From: baez@galaxy.ucr.edu (John Baez) Subject: Re: string theory may not be a QT Date: Thu, 11 Oct 2001 21:54:33 +0000 (UTC) Newsgroups: sci.physics.research Summary: Axiomatic Quantum Field Theory In article , Aaron J. Bergman wrote: >In article <9q06sj$ig4@gap.cco.caltech.edu>, > baez@galaxy.ucr.edu (John Baez) wrote: >> In article , >> Aaron J. Bergman wrote: >> >As has already been mentioned, AQFT appears to be sufficiently general >> >enough so as to be completely useless in formulating a notion of >> >locality. >> I disagree with this: to me, the analysis of "locality" in algebraic >> quantum field theory seems quite perfect in the context of theories >> formulated on a spacetime with a fixed metric. >I'm just reporting what I've heard from others on this group, Dangerous, dangerous... now we see how those completely false rumors get started! Let me attempt to stomp out this wildfire before it spreads. >but the >basic point seemed to be that AQFT allowed way too many fields. With an >infinite number of fields floating around (or an infinite number of >derivatives, for that matter), you can encode an infinite number of things >locally. Right! The basic axioms of algebraic quantum field theory allow for field theories that have "too many degrees of freedom per spacetime point," in some rough sense. These theories are perfectly fine as far as *locality* goes. What makes them "bad" is their *thermodynamic properties*. If your theory has "too many degrees of freedom per spacetime point", it won't have a state of thermal equilibrium at nonzero temperature, even in a box - it'll suffer a version of the "ultraviolet catastrophe". More precisely: it's easy to make up theories which satisfy Axioms A - G in Haag's "Local Quantum Physics", but still violate the Buchholz-Wichmann "nuclearity requirement" which is needed for decent thermodynamic behavior. (The simplest example would be a theory containing infinitely many different free fields, all of the same mass.) The causality axioms (stated in a simplified way on pages 59-60) are axioms E and G. Axiom E, "causality", says roughly that spacelike separated observables commute. Axiom G, "primitive causality", says roughly that no influence travels faster than light. If you look at this book, you'll see Haag takes axioms A-G, restates them more generally using the modern language of C*-algebras, and proves a bunch of basic stuff using them, like the CPT theorem and the spin-statistics theorem. But then, for deeper results, he needs to introduce some extra axiom which says the theory has good thermodynamics properties. He does this in Section V.5, saying: "The principles formulated in Chapter II do not yet entail such basic features of experience as the existence of particles and a reasonable thermodynamics behaviour. [....] Both questions have a common root. We must consider the analogue of classical phase space volumes i.e. the part of the state space corresponding to a simultaneous limitation of energy and space volume. Loosely speaking, finite volumes in classical phase space should correspond to finite dimensional parts of phase space in quantum physics. Starting from the vacuum representation this idea may be implemented in the following way. [....]" ... and then he goes ahead and motivates the Buchholz-Wichmann nuclearity requirement, and proves a bunch of other results using this. I believe the real content of Lubos Motl's earlier remarks on holography is this. We can easily create lower-dimensional field theories from higher-dimensional ones using Rehren's "algebraic holography" idea (cf hep-th/9905179). The lower-dimensional theories should satisfy axioms A - G. However, if we do this starting from a traditional sort of higher-dimensional field theory, e.g. a free field theory, the lower-dimensional theory we get will have "too many degrees of freedom per spacetime point" - it will violate the Buchholz-Wichmann nuclearity requirement. I haven't checked this, and I don't know if anyone has, but it should be possible to check it by explicit calculation if we start with a free field theory, where there are concrete formulas for everything in sight. Clearly the proponents of "algebraic holography" should do this. I don't know if they have or not. If they haven't, well, shame on them! In short, nothing is wrong with locality in algebraic quantum field theory; if something is wrong, it's that proponents of algebraic holography have not investigated the thermodynamic behavior of theories obtained from this construction. >If I'm wrong, please correct me. For example, does a field theory on the >Moyal plane or a noncommutative torus satisfy the axioms of a local AQFT? I'm not an expert on these field theories, but if you are using these noncommutative spaces to play the role of "space" or "spacetime", then they can't satisfy the axioms of AQFT in any obvious way, because AQFT as formulated so far assumes that spacetime is a Lorentzian manifold - and it really uses this assumption in crucial ways. It might be fun to generalize the axioms of algebraic quantum field theory to handle noncommutative spacetimes, but I don't know that anyone has done that yet. >If not, then I withdraw my comment and further submit that, then, string >theory is intrinsically nonlocal. If in some vacua it gives rise to quantum field theories on noncommutative spacetimes, it's even funkier than a nonlocal field theory on an ordinary manifold, which is something people studied long ago.