Timeline for Extracting each field operator as Wightman fields from a set of time-ordered products satisfying Eckmann-Epstein axioms
Current License: CC BY-SA 4.0
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| May 28 at 5:23 | comment | added | Pedro Lauridsen Ribeiro | (continued) Removing the interaction cutoff (the so-called "adiabatic limit") can only be done for the matrix elements of the S matrix, and even so can only be done in the presence of a mass gap, as shown already by Epstein and Glaser. This is consistent with the underlying hypotheses of the LSZ reduction formulae. In the case of QED, one needs to compute inclusive cross sections directly and show that soft photon contributions cancel out in their adiabatic limit, as done e.g. by the Bloch-Nordsieck approximation. This is pointed in Scharf's book as well. | |
| May 28 at 5:15 | comment | added | Pedro Lauridsen Ribeiro | Scharf doesn't address the LSZ reduction formulae directly, but the Bogolyubov S matrix has them somewhat built in. More precisely, the space-time interaction cutoff ("adiabatic switching") evades Haag's theorem, so one can construct the S matrix at the operator level using the interaction picture as long as the cutoff is not removed. Moreover, the fact that the interaction has compact space-time support makes it easier to relate it to time-ordered products. In fact, Bogolyubov's formula states that the cutoff S matrix is the generating function of the (cutoff) time-ordered products. | |
| May 27 at 18:16 | comment | added | Isaac | QED doesn't have a mass gap in the $U(1)$ sector, so that your issue applies. However, causal perturbation approach (as presented in "Finite QED" by Gunter Scharf) at least constructs the S-matrix rigorously at "perturbative level". Of course the perturbative expansion cannot be Borel-summable and I don't remember if LSZ issue is addressed in that book. | |
| May 27 at 17:40 | comment | added | Pedro Lauridsen Ribeiro | One can upgrade the LSZ reduction formulae to the strong sense (i.e. for Hilbert space vectors) in the case of non-collinear momenta for the multiparticle "in" and "out" arrangements using the Haag-Ruelle QFT scattering theory, provided the mass gap hypothesis still holds. If the latter fails, the S matrix no longer exists in the usual sense due to the presence of soft massless particles (i.e. with arbitrarily small 4-momenta), and formulating a proper QFT scattering theory in this scenario in a rigorous way still is a major open problem. | |
| May 27 at 17:34 | comment | added | Pedro Lauridsen Ribeiro | No, LSZ doesn't violate Haag's theorem if formulated correctly. Namely, if the energy-momentum spectum has a mass gap, the LSZ reduction formulae hold true in the weak sense, that is, for the matrix elements of the S matrix. That's why the time-ordered products only enter there as their vacuum expectation values. Haag's theorem prevents these formulae from holding true in the operator sense, that's why so-called "derivations" of LSZ from the interaction picture (as still done e.g. by QFT books like Peskin-Schroeder) are fundamentally flawed. | |
| May 27 at 15:34 | comment | added | Isaac | One last thing: if my understanding is correct, LSZ formula violates Haag's theorem and therefore is not mathematically consistent (unless you start with some IR cutoff and later take limit). Am I correct? | |
| May 27 at 15:33 | comment | added | Isaac | Thank you very much for your clarification. So...it seems that extracting (or reconstructing) individual field operators "directly" from time-ordered products is quite nontrivial.....and moreover, there is the issue of "consistency" among different paths of construction... | |
| May 24 at 23:57 | comment | added | Pedro Lauridsen Ribeiro | More generally, there is no loss of information to restrict oneself to vacuum expectation values thanks to the Wightman-GNS and Osterwalder-Schrader reconstruction theorems, which allow us to recover the field operators and their products. | |
| May 24 at 23:53 | comment | added | Pedro Lauridsen Ribeiro | (continued) ...and from there one can recover the time-ordered field operator products through its matrix elements (at least as sesquilinear forms). These matrix elements are obtained by time ordering subsets of the space-time arguments of the Wightman functions. Also, the main use for time-ordered products is for the S-matrix elements through the LSZ reduction formulae, and there the former only enter as their vacuum expectation values = time-ordered Green functions, so the latter are the objects that really matter in practice. | |
| May 24 at 23:51 | comment | added | Pedro Lauridsen Ribeiro | Moreover, as I said in the edit, it's not a trivial problem to define time-ordered Green functions because of the renormalization problem. A similar problem happens with time-ordered field products, only worse because you're dealing with (possibly unbounded) operator-valued distributions and hence have even less analytic control over the objects you're trying to construct. The renormalization problem for time-ordered products is easier to manage for the Green functions... | |
| May 24 at 23:45 | comment | added | Pedro Lauridsen Ribeiro | I reversed the question because these time-ordered products have to come from somewhere. In other words, one has to see if one can go "both ways" - particularly, if one start from Wightman fields, define time-ordered Green functions in the above fashion and then proceed as in my answer to get back to the Wightman fields, are the latter the same as the former? | |
| May 24 at 17:07 | comment | added | Isaac | Plus, time-ordered products are "operator-valued" themselves, so I guessed that we can obtain Wightman "fields" (again, operator-valued) from the time-ordered products directly. I wonder why you only consider Wightman "functions" (or VEVs), which are just c-numbered. | |
| May 24 at 17:02 | comment | added | Isaac | Hmm...I do not understand what you mean by "how to define these from the Wightman axioms alone". As in the sentence previous to this quoted one, we start from time-ordered products and want to extract field operators satisfying Wightman axioms. I don't see why you suddenly reverse the direction. | |
| May 19 at 19:07 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
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| May 19 at 19:01 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
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| May 19 at 18:38 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Added explanation addressing point raised in comment
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| May 18 at 22:35 | vote | accept | Isaac | ||
| May 18 at 22:32 | comment | added | Isaac | I see...I thought of any possibility to use the time-ordered products directly since they are already operator-valued distributions. Your approach looks perfectly correct, but seems a bit like a by-pass too. | |
| May 18 at 20:25 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Minor stylistic improvement
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| May 18 at 20:15 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |
Minor stylistic improvement
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| May 18 at 20:09 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 4.0 |