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A. Neumaier
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I'd be satisfied with a proof at the level of rigor of Peskin and Schroeder, say.Demystifier said:I don't know a reference with a rigorous proof
Only superficially similar! Wolff first considers a much more general context of QFTs that are somehow regularized by a cutoff. This includes standard continuum QED with the usual textbook cutoff (i.e., not the causal approach). The specialization to the lattice comes later (in scenario B of ##\phi^4## theory). There he saysDemystifier said:but I know references which make claims similar to mine.
which means (unlike your earlier claim) that the tuning of the constants is already taken into account.Ulli Wolff said:in D⩾4 we encounter the phenomenon of triviality: we find that ##g_R\to 0## whenever we tune for the continuum limit ##am_R→0##. Thus only free non-interacting theories are reached in the strict continuum limit.
This carefully formulated statement is true since it is only a conditional statement. The problem is that the condition can apparently not be satisfied for lattice QED, while you assume without proof that it can. You mix things up with standard renormalized continuum QED, which is trivial when renormalized with an explicit cutoff, but nevertheless matches experiment quite well. But one may not apply it to lattice QED without first having shown that it also matches experiments quite well!Demystifier said:For instance http://www.scholarpedia.org/article/Triviality_of_four_dimensional_phi^4_theory_on_the_lattice
says the following:
"Trivial theories can still be useful as effective theories, if there are parameter values with enough interaction to match experiments and at the same time small enough that effects of the unremoved lattice (or other UV cutoff) are so small as to not contradict experiment.
The paper cited by Wolff for details for ##\phi^4## is
- U. Wolff, Precision check on the triviality of the ##\phi^4## theory by a new simulation method. Physical Review D, 79(2009), 105002.
To get a positive result for QED one would need to do similar simulations and check in which range of lattice spacings one can realize the physical values of electron mass and charge, and which accuracy limit this entails.
''presumably'' means ''it is conjectured'' or "I believe", not "it is justified by solid arguments".Demystifier said:In particular, note the statement above: "Presumably the status of being an effective theory in the above sense is also true for the Standard Model of particle physics". If we ignore the word "presumably", it says that I was essentially right. The word "presumably" indicates that I could have been wrong (mea culpa!), but that there is no proof that I was wrong and that there is a good reason to think that I was right.
You could have looked up the references instead of believing such crucial things upon a word of authority phrased in the subjunctive! And you shouldn't have generalized to statements that are simply not there - making claims for arbitrarily small lattice spacings...Demystifier said:The word "presumably" indicates that I could have been wrong (mea culpa!), but that there is no proof that I was wrong and that there is a good reason to think that I was right.
Since the viability of relativistic Bohmian lattice model depend on the answer of this question, this fits the present thread; so we don't need a new one.Demystifier said:Perhaps we need a new thread about whether trivial theories can serve as effective theories, because it seems that there are arguments for both "yes" and "no".
What are the arguments for "yes"? So far you haven't given any, only assertions that it is so.
I only know arguments for "no", except in the asymptotically free case. The latter applies to QCD, where lattice models are indeed heavily used, but neither to QED nor to the standard model, where they are not used at all, except to demonstrate triviality.