Timeline for What is the relationship between different types of quantum field theories?
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| Aug 15, 2017 at 5:33 | history | bounty ended | CommunityBot | ||
| Aug 9, 2017 at 13:27 | comment | added | Arnold Neumaier | @tparker: I simply took 2D examples, since they are best understood. | |
| Aug 8, 2017 at 6:06 | comment | added | tparker | (b) Okay, but why are you only considering 2D? General relativity in 2D is highly exceptional, because the Ricci scalar alone captures all the intrinsic curvature information. And TQFTs do have dynamical fields - think about Chern-Simons theory (the Wikipedia article literally has a section titled "dynamics"). (c) Again, I'm not sure why you're only considering 2D. | |
| Aug 8, 2017 at 5:33 | comment | added | Arnold Neumaier | (e) Conventional Lagrangian descriptions are essentially (modulo gauge complications) perturbed free theories. Perturbing CFTs is actually being done and is in this sense a generalization. (c+) In CFT one needs space-time plus points at infinity (or appropriate covers) to have a global action of the confomal group. Wightman's axioms apply only to the restriction to the finite part. | |
| Aug 8, 2017 at 5:31 | comment | added | Arnold Neumaier | @tparker: (a) I required volume preserving diffeomorphisms just to be cautious. It is not clear to me whether scale invariance is preserved or broken in TQFTs; I didn't study it closely enough. (b) In 2D general relativity is topological, I believe. In my view, topological field theories are just those wihout dynamical fields. (c) complex curve = real surface. (d) Possibly, but all questions look very different for different groups. Note that there are also many Galilei invariant nonrelativistic QFTs, for which Wightman's axioms are violated. Also for field theoies on curved backgrounds. | |
| Aug 7, 2017 at 18:46 | comment | added | tparker | Following up on (c): it seems to me that in higher than 2D, the spacetime backgrounds for both "standard" QFTs and CFTs are the same (Minkowski spacetime). | |
| Aug 7, 2017 at 18:44 | comment | added | tparker | (e) Since the conformal group contains the Poincare group, under your definition any CFT should be a special case of a "standard" QFT. If indeed there are CFTs with no Lagrangian description, then by perturbing them to explicitly break the conformal symmetry while preserving the Poincare symmetry, we could presumably come up with a non-corformal, Poincare-invariant QFT with no Lagrangian description. How would we describe such a theory? | |
| Aug 7, 2017 at 18:44 | comment | added | tparker | (d) Since the conformal and homeomorphism groups contain the Poincare group, the ground states of CFTs and TQFTs are Poincare-invariant, so shouldn't the Wightman axioms characterize them as well? | |
| Aug 7, 2017 at 18:38 | comment | added | tparker | A few questions: (a) why must the diffeomorphisms preserve volume in order for a TQFT to remain invariant? (b) Diffeomorphism invariance naturally makes me think of general relativity (which is not just topological) rather than topological QFT. It seems to me that TQFT's are invariant under arbitrary homeomorphisms, which is a much stronger symmetry. Do you agree? (c) By "Riemann curve", do you mean "Riemann surface"? I believe that those are only the natural background spacetimes for the special case of 2D CFTs. | |
| Aug 7, 2017 at 18:29 | history | edited | tparker | CC BY-SA 3.0 |
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| Aug 7, 2017 at 10:39 | history | answered | Arnold Neumaier | CC BY-SA 3.0 |