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- Books and review articles on "Lattice Quantum Field Theory"
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I suppose it could be so, but precedent suggests not. There are some physical phenomena in classical mechanics or statistical mechanics where proving theorems really settled physical questions, so you could be right.
The field theories we have solved, however, were first solved by less rigorous methods more familiar to theoretical physicists. There is a long list of field-theory models the Lee model, integrable theories, QED with monopoles which were well understood by which I mean some observables were calculated without rigorous methods. Rigorous tools were applied, only afterwards, to a small subset of these models. From a conceptual point of view, however, lattice theories have the distinct disadvantage, that you have to redo the proof for every observable you investigate. I digress. My point is, that from the outset it should not matter where we start, as long as the starting point is mathematically sound.
And as such it does have a lot of information not only on the ultraviolet behaviour but also on the infrared behaviour of YM. Anyway, we have to find an approximation of SM where we can control the infrared behaviour. So we should try a few. Your WZW example is just one of many indicators, showing that there are a very large amount of bad approximations. Hence what I said before, that we should throw everything in the ring we have, everything from algebra to analysis, groups to spaces.
If not, can you point me to a document I can read to understand this? I would like to understand the very basic idea you have. We know the S matrix and some off-shell information exactly. I think that most people working on these expect that these results will eventually lead to a rigorous definition. This is done without the benefit of a functional measure or a Lagrangian.
I shall gather my wits and reply to yours asap. In particular, in lattice SU 2 Yang-Mills, everything physically relevant looks real-valued to me. The Wilson action is real, and the Haar measure on gauge fields is real, so the lattice path integral measure is real. The observables should all be generated from characters of products of holonomies, and the characters of SU 2 are real valued. Forgive the restriction to SU 2.
I thought it best to get concrete. So it looks to me like the relevant generating functional is also real-valued. As a side comment: I doubt the Standard Model can be constructed mathematically. The Higgs sector is going to cause trouble. Do you contest, that perturbative calculations about QCD — strongly interacting as it is — are getting harder by the loop? I am just saying that QFT is a complicated problem and we need to view it from with many different angles. Usually insight comes first, then theorems. Constructive FT is usually better for the latter, which I agree is important.
We have to able to do everything. That is, I can see that all our non-rigorous QFT methods can be justified for the SM because it fits with experimental data. But, is there any justification to using the same methods for things like SuperYM? What if, the SM is a very special case where the non-rigorous methods just happen to give the same answer as the hypothetical rigorous formulation of QFT? Our discrepancy shows how differently we think, even if we both seem to both come from the mathematical physics community.
Of course, starting from an approximation which is already obviously supporting a measure will have certain advantages. But other approximations not supporting a measure might have other advantages. While writing this it ocurrs to me, that calculations for the LHC are probably done using lattice gauge theory. Maybe I should reevaluate my starting point. AJ For the sake of being concrete, how about path integrals? So, we know at least one situation in which the path integral is ill-defined but QFT supposedly exists.
I want to make a technical remark that may seem like a triviality, but I think is important for the perception of the mathematical foundations of QFT.
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It is of course true that physicists often use heuristic mathematical manipulations in their calculations. And just as Peter Orland has pointed out, these calculations are replete with insights that are then routinely turned into completely rigorous proofs by mathematical physicists and mathematicians it is also not excluded that the same person could wear more than one of these hats. So there is no particular lack of rigor in the mathematical treatment of QFT. And there are examples where I think rigorous methods do correct erroneous thinking that was guided by more naive heuristic methods.
What is true is that these rigorous proofs do not necessarily establish the exact results that we would want.
Particle Physics, Gauge Theory
By that, I mean that we are not currently able with a few low dimensional exceptions to replace the above mentioned formal power series by actual functions, whose asymptotics these series represent. And until we do, as already brought up by deconstructed and KP, we do not have reliable estimates on the errors that we get by truncating the asymptotic power series. That is, until then, we do not have reliable error bars around our theoretical predictions.
Unfortunately, I think it is rather hard to predict at the moment whether it will be solved by a satisfactory construction of field-theoretic path integrals or by some other means.
Books and review articles on "Lattice Quantum Field Theory"
I think it would pay not to be dogmatic on this point. The path integral is one of these.
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The continuum limit of these lattice approximations has been constructed rigorously in a number of low-dimensional examples, and in these examples, everything works pretty much as the physicists think it should. On grounds of the intransparent and confusing aggregation of definitions and princibles in mechanics the english civil engineer Walter Raleigh Browne tried to accomplish a more specific and clear guideline.
This book introduces the subjects fundamentals that were valid Group Theory and Quantum Mechanics. The German edition of this book appeared in under the title Die gruppentheoretische Methode The German edition of this book appeared in under the title Die gruppentheoretische Methode in der Quantenmechanik. Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory Industrial Mechanics and Maintenance.
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Aldrovandi : Curvature and characteristic classes G. Boffi Trieste Univ. Dabrowski : Introduction to non Commutative Geometry B. Reina: Algebraic geometry and String Theory F. Bruzzo: Introduction to algebraic geometry M. Carfora: Geometry and statistical mechanics of random surfaces G. Dell'Antonio Rome : Functional analysis and quantum mechanics L. Dubrovin : Algebraic aspects of integrable systems G.
Falqui: Lie Algebras F. Bruzzo: Introduction to Algebraic Geometry M. Carfora: Introduction to Differential Geometry L. Dabrowski and G. Falqui: Introduction to Lie Algebras G. Dell'Antonio Rome : Functional analysis and quantum mechanics B. Reina: Infinite dimensional homogeneous spaces G. Bruzzo: Introduction to non commutative geometries. Carfora: Simplicial gravity and dynamical triangulations G. Dubrovin: Topics in analytic theory of differential equations C. Reina: Geometrical aspects of representation theory F.
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