4 Locally Covariant Quantum Field Theories

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1 4 Locally Covariant Quantum Field Theories Romeo Brunetti II. Institut für Theoretische Physik, Luruper Chaussee 149, D Hamburg, Germany Ausculta fili verba magistri Benedetto ( ), incipit from Regola. Dedicated to Jacques Bros 4.1 Locality and general covariance In his famous Missed Opportunities paper [8], Freeman Dyson attempted to establish a generally covariant approach to quantum field theory. His aim was to unify the different perspectives on QFT of Tomonaga, Schwinger and Feynman, but the question remained unsettled, mainly because he made some strong assumtpions that were unjustified. But the important point is that he considered the framework of algebraic QFT of Araki, Haag, Kastler [1, 9] as the right one for implementation. A main aim of this article is to convince the reader that, with the appropriate generalizations, Dyson s aim can indeed be fullfilled. The core of the generalization originates from a collaboration with Klaus Fredenhagen [5], dating back to the mid 1990s. The main tool that we shall be using is the language of category theory. The idea that QFT can be described by the use of category theory is certainly not new. In the eighties, something similar was advocated by Sir Atiyah [2], for the sake of topological QFT, and along the same lines, by G. Segal [13] for conformal QFT. Moreover, Dimock has perhaps been the pioneer of the use of categories for the algebraic description of free QFT [7]. Iwishtosubstantiatetheideawithaclaimthat,unlikeotherattempts, originates from physics [6]. The claim is that it would be better to work locally, to describe (relativistic) quantum theories, where here locality is intended in the geometric sense, not as causality. The result is that the best things one can do for QFT in a general covariant setting is to quantize simultaneously

2 40 Romeo Brunetti in all space-times of a certain family. The recipe can be condensed into the following Categorical Imperative: Donothingthatyoucannotdoonanymanifold. Iwillbedealingwiththegeneralstructureatthealgebraicleveland present some examples. For stronger results dealing with the Wightman approach on specific space-times such as de Sitter, you should better look at the papers written by Jacques Bros [3]. 4.2 Quantum Field Theory as a Functor Rigorous implementation of the generally covariant locality principle uses the language of category theory. The following two categories are used : Loc The class of objects obj(loc) is formed by all (smooth) d-dimensional (d 2 is held fixed), globally hyperbolic Lorentzian space-times M which are oriented and time-oriented. Given any two such objects M 1 and M 2,the morphisms ψ hom Loc (M 1,M 2 ) are taken to be the isometric embeddings ψ : M 1 M 2 of M 1 into M 2 but with the following constraints; (i) if γ :[a, b] M 2 is any causal curve and γ(a),γ(b) ψ(m 1 ),then the whole curve must be in the image ψ(m 1 ),i.e.,γ(t) ψ(m 1 ) for all t ]a, b[; (ii) any morphism preserves orientation and time-orientation of the embedded space-time. Composition is composition of maps; the unit element in hom Loc (M,M) is given by the identical embedding id M : M M for any M obj(loc). Obs The class of objects obj(obs) is formed by all C -algebras possessing unit elements, and the morphisms are faithful (injective) unit-preserving -homomorphisms. The composition is again defined as the composition of maps; the unit element in hom Obs (A, A) is for any A obj(obs) given by the identical map id A : A A, A A. One may change categories according to particular needs, as for instance in perturbation theory where instead of C*-algebrasgeneraltopological - algebras are better suited. Or one may use von Neumann algebras, in case particular states are selected. On the other hand, one might consider for Loc bundles over space-times, or one might (in conformally invariant theories) admit conformal embeddings as morphisms. In case one is interested in spacetimes which are not globally hyperbolic, i.e. AdS, one could look at the globally hyperbolic subregions (where attention has to be payed to the causal convexity condition (i) above), and fix the target. Otherwise one might look at a larger family, as for instance the family composed of all asymptotically AdS spacetimes of the same dimension and look for embeddings that remain between globally hyperbolic subregions.

3 4 Locally Covariant Quantum Field Theories 41 Now we define the concept of locally covariant quantum field theory. Definition 1 (i) A locally covariant quantum field theory is a covariant functor A from Loc to Obs and (writing α ψ for A (ψ)) withthecovariance properties α ψ α ψ = α ψ ψ, α idm =id A (M), for all morphisms ψ hom Loc (M 1,M 2 ),allmorphismsψ hom Loc (M 2,M 3 ) and all M obj(loc). (ii) Alocallycovariantquantumfieldtheorydescribedbyacovariantfunctor A is called causal if the following holds: Whenever there are morphisms ψ j hom Loc (M j,m), j =1, 2, sothatthesetsψ 1 (M 1 ) and ψ 2 (M 2 ) are causally separated in M, thenonehas [α ψ1 (A (M 1 )),α ψ2 (A (M 2 ))] = {0}, where the element-wise commutation makes sense in A (M). (iii) We say that a locally covariant quantum field theory given by the functor A obeys the time-slice axiom if α ψ (A (M)) = A (M ) holds for all ψ hom Loc (M,M ) such that ψ(m) contains a Cauchysurface for M. Thus, a quantum field theory is an assignment of C -algebras to (all) globally hyperbolic space-times so that the algebras are identifiable when the spacetimes are isometric, in the indicated way. This is a precise description of the generally covariant locality principle The traditional approach in Minkowski space-time The traditional framework of algebraic quantum field theory, in the Araki- Haag-Kastler sense, on a fixed globally hyperbolic space-time can be recovered from a locally covariant quantum field theory, i.e. from a covariant functor A with the properties listed above. The change is done at the level of the first category by choosing a full subcategory in which a target manifold is fixed and one considers all globally hyperbolic space-times of the target. Indeed, let M be an object in obj(loc). WedenotebyK(M) the set of all open subsets in M which are relatively compact and contain with each pair of points x and y also all g-causal curves in M connecting x and y (cf. condition (i) in the definition of Loc). O K(M), endowedwiththemetricof M restricted to O and with the induced orientation and time-orientation, is a member of obj(loc), and the injection map ι M,O : O M, i.e.theidentical map restricted to O is an element in hom Loc (O, M). Withthisnotationitis easy to prove the following assertion:

4 42 Romeo Brunetti Theorem 1. Let A be a covariant functor with the above stated properties, and define a map K(M) O A(O) A (M) by setting Then the following statements hold: (a) The map fulfills isotony, i.e. A(O). = α ιm,o (A (O)). O 1 O 2 A(O 1 ) A(O 2 ) for all O 1,O 2 K(M). (b) The group G of isometric diffeomorphisms κ : M M (so that κ g = g) preserving orientation and time-orientation, is represented by C -algebra automorphisms α κ : A (M) A (M) such that α κ (A(O)) = A(κ(O)), O K(M). (c) If the theory given by A is additionally causal, then it holds that [A(O 1 ), A(O 2 )] = {0} for all O 1,O 2 K(M) with O 1 causally separated from O 2. These properties are just the basic assumptions of the Araki-Haag-Kastler framework. It is certainly not a big advance to replace three axioms by two. We are confident that the more general description may open the way to a sharper understanding. 4.3 Beyond simple functoriality: Equivalence, Dynamics, Fields, Scattering, and more. The real substance in category theory is the notion of natural transformations. So far, we have scratched only the surface of the new framework, and we will see now that introduction of the natural transformations allows us to reach a higher descriptive level. Let me then remind you of the general definition of natural trasformations: Consider two categories, say A and B, and two functors between them F and G ;thenanaturaltransformationn is a functor over functors, i.e., a function N : F G that assigns to each object a in A an arrow N a of B and such that any arrow A : a a in A yields acommutativediagramsuchas a A a FA F a N a G a G A F a N a G a. The fact that the diagram is commutative is the heart of the matter. One may equivalently describe the naural transformation as a family of functors, i.e. {N a } a obj(a ). Let us then see the usefulness of this notion.

5 4 Locally Covariant Quantum Field Theories It allows a neat description of equivalence between theories, something which is certainly beyond reach in the more traditional formulations. Indeed, if we have two theories F 1, F 2,inthesensedescribedaboveof covariant functors, then we may say the theories are equivalent whenever there exists a natural tranformation {N M } M obj(loc) such that each element of the family is a -bijective map. Examples and counterexamples are easily given, see e.g. [6]. Another, yet to be proved, equivalence may be discussed using the concept of PCT operator as a natural transformation. Here, we say that a theory A satisfies the PCT symmetry whenever a natural transformation Θ exists (again composed of a family of -bijective maps) and links A to the theory A for which the space-times M have opposite time-orientations, notationally M, andwherethec -algebras are complex conjugates, i.e. Θ(A (M)) = A (M). 2. The most important use of natural transformations comes with the definition of locally covariant fields. Here,suchafieldisdescribedbyafamily of (linear) continuous maps A M : D(M) A (M), whered is the functor of a test-function space, and now A has to be interpreted in the larger sense of topological algebras, i.e., such that they conform to the definition above and therefore satisfy a commutativity rule A ψ A M = A N Dψ, where Dψ denotes the push-forward on test function spaces. The importance of such a definition is that it provides a way for any theory to compare observations based on different portions of a single space-time or even better on different space-times. 3. In the general setting described in these notes one natural question is the description of dynamics. Usuallythisisdoneatthealgebraiclevelby invoking space-time isometries, and especially time translations, whose generator is considered to be the Hamiltonian. Here, we do not have such apossibility.indeed,inagenericspace-timewedonothaveanyoneparameter group of time translation. Hence, what can be done? An answer to this question takes advantage of the timeslice property, invoked in the first part, and the notion of a locally covariant field. Indeed, this allows us to compare time-evolutions on different space-times. Namely, let N ± be two isometric neighbourhoods of two Cauchy surfaces in M 1,resp. M 2,andletψ i,± be the corresponding embeddings. Then, using that the. corresponding algebraic embeddings A ψ i,± = αi,± are surjective, one can define an automorphism of the first algebra A (M 1 ) with β = α 1,+ α 1 2,+ α 2, α 1 1,. This automorphism describes the effect of a change of the metric between the two Cauchy surfaces and, because of that, called the relative Cauchy evolution. To see that this is related to the field definition, one may argue as follows. Suppose we introduce, for any space-time M, thesetl (M)

6 44 Romeo Brunetti of Lorentzian metrics that differ from the metric of M only in a compact region. Let then β Mh be the relative Cauchy evolution between M and the space-time obtained by replacing the metric with h L (M). Letnow, β M : L (M) Aut(A (M)) be defined as β M (h) =β Mh. The family {β M } M obj(loc) is then a natural transformation, as the commutation relation can be easily checked. 4. A last example, also in the nonlinear case, is related to the scattering theory. Here we mean scattering in the sense of the local S-matrices approach of Stückelberg-Bogoliubov-Epstein-Glaser, although not necessarily in the perturbative sense. These are unitaries S M () with M Loc and D(M) which satisfy the conditions S M (0) = 1, S M ( + µ + ν) =S M ( + µ)s M (µ) 1 S M (µ + ν) for, µ, ν D(M) such that the supports of and ν can be separated by acauchysurfaceofm with supp in the future of the surface. These S- matrices can be used to define a new quantum field theory. The new theory is locally covariant if the original theory was and if the local S-matrices satisfy the local covariance condition. A perturbative construction was completed in this way by Hollands and Wald [10], based on previous work done by me and Fredenhagen [5] Prolegomena to Generally Covariant Scaling Algebras At this point it is a rather trivial task to generalize to the generally covariant situation the nice description of the renormalization group idea in the algebraic setting pioneered by Buchholz and Verch. Here, as in the previous section, we use natural transformations. One needs a slightly more general version of the locality category to include also a set of physical parameters, ageneralizationthatisrathertrivialandthatwillnotbediscussedindetail. So our starting point will be the family of all pairs given by a globally hyperbolic space-time and a set of physical parameters like masses, charges, couplings etc. as (M,p) (we assume that the parameters are the same for all space-times). The main idea is that we are furnished with a one-parameter family A of covariant functors between the same categories as before (with the locality category slightly generalized according to the above) where ]0, 1], for instance, and is such that it acts as follows: A :(M,p) A (M,p ) where M means that the manifold is still M but its metric gets scaled by. afactor, i.e.,g = 2 g,andtheparametersp are scaled accordingly. Now,

7 4 Locally Covariant Quantum Field Theories 45 we may define scaling transformations as natural ones such that given any element in the supplemented locality category (M,p) we may define maps R (M,p) such that R (M,p) : A (M,p) A (M,p ) is a C -isomorphism, and choosing a different element in Loc the appropriate commutative diagrams for the scaling w.r.t. the isometric embeddings are obtained, i.e., in functional form R (M,p) α ψ = α ψ R (M,p) where α ψ isometrically embeds (M,p ) into (M,p ). In order to compare with the Buchholz-Verch analysis we need some form of dynamics and to this extent we assume the time-slice axiom and make use of the relative Cauchy evolution. It is now very easy to check that we have scaling transformations acting as intertwiners of the relative Cauchy evolutions, R (M,p) β g = β g R (M,p). If we call g the compactly supported perturbation to the metric g and δg = g g their difference, then we may use the following as a substitute for the uniformity condition of Buchholz-Verch, namely lim sup ]0,1] β g (R (M,p) (A)) R (M,p) (A) δg 0 0 (4.1) as long as δg tends to zero in the topology of test functions and for any element A A (M,p). The scaling algebras will be formed by all uniformly bounded elements R (M,p) (A) for which the latter condition holds true. Hence, we consider the functions A (M,p) : ]0, 1] A (M,p ) (4.2) and make all these functions into unital C -algebras denoted by A (M,p) by the usual definitions for linear and multiplicative properties and definition of the norm. We have now again a covariant functor between the same categories as before, but which represents the generally covariant scaling algebras of Buchholz and Verch A and that underlies the algebraic description of the renormalization group, i.e., A :(M,p) A (M,p) where each unital C -algebra contains the above functions (4.2) obeying the uniformity relation (4.1) and to each isometric embedding there is associated aliftingtothenewalgebrasofthepreviousc -monomorphisms. Accordingly, state spaces can be defined in the usual manner and the treatment of the scaling limit proceeds along the same line of the Buchholz and Verch treatment. More details will be presented elsewhere.

8 46 Romeo Brunetti 4.4 Conclusions and Outlook The generally covariant treatment of QFT discussed in this paper is based on the first principle that ensures equivalence of observable algebras based on isometric regions of different space-times. That s all one needs to proceed, at the conceptual level. Important developments are those connected to the works of Hollands and Wald, Verch, Hollands, Ruzzi, and one easily foresees applications of the framework to interesting situations, such as those related to AdS space-time, or in general theories on space-times with boundaries, to the exploitation of the renormalization group at the algebraic level and its possible use towards a clarification of the role of the conformal anomaly in the treatment of theories on asymptotically AdS space-times. Another, perhaps more important topic, is that related to background independent formulation of perturbative quantum gravity. We hope to report on these soon. References 1. H. Araki: Mathematical theory of quantum fields. OxfordUniversityPress,Oxford, M. Atiyah: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68: (1989). 3. J. Bros, H. Epstein and U. Moschella, Towards a General Theory of Quantized Fields on the Anti-de Sitter Space-Time. Communications in Mathematical Physics 231: (2002). 4. R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Communications in Mathematical Physics 208: (2000). 5. R. Brunetti, K. Fredenhagen and M. Köhler, The microlocal spectrum condition and Wick polynomials on curved spacetime. Communications in Mathematical Physics 180:633 (1996). 6. R. Brunetti, K. Fredenhagen and R. Verch, The generally covariant locality principle A new paradigm for local quantum field theory. Communications in Mathematical Physics 237:31-68 (2003). 7. J. Dimock, Algebras of local observables on a manifold. Communications in Mathematical Physics 77:219 (1980) 12; and, Dirac quantum fields on a manifold. Transactions of the American Mathematical Society 269:133 (1982). 8. F. Dyson, Missed Opportunities, Bulletin of the American Mathematical Society 78:635 (1972). 9. R. Haag, Local Quantum Physics. Springer-Verlag, Berlin, Heidelberg, New York, 2nd ed., S. Hollands and R.M. Wald, Local Wick polynomials and time ordered products of quantum fields in curved spacetime. Communications in Mathematical Physics 223: (2001); and, Existence of local covariant time ordered products of quantum field in curved spacetime. Communications in Mathematical Physics 231: (2003). 11. S. Hollands, PCT theorem for the operator product expansion in curved spacetime. Communications in Mathematical Physics 244: (2004).

9 4 Locally Covariant Quantum Field Theories G. Ruzzi, Punctured Haag duality in locally covariant quantum field theories, to be published in Communications in Mathematical Physics (2005), mathph/ G. Segal, The definition of conformal field theory. Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 308, Cambridge Univ. Press, Cambridge, pp , R. Verch, A spin-statistics theorem for quantum fields on curved spacetime manifolds in a generally covariant framework. Communications in Mathematical Physics 223:261 (2001).