tag:blogger.com,1999:blog-8883034.post115392701434352155..comments2024-08-24T00:41:35.396+02:00Comments on atdotde: Mastering anomalies?Roberthttp://www.blogger.com/profile/06634377111195468947noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-8883034.post-50913546044597458302022-01-17T12:09:46.889+01:002022-01-17T12:09:46.889+01:00Hi thankss for sharing thisHi thankss for sharing thisAndrew Lacehttps://www.andrewlace.com/noreply@blogger.comtag:blogger.com,1999:blog-8883034.post-1154232062330581562006-07-30T06:01:00.000+02:002006-07-30T06:01:00.000+02:0007 28 06Good post. I am always suspicious when I s...07 28 06<BR/><BR/>Good post. I am always suspicious when I see things like: Master Constraint etc. Nice blog. I just finished reading the seminal paper you and Policastro did a couple of years back on the Fock v. LQG quantization of the string. I really liked the way you explained the singular nature of the state he chose and why that led to a non separable Hilbert space, and non definable momenta etc. Pardon my lateness, I have just began to seriously read LQG and some string theory. Just figured I would stop by to say thanks:)Mahndisa S. Rigmaidenhttps://www.blogger.com/profile/08507292526980604567noreply@blogger.comtag:blogger.com,1999:blog-8883034.post-1154095678813349832006-07-28T16:07:00.000+02:002006-07-28T16:07:00.000+02:00A completely trivial observation, which I made in ...A completely trivial observation, which I made in math-ph/0603024, is this: the class of gauge transformation matters. The existence of anomalies depends on whether you consider polynomials or Laurent polynomials, or more generally on whether your algebra admits a one-sided or two-sided grading (or filtration). <BR/><BR/>If you have a two-sided grading with non-zero global charge generators sitting in the middle, neither of the nilpotent subalgebras can be represented trivially. If you consider a one-sided grading, OTOH, you can combine non-zero charge with local gauge transformations being trivial.<BR/><BR/>E.g., the charge corresponding to conformal symmetry is the anomalous dimension L_0. You can combine L_0 = h != 0 with all L_m with m > 0 acting trivially, but only if you restrict attention to m >= -1. If you consider the Laurent polynomial version of the Virasoro algebra, a non-zero charge L_0 is incompatible with conformal invariance.<BR/><BR/>The same is true for Yang-Mills theory: with Laurent polynomials (in r rather than z), non-zero charge implies that no gauge transformations act trivially and thus that there are anomalies.<BR/><BR/>So why are Laurent polynomials forbidden in Yang-Mills theory but allowed in string theory?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8883034.post-1154033141023368352006-07-27T22:45:00.000+02:002006-07-27T22:45:00.000+02:00Reminds me a bit of a quip which I've heard attrib...Reminds me a bit of a quip which I've heard attributed to Feynman. You could ask: is there "one" equation of the universe or "many"? Feynman remarked that since all equations can be written as E_i=0, then the equation of the universe could be expressed as \sum_i E_i^2=0, i.e as just one equation.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8883034.post-1153996003044433502006-07-27T12:26:00.000+02:002006-07-27T12:26:00.000+02:00Sorry, I take that back. I missed that the C_i ent...Sorry, I take that back. I missed that the C_i enter as squares into M.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-8883034.post-1153995756758627082006-07-27T12:22:00.000+02:002006-07-27T12:22:00.000+02:00Robert wrote:"[...] M annihilates a state exactly ...Robert wrote:<BR/><BR/>"[...] M annihilates a state exactly iff all C_i annihilate the state. [...]"<BR/><BR/>I am guessing you want to demand that M annihilates states not for _some_ a_i but for _all_ a_i. Otherwise the implication does not follow.<BR/><BR/>So there is not just one constraint M instead of n of them, but an n-parameter family of operators M(a_i).<BR/><BR/>No? Maybe I should go and read the paper...Anonymousnoreply@blogger.com