Woit is a typical failing-grade pupil who only isn't too has never been the right textile for college. His inability to larn string theory is a well-known aspect of this fact. But most people inward the basis – too maybe fifty-fifty most of the physics students – misunderstand string theory. But his depression math-related intelligence is oftentimes manifested inward things that are comprehensible to all average or improve students of physics.
Two years ago, Woit argued that
the West Coast metric is the incorrect one.Now, unless yous are a consummate idiot, yous must empathise that the alternative of the metric tensor – either \(({+}{-}{-}{-})\) or \(({-}{+}{+}{+})\) – is a pure convention. The metric tensor \(g^E_{\mu\nu}\) of the kickoff civilization is only equal to minus the metric tensor of the 2nd civilization \(g^W_{\mu\nu}\), i.e. \(g^E_{\mu\nu} = - g^W_{\mu\nu}\), too every controversy or formula written amongst 1 laid of conventions may plainly endure translated to a controversy written inward the other, too vice versa. The equations or statements basically differ exactly yesteryear some signs. The translation from 1 convention to some other is e'er possible too is no to a greater extent than mysterious than the translation from British to US of America English linguistic communication or vice versa.
How stupid do yous accept to endure to misunderstand this point, that at that topographic point can't endure whatsoever "wrong" convention for the sign? And how many people are willing to believe that someone's inability to acquire this simple indicate is compatible amongst the credibility of his comments nearly string theory?
Well, this private has brought us a novel ludicrous triviality of the same type,
Two Pet PeevesWe're told that nosotros mustn't work the same annotation for a Lie grouping too a Lie algebra. Why? Because Tony Zee, Pierre Ramond, too partially Howard Georgi were using the unified annotation too Woit "remember[s] beingness rattling confused nearly this when I kickoff started studying the subject". Well, Mr Woit, yous were confused only because yous accept never been college material. But it's easier to expect for flaws inward Lie groups too Lie algebras than inward your ain worthless existence, right?
Many physicists work the same symbols for Lie groups too the corresponding Lie algebras for a simple reason: they – or at to the lowest degree their demeanour close the identity (or whatsoever other indicate on the grouping manifold) – is completely equivalent. Except for some global behavior, the data nearly the Lie grouping is completely equivalent to the data nearly the corresponding Lie algebra. They're exactly 2 languages to verbalize nearly the same thing.
Just to endure sure, inward my too Dr Zahradník's textbook on linear algebra, nosotros used the separate symbols too I honey the fraktur fonts. In Czech Republic too maybe elsewhere, most people who are familiar amongst similar fonts at all telephone band them "Schwabacher" but strictly speaking, Textura, Rotunda, Schwabacher, too Fraktur are 4 dissimilar typefaces. Schwabacher is older too was replaced yesteryear Fraktura inward the 16th century. In 1941, Hitler decided that at that topographic point were also many typos inward the newspapers too that foreigners couldn't decode Fraktura which diminishes the importance of Deutschland abroad, so he banned Fraktura too replaced it amongst Antiqua.
When nosotros published our textbook, I was bragging nearly the extensive index that was automatically created yesteryear a \({\rm \LaTeX}\) macro. I told somebody: Tell me whatsoever give-and-take too yous volition reckon that nosotros tin detect it inward the index. In front end of several witnesses, the kickoff someone wanted to humiliate me so he said: "A broken bone." So I abruptly responded: "The index doesn't include a 'broken bone' literally but there's a fracture inward it!" ;-) Yes, I did include a comment nearly the font inward the index. You know, the composition of the index was equally simple equally placing the dominance similar \placeInTheIndex{fraktura} inward a given house of the source. After several compilations, the right index was automatically created. I recollect that inward 1993 when I began to type it, 1 compilation of the mass took xv minutes on the PCs inward the calculator lab of our hostel! When nosotros received novel ninety GHz frequency PCs, the speed was almost doubled. ;-)
OK, I don't desire to review uncomplicated things because some readers know them too wouldn't larn anything new, spell others don't know these things too a brief introduction wouldn't assist them. But at that topographic point is a simple human relationship betwixt a Lie algebra too a Lie group. You may obtain the elements of the grouping yesteryear a simple exponentiation of an chemical ingredient of a Lie algebra. For this reason, all the "structure coefficients" \(f_{ij}{}^k\) that recollect the construction of commutators\[
[T_i,T_j] = f_{ij}{}^k T_k
\] comprise the same data equally all the curved shape data nearly the grouping manifold close the identity. The Lie algebra only is the tangent infinite of the grouping manifold some the identity (or whatsoever element) too all the commutators inward the Lie algebra are equivalent to the data nearly the distortions that a projection of the neighborhood of the identity inward the grouping manifold to a apartment infinite causes.
We oftentimes work the same symbols because it's harder to write the gothic fonts. More importantly,
whenever a theory, a solution, or a province of affairs is connected amongst a particular Lie group, it's also connected amongst the corresponding Lie algebra, too vice versa!That's the existent ground why it doesn't affair whether yous verbalize nearly a Lie grouping or a Lie algebra. We work their labels for "identification purposes" too the identification is the same whether yous accept a Lie grouping or a Lie algebra inward mind. Influenza A virus subtype H5N1 rattling simple example:
There be 2 rank-8, dimension-496 heterotic string theories whose approximate groups inward the 10-dimensional spacetime are \(SO(32)\) too \(E_8\times E_8\), respectively.I wrote the judgement inward 2 ways. The kickoff 1 sort of talks nearly the grouping manifolds spell the 2nd talks nearly Lie algebras. The data is plainly almost completely equivalent.
There be 2 rank-8, dimension-496 heterotic string theories whose approximate groups inward the 10-dimensional spacetime are (or accept the Lie algebras) \({\mathfrak so}(32)\) too \({\mathfrak e}_8\oplus {\mathfrak e}_8\), respectively.
Well, except for subtleties – the global choices too identifications inward the grouping manifold that don't touching the demeanour of the grouping manifold inward the vicinity of the identity element. If yous desire to endure careful nearly these subtleties, yous demand to verbalize nearly the grouping manifolds, non exactly Lie algebras, because the Lie algebras "forget" the data nearly these global issues.
So yous mightiness desire to endure accurate too verbalize nearly the Lie groups inward 10 dimensions – too say that the allowed heterotic approximate groups are \(E_8\times E_8\) too \(SO(32)\). However, this seek of yours would truly brand things worse because when yous work a linguistic communication that has the ambition of beingness right nearly the global issues, it's your responsibleness to endure right nearly them, indeed, too chances are that your kickoff guess volition endure wrong!
In particular, the "\(SO(32)\)" heterotic string also contains spinors. So a somewhat smart someone could say that the approximate grouping of that heterotic string is truly \(Spin(32)\), non \(SO(32)\). However, that would endure nearly equally incorrect equally \(SO(32)\) itself – almost no improvement – because the actual perturbative approximate grouping of this heterotic theory is isomorphic to\[
Spin(32) / \ZZ_2
\] where the \(\ZZ_2\) is chosen inward such a way that the grouping is not isomorphic to \(SO(32)\). It's some other \(\ZZ_2\) from the middle isomorphic to \(\ZZ_2\times \ZZ_2\) that allows left-handed spinors but non the right-handed ones! By the way, funnily, the S-dual theory is type I superstring theory whose approximate grouping – arising from Chan-Paton factors of the opened upwards strings – seems to endure \(O(32)\). However, the global shape of the approximate grouping gets modified yesteryear D-particles, the other one-half of \(O(32)\) beyond \(SO(32)\) is broken, too spinors of \(Spin(32)\) are allowed yesteryear the D-particles so non-perturbatively, the approximate grouping of type I superstring theory agrees amongst that of the heterotic S-dual theory including the global subtleties.
(Peter Woit also ludicrously claims that physicists entirely demand iii groups, \(U(1),SU(2), SO(3)\). That may accept been almost right inward the 1920s but it's for certain non truthful inward the 21st century particle physics. If you're an undergraduate amongst plans to do particle physics too someone offers yous to apace larn nearly symplectic or exceptional groups, too maybe a few others, yous shouldn't decline it.)
You don't demand to verbalize nearly string theory to run across similar subtleties. Ask a simple question. What is the approximate grouping of the Standard Model? Well, people volition usually respond \(SU(3)\times SU(2)\times U(1)\). But what they truly hateful is exactly the controversy that the Lie algebra of the approximate grouping is\[
{\mathfrak su}(3) \oplus {\mathfrak su}(2) \oplus {\mathfrak u}(1).
\] Note that the simple, Cartesian \(\times\) production of Lie groups gets translated to the straight \(\oplus\) amount of the Lie algebras – the latter are linear vector spaces. OK, so the controversy that the Lie algebra of the approximate grouping of the Standard Model is the displayed appear inward a higher house is correct.
But if yous accept the ambition to verbalize nearly the precise grouping manifolds, those know nearly all the "global subtleties" too it turns out that \(SU(3)\times SU(2)\times U(1)\) is not isomorphic to the Standard Model approximate group. Instead, the Standard Model approximate grouping is\[
[SU(3)\times SU(2)\times U(1)] / \ZZ_6.
\] The quotient yesteryear \(\ZZ_6\) must endure introduce because all the fields of the Standard Model accept a correlation betwixt the hypercharge \(Y\) modulo \(1/6\) too the spin nether the \(SU(2)\) equally good equally the representation nether the \(SU(3)\). It is thence impossible to gear upwards states that wouldn't endure invariant nether this \(\ZZ_6\) fifty-fifty a priori which agency that this \(\ZZ_6\) acts trivially fifty-fifty on the master copy Hilbert infinite too "it's non there".
The \(\ZZ_6\) must endure divided yesteryear for the same reasons why nosotros usually say that the Standard Model approximate grouping doesn't comprise an \(E_8\) factor. You could also say that there's also an \(E_8\) factor except that all fields transform equally a singlet. ;-) We don't do it – when nosotros say that at that topographic point is a symmetry or a approximate group, nosotros desire at to the lowest degree something to transform nontrivially.
OK, yous reckon that the analysis of the correlations of the discrete charges modulo \(1/6\) may endure subtle. We usually don't tending nearly these details when nosotros desire to determine much to a greater extent than of import things – how many approximate bosons at that topographic point are too what their couplings are. These of import things are given purely yesteryear the Lie algebra which is why our statements nearly the identity of the approximate grouping should generally endure understood equally statements nearly Lie algebras.
At some level, yous may desire to endure picky too verbalize over the global properties of the approximate grouping too correlations. But yous usually don't demand to know these answers for anything else. The noesis of these facts is usually entirely practiced for its ain sake. You can't calculate whatsoever couplings from it, too so on. That's why our sentences should endure assumed non to verbalize nearly these details at all – and/or endure sloppy nearly these details.
(Just to endure sure, the global subtleties, centers of the group, differences betwixt \(SO(N)\) too \(O(N)\) too \(Spin(N)\), differences for fifty-fifty too strange \(N\), or dependence on \(N\) modulo 8, may all the same Pb to interesting physical consequences too consistency checks too several papers of mine, peculiarly nearly the heterotic matrix models, were obsessed amongst these details, too. But this variety of concerns entirely represents a minority of physicists' interests, peculiarly inward the instance of beginners.)
By the way, the 2nd "pet sleeve" yesteryear Woit is that 1 should distinguish existent too complexified versions of the same Lie algebras (and groups). Well, I concur yous should distinguish them. But at some full general analytic or algebraic level, all algebras too other structures should e'er endure understood equally the complexified ones – too entirely afterwards, nosotros may impose some reality weather condition on fields (and thence the allowed symmetries, too). So I would say that to a large extent, fifty-fifty this complaint of Woit reflects his misunderstanding of something of import – the fact that the most of import data nearly the Lie groups is hiding inward the construction constants of the corresponding Lie algebra, too those are identical for all Lie groups amongst the same Lie algebra, too they're also identical for existent too complex versions of the groups.
(By the way, he pretends to endure rattling careful nearly the complexification, but he writes the status for matrix elements of an \(SU(2)\) matrix equally \(\alpha^2+\beta^2=1\) instead of \(|\alpha|^2+|\beta|^2 = 1\). Too bad. You exactly shouldn't insist on people's distinguishing non-essential things nearly the complexification if yous can't fifty-fifty write the essential ones correctly yourself.)
In the futile conversations nearly the foundations of quantum mechanics, I oftentimes hear or read comments like:
Please, don't work the confusing give-and-take "observation" which makes it expect similar quantum mechanics depends on what is an observation too what isn't etc. too it's scary.Well, the ground why my – too Heisenberg's – statements expect similar nosotros are proverb that quantum mechanics depends on observations is that quantum mechanics depends on observations, indeed. So the dissatisfied laymen or beginners truly enquire the physicists to work the linguistic communication that would strengthen the listeners' belief that classical physics is all the same basically right. Except that it's not! We generally work this linguistic communication – including the give-and-take "observation" – because it truly is essential inward the novel framework of physics.
In the same way, failing-grade students such equally Peter Woit may endure constantly bespeak whether a physicist talks nearly a Lie grouping or the corresponding Lie algebra. They are basically complaining:
Georgi, Ramond, Zee, don't work this annotation that looks similar it suggests that the Lie grouping too the Lie algebra are basically the same thing fifty-fifty though they are something completely different.The job is, of course, that the failing-grade students such equally Peter Woit are wrong. Georgi, Ramond, Zee, too others oftentimes work the same symbols for the Lie groups too the Lie algebras because they really are basically the same thing. And it's exactly also bad if yous don't empathise this tight human relationship – basically an equivalence.
I think that at that topographic point be many lousy teachers of mathematics too physics that are similar to Peter Woit. Those don't empathise the substance – what is truly important, what is true. So they focus on what they empathise – arbitrarily invented rules what the students are obliged to parrot for the instructor to experience to a greater extent than important. So the pitiful students who accept such teachers are oftentimes beingness punished for using a dissimilar metric tensor convention in 1 lawsuit or for using a incorrect font for a Lie algebra. These teachers don't empathise the powerfulness too beauty of mathematics too physics too they're working difficult to brand certain that their students won't empathise them, either.
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