Alexander-Yorke Continuation Numerically finding span class=a-plus-plus emphasis type-underlineallspan the stationary solutions in a spectral model.pdf

Chapter 2Perturbative Construction of Modelsof Algebraic Quantum Field TheoryKlaus Fredenhagen and Katarzyna RejznerAbstract The construction of models of algebraic quantum field theory by renor-malized perturbation theory is reviewed.2.1 IntroductionTheaxiomaticframeworkofAQFTallowsforaqualitativedescriptionofalargeclassof phenomena occurring in particle physics and some parts of solid state physics. Itdoes not, however, yield quantitative predictions, and there is a widespread impres-sionthatonehastoabandontheformalismofAQFTifonewantstomakerealcontactwith experiments. Actually, as explained in Chap.1, up to now no single model ofan interacting AQFT in 4d Minkowski space has been constructed.Butwhatarethealternatives?StandardtextbooksonQFTeitherstartfromcanon-icalquantizationoffreefieldtheoryonFockspaceandtrytoconstructtheinteractingtheory in the interaction picture, or they use the path integral formalism. The canon-ical approach ends up in the Gell-Mann Low formula for the vacuum expectationvalues of time ordered products of fields,0(T(x1).(xn) =?,T0(x1).0(xn)ei?LI(x)d4x?,Tei?LI(x)d4x?,(2.1)where0isthefreefieldtreatedasanoperatorvalueddistributionontheFockspace,LIis theinteraction density treated asa Wickpolynomial of0and isthe vacuumvector of the free theory. The time ordering symbol T means that the products haveto be performed after ordering of the factors according to their time arguments.K. Fredenhagen (B)II. Institut fr Theoretische Physik, 22761 Hamburg, Germanye-mail: klaus.fredenhagendesy.deK. RejznerUniversity of York, Heslington, York YO10 5DD, UKe-mail: kasia.rejzneryork.ac.uk Springer International Publishing Switzerland 2015R. Brunetti et al. (eds.), Advances in Algebraic Quantum Field Theory,Mathematical Physics Studies, DOI 10.1007/978-3-319-21353-8_23132K. Fredenhagen and K. RejznerThepathintegralapproachreinterpretstheGell-MannLowformulaasanintegralover all classical field configurations 0(T(x1)(xn) = Z1?(x1)(xn)ei?L(x)d4xD(2.2)where now L is the full classical Lagrangian, Z is a normalization factor, and D isthought of as the Lebesgue integral over field space.Both versions are only heuristic, and it required the hard and ingenious work ofseveral generations of physicists to turn these formal expressions into unambiguouscomputations. The state of the art is that one can create a formal power series in ?whereeverytermiswelldefined,uptosomeremaininginfraredproblemsoriginatingfrom the integral over Minkowski space in the exponent. The great success of QFTrelies on the fact that already the first few terms of this series yield a good and ofteneven excellent agreement with experimental data.Thepathintegralapproachhastheadvantagethatitisformallysimilartoprobabil-ity theory. Actually, by passing to imaginary time (Wick rotation), one can interpretthe vacuum expectation values of time ordered products of fields as correlation func-tions of a probability distribution (euclidean QFT). In particular, counter-intuitiveproperties of quantum physics as e.g., entanglement do not occur. Moreover, themomentumspaceintegralsintheevaluationofFeynmandiagramshavebetterconver-gence properties. Finally, due to the Osterwalder-Schrader theorem , a Wick rotationback to real time is possible under very general conditions.Thedisadvantageofthepathintegralapproachisthatthenoncommutativeproductof operators, which is crucial for the structure of quantum physics, appears onlyindirectlyintermsofdifferentboundaryvaluesofanalyticfunctions.Inthecanonicalapproach, the operator product is given from the beginning, but there the definitionof the time ordered product is problematic. First of all, it is not well defined as aproduct of operators, since, by the existence of a deterministic time evolution, fieldsatagiventimecanbeexpressedintermsoffieldsatanearliertime,andthusthetimeordering prescription is ambiguous. One may instead define time ordered productsT A(t1) A(tn) of an operator valued function of time t ? A(t) as a symmetricoperator valued function of n time variables such thatT A(t1) A(tn) = A(t1) A(tn) if t1 tn.(2.3)This, however, does not work since the quantum fields are distributions, and thetime ordering prescription would amount to multiplying them with a discontinuousfunction.But there is a way out, as first observed by Stckelberg, further elaborated byBogoliubov and collaborators and finally worked out by Epstein and Glaser (causalperturbation theory). Namely, one may define the time ordered product of n fields asan operator valued distribution which is already known for non-coinciding points.Due to the UV divergences of QFT, the extension to coinciding points is ambiguous,but the crucial observation is that this ambiguity is the same ambiguity which occurs2Perturbative Construction of Models 33in the removal of infinities in approaches where the theory is regularized by theintroduction of a momentum cutoff, and where the theory without cutoff has to befixed by renormalization conditions.Originally,theinsertio。