exp(g) (2-15)

then G is called an exponential Lie group. It is a well known fact [10] that every connected

compact Lie group is exponential. For these groups the proof of the previous corollary can

hence be simplified.

The question as to when a Lie group is exponential is still not answered in its full

generality (see [10] for details).

The following lemma shows that if there is one essential element in a Lie algebra for a

Lie group, then there are indeed many of them.

Lemma 3. If M E G is essential then so is exp(N)Mexp(-N) E G for all NE G.

Proof. Since exp(N)Mexp(-N) is the (-1., v- iloint) generator of

t exp(N) exp(tM) exp(-N), (2-16)

it is indeed in g. Also if K / 0 and A / 0 such that [M, K] = AK, then


[exp(N)Mexp(-N),exp(N)Kexp(-N)] Aexp(N)Kexp(-N) (2-17)









and exp(N)Kexp(-N) : 0. As M is essential, we conclude that exp(N)Mexp(-N)

together with the eigenvectors of its adjoint action for nonzero eigenvalues generate all of

exp(N)g exp(-N) g O

Finally we want to remark, that due to the isomorphism between g and g the

following corollary a is direct consequence of Corollary 1.

Corollary 2. If m E g is essential and U(exp(tm))O = for all real t, then U(A)Q = for

all Ae G.

2.2 Real Lie Algebras and Essential Elements

A Lie algebra is called simple if it has no nontrivial twosided ideals and a Lie algebra

g is called semisimple if it is equal to a finite sum of simple ideals (any ideal is a Lie

subalgebra).

Also a Lie group G is called simple (semisimple) if its Lie algebra has the corresponding

property.

Most of the Lie groups that appear in physical applications are semisimple. One

important feature about semisimple Lie algebras is the fact that their representation

theory is well understood, which is connected to the fact that they admit an invariant

bilinear form, the so called Killing form, which is defined as


K(m, n) = trace(Ad(m) o Ad(n)) (2-18)


for m, n e g. The invariance here means that for all m, n,p c g the following equality

holds:

K([m, n], p) K(m, [n,p]). (2-19)

2.2.1 Compact Real Lie Algebras and Essential Elements

We remind the reader that a real semisimple Lie algebra g is called compact if its

Killing form is negative definite. According to a theorem of Weyl (see for instance [15,

Theorem 2.4]) this is equivalent to the fact that every connected Lie group G having g as

Lie algebra is compact. Now the following is true.









Theorem 1. Let g be a semisimple, compact real Lie il/, 1.',,. Then g has no essential

elements.

Proof. Assume m E g is essential. Then Ad(m) is R-diagonalizable. Hence with respect to

a suitable basis we have Ad(m) = diag(Ai, A2,..., An) with Ai E R. Let K be the Killing

form for g.

Then K(m, m) trace(Ad(m)2) ~1 A2 > 0. This contradicts the fact that K is

negative definite. Hence there is no such m. D

From this we get the following easy corollary.

Corollary 3. Neither so(n), the Lie ,l. 1',.,, of SO(n) (for n > 3) nor su(n) the real

Lie ,l/,. 1ra of the Lie ,. "1' of special unitary complex n x n matrices (for n > 2) have

essential elements.

Proof. For so(n) one has ([15])

K(m) = -(n 2) trace(tm o m) (2-20)

which is negative definite since trace(tm o m) = I|m| 2 > 0 for m / 0 (the norm here is the

matrix sup norm). Thus so(n) is compact. Similarly for su(n) one has

K(m) = -2n trace(tm o m) (2-21)

and again there holds trace(tm o m) = ||m||2 > 0 for nonzero m. Hence also su(n) is

compact. D

Another consequence of the theorem above is the following.

Lemma 4. Let g be compact and let 0 be abelian. Then gQ 0 has no essential element.

Proof. For each m E a E g 0 there holds

Ko 0(m E a) = Kg(m) + KO(a) = K(m) < 0. (2-22)

Thus as before Ad(m E a) cannot be R-diagonalizable if it is not zero. E









2.2.2 Noncompact Real Lie Algebras and Essential Generators

According to the previous section we will only find examples of real Lie algebras with
essential elements among the noncompact ones (or among the non semisimple ones).
In the following we will give examples of noncompact real Lie algebras g having
essential generators.

g = gl(n, R) The Lie algebra g has dimension n2 and generators e,, = E(p, v). Here

E(i,j) is the n x n matrix having a 1 in row i and column j and zeros elsewhere.
Then each e, (1 < v < n) is essential. To see this we observe that

[E(i, j), E(k,1)] = 6jkE(, 1) uE(k, j). (2-23)

A basis of g consisting of eigenvectors for Ad(e,,) for real eigenvalues is then just the

set of all these generators:

{e p,}1~j_,pn

Thus Ad(e,,) is R-diagonalizable. Also g, = [e, g] = span({e,,, ep.v},#) and so also

ep, = [e,, ep] e [g*, g,] for p, p / v. Therefore Re,, + g, + [g, g,] = g and hence
e, is essential.
S= sl(n, R) The Lie algebra g has dimension n2 1 and generators e, = E(v, v) E(v +

1, v + 1) (1 < v < n 1) and fl, = El,, (p / v, 1 < /, v < n).
Then each e, is an essential element. From the commutator (2-23) we get


{e ~p}<- < U {l fl _p,ipp< (2-24)

is a generating set for g consisting of eigenvalues for Ad(e,) (for real eigenvalues).

Hence Ad(e,) is R-diagonalizable.
Also g, [e,,g] = span ({ff, fv}p U {f(.-+l), f.(v+1)}4+I1) and so for p / p

(and both / v) f4p = [f,, fp] e [g, g*].
But also for p < v one has e. + ep+l + ... + e,- = [f1,, f,] E [*, g*] and for p > v

one has e, + e+i +... + epl= [f, f,1u] C [g*, g*]. Thus also all e, belong to [g,,g,]









and therefore we finally conclude that Re, + g, + [g, g,] = So indeed e, is an

essential element.

g = p(2n, R) This is a 2n2+n dimensional real Lie algebra with generators fp, = E(p, v+

n) + E(v, + n), gp, = E(p + n, v) + E(v + n, p) and hp, = E(pv) E(p + n,v + n)

where 1 < p, v < n. Using the relation (2-23) one verifies that for instance any h, is

essential. A (linearly) generating set for g consisting of eigenvectors for Ad(h,,) for

real eigenvalues is just the set of all the above mentioned generators, thus Ad(h,,) is

R-diagonalizable. Also the space spanned by the eigenvectors for nonzero eigenvalues

is

g, = [h,.,g] = span({g,,} U {f,} U {h,,p}p, U {h,,}- ,-). (2-25)

Furthermore it then follows that f,4 = [h,, fp] E [g0, g,], g, =,] [o., h g,]

and hp, = [gy,, f,p] E [*,,g,] (p, p / v). Therefore Rh,, + g, + [g*,g,] = which

means h,, is essential.

q = o(1, n). This is the Lie algebra of the identity component of the Lorentz group and 0
has a generating set mp, with 0 < p, v < n fulfilling the Lie algebra relations


[mpv, mpo] = gppmy, + g,,mp, gpmvp gpm,, (2-26)

where g = diag(, -1, -1,...,-1, -1) and mp, = -m,,. Then any of the elements

mov with 1 < v < n is essential. A generating set for g of eigenvectors for real

eigenvectors of Ad(mo,) is given by


{mo, m, i}po,v U {mpp}p,po,v} U {mov}. (2-27)

Thus Ad(mo,) is R-diagonalizable.

Also g, = [mo, ] = span ({mo, m'p},jpo,.) = span ({nmop, mPI},Po,1) and as

furthermore [mop, mop] = mp, for p, p {0, v} we have Rmon + g, + [g*, g] = g.

Poincar6 algebra. Here g is the Lie algebra of the identity component of the Poincard

group G = SO(1, n)+ x R"'+1 and it has in addition to the generators m,, above the









translation generators p& for 0 < p < n with the additional Lie algebra relations


[pP, p] = 0 (2-28)

[mP, P] = g9,Pv 9vpp. (2-29)

Still the elements mov with 1 < v < n are essential. We can simply prolong the list

of eigenvectors for Ad(mo,) generating g from above by {po pv} U {pp}I{o,v} -

hence Ad(mo,) is again diagonalizable and

0g, span ({mop, mT }pIpo,v U {po, P,}) (2-30)

As also [mo, po] = p, we again get Rmov + g* + [0*, 0] = 9.

S= so(p, q). The Lorentz algebra example above can be easily generalized to any Lie

algebra so(p, q) with p, q > 1. If the generators are labelled as before by m/, with

1 < p,v < p+ q but now g = diag(1, 1,..., 1,-1,..., -1) with p entries 1 and q

entries -1 then every element m., with p < p and v > p + 1 will be essential by an

analogous calculation as above.









CHAPTER 3
ON /-KMS-STATES

The KMS-condition for a state w (a normalized, positive linear functional) on a

C*-dynamical system (A, at) describes a state which is in thermodynamical equlibrium at

an inverse temperature 3 > 0 ([12]). The mathematical description of that condition reads

as follows:

For all A, B E A there is a function f continuous in the complex strip Sp {z 0 <

-(z) < 3} and analytic in the interior of that strip such that for all real t


f(t) = w(Aa,(B)) and f(t + iP) = w(Ba_,(A)). (3-1)


In [2], [5] and [8] the authors show how the geometry of the de Sitter space-time and

that of Anti-de-Sitter space-time, in particular the specific commutation relations in

the corresponding symmetry groups, determine the value 3 for a 3-KMS-state (see [12])

with respect to the dynamics given by a boost subgroup uniquely in each of the two

space-times. Here de Sitter space-time (in four space-time dimensions) can be thought of

as the hyperboloid
dS {xJ x_ 2 1} c Ri (3-2)
uO ~- lO 1 2 3 4 J \ /

inheriting the Minkowski metric from the surrounding R5 and Anti-de-Sitter space-time

(again four-dimensional) can be identified with the hyperboloid
AdS {x 2 { + x2_- a X2 ax 1} c R5 (3 3)
Aiuc, ~- LO a2 43 4 -

again having the metric induced from the ambient space.

The results in [2], [5] and [8] rely heavily on concrete calculations in the corresponding

Lie algebras. By generalizing their arguments, we show in the following that the value of 3

is directly related to certain structure constants in the Lie algebra of the isometry group of

the given general space-time.


















Figure 3-1. de Sitter and Anti-de-Sitter space-time


Theorem 2. Let E c R be a R-KMS-state for the 1;,iiii. given by a skew-adjoint

generator M on a von Neumann i/l. Ira A C B(H) and let 3 > 0. Let furthermore N be

skew adjoint such that

(a) Ad(M)(N) = AN for some nonzero A and N;

(b) there is a -,l.-,l. 1I,, B C A such that

exp(tN) exp(rM)B exp(-rM) exp(-tN) C A (3-4)

for Irl + Itl < 6 for some 6 > 0, and i is ;. 1.: for B.

Then 3 2.

Proof. As 0 is cyclic for B C A, 0 is also cyclic for A. To see that it is also separating

consider A E A such that AO = 0. Now as 0 is a KMS state there is a function f

continuous in the complex strip Sp {z | 0 < S(z) < p3} and analytic in the interior of

that strip such that for real t and B, C c A

f(t) = (0, C*A exp(tM)BO) and f(t + iP) = (0, B exp(-tM)C*A ) = 0. (3-5)

Hence f vanishes everywhere in Sp, and we have in particular f(0) = (C(, ABO) = 0. As

this holds for arbitrary B, C E A and ) is cyclic for A, we get A = 0.

As 0 is cyclic and separating for A, we can consider the modular operator A and the

modular conjugation J associated with the pair (A, 0).









The fact that the adjoint action of exp(tM) leaves A invariant and fulfills the KMS

property entails ([23]) that exp(tM) = for all t and


A" exp(-3tM). (3-6)


Consequently, we can also compute that for all A E A


JA = J J A*)A = exp -M )A*.


As a consequence of the commutation relation (a), we have for all real t, s:


exp(sM) exp(tN) = exp(t exp(As)N) exp(sM), (3-7)


and we also know (Lemma 1) that A E R and exp(tN)o = for all t e R.

Now pick any B e B. Then one has for any be c H:


(, exp(sM)exp(tN)Bexp(-tN)O) = (, exp(t exp(As)N) exp(sM)BO). (3-8)

By assumption, exp(tN)B exp(-tN) E A for It < 6 and hence we conclude that

exp(tN)B exp(-tN)) is in the domain of A1/2 = exp(MiM). Thus the left hand side can

2
be analytically continued in s into the strip S, and has a continuous limit at the upper

end of that strip.

For the right hand side observe that there is a dense set of Nelson vectors b for which

z -+ exp(zN)b can be analytically continued in z inside a ball B(0, p) C C. Thus for a

Nelson vector b the function s v-+ exp(-texp(As)N)b can be analytically continued in s

into the ball B(0, p(t)) with p(t) = log(p |t|- ) A-1 oo as |t| 0.

As also s v- exp(sM)BO allows an analytic continuation into S_ by the same
2
argument as above, we deduce that both sides of the last equation can be analytically

continued into the region S3 n B(0, p(t)) with continuous boundary values and are hence
2
equal there. Thus for sufficiently small \t\ we can set in particular s = to get the








equality


(, exp M exp(tN)B exp(-tN) ,exp t(exp ( NA)N)exp (Mi B (3-9)

This is equivalent to

( J exp(tN)B* exp(-tNO) (= exp t exp N JB* O. (3-10)

Now since this is true for a dense set of vectors b and since 0 is cyclic for B by assumption
and 0 is fixed by exp(-tN) for all t, we get

J exp(tN) -exp t exp( ) N )J (3-11)

for small Itl. After iterating this equation suitably often, we see that it actually holds for
all real t.
As J is anti-unitary and exp(tN) is unitary it then follows that exp(A) E R, i.e.

3 = for some positive integer k.
Now suppose k > 2. Setting B = exp(rM)Cexp(-rM) for C E B in equation (3-8),
we see that for |r| + Itl < 6 both sides of the equation allow an analytic continuation in s
into the region S3 n B(0, p(t)). Setting first s < j yields
S7i2)
,exp ( M)exp(tN)exp(rM)C) (exp(-tN)exp AM exp(rM)C) (3-12)
bI ( )4b| ( M) \|A /

for small Itl and Irl. Now consider any compact set A and let P(A) be the projection onto
the corresponding spectral subspace of the (-, ifldioint) operator iM. Then multiplying
the previous equation with P(A) from the left gives

Ti T i
exp M) P(A)exp(tN)exp(rM)C( -P(A)exp(-tN)exp ( M) exp(rM)C.
(3-13)
Since exp (FM) P(A) is a bounded operator and since (as 1 > 7) Cf is in the domain
of exp ( -M ) we can again continue both vector-valued sides analytically in r into S .
Hence (by the Edge-of-the-Wedge theorem) the last equality does not only hold for small








Irl, but for all real r and small Itl. This implies


exp M) P(A)exp(tN)P(A)C P(A)exp(-tN)exp MP(A)Co. (3-14)

Now, as we are dealing only with bounded operators, the fact that 0 is cyclic for B entails

exp M)P(A) exp(t)P(A) P(A)exp(-t)P(A)exp ( M) (3-15)

for small It|. Hence P(A) exp (~M P(A) commutes with P(A) exp(tN)P(A) for
small It|. This entails ([16, Lemma 5.6.13, 5.6.17]) that the spectral projections of the
-. Iidioint operators P(A)iMP(A) and P(A)iNP(A) commute. As A was arbitrary this
in particular implies that M and N commute, contradicting the assumptions.
Consequently we must have = 2. D
FA"









CHAPTER 4
APPLICATION TO ALGEBRAIC QUANTUM FIELD THEORY

In this chapter we will now show how the previous results can be applied in quantum

field theoretic problems.

We will examine the problem of how to choose a physically relevant, fundamental

state in a given quantum field theoretic model on an a priori general curved space-time.

As a main result we will show that the requirement for such a state to be passive [23]

with respect to the dynamics of an essential element implies that such a state will share

many properties with vacuum states that were constructed on Minkowski, de Sitter and

Anti-de-Sitter space-time.

4.1 Basic Setup

We will make use of the algebraic formulation of quantum field theory as introduced

by Haag and Kastler (see [12] for more details).

In particular we will be considering an n-dimensional manifold M together with

a Lorentzian metric that models our space-time. Whereas a generic space-time M will

have a trivial isometry group, for our approach it is crucial that M has indeed nontrivial

symmetries. We consider a connected subgroup G of the isometry group of M and assume

that it is strongly continuously, unitarily and faithfully represented on some separable

Hilbert space iH via the representation U.

The observables of the theory form an isotonous net of von Neumann algebras A(O)

indexed by open subsets O C M, i.e. we have an assignment 0 v-+ A(O) such that

01 C 02 implies A(01) C A(02). The (global) observable algebra VocM A(O) -

(UocM A(O))" is denoted by A. Also G is assumed to act covariantly upon the net, i.e.
for every g E G and every open ( C M we have


U(g)A(O)U(g)* A(go). (4-1)









4.2 Observers and Wedges

We are looking at observers travelling along worldlines generated by a one-parameter

group of isometries. To be precise let {A(t)}teR be a one-parameter subgroup of G. If

for some x cE M the curve t v- A(t)x is timelike everywhere, we regard it as a possible

worldline of an observer.

Let W(A) be the open set of all x for which t v- A(t)x is a timelike curve. The

connected component of W(A) that contains the given worldline, i.e. the set of all

neighboring worldlines, will be called the wedge W(A, x) associated to the observer,

respectively associated to the worldline. This is typically the set of events that can

influence or can be influenced by our observer. In any case we regard W(A, x) as the

maximal localization region of observables that can be measured by the observer.

In Minkowski space-time, for instance, wedges for the boost-subgroup (2-7) are

precisely the wedge shaped regions WR {x cE I4 X < xl} and WL = -WR.










Figure 4-1. Wedges in Minkowski-, de Sitter and Anti-de-Sitter space-time


A technical requirement on the size of the wedges and the size of G is the following:

Let S be the set of 0 C MA for which VEG A(gO) = A. Also we call an inclusion

01 C 02 of open subsets of MA proper if there is an open neighborhood N of 1 in G such

that NO1 C 02. Then we require

(WA) (Weak Additivity): Each wedge W(A, x) has a proper subset in S.

In Minkowski space-time this holds under very general assumptions [25]. It even holds in

models in which the local algebras localized in sufficiently small regions are trivial, e.g. [6].









Lemma 5. The set of wedges is invariant under the action of G. If W(A,x) 1f(Ill7 (WA)

then so does gW(A, x) for all g CG.

Proof. When t v-+ A(t)x is a timelike curve then so is t v-+ gA(t)x for all g E G. This

implies the result. O

4.3 States

One of the key problems in quantum field theory on general space-times is to pick

states of interest out of the abundance of possible states. In the following we will

introduce a list of properties which can be used to characterize fundamental states for

quantum systems vacuaa). A similar approach was taken in [5]. We do not assume our

state to have all these properties; instead we will show in the following sections how these

properties are related in our special situation.

First of all, we can assume that such a state will be represented by a normalized

vector Q E CT (by considering the GNS-representation associated to our state). We can

also assume that Q is cyclic for A since otherwise we could just restrict ourselves to a

smaller Hilbert space.

Furthermore it is well known, that the vacuum state in quantum field theories

constructed on Minkowski, de-Sitter and Anti-de-Sitter space-time is invariant under

symmetries of the respective space-times. Therefore it is in general desirable for such

a fundamental vector state to be invariant under isometries. Hence we introduce the

following notion:

(I) (Invariance): U(g)Q = Q for all g E G.

Also an observer freely falling along a worldline described above should see this potential

vacuum Q as energetically stable in the sense that the expected value of the energy in

this state is minimal among the energy expectations in small perturbations of Q. The

mathematical description of this property of a state is as follows (see also [23]):

(P) (Passivity): Q is a passive state for observers travelling along certain

worldlines t v- A(t)x. This means that for all unitary V E A(W(A, x)) and for









the -, ifdioint generator M of U(A(t)) we have

(VQ, MV) > (Q,MQ). (4-2)


We will ah-i--, make clear what exactly we mean by certain worldlines when we impose

the passivity condition on a state.

Another requirement for l (and for the net of observables) is that Q is fundamental

in the sense that each other state can be at least approximately prepared out of Q by

operations performed just in some open region O properly contained in the maximal

laboratory W(A, x) of an observer. Mathematically speaking this is the

(RS) (Reeh-Schlieder-property): Each wedge contains properly an open 0

such that f is cyclic for A(0), i.e.


A(0) = -H. (4-3)

The Reeh-Schlieder-property was first shown under very general assumptions to be a

feature of vacuum states for quantum field theories on Minkowski space-time in [24]. In

[2] and [8] the Reeh-Schlieder-property was shown to be a consequence of certain stability

conditions on a state in de-Sitter and Anti-de-Sitter space-time.

Another property of states describing pure thermodynamical phases (see [23], [13]) is

the following. It describes the fact that in a pure phase in mean the correlation between

observables respectively localized in two regions decays suitably fast as a function of their

timelike separation with respect to the dynamics given by M.

(WM) (Weak Mixing): Q is weakly mixing for an observer travelling along
t -+ A(t)x if for all A, B E A(W(A, x)) the expression


S' ((Q, Ad(exp(itM))(A)B)- (Q, Ad(exp(itM))(A))(, B)) dt (4-4)
vanishes in the limit T
vanishes in the limit T -- oo.









Finally, a state Q is called central for some observer travelling along t v-+ A(t)x if

(Q, ABQ) = (Q, BA) for all A,B e A(W(A, x)). In this case either Q is annihilated by

most of the elements in A(W(A, x)) or this algebra is of finite type (see [17] for details).

These are (from the view of quantum field theory) pathological circumstances that we

want to avoid. Therefore one introduces the following notion:

(NC) (Noncentrality): Q is not central for certain observers.
Again, it will be made clear with respect to which observer we want Q to be noncentral

when we impose this condition on a state.

4.4 Invariance and Reeh-Schlieder Property

We now want to investigate some relations among these properties when we deal with

subgroups generated by essential elements. The arguments presented, as well as the idea

of taking the assumption of passivity as a starting point of the investigation, were first

published for the special case of Anti-de-Sitter space-time in [5] and [8].

Theorem 3. Let Q fulfill properties (P), (WM) and (NC) for an observer travelling il...,

t v- A(t)x with U(A(t)) = exp(tM) for some (sk., -.,;i. /.:,) essential M. Then

(a) l fulfills (I);

(b) if -iM is not a positive operator, then Q fulfills (RS) for all wedges gW(A, x) with

g E G as well.

Proof. (a) Using deep results of Pusz and Woronowicz ([23]), the passivity and the

weak mixing property of Q entail that M =- 0 and Q is either a ground state or a

KMS-state at some inverse temperature 3 > 0 for M. Thus Corollary 1 implies that

Q is invariant under the whole group action.

(b) It suffices to show the result for W(A,x), since


A(gW(A, x)) U(g)A((A, x))U(g)*. (4-5)

Since W(A, x) fulfills (WA), there is an open O E S properly included in W(A, x), i.e.

there is an open neighborhood N of 1 E G such that NO C W(A, x). The preimage









of N under the continuous product map G x G -- G contains an open rectangle

L1 x L2 containing (1, 1). Then L L1 n L2 is a second neighborhood of 1 E G such

that L2 C N. Set K L N.

Then KO C NO C W(A, x) and K(KO) = K20 C NO C W(A, x). Hence KO is

properly included in W(A, x).

Consider a vector E T-H such that

(Q, A) 0 (4-6)

for all A E A(KO). We are going to show that 0 = 0.

Pick a B A(O) and any g e K n N-1. Then for small t| < e we have

Ad(U(gA(t)g-1))(B) e A(KO) (4-7)

Hence we have that


f(t) (0, U(gA(t)g-1)B) = (Q, U(g) exp(tM)U(g-1)BQ) (4-8)

vanishes on It| < e. Since -iM is not a positive operator, Q is not a ground state

for the dynamics exp(tM). Hence according to the results of Pusz and Woronowicz

mentioned above, Q is a KMS state for some inverse temperature 3 > 0. In fact as the

representation U is assumed to be faithful and Q is noncentral, we must have 3 > 0.

Furthermore we know that for g c K n N-1 we have U(g-1)BU(g) e A(g-'O) C

A(NO) c A(W). Thus the function f : t U(g) exp(tM)U(g-1)B2 is a boundary

value of an analytic vector valued function in a strip in the complex plane, hence it

vanishes everywhere in that strip. In particular, f(t) vanishes for all t e R.

Hence we have

(U(gA(t)g- 1), B) =0 (4-9)

for all t e R. Repeating the same argument several times we get that











(U(gA(t,)g, )U(g2(t2)g2 1) ... U(gkA(tk)g 1), B) = 0 (4-10)

for all t E IR and gi E K n N-1.

Now we prove the following small lemma.

Lemma 6. Let N C G be i,.;, open neighborhood of the .:, 1.:;'/;/ in G. Then the

strong operator closure H of the ij i,'n generated by the unitaries U(AA(t)A-1) for

t E R, A E N coincides with all of U(G).

Proof. Let g' = {K e g exp(tK) e H, Vt e R}. Then using Trotter formulae as in

the Lemma 1 (a), we readily see that g' is a Lie subalgebra of g.

Hence due to the essentiality of M, it suffices to show that M cE and K E G',

if [M, K] = pK for real nonzero p and K e Q. While the first is obvious, for the

second we argue as follows: As N is an open neighborhood of 1 in G, we find no E N

such that exp(K/n) E N and exp(-K/n) E N for all n > no and hence

exp(-K/n) exp(-tM/n) exp(K/n) H

for all n > no and all real t. Thus we have (again by the Trotter formula)

s-lim (exp(-K/n)exp(-tM/n)exp(K/n)exp(tM/n))"2 = exp([tM, K])

exp(tpK) E H

for all real t and as p : 0. So we have K E g', which finishes the proof of the

lemma. E

Hence we conclude

(0, U(g)BQ)= (U(g- ), BQ) = 0 (4-11)









for all g e G. From this and the fact that B E A(O) was arbitrary, we finally deduce

that then also


(V U)A(o)U1g-)) Q (V (O A( O) = AK (4-12)
gEG / \g6G /
is perpendicular to Q which implies = 0 as Q is cyclic for A.



4.5 Modular Objects and Unruh-Temperature

In a classic paper ([1]) Bisognano and Wichmann showed that the modular objects

associated to a vacuum state and the algebra of observables of a wedge region in

Minkowski space-time (generated by Wightman fields) act geometrically upon the net

of observable algebras. This result has been extended to various other space-times

([2],[8],[21]). Also, in light of these results, the property of a state (and a net of observable
algebras), that certain modular objects act geometrically, was proposed as a selection

criterion for physically relevant states ([7],[4]). In this section we show that also under

our general assumptions the modular objects have a geometric interpretation and that the

corresponding Unruh-Temperature can be determined.

The following holds as long as the Lie group G has at least dimension 2.

Theorem 4. Let Q fulfill properties (P), (WM) and (NC) for an observer travelling 1..".,

t v- A(t)x with U(A(t)) =exp(tM) for some (skew-adjoint) essential M such that -iM is

not positive. Then

(a) there is a nonzero .:. '.1 ,,.;,. A for the adjoint action of M on g; all such .:, ,.:; l-

ues have the same modulus and 2 is a M-KMS state for the hi;,im.. exp(tM) on

A(W(A, x));

(b) 2 is ;/. V.- and separating for A(W(A, x)) and hence for each A(gW(A, x)) for all

g E G. The modular operators for the pair (A(W, A), Q) are given as

A = exp(- tM) and (4-13)
JAI









JAf = exp ( M) A*Q. (4-14)

Furthermore the commutation relations of J with the p ,-q, representations are fixed

as follows: J commutes with exp(tN) if [M, N] = 0 and if [M, N] = AN then

Jexp(tN) exp(-tN)J. (4-15)

Proof. (a) As M is essential and g has dimension greater than 2 there must be some

nonzero N E Q and some nonzero A with [M, N] = AN. As seen before, properties

(P), (WM) and (NC) entail that Q is a /3-KMS state for some f3 > 0. Also, from the
previous theorem the Reeh-Schlieder property holds for A(W(A, x)), and hence there

is an 0 properly included in A(W(A, x)) such that Q is cyclic for A(O).

Therefore all the assumptions for Theorem 2 are fulfilled, and we conclude that

indeed f = and hence |A| is uniquely given.

(b) This is proved in the first part of the proof of Theorem 2. The commutation

relations follow from equation (3-11) by plI ii :in; i 3 2= 2



In the special case of Minkowski space-time, the derived condition (4-13) is known as

modular covariance. With modular covariance as one of the assumptions, the authors of [3]

derive a representation of the Poincare group which acts covariantly upon the net.

4.6 Weak Locality

The stated assumptions on 2 seem not to suffice to deduce strong locality relations

in general, see for instance the examples constructed in [8] and [9] on AdS and Minkowsi

space-time respectively. But one can at least formulate the following result on weak

locality. Similar results in the special case of Anti-de-Sitter space-time had first been

published in [8].









Theorem 5. Let Q fulfill properties (P), (WM) and (NC) for an observer travelling il..'.
t ) A(t)x with U(A(t)) = exp(tM) for some (skew-adjoint) essential M such that -iM is

not positive.

Suppose furthermore that there is a I'';, of s(2,R) generated by {M, N+, N_} inside
G, i.e. [M, N] = AN and [N+, N] = AM with A > 0. If now N (N+ + N_)

generates a compact -il' ;g.' of G, i.e. U(p(t)) exp(tN) and p(2) 1, then
observables in A(W(A,x)) and A(p(,)W(A,x)) commute Ii, .l;1ii ,n .,g that

(Q, ABQ)= (Q, BAQ) (4-16)

for each A E A(W(A,x)) and B e A(p(')W(A,x)). The same holds then also for the

wedge pairs A(gW(A,xx)) and A(gpQ()W(A,x)) for each g C G.

Proof. Observe first that [N, M] = A(N N_) and [N, >A(N N_)] = -A2M. Hence

exp(tN)Mexp(-tN) = exp(Ad(tN))(M) = cos(At)M + sin(t)(N+ N_). (4-17)
2

Setting t = 2 and expc(.,. ,i ili .Ir we get

e N) N exp ( M exp (-N exp ( M (4-18)

wherever these operators are defined. Also we know from equation (4-15) that

Jexp(tN) Js-lim (exp(tN+/2) exp(tN /2))"
n- 0oo
Ss-lim (exp(-tN+/2) exp(-tlN/2)) J = exp(-tN)J.
n-oo

Now for all B c A(p(')W(A, x)) we have


exp( N)B exp(- N) E A(p(2 )W(A, x)) = A(W(A, x)). (4-19)

Hence we can conclude

JB = Jexp( N)exp( N)Bexp(- N)Q exp( N)Jexp( N)Bexp(- N)Q (4-20)
A A A A A A








where we used exp( N) = exp(- N). Going on using equations (4-14) and (4-18) we get

JBQ = )xp( exp ) exp(N)B*Q exp ( M) B*Q. (4-21)
A X A X (

This then finally gives for all A A(W(A, x)) and B e A(p( )W(A, x)) that

(Q, ABQ) (Q, AJJB)

(A* Jexp ( 7 M) B*Q)

S(JA* ,exp (-iM) B*Q)

= (exp (f M) AQ,exp (-aM) B*Q)

S(exp (LM B*Q,exp ( M) AQ)

(Q, BA4). E

4.7 Concrete Examples
The theorems presented above can be, in particular, applied to the situations of
Minkowski space-time, de-Sitter space-time and Anti-de-Sitter space-time, each of them at
least 3-dimensional, and their corresponding (identity components of the) isometry group.
In each of these cases the (-1: .--- dijoint) generator M of a boost subgroup serves as an
essential element for which the slrf ,lioint operator -iM is not positive. The fact that
these generators are essential was shown in section 2.2.2; to see the positivity observe that
because the dimension of the space-times is at least 3, there is a (-1: .--- dijoint) rotation
generator N in the respective Lie algebras that does not commute with M. From the
concrete commutation relations (2-26) one has then

exp(rN)Mexp(-rN)= --M (4-22)

and thus -iM can not be positive. Also one easily sees by direct inspection of the
commutation relations (2-26) that there ahv--, is a second boost generator N' such that









{M, N + N', N N'} generates a copy of sl(2, R) inside g. If M = .1,, then one can for

instance pick N M= Mj and N' = 1,,. for j = 2, 3,..., n 1. In each of these cases the

rotation 1((N + N') + (N N')) = N generates a compact group. While the resulting

concrete statements for the cases of de-Sitter and Anti-de-Sitter space can be found in [2]

and [8] as an example we state the resulting theorem for Minkowski space-time here.

Theorem 6. Let an i'A. braic quantum field theory on n-dimesnional Minkowski space-

time in the form ,. ; .:' ;,-l;i presented be given. Let in particular Q be a passive, a..' l.;

mixing and noncentral state for each observer travelling along the worldline generated by

(a conjugate of) a boost -ilgi.,'up. Then one has

(a) Q is invariant under the whole Poincared ,.'.,,';

(b) Q is i,. 1.. for each wedge 'l. ira A(W) with W gWR for some g C G;

(c) an observer travelling il. ,.i a worldline generated by a boost -i;,1,g,.,', sees Q2 as a

27-KMS state;

(d) the action of the modular ,., -', for the right wedge WR coincides with the corre-

-1.*!,]'o,:, boost -., ,1'.. ', action.

(e) Observables in a wedge 'il 1, ,, commute with the observables in the opposite wedge

il, 1,,, i ,.l. ;lj here the opposite wedge of the right wedge is the .;,,,.,' of this wedge

under a rotation in the (1, 2)-plane by 7r also called the left wedge WL. In general

gWp has opposite wedge gWL for g c G.

We do not want to forget to mention that such a result (for the case of Minkowski

space-time) had also been obtained in [18] under different assumptions. There the author

shows that if a state is passive with respect to all generators of time evolutions of systems

that move at arbitrary constant velocities and if in addition the unitaries implementing

the translation symmetry belong to the observable algebra, then the spectrum condition

holds ([18, Prop. 5.1.]) and later he shows (using the spectrum condition, see [18, Prop.

6.1.]) that uniformly accelerated observers see the vacuum as a KMS-state at a fixed

Unruh-temperature.









4.8 Further Examples Using Conformally Covariant Theories

It is an easy observation that all the previous results remain true if G is assumed

to be a subgroup of the conformal group (instead of the isometry group) of the given

spacetime manifold M. In [6] the authors show how to construct among other things

conformally covariant nets of local algebras on a special class of Robertson-Walker

spacetimes. These spacetimes are Lorentzian warped products topologically equivalent to

R x S3 where the metric in the usual cylindrical coordinates (t, X, 0, 0) is of the form

dS2 dt2 S(t)2 (dx2 + sin2)X (d2 + sin2 ()d2)).


Here S is assumed to be a positive, smooth (warping) function. Then following [14] one

can define a new time variable r via

dr7 1
dt S(t)

and in these new coordinates the metric takes the form

ds22 2() (d2 dX2 + sin2 ) (do2 + sin2 (0)d2))


Now T has, as a strictly increasing continuous function of t, some open interval as its

range. In the special case when this range is of the form (- ) the corresponding

Robertson-Walker spacetime has SO(4, 1) as its conformal group and is conformally

isomorphic to four-dimensional de-Sitter spacetime. In [6] a method called transplanta-

tion is then used to construct conformally covariant nets on such a Robertson-Walker

spacetime.

As discussed in section 2.2.2 the group SO(4, 1) has essential elements and due to the

conformal equivalence to the de-Sitter case all our results are applicable for these special

Robertson-Walker spaces as well.









4.9 Einstein Static Universe

In this section we want to collect various results we obtained when we examined the

Einstein static space-time as a possible further example that could fit into the presented

scheme. This seems promising as the isometry group of this space-time is R x SO(n) x Z2

which is relatively big, acting transitively on the space-time.

The n-dimensional Einstein static universe S" is a globally hyperbolic space time

having the topological structure of R x S"-1. We can (and will in the following) view S"

as the cylinder

(xx,.,X x+x +...+x2 R2} (423)

in n + 1-dimensional Minkowski space where R > 0 is a fixed radius. It inherits the

Minkowski metric

ds2 dxo dx2 -... dx. (4-24)

In the case of S4 changing to local cylinder coordinates (t, p, 0, 0) the metric takes the

form

ds2 dt R2(dp2 + sin2 p(d02 + sin2 Od2)). (4-25)

Here t E R, 0 < p, 0 < 7 and 0 < Q < 27. So in particular Einstein static space-time is a

special Robertson-Walker space-time but it does not fit into the conformal scheme of the

previous section as the warping function S is constant here and thus the function has all

reals as its range.

4.9.1 Wedges in Einstein Static Universe

A first question that one can address when we investigate the possibility of our

scheme to be applicable in the Einstein universe case is the following: If there was indeed

a one parameter subgroup of the isometry group of 4 generated by an essential element

and fulfilling all the assumptions of Theorem 4 then the corresponding modular groups

would act geometrically and would leave the corresponding wedge region invariant.









Therefore it is of interest to investigate which subsets of S4 or in general S" (n > 2)

can appear as wedges.

Theorem 7. The only wedges in S2 are the mpil', set and S2 itself (trivial wedges).

The only wedges in S3 are the trivial ones and in addition the sets of the form R x C

where C is an open spherical cap of opening :u1i- 0 < r.

The only wedges in S4 are the trivial ones and the connected components of the sets


Rx{y y2+y < r}cRxS3 (426)

where 0 < r < 1 and (yi, y2, Y3, Y4) are ,i, set of orthonormal coordinates for R4.

Proof. The identity component of the isometry group of S' is R x SO(n) and its Lie

algebra is R so(n). We consider the one parameter subgroups of R x SO(n) generated by

general Lie algebra elements a E M where a E R+ and M is a skew symmetric real n x n

matrix in so(n). This group is then given by


A(t)= (ta,exp(tM)). (4-27)

Now for every skew symmetric real n x n matrix M we find a matrix S E O(n), a

nonnegative integer k and real numbers al,... ak such that ([20],[26])

SMS- = diag (aiH,..., akH, 0,..., 0), (4-28)

where

H (4-29)
-1 0

That means that


A(t) (ta, Sdiag (exp(taiH),..., exp(takH), ,...,1) S-1)

(ta, Sdiag(R(tal),..., R(tak), 1,... ,1) S-1), (4-30)









where

R(a) ( cos(a) sin(a) (4-31)
sin(a) cos(c)

Now by definition a wedge generated by this one-parameter group is a connected

component of all those x = (xo, xl,..., x,) E S' for which t v-+ A(t)x is a timelike

curve.

Now there holds


d(A(t)x)2 (a2 IS diag (R(tal),..., R(tak), 1,...,) S-1 (x, 2,..., X 2) dt2

(4-32)

That means we have in the new orthonormal coordinates (yo, Yl,..., y n) given by the base

change (observe that the time direction stays fixed)


Yo Xo

1 0 ) X' (4-33)

i [0 S-1

yn \ xn

the expression
k
d(A(t)x)2 a2 2i (1-1 + Y) (4 34)
i= 1
If there are no points y = (yi, y2,..., Yn) E S"-1 for which

k
i (2i-I + 2i) < a2,
i 1

then there are no wedges corresponding to this one-parameter subgroup. Otherwise the

wedges are the connected components of the set R x C(a, al,..., ak), where

k
C(a, a,..., ak) = {y i i<
i= 1

Since on the other hand all possible choices of a and ci give rise to a one-parameter

subgroup in (4-30) each set of parameters also gives a possible set of wedges.









Now in the case n = 2 we see that either k = 0, in which case we get


R x C(a) = a (4-35)
[ 2, a/ 0

or we have k = 1, and then we have (since y + y? = ||l2 1)


Rx C(a, a {)= a, 2 (4-36)


Thus the only wedges in S2 are the empty set and S2 itself.

For n = 3 we again have two possibilities. Either k = 0, which yields


s0, a 0
Sx C ) j= 0o (4 37)


or we have k = 1, in which case we get the inequality a2 > a(y/ + 4) a(1 /). In

the case al = 0 this again has either all of S3 as solution (if a / 0) or is the empty set (if

a =0).

If on the other hand al / 0 the inequality becomes

2 a2
y3 >1- 2
Y32 2
a1

This again yields all of 8 if 2 > 1, the empty set if a 0 or, in the case 0 < < 1 we

get two di -i, iiil wedges of the form


Rx y y3> -- and Rx y I<- -1 (438)

These correspond to cylinders with spherical caps as bases.

Finally for n = 4 there are three cases. Again k = 0 yields all of S4 or the empty

set. The case k = 1 gives again the inequality a (y2 + y2) < a2, which yields as before a





















Figure 4-2. Spherical caps corresponding to a pair of wedges in S3


nontrivial set of wedges if 0 < 2 < 1, namely the connected components of the set


Rx y y+y < (4-39)


Lastly there is the possibility that k = 2 and then the inequality reads


a (Y2 + y2) + 2 (y32 + y2) < 2. (440)
"1 y1 2 2 3 (4 40)


Without loss of generality (due to symmetry) we can assume that at > a2. While the

case a= ca yields once again only trivial wedges, the case of aC > a~ turns out to be

equivalent to the previous one because using the fact that y3 + y = 1 y y2 we see,

that then the inequality becomes


Y1 + Y2 a< a2 (4-4
1 2

This is hence of the same form as in the case k = 1 and therefore yields the same types of

wedges. E

In all the cases where there are nontrivial wedge regions, these are not spacelike

separated and are not causally closed. In fact, the causal closure, even if one only

considers geodesic timelike curves, is the whole space again.

These properties make it unlikely that the found wedges and hence the corresponding

observers can serve our purpose.









4.9.2 Essential Elements in Einstein Static Universe

Unfortunately the question as to whether there are essential elements in the isometry

group of S4 has a negative answer.

Lemma 7. The (connected component of the .:,. ,.l.:;i) of the isometry 'i"',r' of S" has no

essential elements for n > 3.

Proof. The connected component of the identity of the isometry group is R x SO(n). Its

Lie algebra is hence R x so(n). According to Lemma 4 there are no essential elements in

this algebra. O

Also, since for n = 2 the Lie algebra Rxso(2) is abelian, there can also be no essential

elements in 2.









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BIOGRAPHICAL SKETCH

Robert Strich was born on May 9, 1978 in Merseburg, Germany. He grew up

mostly in Halle/Saale, Germany, where he attended the Georg-Cantor-Gymnasium

(High school), graduating in 1996. In 1996 and 1997, Robert served in a nursing

home in Naumburg/Saale, Germany, in fulfillment of his alternative civilian service

(army service replacement). In October 1997 he started his undergraduate studies at

Georg-August-University Goettingen, Germany, in mathematics and physics. Robert

received his "Vordiplom" (bachelor's degree) in Mathematics in April 1999 and he passed

his "Vordiplom" in physics three months later. He started his graduate work under Prof.

Dr. D. Buchholz in theoretical physics in October 1999 and graduated with a "Dilp!i. 1

(master's degree) in Theoretical Physics in November 2002. The title of his master's thesis

was "Symetrien im Skalenlimes der Quantenfeldtheorie" ("Symmetries in the scaling

limit of quantum field theory"). Since January 2003 Robert completed doctoral studies

in mathematics at the University of Florida, partly supported by an Alumni Fellowship

of the University. His Ph.D.-adviser is Prof. Dr. S. J. Summers. Upon completion of his

Ph.D. program, Robert will return to Germany to work as a (High school) teacher.





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IwouldliketothankmyadviserProf.StephenJ.Summersforsuggestingthetopicofthisdissertationforhispatienceandfornumeroushelpfuldiscussionsthroughoutmywork.IwouldalsoliketothankProf.JohnKlauder,Prof.ScottMcCollough,Prof.PaulRobinsonandProf.BernardWhitingforagreeingtoserveonmysupervisorycommittee.FinallyIthankKristinStrothandKarstenRoeselerforproofreadingthisworkatvariousstagesofitsbecoming. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 6 ABSTRACT ........................................ 7 CHAPTER 1INTRODUCTION .................................. 8 2REPRESENTATIONSOFLIEGROUPS ..................... 12 2.1LieGroupRepresentationsandInvariantVectors .............. 12 2.2RealLieAlgebrasandEssentialElements .................. 19 2.2.1CompactRealLieAlgebrasandEssentialElements ......... 19 2.2.2NoncompactRealLieAlgebrasandEssentialGenerators ...... 21 3ON-KMS-STATES ................................. 24 4APPLICATIONTOALGEBRAICQUANTUMFIELDTHEORY ........ 29 4.1BasicSetup ................................... 29 4.2ObserversandWedges ............................. 30 4.3States ...................................... 31 4.4InvarianceandReeh-SchliederProperty .................... 33 4.5ModularObjectsandUnruh-Temperature .................. 36 4.6WeakLocality .................................. 37 4.7ConcreteExamples ............................... 39 4.8FurtherExamplesUsingConformallyCovariantTheories .......... 41 4.9EinsteinStaticUniverse ............................ 42 4.9.1WedgesinEinsteinStaticUniverse .................. 42 4.9.2EssentialElementsinEinsteinStaticUniverse ............ 47 REFERENCES ....................................... 48 BIOGRAPHICALSKETCH ................................ 50 5

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Figure page 3-1deSitterandAnti-de-Sitterspace-time ....................... 25 4-1WedgesinMinkowski-,deSitterandAnti-de-Sitterspace-time .......... 30 4-2SphericalcapscorrespondingtoapairofwedgesinE3 46 6

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WeshowthatifarealLiegroupisrepresentedunitarilyandfaithfullyonaHilbertspaceHandifavector2Hisxedbytheactionofanessentialone-parametersubgroupthenthisvectorisstabilizedbythewholeLiegroupaction.WeexhibitLiegroupshavingessentialone-parametersubgroupsandshowthatcompact,semisimpleLiealgebrasdonothaveessentialelements. Thenweshowthatthe"temperature"ofaKMS-statewithrespecttothedynamicsgeneratedbyasuitableLiealgebraelementisxedbythecommutationrelationsofthisLiealgebraelement. Furthermoreweinvestigateaselectioncriterionforphysicallysignicant,fundamentalstatesonquantumsystemsonapriorigeneralspace-times,namelytherequirementofastatetobepassivewithrespecttothedynamicsofanessentialobserver.Weshowthatthisrequirementhasstrongconsequencesasforinstancetheinvarianceofthestateunderspace-timesymmetriesandtheReeh-Schliederpropertyforwedgealgebras. 7

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ThealgebraicformulationofquantumeldtheoryasintroducedbyHaagandKastler(see[ 12 ])providesamodelindependentmathematicallyrigorousapproachtoconceptualquestionsinquantumphysics.Itreliesontheassumptionthatallphysicalinformationofaquantumsystemisencodedinanassignment ofopenspace-timeregionsOtoC-algebrasA(O)whoseselfadjointelementsareregardedastheobservablescorrespondingtomeasurementsthatcanbedoneinthegivenspace-timeregionO.Withinthisframeworkinrecentyearsmuchworkhasbeendoneononeofthekeyproblemsoneencounterswhendealingwithquantumeldtheoryongeneralspace-times,namelyhowtochoosephysicallyrelevant,fundamentalstatesforaquantumsystem.InMinkowskispace-timeavacuumstatecanbecharacterizedasastatethatadmitsaunitaryrepresentationofthespace-timetranslationsfulllingthespectrumcondition,i.e.thejointspectrumofthefourmomentumoperatorsliesintheforwardlightcone,andleavingthevacuumstateinvariant.Theisometrygroupofagenericgeneralspace-timehoweveristrivialorinanycasenotbigenoughtocontainacounterparttothespace-timetranslationsinMinkowskispace. Onerecentapproachtothisconceptualproblem([ 4 ])usesanobservationthatwasmadebyBisognanoandWichmannintheirclassicalpaper[ 1 ],namelythefactthatthemodularobjectsassociatedtothealgebrasofcertainwedgeshapedregionsinMinkowskispace-timeandthevacuumvectoractgeometricallyuponthenetoflocalalgebras,ifthisnetisgeneratedbya(nitecomponent)quantumeldfulllingtheWightmanaxioms([ 12 ]).Theauthorsof[ 4 ]proposetotakethisgeometricactionasaselectioncriterionforphysicallyrelevantstates.TheirConditionofGeometricModularAction(CGMA)canbeformulatedinanyspace-timeanddoesnotrequirethepresenceofspace-timesymmetries. 8

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4 ],[ 6 ]),itisnotclearastohowapplicabletheirapproachisingeneral. Anotherproposalofhowtotackletheproblemofchoosingafundamentalreferencestatehasbeenmadein[ 2 ],[ 5 ]and[ 8 ].Theretheauthorsshowthatifoneimposescertainstabilityconditionsonaquantumstateforaquantumsystemonde-Sitterspace-time(dS)orAnti-de-Sitterspace-time(AdS),thenthishasstrongconsequencesforthequantumsystem. OnesuchstabilityconditionistheKMS-condition(see[ 2 ]fordetails)whichcharacterizesastatethatisinthermodynamicequilibriumatacertain"temperature".Anotherrelatedbutmorefundamentalconditionistheconditionofpassivitywhichwasrstdiscussedin[ 23 ].Thisconditionisamathematicalformulationofthesecondlawofthermodynamicswhichinprinciplesaysthattherecannotbeaperpetuummobile. Soforinstancein[ 8 ]theauthorsshowthatifastateispassive([ 23 ])withrespecttothedynamicsofauniformlyacceleratedobserverinAdS,thenthisobserverseesthegivenstateasanequilibriumstate(KMS-state)atacertainxedtemperature,thestateisinvariantundertheisometrygroupofAdSand,whatismore,onecandeduceweaklocalityrelationsamongthemeasurementsthattheobserverinquestioncanperforminhismaximallaboratoryandthemeasurementsthatcanbeperformedinanoppositelaboratoryregion. Similarresultswereshowntoholdinde-Sitterspace-time([ 2 ])underslightlydierentassumptions,whereofcoursetheprecisenotionsofwhatismeantbymaximallaboratoryandoppositehavetobeadaptedtotherespectivegeometries.Also,relatedworkhasbeendoneforthecaseofMinkowskispace-timein[ 18 ],buttheretheauthorimposesadierentsetofassumptions. 9

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Inchapter2wetrytoovercomethisdicultybyintroducingasomewhatadhocbutusefulreplacementfortheuniformlyacceleratedobserversusedin[ 2 ]and[ 8 ]{calledessentialobservers.Theseareobserversthatmovealongworldlinesinourspace-timewhicharegeneratedbycertainessentialelementsintheLiealgebraoftheisometrygroupofthespace-time,whereanessentialelementisonethatgeneratesenoughoftheLiealgebrainacertainsense(seeDenition 1 fordetails).Oneofthemainresultsofthischapteristhenthefollowing(seeCorollary 2 ):IfaLiegroupisunitarily,faithfullyrepresentedonaHilbertspaceHandifavector2HisinvariantunderaoneparametersubgroupoftheLiegroupthatisgeneratedbyanessentialLiealgebraelement,thenthevectorisstabilizedbythewholegroup. ThispropertyislaterusedinphysicalapplicationsbutduetothisresultitisalsoofpurelymathematicalinteresttoinvestigatewhichLiegroupsrespectivelywhichLiealgebrashaveessentialelements.Thisisdonetosomeextentinchapter2aswell.Weexhibitalistof(noncompact)realLiealgebrashavingessentialelementsandwealsoprovethatcompactsemisimplerealLiealgebrascannothaveessentialelements(Theorem 1 ). Chapter3dealswithKMS-states.Aspointedoutbefore,thestabilityassumptionsonastateimplyasoneofmanyconsequencesaxedHawking-Unruhtemperature(see[ 27 ],[ 11 ]fordetails)thatauniformlyacceleratedobserver(indSorAdS)ndsthatstatein.Weshowthatinourgeneralspace-timeapproachunderrathergeneralassumptionsthistemperatureisdirectlyrelatedtocertainstructureconstantsoftheLiealgebraofthesymmetrygroupinquestion. 10

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forourgivenstate!. FurthermoreweextendtheresultstoconformallyinvarianttheoriesoncertainRobertson-Walkerspace-times.TheseareLorentzianwarpedproductsthataretopologicallyequivalenttoRS3. FinallywegiveaselectionofresultsconcerningthecaseoftheEinsteincylindricalspace-time.Weshowthatinatleastthreespace-timedimensionsEinsteinspace-timecontainsnontrivialwedges(accordingtoourdenitionofawedge).SimilartotheMinkowskianddeSitterspace-timecasethesewedgescomeinpairsinthecaseofthreedimensionalEinsteinspace-timebutunlikeintheprevioustwoexamplestheyarenotcausallydisconnected. WealsoshowthatEinsteinspace-timedoesnotadmitessentialgenerators. 11

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ThischapterdealswithquestionsintherepresentationtheoryofrealLiegroupswhichontheonehandareofmathematicalinterestintheirownrightandwhichontheotherhandwillbeusedinchapterfourtoanswerquestionsinphysicalapplications. Firstinsection 2.1 weinvestigatewhatandhowmuchinformationonaunitaryrepresentationofarealLiegroupcanbegainedfromthefactthatthereisaninvariantvectorforaoneparametersubgroupofourLiegroup.WeintroducethenotionofanessentialLiegroupelementandshowitsusefulnessinthepresentsetting.Wealsoexhibitconditionsonthe(non)existenceofessentialelementsinrealLiealgebrasanddeterminewhichoftheclassicalrealLiealgebrashaveessentialelements. Insection 3 wethenturntoaquestionconcerningthe"temperature"ofa-KMSstatewithrespecttoadynamicalsystemthatisgivenbyageneratorinourLiegrouprepresentation.WeshowunderverygeneralassumptionsthatthisisuniquelyxedanditsvalueistightlyboundtothestructureconstantsofourLiealgebra. denesasmoothmap. Letfurthermore beastronglycontinuous,unitary,faithfulrepresentationofGonsomeHilbertspaceH.AsusualU(H)denotesthespaceofunitaryoperatorsonH. 12

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Ingeneral,thiscertainlyhasnoimplicationfortheactionoftherestofthegroupGon. Asanexample,considerthestandardrepresentationofthespecialorthogonalgroupinthreedimensionsSO(3),i.e.thegroupofreal,orthogonal33matriceswithdeterminant1,onL2(R3)givenby foreveryg2SO(3)andevery2L2(R3). Foreveryoneparametersubgroupofrotationsthereisanabundanceofstatesthatareinvariantunderthatparticularsubgroupbutnotinvariantunderanyotherrotation.Onecanforexampleconsidertheoneparametergroup ofrotationsinthe1-2-coordinateplane.Undertheabovedenedactionthisone-parametergroupwillxthefunction Andcertainlynootheroneparameterrotationgroupwillkeepthisxed. But,asforinstanceshownin[ 2 ]and[ 8 ]usingdirectcalculationsintherespectiveLiegroups,foranystronglycontinuousunitaryrepresentationofSO(1;n1)orSO(2;n2) 13

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mustautomaticallybeinvariantunderthewholegroup.Wewanttogiveageneralizationoftheargumentspresentedthere. LetgbetheLiealgebracorrespondingtoG,i.e.gisthetangentspaceattheidentityofGequippedwiththenaturalLiebracketoperation. ToanycoordinatesysteminaneighborhoodoftheidentityonourLiegroupGwehaveasetofgeneratorsoftranslationsinthecoordinatedirectionsm1;m2;:::;mn2gandwecan(viaStone'stheorem)ndasetofskewadjointgeneratorsM1;M2;:::;MnofU(G)suchthatU(exp(tmi))=exp(tMi)forallrealtandalli. LetGbetherealLiealgebrageneratedbythesetfMig1in.ThenGconsistsofskewadjointoperatorsactingonsomecommondenseinvariantdomainofanalyticvectorsinH.SinceUisfaithfulGisisomorphictog. Nowthefollowingistrue. (a) ThesetG=fN2Gjexp(tN)=8t2RgisaLiesubalgebraofGcontainingM. (b) IfAd(M)(N)=NforsomeN2Gand6=0thenN2G. Proof. UsingoneoftheTrotterproductformulas,givenN1;N22Gwehave exp(t(N1+N2))=slimn!1(exp(tN1=n)exp(tN2=n))n=(2{8) 14

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exp(t[N1;N2])=slimn!1(exp(tN1=n)exp(N2=n)exp(tN1=n)exp(tN2=n))n2= guaranteesthatGisaclosedunderthebracketoperationandhenceisindeedaLiesubalgebra. (b) ExcludingtheobviouscaseN=0werstobservethat2Ras Sets=retforsomexedbutarbitraryr2Rtoget exp(tM)exp(retN)exp(tM)=exp(rN):(2{11) Henceif>0weget,usingthefactthatM2Gandthatexp(tM)isunitaryforallrealt:jjexp(rN)jj=limt!1jjexp(rN)jj=limt!1exp(tM)exp(retN)exp(tM)=limt!1exp(retN)=0: Henceitisusefultoobservethefollowing. 15

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(a) (b) Ontheotherhand,if(a)isfullledthenGisgeneratedasLiealgebrabyasetofeigenvectorsofAd(M)belongingtononzeroeigenvaluestogetherwithM,whichitselfisaneigenvectorforAd(M)witheigenvalue0.ButifN1andN2areeigenvectorsforAd(M)withrealeigenvalues1and2respectively,onehas Ad(M)([N1;N2])=[M;[N1;N2]]=[N2;[M;N1]][N1;[N2;M]]=(1+2)[N1;N2]:(2{13) Thusthecommutator[N1;N2]iseitherzerooraneigenvectorfortheactionofAd(M)withrealeigenvalue1+2whichentailsthatactuallyGisalreadyspannedbytheeigenvectorsofAd(M)asavectorspace.HenceAd(M)isdiagonalizableoverR. Toproveequation( 2{12 )letN1=M;N2;:::;NkbeabasisofGconsistingofeigenvectorsofAd(M)belongingtorealeigenvalues1;2;3;:::;kwherei=0fori=1;:::;r1andi6=0forir. ThenG=[M;G]isjustthelinearspanofthesetfNigir. AsMtogetherwiththeAd(M)-eigenvectorsfornonzerorealeigenvaluesgenerateGasaLiealgebra,everyelementinGisanitelinearcombinationofbasicnested

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WehavetoshowthateachsuchcommutatorisinRM+[M;G]+[[M;G];[M;G]]=RM+G+[G;G]. Thiswillbedonebyinductiononthelengthnofthecommutator.Forn=1wehaveX=X1,whichequalseitherMoroneoftheNiandthusisinRM+G. NowletthosecommutatorslieinRM+G+[G;G]forallnn0andconsideranX=[X1;[X2;[:::[Xn0;Xn0+1]:::]]]oflengthn0+1.IfX1=M,thenX2Gandwearedone.OtherwiseX1=Niforsomeir.Byinductivehypothesis[X2;[:::[Xn0;Xn0+1]:::]]isalinearcombinationofelementsoftheformM+Nj+[Nk;Nl]withj;k;lrandreal;;.ThusXisalinearcombinationofelementsoftheform[Ni;M]+[Ni;Nj]+[Ni;[Nk;Nl]]: SoconsiderN=[Ni;[Nk;Nl]]withi;k;lr.Accordingtothecomputationabove,NiseitherzerooraneigenvalueforAd(M)witheigenvalue=i+k+l.If6=0thenN2GbydenitionofG.Otherwise=0butthenasi6=0wehavej+l6=0andthus[Nk;Nl]2GimplyingN2[G;G].Thisprovesthestatement.

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1 andLemma 2 ,MtogetherwithalltheeigenvectorsofAd(M)fornonzerorealeigenvaluesbelongtotheLiesubalgebraGofG,butalsogenerateGasaLiealgebra.HencewemusthaveG=G. Now([ 28 ],[ 22 ])foreveryelement2Gwendn1;n22gwith=exp(n1)exp(n2).HencewendN1;N22GwithU()=exp(N1)exp(N2).SinceN1;N22G=G,weconcludeU()=. InthelastpartoftheproofweusedthefactthateachLiegroupisequaltothesquareoftheexponentialimageofitsLiealgebra([ 28 ])i.e. IfaLiegroupGfulllsthemorerestrictiveequation thenGiscalledanexponentialLiegroup.Itisawellknownfact[ 10 ]thateveryconnectedcompactLiegroupisexponential.Forthesegroupstheproofofthepreviouscorollarycanhencebesimplied. ThequestionastowhenaLiegroupisexponentialisstillnotansweredinitsfullgenerality(see[ 10 ]fordetails). ThefollowinglemmashowsthatifthereisoneessentialelementinaLiealgebraforaLiegroup,thenthereareindeedmanyofthem. Proof. itisindeedinG.AlsoifK6=0and6=0suchthat[M;K]=K,then [exp(N)Mexp(N);exp(N)Kexp(N)]=exp(N)Kexp(N)(2{17) 18

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Finallywewanttoremark,thatduetotheisomorphismbetweenGandgthefollowingcorollaryaisdirectconsequenceofCorollary 1 AlsoaLiegroupGiscalledsimple(semisimple)ifitsLiealgebrahasthecorrespondingproperty. MostoftheLiegroupsthatappearinphysicalapplicationsaresemisimple.OneimportantfeatureaboutsemisimpleLiealgebrasisthefactthattheirrepresentationtheoryiswellunderstood,whichisconnectedtothefactthattheyadmitaninvariantbilinearform,thesocalledKillingform,whichisdenedas form;n2g.Theinvarianceheremeansthatforallm;n;p2gthefollowingequalityholds: 15 ,Theorem2.4])thisisequivalenttothefactthateveryconnectedLiegroupGhavinggasLiealgebraiscompact.Nowthefollowingistrue. 19

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Proof. ThenK(m;m)=trace(Ad(m)2)=Pni=12i0.ThiscontradictsthefactthatKisnegativedenite.Hencethereisnosuchm. Fromthiswegetthefollowingeasycorollary. Proof. 15 ]) whichisnegativedenitesincetrace(tmm)=jjmjj2>0form6=0(thenormhereisthematrixsupnorm).Thusso(n)iscompact.Similarlyforsu(n)onehas andagainthereholdstrace(t Anotherconsequenceofthetheoremaboveisthefollowing. Proof. ThusasbeforeAd(ma)cannotbeR-diagonalizableifitisnotzero. 20

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InthefollowingwewillgiveexamplesofnoncompactrealLiealgebrasghavingessentialgenerators. TheLiealgebraghasdimensionn2andgeneratorse;=E(;).HereE(i;j)isthennmatrixhavinga1inrowiandcolumnjandzeroselsewhere. Theneache(1n)isessential.Toseethisweobservethat [E(i;j);E(k;l)]=jkE(i;l)liE(k;j):(2{23) AbasisofgconsistingofeigenvectorsforAd(e)forrealeigenvaluesisthenjustthesetofallthesegenerators:feg1;n TheLiealgebraghasdimensionn21andgeneratorse=E(;)E(+1;+1)(1n1)andf=E;(6=,1;n). Theneacheisanessentialelement.Fromthecommutator( 2{23 )weget isageneratingsetforgconsistingofeigenvaluesforAd(e)(forrealeigenvalues).HenceAd(e)isR-diagonalizable. Alsog=[e;g]=spanff;fg6=[ff(+1);f(+1)g6=+1andsofor6=(andboth6=)f=[f;f]2[g;g]. Butalsoforonehase+e+1+:::+e1=[f;f]2[g;g].Thusalsoallebelongto[g;g] 21

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Thisisa2n2+ndimensionalrealLiealgebrawithgeneratorsf=E(;+n)+E(;+n),g=E(+n;)+E(+n;)andh=E()E(+n;+n)where1;n.Usingtherelation( 2{23 )oneveriesthatforinstanceanyhisessential.A(linearly)generatingsetforgconsistingofeigenvectorsforAd(h)forrealeigenvaluesisjustthesetofalltheabovementionedgenerators,thusAd(h)isR-diagonalizable.Alsothespacespannedbytheeigenvectorsfornonzeroeigenvaluesis Furthermoreitthenfollowsthatf=[h;f]2[g;g],g=[g;h]2[g;g]andh=[g;f]2[g;g](;6=).ThereforeRh+g+[g;g]=gwhichmeanshisessential. [m;m]=gm+gmgmgm(2{26) whereg=diag(1;1;1;:::;1;1)andm=m.Thenanyoftheelementsm0with1nisessential.AgeneratingsetforgofeigenvectorsforrealeigenvectorsofAd(m0)isgivenby ThusAd(m0)isR-diagonalizable. Alsog=[m0;g]=span(fm0mg6=0;)=span(fm0;mg6=0;)andasfurthermore[m0;m0]=mfor;=2f0;gwehaveRm0+g+[g;g]=g. 22

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(2{28)[m;p]=gpgp: Stilltheelementsm0with1nareessential.WecansimplyprolongthelistofeigenvectorsforAd(m0)generatinggfromabovebyfp0pg[fpg=2f0;g-henceAd(m0)isagaindiagonalizableand Asalso[m0;p0]=pweagaingetRm0+g+[g;g]=g. 23

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TheKMS-conditionforastate!(anormalized,positivelinearfunctional)onaC-dynamicalsystem(A;t)describesastatewhichisinthermodynamicalequlibriumataninversetemperature>0([ 12 ]).Themathematicaldescriptionofthatconditionreadsasfollows: ForallA;B2AthereisafunctionfcontinuousinthecomplexstripS:=fzj0=(z)gandanalyticintheinteriorofthatstripsuchthatforallrealt f(t)=!(At(B))andf(t+i)=!(Bt(A)):(3{1) In[ 2 ],[ 5 ]and[ 8 ]theauthorsshowhowthegeometryofthedeSitterspace-timeandthatofAnti-de-Sitterspace-time,inparticularthespeciccommutationrelationsinthecorrespondingsymmetrygroups,determinethevaluefora-KMS-state(see[ 12 ])withrespecttothedynamicsgivenbyaboostsubgroupuniquelyineachofthetwospace-times.HeredeSitterspace-time(infourspace-timedimensions)canbethoughtofasthehyperboloid dS=fx20x21x22x23x24=1gR5(3{2) inheritingtheMinkowskimetricfromthesurroundingR5andAnti-de-Sitterspace-time(againfour-dimensional)canbeidentiedwiththehyperboloid AdS=fx20+x21x22x23x24=1gR5(3{3) againhavingthemetricinducedfromtheambientspace. Theresultsin[ 2 ],[ 5 ]and[ 8 ]relyheavilyonconcretecalculationsinthecorrespondingLiealgebras.Bygeneralizingtheirarguments,weshowinthefollowingthatthevalueofisdirectlyrelatedtocertainstructureconstantsintheLiealgebraoftheisometrygroupofthegivengeneralspace-time. 24

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deSitterandAnti-de-Sitterspace-time (a) (b) thereisasub-algebraBAsuchthat Then=2 Proof. HencefvanisheseverywhereinS,andwehaveinparticularf(0)=(C;AB)=0.AsthisholdsforarbitraryB;C2AandiscyclicforA,wegetA=0. AsiscyclicandseparatingforA,wecanconsiderthemodularoperatorandthemodularconjugationJassociatedwiththepair(A;). 25

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23 ])thatexp(tM)=foralltand it=exp(tM):(3{6) Consequently,wecanalsocomputethatforallA2AJA=JJ1 2A=expi exp(sM)exp(tN)=exp(texp(s)N)exp(sM);(3{7) andwealsoknow(Lemma 1 )that2Randexp(tN)=forallt2R. NowpickanyB2B.Thenonehasforany2H: (;exp(sM)exp(tN)Bexp(tN))=(;exp(texp(s)N)exp(sM)B):(3{8) Byassumption,exp(tN)Bexp(tN)2Aforjtj
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Thisisequivalentto (;Jexp(tN)Bexp(tN))=;exptexpi NowsincethisistrueforadensesetofvectorsandsinceiscyclicforBbyassumptionandisxedbyexp(tN)forallt,weget forsmalljtj.Afteriteratingthisequationsuitablyoften,weseethatitactuallyholdsforallrealt. AsJisanti-unitaryandexp(tN)isunitaryitthenfollowsthatexp(i Nowsupposek2.SettingB=exp(rM)Cexp(rM)forC2Binequation( 3{8 ),weseethatforjrj+jtj
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expi Now,aswearedealingonlywithboundedoperators,thefactthatiscyclicforBentails expi forsmalljtj.HenceP()exp2i 16 ,Lemma5.6.13,5.6.17])thatthespectralprojectionsoftheselfadjointoperatorsP()iMP()andP()iNP()commute.AswasarbitrarythisinparticularimpliesthatMandNcommute,contradictingtheassumptions. Consequentlywemusthave=2 28

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Inthischapterwewillnowshowhowthepreviousresultscanbeappliedinquantumeldtheoreticproblems. Wewillexaminetheproblemofhowtochooseaphysicallyrelevant,fundamentalstateinagivenquantumeldtheoreticmodelonanapriorigeneralcurvedspace-time.Asamainresultwewillshowthattherequirementforsuchastatetobepassive[ 23 ]withrespecttothedynamicsofanessentialelementimpliesthatsuchastatewillsharemanypropertieswithvacuumstatesthatwereconstructedonMinkowski,deSitterandAnti-de-Sitterspace-time. 12 ]formoredetails). Inparticularwewillbeconsideringann-dimensionalmanifoldMtogetherwithaLorentzianmetricthatmodelsourspace-time.Whereasagenericspace-timeMwillhaveatrivialisometrygroup,forourapproachitiscrucialthatMhasindeednontrivialsymmetries.WeconsideraconnectedsubgroupGoftheisometrygroupofMandassumethatitisstronglycontinuously,unitarilyandfaithfullyrepresentedonsomeseparableHilbertspaceHviatherepresentationU. TheobservablesofthetheoryformanisotonousnetofvonNeumannalgebrasA(O)indexedbyopensubsetsOM,i.e.wehaveanassignmentO7!A(O)suchthatO1O2impliesA(O1)A(O2).The(global)observablealgebraWOMA(O):=SOMA(O)00isdenotedbyA.AlsoGisassumedtoactcovariantlyuponthenet,i.e.foreveryg2GandeveryopenOMwehave 29

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LetW()betheopensetofallxforwhicht7!(t)xisatimelikecurve.TheconnectedcomponentofW()thatcontainsthegivenworldline,i.e.thesetofallneighboringworldlines,willbecalledthewedgeW(;x)associatedtotheobserver,respectivelyassociatedtotheworldline.Thisistypicallythesetofeventsthatcaninuenceorcanbeinuencedbyourobserver.InanycaseweregardW(;x)asthemaximallocalizationregionofobservablesthatcanbemeasuredbytheobserver. InMinkowskispace-time,forinstance,wedgesfortheboost-subgroup( 2{7 )arepreciselythewedgeshapedregionsWR=fx2R4jjx0j
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Proof. 5 ].Wedonotassumeourstatetohavealltheseproperties;insteadwewillshowinthefollowingsectionshowthesepropertiesarerelatedinourspecialsituation. Firstofall,wecanassumethatsuchastatewillberepresentedbyanormalizedvector2H(byconsideringtheGNS-representationassociatedtoourstate).WecanalsoassumethatiscyclicforAsinceotherwisewecouldjustrestrictourselvestoasmallerHilbertspace. Furthermoreitiswellknown,thatthevacuumstateinquantumeldtheoriesconstructedonMinkowski,de-SitterandAnti-de-Sitterspace-timeisinvariantundersymmetriesoftherespectivespace-times.Thereforeitisingeneraldesirableforsuchafundamentalvectorstatetobeinvariantunderisometries.Henceweintroducethefollowingnotion: Alsoanobserverfreelyfallingalongaworldlinedescribedaboveshouldseethispotentialvacuumasenergeticallystableinthesensethattheexpectedvalueoftheenergyinthisstateisminimalamongtheenergyexpectationsinsmallperturbationsof.Themathematicaldescriptionofthispropertyofastateisasfollows(seealso[ 23 ]): 31

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(V;MV)(;M):(4{2) Wewillalwaysmakeclearwhatexactlywemeanbycertainworldlineswhenweimposethepassivityconditiononastate. Anotherrequirementfor(andforthenetofobservables)isthatisfundamentalinthesensethateachotherstatecanbeatleastapproximatelypreparedoutofbyoperationsperformedjustinsomeopenregionOproperlycontainedinthemaximallaboratoryW(;x)ofanobserver.Mathematicallyspeakingthisisthe TheReeh-Schlieder-propertywasrstshownunderverygeneralassumptionstobeafeatureofvacuumstatesforquantumeldtheoriesonMinkowskispace-timein[ 24 ].In[ 2 ]and[ 8 ]theReeh-Schlieder-propertywasshowntobeaconsequenceofcertainstabilityconditionsonastateinde-SitterandAnti-de-Sitterspace-time. Anotherpropertyofstatesdescribingpurethermodynamicalphases(see[ 23 ],[ 13 ])isthefollowing.ItdescribesthefactthatinapurephaseinmeanthecorrelationbetweenobservablesrespectivelylocalizedintworegionsdecayssuitablyfastasafunctionoftheirtimelikeseparationwithrespecttothedynamicsgivenbyM. 1 vanishesinthelimitT!1. 32

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17 ]fordetails).Theseare(fromtheviewofquantumeldtheory)pathologicalcircumstancesthatwewanttoavoid.Thereforeoneintroducesthefollowingnotion: Again,itwillbemadeclearwithrespecttowhichobserverwewanttobenoncentralwhenweimposethisconditiononastate. 5 ]and[ 8 ]. (a) (b) ifiMisnotapositiveoperator,thenfullls(RS)forallwedgesgW(;x)withg2Gaswell. Proof. UsingdeepresultsofPuszandWoronowicz([ 23 ]),thepassivityandtheweakmixingpropertyofentailthatM=0andiseitheragroundstateoraKMS-stateatsomeinversetemperature0forM.ThusCorollary 1 impliesthatisinvariantunderthewholegroupaction. (b) ItsucestoshowtheresultforW(;x),since SinceW(;x)fullls(WA),thereisanopenO2SproperlyincludedinW(;x),i.e.thereisanopenneighborhoodNof12GsuchthatNOW(;x).Thepreimage 33

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ThenKONOW(;x)andK(KO)=K2ONOW(;x).HenceKOisproperlyincludedinW(;x). Consideravector2Hsuchthat (;A)=0(4{6) forallA2A(KO).Wearegoingtoshowthat=0. PickaB2A(O)andanyg2K\N1.Thenforsmalljtj0.Furthermoreweknowthatforg2K\N1wehaveU(g1)BU(g)2A(g1O)A(NO)A(W).Thusthefunctionf:t7!U(g)exp(tM)U(g1)Bisaboundaryvalueofananalyticvectorvaluedfunctioninastripinthecomplexplane,henceitvanisheseverywhereinthatstrip.Inparticular,f(t)vanishesforallt2R. Hencewehave (U(g(t)g1);B)=0(4{9) forallt2R.Repeatingthesameargumentseveraltimeswegetthat 34

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forallti2Randgi2K\N1. Nowweprovethefollowingsmalllemma. Proof. HenceduetotheessentialityofM,itsucestoshowthatM2G0andK2G0,if[M;K]=KforrealnonzeroandK2G.Whiletherstisobvious,forthesecondweargueasfollows:AsNisanopenneighborhoodof1inG,wendn02Nsuchthatexp(K=n)2Nandexp(K=n)2Nforallnn0andhenceexp(K=n)exp(tM=n)exp(K=n)2H Henceweconclude (;U(g)B)=(U(g1);B)=0(4{11) 35

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isperpendiculartowhichimplies=0asiscyclicforA. 1 ])BisognanoandWichmannshowedthatthemodularobjectsassociatedtoavacuumstateandthealgebraofobservablesofawedgeregioninMinkowskispace-time(generatedbyWightmanelds)actgeometricallyuponthenetofobservablealgebras.Thisresulthasbeenextendedtovariousotherspace-times([ 2 ],[ 8 ],[ 21 ]).Also,inlightoftheseresults,thepropertyofastate(andanetofobservablealgebras),thatcertainmodularobjectsactgeometrically,wasproposedasaselectioncriterionforphysicallyrelevantstates([ 7 ],[ 4 ]).InthissectionweshowthatalsounderourgeneralassumptionsthemodularobjectshaveageometricinterpretationandthatthecorrespondingUnruh-Temperaturecanbedetermined. ThefollowingholdsaslongastheLiegroupGhasatleastdimension2. (a) thereisanonzeroeigenvaluefortheadjointactionofMonG;allsucheigenval-ueshavethesamemodulusandisa2 (b) 36

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MA:(4{14) AsMisessentialandGhasdimensiongreaterthan2theremustbesomenonzeroN2Gandsomenonzerowith[M;N]=N.Asseenbefore,properties(P),(WM)and(NC)entailthatisa-KMSstateforsome>0.Also,fromtheprevioustheoremtheReeh-SchliederpropertyholdsforA(W(;x)),andhencethereisanOproperlyincludedinA(W(;x))suchthatiscyclicforA(O). ThereforealltheassumptionsforTheorem 2 arefullled,andweconcludethatindeed=2 (b) ThisisprovedintherstpartoftheproofofTheorem 2 .Thecommutationrelationsfollowfromequation( 3{11 )bypluggingin=2 InthespecialcaseofMinkowskispace-time,thederivedcondition( 4{13 )isknownasmodularcovariance.Withmodularcovarianceasoneoftheassumptions,theauthorsof[ 3 ]derivearepresentationofthePoincaregroupwhichactscovariantlyuponthenet. 8 ]and[ 9 ]onAdSandMinkowsispace-timerespectively.Butonecanatleastformulatethefollowingresultonweaklocality.SimilarresultsinthespecialcaseofAnti-de-Sitterspace-timehadrstbeenpublishedin[ 8 ]. 37

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Supposefurthermorethatthereisacopyofsl(2;R)generatedbyfM;N+;NginsideG,i.e.[M;N]=Nand[N+;N]=Mwith>0.IfnowN:=1 2(N++N)generatesacompactsubgroupofG,i.e.U((t))=exp(tN)and(2 )=1,thenobservablesinA(W(;x))andA(( )W(;x))commuteweakly,meaningthat )W(;x)).ThesameholdsthenalsoforthewedgepairsA(gW(;x))andA(g( )W(;x))foreachg2G. Proof. 2(N+N)and[N;1 2(N+N)]=2M.Hence exp(tN)Mexp(tN)=exp(Ad(tN))(M)=cos(t)M+1 2sin(t)(N+N):(4{17) Settingt= andexponentiating,weget exp Nexpi Mexp N=expi M(4{18) wherevertheseoperatorsaredened.Alsoweknowfromequation( 4{15 )thatJexp(tN)=Jslimn!1(exp(tN+=2)exp(tN=2))n=slimn!1(exp(tN+=2)exp(tN=2))nJ=exp(tN)J: )W(;x))wehave exp( N)Bexp( N)2A((2 )W(;x))=A(W(;x)):(4{19) Hencewecanconclude N)exp( N)Bexp( N)=exp( N)Jexp( N)Bexp( N)(4{20) 38

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N)=exp( N).Goingonusingequations( 4{14 )and( 4{18 )weget N)expi Mexp( N)B=expi MB:(4{21) ThisthennallygivesforallA2A(W(;x))andB2A(( )W(;x))that(;AB)=(;AJJB)=(A;Jexpi MB)= (JA;expi MB)= (expi MA;expi MB)=(expi MB;expi MA)=(;BA): 2.2.2 ;toseethepositivityobservethatbecausethedimensionofthespace-timesisatleast3,thereisa(skew-adjoint)rotationgeneratorNintherespectiveLiealgebrasthatdoesnotcommutewithM.Fromtheconcretecommutationrelations( 2{26 )onehasthen exp(N)Mexp(N)=M(4{22) andthusiMcannotbepositive.Alsooneeasilyseesbydirectinspectionofthecommutationrelations( 2{26 )thattherealwaysisasecondboostgeneratorN0suchthat 39

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2((N+N0)+(NN0))=Ngeneratesacompactgroup.Whiletheresultingconcretestatementsforthecasesofde-SitterandAnti-de-Sitterspacecanbefoundin[ 2 ]and[ 8 ]asanexamplewestatetheresultingtheoremforMinkowskispace-timehere. (a) (b) (c) anobservertravellingalongaworldlinegeneratedbyaboostsubgroupseesasa2-KMSstate; (d) theactionofthemodulargroupfortherightwedgeWRcoincideswiththecorre-spondingboostsubgroupaction. (e) Observablesinawedgealgebracommutewiththeobservablesintheoppositewedgealgebraweakly;heretheoppositewedgeoftherightwedgeistheimageofthiswedgeunderarotationinthe(1;2)-planebyalsocalledtheleftwedgeWL.IngeneralgWRhasoppositewedgegWLforg2G. 18 ]underdierentassumptions.Theretheauthorshowsthatifastateispassivewithrespecttoallgeneratorsoftimeevolutionsofsystemsthatmoveatarbitraryconstantvelocitiesandifinadditiontheunitariesimplementingthetranslationsymmetrybelongtotheobservablealgebra,thenthespectrumconditionholds([ 18 ,Prop.5.1.])andlaterheshows(usingthespectrumcondition,see[ 18 ,Prop.6.1.])thatuniformlyacceleratedobserversseethevacuumasaKMS-stateataxedUnruh-temperature. 40

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6 ]theauthorsshowhowtoconstructamongotherthingsconformallycovariantnetsoflocalalgebrasonaspecialclassofRobertson-Walkerspacetimes.ThesespacetimesareLorentzianwarpedproductstopologicallyequivalenttoRS3wherethemetricintheusualcylindricalcoordinates(t;;;)isoftheformds2=dt2S(t)2d2+sin2()d2+sin2()d2: 14 ]onecandeneanewtimevariableviad dt=1 andinthesenewcoordinatesthemetrictakestheformds2=S2()d2d2+sin2()d2+sin2()d2: 6 ]amethodcalledtransplanta-tionisthenusedtoconstructconformallycovariantnetsonsuchaRobertson-Walkerspacetime. Asdiscussedinsection 2.2.2 thegroupSO(4;1)hasessentialelementsandduetotheconformalequivalencetothede-SittercaseallourresultsareapplicableforthesespecialRobertson-Walkerspacesaswell. 41

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Then-dimensionalEinsteinstaticuniverseEnisagloballyhyperbolicspacetimehavingthetopologicalstructureofRSn1.Wecan(andwillinthefollowing)viewEnasthecylinder inn+1-dimensionalMinkowskispacewhereR>0isaxedradius.ItinheritstheMinkowskimetric InthecaseofE4changingtolocalcylindercoordinates(t;;;)themetrictakestheform Heret2R,0<;
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TheonlywedgesinE3arethetrivialonesandinadditionthesetsoftheformRCwhereCisanopensphericalcapofopeningangle<. TheonlywedgesinE4arethetrivialonesandtheconnectedcomponentsofthesets Proof. NowforeveryskewsymmetricrealnnmatrixMwendamatrixS2O(n),anonnegativeintegerkandrealnumbers1;:::;ksuchthat([ 20 ],[ 26 ]) where Thatmeansthat(t)=t;Sdiag(exp(t1H);:::;exp(tkH);1;:::;1)S1=t;Sdiag(R(t1);:::;R(tk);1;:::;1)S1; 43

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Nowbydenitionawedgegeneratedbythisone-parametergroupisaconnectedcomponentofallthosex=(x0;x1;:::;xn)2Enforwhicht7!(t)xisatimelikecurve. Nowthereholds dtdiag(R(t1);:::;R(tk);1;:::;1)S1t(x1;x2;:::;xn)jj2dt2(4{32) Thatmeanswehaveintheneworthonormalcoordinates(y0;y1;:::;yn)givenbythebasechange(observethatthetimedirectionstaysxed) theexpression Iftherearenopointsy=(y1;y2;:::;yn)2Sn1forwhichkXi=12iy22i1+y22i<2; 4{30 )eachsetofparametersalsogivesapossiblesetofwedges. 44

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orwehavek=1,andthenwehave(sincey21+y22=jjyjj2=1) ThustheonlywedgesinE2aretheemptysetandE2itself. Forn=3weagainhavetwopossibilities.Eitherk=0,whichyields orwehavek=1,inwhichcasewegettheinequality2>21(y21+y22)=21(1y23).Inthecase1=0thisagainhaseitherallofE3assolution(if6=0)oristheemptyset(if=0). Ifontheotherhand16=0theinequalitybecomesy23>12 Thesecorrespondtocylinderswithsphericalcapsasbases. Finallyforn=4therearethreecases.Againk=0yieldsallofE4ortheemptyset.Thecasek=1givesagaintheinequality21(y21+y22)<2,whichyieldsasbeforea 45

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SphericalcapscorrespondingtoapairofwedgesinE3 Lastlythereisthepossibilitythatk=2andthentheinequalityreads Withoutlossofgenerality(duetosymmetry)wecanassumethat2122.Whilethecase21=22yieldsonceagainonlytrivialwedges,thecaseof21>22turnsouttobeequivalenttothepreviousonebecauseusingthefactthaty23+y24=1y21y22wesee,thatthentheinequalitybecomes Thisishenceofthesameformasinthecasek=1andthereforeyieldsthesametypesofwedges. Inallthecaseswheretherearenontrivialwedgeregions,thesearenotspacelikeseparatedandarenotcausallyclosed.Infact,thecausalclosure,evenifoneonlyconsidersgeodesictimelikecurves,isthewholespaceagain. Thesepropertiesmakeitunlikelythatthefoundwedgesandhencethecorrespondingobserverscanserveourpurpose. 46

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Proof. 4 therearenoessentialelementsinthisalgebra. Also,sinceforn=2theLiealgebraRso(2)isabelian,therecanalsobenoessentialelementsinE2. 47

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[1] J.J.BisognanoandE.H.Wichmann.OnthedualityconditionforaHermitianscalareld.J.Math.Phys.,16:985-1007,1975 [2] H.J.BorchersandD.Buchholz.GlobalpropertiesofvacuumstatesindeSitterspace.Ann.Inst.H.PoincarePhys.Theor.,70(1):23-40,1999 [3] R.Brunetti,D.GuidoandR.Longo.Groupcohomology,modulartheoryandspace-timesymmetries.Rev.Math.Phys.7(1):57-71,1995 [4] D.Buchholz,O.Dreyer,M.FlorigandS.J.Summers.Geometricmodularactionandspacetimesymmetrygroups.Rev.Math.Phys.12(4):475-560,2000 [5] D.Buchholz,M.FlorigandS.J.Summers.Thesecondlawofthermodynamics,TCPandEinsteincausalityandanti-deSitterspacetime.ClassicalQuantumGravity,17(2):L31-L37,2000 [6] D.Buchholz,J.MundandS.J.Summers.TransplantationoflocalnetsandgeometricmodularactiononRobertson-Walkerspace-times.Mathematicalphysicsinmathematicsandphysics(Siena,2000),65-81,FieldsInst.Commun.,30,Amer.Math.Soc.,Providence,RI,2001 D.BuchholzandS.J.Summers.AnalgebraiccharacterizationofvacuumstatesinMinkowskispace.Comm.Math.Phys.155:449-458,1993 [8] D.BuchholzandS.J.Summers.Stablequantumsystemsinanti-deSitterspace:causality,independenceandspectralproperties.J.Math.Phys.,45(12):4810-4831,2004 [9] D.BuchholzandS.J.Summers.String-andbrane-localizedcausaleldsinastronglynonlocalmodel.J.Phys.A,40:2147-2163,2007 [10] D.Z.DokovicandK.H.Hofmann.ThesurjectivityquestionfortheexponentialfunctionofrealLiegroups:Astatusreport.J.LieTheory,7:171-199,1997 [11] G.W.GibbonsandS.W.Hawking.Cosmologicaleventhorizons,thermodynamicsandparticlecreation.Phys.Rev.D15,2738-2751,1977 [12] R.Haag.LocalQuantumPhysics.Springer,Berlin,1992. [13] R.Haag,D.KastlerandB.Trych-Pohlmeyer.Stabilityandequlibriumstates.Comm.Math.Phys.38:172-193,1973 [14] S.W.HawkingandG.F.R.Ellis,Thelargescalestructureofspace-time,CanbridgeUniversityPress,Cambridge,1973 [15] M.IseandM.Takeuchi.LieGroupsIandII.AmericanMathematicalSociety.Volume85,1991 48

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R.V.KadisonandJ.R.Ringrose.FundamentalsoftheTheoryofOperatorAlgebras.VolumeI:ElementaryTheory,volume15ofGraduateStudiesinMathematics.AmericanMathematicalSociety,Providence,1997 [17] R.V.KadisonandJ.R.Ringrose.FundamentalsoftheTheoryofOperatorAlgebras.VolumeII:AdvancedTheory,volume16ofGraduateStudiesinMathematics.AmericanMathematicalSociety,Providence,1997 [18] B.Kuckert.Covarientthermodynamicsofquantumsystems:Passivity,semipassivityandtheUnruheect.Ann.Physics,295(2):216-229,2002 [19] B.Kuckert.Movingquantumsystems:particlesversusvacuum.InOperatoralgebrasandmathematicalphysics(Constanta,2001),pages235-241.Theta,Bucharest,2003 [20] E.Maus.AnalytischeGeometrieundLineareAlgebra1.SkriptsammlungUniversitatGottingen,1992 [21] W.MorettiandS.Steidl.ABisognano-Wichmann-liketheoreminacertaincaseofanon-bifurcateeventhorizonrelatedtoanextremeReissner-Nordstromblackhole.Class.QuantumGrav.13:2121-2143,1996 [22] M.MoskowitzandR.Sacksteder.TheexponentialmapanddierentialequationsonrealLiegroups.J.LieTheory,13:291-306,2003 [23] W.PuszandS.L.Woronowicz.PassivestatesandKMSstatesforgeneralquantumsystems.Comm.Math.Phys.,58(3):273-290,1978. [24] H.ReehandS.Schlieder.BemerkungenzurUnitaraquivalenzvonLorentzinvariantenFeldern.NuovoCimemento,22:1051-1068,1961 [25] S.J.SummersandE.H.Wichmann.Concerningtheconditionofadditivityinquantumeldtheory.Ann.Inst.H.PoincarePhys.Theor.47no.2:113-124,1987 [26] G.Thompson.Normalformsforskew-symmetricmaricesandhamiltoniansystemswithrstintegralslinearinmomenta.Proc.Am.Math.Soc.104no.3:910-916,1988 [27] W.G.Unruh.Notesonblackholeevaporation.Phys.Rev.,D14:870-892,1976. [28] M.Wustner.AconnectedLiegroupequalsthesquareoftheexponentialimage.J.LieTheory,13:307-309,2003 49

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RobertStrichwasbornonMay9,1978inMerseburg,Germany.HegrewupmostlyinHalle/Saale,Germany,whereheattendedtheGeorg-Cantor-Gymnasium(Highschool),graduatingin1996.In1996and1997,RobertservedinanursinghomeinNaumburg/Saale,Germany,infulllmentofhisalternativecivilianservice(armyservicereplacement).InOctober1997hestartedhisundergraduatestudiesatGeorg-August-UniversityGoettingen,Germany,inmathematicsandphysics.Robertreceivedhis\Vordiplom"(bachelor'sdegree)inMathematicsinApril1999andhepassedhis\Vordiplom"inphysicsthreemonthslater.HestartedhisgraduateworkunderProf.Dr.D.BuchholzintheoreticalphysicsinOctober1999andgraduatedwitha\Diplom"(master'sdegree)inTheoreticalPhysicsinNovember2002.Thetitleofhismaster'sthesiswas"SymetrienimSkalenlimesderQuantenfeldtheorie"("Symmetriesinthescalinglimitofquantumeldtheory").SinceJanuary2003RobertcompleteddoctoralstudiesinmathematicsattheUniversityofFlorida,partlysupportedbyanAlumniFellowshipoftheUniversity.HisPh.D.-adviserisProf.Dr.S.J.Summers.UponcompletionofhisPh.D.program,RobertwillreturntoGermanytoworkasa(Highschool)teacher. 50