The Landscape at Princeton

Posted on November 24, 2008 by woit

The Princeton Center for Theoretical Science has been having a mini-symposium on the string theory Landscape, and as part of this today hosted a “panel discussion” on the topic. It turns out that there’s not a lot of support for the Landscape in Princeton.

Michael Douglas was the only real Landscape proponent in evidence. He gave a presentation on the state of Landscape studies, beginning by noting that landscapeologists keep finding more possible string vacua. Evidently the 10^500 number always quoted for the number of semi-realistic vacua is no longer operative, with latest estimates more like 10^(10^5) or higher. Douglas acknowledged that this pretty much removes any hope of making predictions by using experiment to fix this freedom and end up with non-trivial constraints. All that’s left is the idea of doing statistical calculations, but there the problem is that you don’t know the measure. He ended up mainly talking about cosmology, partly about the hope that maybe cosmology would constrain the possible vacua, as well as going over various ideas for putting a measure on the space of vacua. None of this really seems to lead anywhere, with all proposed measures having a rather ad hoc character. Douglas advocated just trying to count all vacua with the same weight, since at least one might hope to calculate that.

Tom Banks began by claiming that the effective field theory picture used in the landscape is just not valid. He also pointed out that if the landscape arguments were valid, the landscape would be disconfirmed by experiment, since 10-20 of the Standard Model parameters are unconstrained by anthropics, but take unusually small values, not the random distribution one would expect. Banks takes the attitude that the CC probably has an anthropic explanation, but not particle physics or the SM parameters. He also attacked the usual claims that different vacua are all states of the same theory, arguing that they instead correspond to different theories. Finally, he pointed out that the one prediction that landscapeologists had claimed they would be able to make, the scale of SSYM breaking, hadn’t worked out at all (Douglas now acknowledges that this can’t be done).

Nati Seiberg then argued that, as one gets to deeper and deeper levels of understanding of particle physics, one might reach a level where the only explanations are environmental and have to give up. He sees no reason for that to be the case now, with the main problem that of EWSB, and nothing to indicate that anthropics has anything to do with the problem. Rather, the problem is there because we haven’t had high enough energy accelerators (the LHC should change that), and the problem is hard. He ended by saying that the appropriate response at the present time to anthropic arguments like the Landscape is to just ignore them.

The last speaker was Nima Arkani-Hamed, who I suppose was chosen as a proponent of anthropics. He didn’t live up to this, saying that he pretty much agreed with Seiberg. Like Banks, he finds the anthropic explanation of the CC a plausible reason for why no one has come up with a better idea. He did say that thinking about anthropics and the Landscape has led people to look at some possiblilities for particle physics that otherwise would not have been examined. About the cosmological issues brought up by Douglas, his opinion is that there’s probably no point to thinking about these questions now, doing so might be like trying to come up with a theory of superconductivity in 1903. As far as EWSB goes, he believes the LHC will show us a non-anthropic explanation for its scale.

He explicitly attacked the discussion of measures that Douglas had engaged in as “not fruitful”, saying that he didn’t see any “endgame”, that it was wildly improbably that these could predict anything about particle physics. He also doesn’t see why our vacuum should be typical, joking that some of the least typical people in the world (Linde was mentioned) are most devoted to claiming that our universe is typical. He went on to argue for the currently fashionable enterprise of studying S-matrix amplitudes, arguing that looking at the local physics embodied in Lagrangians was no longer so interesting, that instead one should be trying to understand questions where locality is not manifest.

Finally, Arkani-Hamed ended with the statement that string theory is useful as a way to study questions about quantum gravity, but “unlikely to tell us anything about particle physics”. This is an opinion that has become quite widespread among theorists, but news of this has not gotten out to the popular media, where the idea that string theory has something to do with the LHC keeps coming up.

So, all in all, I found myself in agreement with most of the speakers. On another positive note, the math and physics book collection at Labyrinth (which has replaced the U-store bookstore) has improved dramatically.

Posted in Multiverse Mania | 21 Comments

Notes on BRST V: Highest Weight Theory

Posted on November 23, 2008 by woit

In the last posting we discussed the Lie algebra cohomology [tex]H^*(\mathfrak g, V)[/tex] for [tex]\mathfrak g[/tex] a semi-simple Lie algebra. Because the invariants functor is exact here, this tells us nothing about the structure of irreducible representations in this case. In this posting we’ll consider a different sort of example of Lie algebra cohomology, one that is intimately involved with the structure of irreducible [tex]\mathfrak g[/tex]-representations.

Structure of semi-simple Lie algebras

A semi-simple Lie algebra is a direct sum of non-abelian simple Lie algebras. Over the complex numbers, every such Lie algebra is the complexification [tex]\mathfrak g_{\mathbf C}[/tex] of some real Lie algebra [tex] \mathfrak g[/tex] of a compact, connected Lie group. The Lie algebra [tex] \mathfrak g[/tex] of a compact Lie group [tex]G[/tex] is, as a vector space, the direct sum

[tex] \mathfrak g=\mathfrak t \oplus \mathfrak g/\mathfrak t[/tex]

where [tex] \mathfrak t[/tex] is a commutative sub-algebra (the Cartan sub-algebra), the Lie algebra of [tex]T[/tex], a maximal torus subgroup of [tex]G[/tex].

Note that [tex]\mathfrak t[/tex] is not an ideal in [tex]\mathfrak g[/tex], so [tex]\mathfrak g/\mathfrak t[/tex] is not a subalgebra. [tex]\mathfrak g[/tex] is itself a representation of [tex]\mathfrak g[/tex] (the adjoint representation: [tex]\pi(X)Y= [X,Y][/tex]), and thus a representation of the subalgebra [tex]\mathfrak t[/tex]. On any complex representation [tex]V[/tex] of [tex]\mathfrak g[/tex], the action of [tex]\mathfrak t[/tex] can be diagonalized, with eigenspaces [tex]V^\lambda[/tex] labeled by the corresponding eigenvalues, given by the weights [tex]\lambda[/tex]. These weights [tex]\lambda\in\mathfrak t_{\mathbf C}^*[/tex] are defined by (for [tex]v\in V^\lambda,\ H\in \mathfrak t[/tex]):

[tex]\pi(H)v=\lambda(H)v[/tex]

Complexifying the adjoint representation, the non-zero weights of this representation are called roots, and we have

[tex]\mathfrak g_{\mathbf C}=\mathfrak t_{\mathbf C} \oplus ((\mathfrak g/\mathfrak t)\otimes\mathbf C)[/tex]

The second term on the right is the sum of the root spaces [tex]V^\alpha[/tex] for the roots [tex]\alpha[/tex]. If [tex]\alpha[/tex] is a root, so is [tex]-\alpha[/tex], and one can choose decompositions of the set of roots into “positive roots” and “negative roots” such that:

[tex]\mathfrak n^+=\bigoplus_{+\ roots\ \alpha}(\mathfrak g_{\mathbf C})^\alpha,\ \mathfrak n^-=\bigoplus_{-\ roots\ \alpha}(\mathfrak g_{\mathbf C})^\alpha[/tex]

where [tex]\mathfrak n^+[/tex] (the “nilpotent radical”) and [tex]\mathfrak n^-[/tex] are nilpotent Lie subalgebras of [tex]\mathfrak g_{\mathbf C}[/tex]. So, while [tex]\mathfrak g/\mathfrak t[/tex] is not a subalgebra of [tex]\mathfrak g[/tex], after complexifying we have decompositions

[tex](\mathfrak g/\mathfrak t)\otimes \mathbf C=\mathfrak n^+ \oplus \mathfrak n^-[/tex]

The choice of such a decomposition is not unique, with the Weyl group [tex]W[/tex] (for a compact group [tex]G[/tex], W is the finite group [tex]N(T)/T[/tex], [tex]N(T)[/tex] the normalizer of [tex]T[/tex] in [tex]G[/tex]) permuting the possible choices.

Recall that a complex structure on a real vector space [tex]V[/tex] is given by a decomposition

[tex]V\otimes \mathbf C=W\oplus\overline{W}[/tex]

so the above construction gives [tex]|W|[/tex] different invariant choices of complex structure on [tex]\mathfrak g/\mathfrak t[/tex], which in turn give [tex]|W|[/tex] invariant ways of making [tex]G/T[/tex] into a complex manifold.

The simplest example to keep in mind is [tex]G=SU(2),\ T=U(1),\ W=\mathbf Z_2,[/tex] where [tex]\mathfrak g=\mathfrak{su}(2)[/tex], and [tex]\mathfrak g_{\mathbf C}=\mathfrak{sl}(2,\mathbf C)[/tex]. One can choose [tex]T[/tex] to be the diagonal matrices, with a basis of [tex]\mathfrak t[/tex] given by

[tex]\frac{i}{2}\sigma_3=\frac{1}{2}\begin{pmatrix}i&0\\0&-i\end{pmatrix}[/tex]

and bases of [tex]\mathfrak n^+,\ \mathfrak n^-[/tex] given by

[tex]\frac{1}{2}(\sigma_1+i\sigma_2)=\begin{pmatrix}0&1\\0&0\end{pmatrix},\ \frac{1}{2}(\sigma_1-i\sigma_2)=\begin{pmatrix}0&0\\1&0\end{pmatrix}[/tex]

(here the [tex]\sigma_i[/tex] are the Pauli matrices). The Weyl group in this case just interchanges [tex]\mathfrak n^+ \leftrightarrow \mathfrak n^-[/tex].

Highest weight theory

Irreducible representations [tex]V[/tex] of a compact Lie group [tex]G[/tex] are finite dimensional and correspond to finite dimensional representations of [tex]\mathfrak g_{\mathbf C}[/tex]. For a given choice of [tex]\mathfrak n^+[/tex], such representations can be characterized by their subspace [tex]V^{\mathfrak n^+}[/tex], the subspace of vectors annihilated by [tex]\mathfrak n^+[/tex]. Since [tex]\mathfrak n^+[/tex] acts as “raising operators”, taking subspaces of a given weight to ones with weights that are more positive, this is called the “highest weight” space since it consists of vectors whose weight cannot be raised by the action of [tex]\mathfrak g_{\mathbf C}[/tex]. For an irreducible representation, this space is one dimensional, and we can label irreducible representations by the weight of [tex]V^{\mathfrak n^+}[/tex]. The irreducible representation with highest weight [tex]\lambda[/tex] is denoted [tex]V_{\lambda}[/tex]. Note that this labeling depends on the choice of [tex]\mathfrak n^+[/tex].

Getting back to Lie algebra cohomology, while [tex]H^*(\mathfrak g, V)=0[/tex] for an irreducible representation [tex]V[/tex], the Lie algebra cohomology for [tex]\mathfrak n^+[/tex] is more interesting, with [tex]H^0(\mathfrak n^+, V)=V^{\mathfrak n^+}[/tex], the highest weight space. [tex]\mathfrak t[/tex] acts not just on [tex]V[/tex], but on the entire complex [tex]C(\mathfrak n^+, V)[/tex], in such a way that the cohomology spaces [tex]H^i(\mathfrak n^+,V)[/tex] are representations of [tex]\mathfrak t[/tex], so can be characterized by their weights.

For an irreducible representation [tex]V_\lambda[/tex], one would like to know which higher cohomology spaces are non-zero and what their weights are. The answer to this question involves a surprising “[tex]\rho[/tex] – shift”, a shift in the weights by a weight [tex]\rho[/tex], where

[tex]\rho=\frac{1}{2}\sum_{+\ roots} \alpha[/tex]

half the sum of the positive roots. This is a first indication that it might be better to work with spinors rather than with the exterior algebra that is used in the Koszul resolution used to define Lie algebra cohomology. Much more about this in a later posting.

One finds that [tex]dim\ H^*(\mathfrak n^+,V_\lambda)=|W|[/tex], and the weights occuring in [tex]H^i(\mathfrak n^+,V_\lambda)[/tex] are all weights of the form [tex]w(\lambda +\rho)-\rho[/tex], where [tex]w\in W[/tex] is an element of length [tex]i[/tex]. The Weyl group can be realized as a reflection group action on [tex]\mathfrak t^*[/tex], generated by one reflection for each “simple” root. The length of a Weyl group element is the minimal number of reflections necessary to realize it. So, in dimension 0, one gets [tex]H^0(\mathfrak n^+, V_\lambda)=V^{\mathfrak n^+}[/tex] with weight [tex]\lambda[/tex], but there is also higher cohomology. Changing one’s choice of [tex]\mathfrak n^+[/tex] by acting with the Weyl group permutes the different weight spaces making up [tex]H^*(\mathfrak n^+, V)[/tex]. For an irreducible representation, to characterize it in a manner that is invariant under change in choice of [tex]\mathfrak n^+[/tex], one should take the entire Weyl group orbit of the [tex]\rho[/tex] – shifted highest weight [tex]\lambda[/tex], i.e. the set of weights

[tex]\{w(\lambda +\rho),\ w\in W\}[/tex]

In our [tex]G=SU(2)[/tex] example, highest weights can be labeled by non-negative half integral values (the “spin” [tex]s[/tex] of the representation)

[tex]s=0,\frac{1}{2},1,\frac{3}{2}\2,\cdots[/tex]

with [tex]\rho=\frac{1}{2}[/tex]. The irreducible representation [tex]V_s[/tex] is of dimension [tex]2s+1[/tex], and one finds that [tex]H^0(\mathfrak n^+,V_s)[/tex] is one-dimensional of weight [tex]s[/tex], while [tex]H^1(\mathfrak n^+,V_s)[/tex] is one-dimensional of weight [tex]-s-1[/tex].

The character of a representation is given by a positive integral combination of the weights

[tex]char(V)=\sum_{weights\ \omega} (dim\ V^\omega)\omega[/tex]

(here [tex]V^\omega[/tex] is the [tex]\omega[/tex] weight space). The Weyl character formula expresses this as a quotient of expressions involving weights taken with both positive and negative integral coefficients. The numerator and denominator have an interpretation in terms of Lie algebra cohomology:

[tex]char(V)=\frac{\chi(H^*(\mathfrak n^+, V))}{\chi(H^*(\mathfrak n^+, \mathbf C))}[/tex]

Here [tex]\chi[/tex] is the Euler characteristic: the difference between even-dimensional cohomology (a sum of weights taken with a + sign), and odd-dimensional cohomology (a sum of weights taken with a – sign). Note that these Euler characteristics are independent of the choice of [tex]\mathfrak n^+[/tex].

The material in this last section goes back to Bott’s 1957 paper Homogeneous Vector Bundles, with more of the Lie algebra story worked out by Kostant in his 1961 Lie Algebra Cohomology and the Generalized Borel-Weil Theorem. For an expository treatment with details, showing how one actually computes the Lie algebra cohomology in this case, for U(n) see chapter VI.3 of Knapp’s Lie Groups, Lie Algebras and Cohomology, or for the general case see chapter IV.9 of Knapp and Vogan’s Cohomological Induction and Unitary Representations.

Posted in BRST | 4 Comments

The Map of My Life

Posted on November 22, 2008 by woit

Springer has just published an autobiography of Goro Shimura, entitled The Map of My Life. Shimura’s specialty is the arithmetic theory of modular forms, and he’s responsible for a crucial construction generalizing the modular curve, now known as a “Shimura variety”. The book has a long section at the beginning about his childhood and experiences during the war in Japan. The rest deals mostly with his career as a mathematician, including often unflattering commentary on his colleagues. One of those who comes off the best is André Weil, who encouraged and supported Shimura’s work from the beginning. They both ended up at Princeton, with Weil at the Institute, Shimura at the University.

The book contains extensive discussion of the story of what Shimura calls “my conjecture”. This is the conjecture proved by Wiles and others that implies Fermat’s Last Theorem. In the past, it has conventionally been referred to by various combinations of the names of Shimura, Taniyama and Weil, although more recently the convention seems to be to refer to it as the “modularity theorem”. Shimura also claims credit for conjecturing the “Woods Hole formula” that inspired Atiyah and Bott to prove their general fixed-point theorem.

To get a flavor of the unusual nature of the book, here are some extracts from one section:

Jean-Pierre Serre, whom I had met in Tokyo and Paris, was among the audience, and kept asking questions on the most trivial points, which naturally annoyed me…. Somebody told me that he had become frustrated and even sour. Much later I formed an opinion that he had been frustrated and sour for most of his life. As described in my letter to Freydoon Shahidi, included as Section A2 in this book, he once tried to humiliate me, and as a result gave me the chance to state my conjectures about rational elliptic curves. I now believe that his “attack” on me was caused by his jealousy towards my supposed “success” — my conjectural formula and lectures — at Woods Hole….

In spite of the fact that my mathematical work was little understood by the general mathematical public, I was often the target of jealousy by other mathematicians, which I found strange. I can narrate many stories about this in detail, but that would be unpleasant and unnecessary, and so I mention only one interesting case…

(he then describes an encounter in which Harish-Chandra compares favorably Apery’s result on the irrationality of ζ (3) to Shimura’s work.)

Clearly he [Harish Chandra] thought he finally found something with which he could humiliate me: To his disappointment, he failed. Did he do such a thing to other people? Unlikely, though I really don’t know. But why me? To answer that question, let me first note an incident that happened in the fall of 1964. As I already explained, Atiyah and Bott proved a certain trace formula based on my idea. Bott gave a talk on that topic at the Institute for Advanced Study. In this case he clearly acknowledged their debt to me. In the talk he mentioned that Weyl’s character formula could be obtained as an easy application. Harish-Chandra, who said, “Oh, I thought the matter was the other way around; your formula would follow from Weyl’s formula.” Bott, much disturbed, answered, “I don’t see how that can be done.” After more than ten seconds of silence, Harish-Chandra said “It was a joke.” There was half-hearted laughter, and I thought that his utterance was awkward and did not make much sense even as a joke.

It is futile to psychoanalyze him, but such an experience may allow me to express some of my thoughts. He was insecure and hungry for recognition. That much is the opinion shared by many of those who knew him. He did not know much outside his own field, but he was not aware of his ignorance. In addition, I would think he was highly competitive, though he rarely showed his competitiveness. From his viewpoint I was perhaps one of his competitors who must be humiliated, in spite of the fact that I was not working in his field. Here I may have written more than is necessary, but my concluding point is: He did so, even though I did nothing to him.

The book contains quite a few other unpleasant characterizations of other people, together with assurances that everyone else shared his view of the person in question. I know for a fact that in at least one case this is untrue:

A well known math-physicist Eugene Wigner was in our department, and so I occasionally talked with him. He was pompous and took himself very seriously. That is the impression shared by all those who talked with him.

Wigner was still around when I was a student at Princeton and often came to tea. My impression of him was not at all that which Shimura claims to have been universal.

Update: An exchange between Shimura and Bott about the Woods Hole story can be found here.

Posted in Book Reviews | 25 Comments

Shouldn’t Something Be Done?

Posted on November 19, 2008 by woit

The sheer awfulness of last night’s History Channel program on physics is hard to exaggerate. Here’s some of what Clifford Johnson (one of the participants in the program) wrote on his blog while watching it:

Oh, right… I remember “there are dinosaurs in your living room” thing. Oh dear. It is coming on in 8 minutes here, and so I guess I’ll pour myself a long single malt and prepare myself. I’ve still got faith in Andy, though…

Got to first commercial break. Er… need more whiskey. There’s some good science embedded in there somewhere (e.g., Tegmark talking about inflation, and WMAP results and flatness and so forth (but the laser beams!?)), but the voice-over (among others) is taking serious liberties (like claiming right at the beginning of the show that scientists have evidence that there may be parallel universes…sigh. No, No, No, No. That was really not necessary.)…

Need. More. Whiskey*.

Ok… That’s it. I had a lot of fun shooting my stuff for this, and while I know that it is maybe really not polite to say this, and I really like Andy and the crew who put this together…but I can’t really defend this. They really really should have sent this out in time for us contributors to comment on. By time I saw the rough cut and sent in suggestions it was too late… I presume other sensible people contributing to this such as Ovrut, Lykken, etc, would have liked to have seen a rough cut of this and made remarks. It is really clear that the VO and script was written without a very good understanding of some of the basic concepts in place, and certainly not a careful regard for what’s accurate and what is blatantly misleading. Anyone watching this would think that string theory or M-theory is experimentally verified and a working tool used to study the early universe… I spilled my whiskey when they showed pictures of people working in (what looked like optics) laboratories while talking about “years of research into string theory…”.

I have never ever heard of this “level x” business. I don’t know who says that. But what was with the laser beams?! Where did that come from? Not the burning a hole in the fabric of spacetime and escaping a dying universe to go to another (WHAT?!), but the shooting them out from WMAP in order to measure the flatness of the universe. What was that?! And did you see the red struts between the blue branes that were supposed to be the “extra dimensions holding the branes in place”? What was that?!

This is all so sad because there’s so much, as we say above, good TV that could be made of this material if done right.

Ok. I’m done with this. It’s very sad.

One would like to just ignore something like this and let it fade into obscurity, but the problem is that the History Channel is likely to keep rebroadcasting it for years and years, doing continuing damage to the public understanding of science and the public image of physicists. I don’t really see how an intelligent person can watch this thing and not come away with the impression that theoretical physicists are a bunch of idiots. It seems to me that it would be a good idea for people in general, and the scientists involved in this in particular (Clifford Johnson, Max Tegmark, Michio Kaku, Joe Lykken and Alex Filippenko) to contact the History Channel with a polite request that this program not be rebroadcast, and that steps be taken to avoid creating more disasters of the same kind.

Update: Chad Orzel also saw the program and has some comments about it one of its dumber aspects, beginning with:

Yeesh. That was so actively irritating that I don’t know where to start.

Posted in Multiverse Mania | 39 Comments

Notes on BRST IV: Lie Algebra Cohomology for Semi-simple Lie Algebras

Posted on November 18, 2008 by woit

In this posting I’ll work out some examples of Lie algebra cohomology, still for finite dimensional Lie algebras and representations.

If [tex]G[/tex] is a compact, connected Lie group, it can be thought of as a compact manifold, and as such one can define its de Rham cohomology [tex]H^*_{deRham}(G)[/tex] as the cohomology of the complex

[tex]0\longrightarrow \Omega^0(G)\stackrel{d}\longrightarrow \Omega^1(G)\stackrel{d}\longrightarrow\cdots\stackrel{d}\longrightarrow\Omega^{dim\ G}(G)\longrightarrow 0[/tex]

where [tex]\Omega^i(G)[/tex] are the differential i-forms on [tex]G[/tex] (note, we’ll use complex-valued forms), and [tex]d[/tex] is the deRham differential.

For a compact group, one has a bi-invariant Haar measure [tex]\int_G[/tex], and can use this to “average” over an action of the group on a space. For a representation [tex](\pi, V)[/tex], we get a projection operator [tex]\int_g \Pi (g)[/tex] onto the invariant subspace [tex]V^G[/tex]. This projection operator gives explicitly the invariants functor on [tex]\mathcal C_{\mathfrak g}[/tex]. It is an exact functor, taking exact sequences to exact sequences.

The differential forms [tex]\Omega^*(G)[/tex] give a representation of [tex]G[/tex] in two ways, taking the induced action on forms by pullback, using either left or right translation on the group. If [tex](\Pi(g), \Omega^*(G))[/tex] is the representation by left translations, we can use this to apply our “averaging over [tex]G[/tex]” projection operator to the de Rham complex. This action commutes with the de Rham differential, so we get a sub-complex of left-invariant forms

[tex]0\longrightarrow \Omega^0(G)^G\stackrel{d}\longrightarrow \Omega^1(G)^G\stackrel{d}\longrightarrow\cdots\stackrel{d}\longrightarrow\Omega^{dim\ G}(G)^G\longrightarrow 0[/tex]

Since elements of the Lie algebra [tex]\mathfrak g[/tex] are precisely left-invariant 1-forms, it turns out that this complex is nothing but the Chevalley-Eilenberg complex considered last time to represent Lie algebra cohomology, for the case of the trivial representation. This means we have [tex]C^*(\mathfrak g, \mathbf R)= \Lambda^*(\mathfrak g^*)=\Omega^*(G)^G[/tex], and the differentials coincide. So, what we have shown is that

[tex]H^*(\mathfrak g, \mathbf C)= H^*_{de Rham}(G)[/tex]

If one knows the cohomology of [tex]G[/tex], the Lie algebra cohomology is thus known, but this identity is normally used in the other direction, to find the cohomology of [tex]G[/tex] from that of the Lie algebra. To compute the Lie-algebra cohomology, we can exploit the right-action of G on the group, averaging over the induced action on the left-invariant forms [tex]\Lambda^*(\mathfrak g)[/tex], which again commutes with the differential. We end up with a complex
[tex]0\longrightarrow (\Lambda^0(\mathfrak g^*))^G \longrightarrow (\Lambda^1(\mathfrak g^*))^G\longrightarrow\cdots\longrightarrow (\Lambda^{\dim\ \mathfrak g}(\mathfrak g^*))^G\longrightarrow 0[/tex]

where all the differentials are zero, so the cohomology is given by

[tex]H^*(\mathfrak g,\mathbf C)=(\Lambda^*(\mathfrak g^*))^G=(\Lambda^*(\mathfrak g^*))^{\mathfrak g}[/tex]

the adjoint-invariant pieces of the exterior algebra on [tex]\mathfrak g^*[/tex]. Finding the cohomology has now been turned into a purely algebraic problem in invariant theory. For [tex]G=U(1)[/tex], [tex]\mathfrak g=\mathbf R[/tex], and we have shown that [tex]H^*(\mathbf R, \mathbf C)=\Lambda^*(\mathbf C)[/tex], this is [tex]\mathbf C[/tex] in degrees 0, and 1, as expected for the de Rham cohomology of the circle [tex]U(1)=S^1[/tex]. For [tex]G=U(1)^n[/tex], we get

[tex]H^*(\mathbf R^n, \mathbf C)=\Lambda^*(\mathbf C^n)[/tex]

Note that complexifying the Lie algebra and working with [tex]\mathfrak g_{\mathbf C}=\mathfrak g\otimes \mathbf C[/tex] commutes with taking cohomology, so we get

[tex]H^*(\mathfrak g_{\mathbf C},\mathbf C)= H^*(\mathfrak g,\mathbf C)\otimes \mathbf C[/tex]

Complexifying the Lie algebra of a compact semi-simple Lie group gives a complex semi-simple Lie algebra, and we have now computed the cohomology of these as

[tex]H^*(\mathfrak g_{\mathbf C}, \mathbf C) = (\Lambda^*(\mathfrak g_{\mathbf C}))^{\mathfrak g_\mathbf C}[/tex]

Besides [tex]H^0[/tex], one always gets a non-trivial [tex]H^3[/tex], since one can use the Killing form [tex]< \cdot,\cdot>[/tex] to produce an adjoint-invariant 3-form [tex]\omega_3(X_1,X_2,X_3)=[/tex]. For [tex]G=SU(n)[/tex], [tex]\mathfrak g_{\mathbf C}=\mathfrak{sl}(n,\mathbf C})[/tex], and one gets non-trivial cohomology classes [tex]\omega_{2i+1}[/tex] for [tex]i=1,2,\cdots n[/tex], such that

[tex]H^*(\mathfrak{sl}(n,\mathbf C))=\Lambda^*(\omega_3, \omega_5,\cdots,\omega_{2n+1})[/tex]

the exterior algebra generated by the [tex]\omega_{2i+1}[/tex].

To compute Lie algebra cohomology [tex]H^*(\mathfrak g, V)[/tex] with coefficients in a representation [tex]V[/tex], we can go through the same procedure as above, starting with differential forms on [tex]G[/tex] taking values in [tex]V[/tex], or we can just use exactness of the averaging functor that takes [tex]V[/tex] to [tex]V^G[/tex]. Either way, we end up with the result

[tex]H^*(\mathfrak g, V)=H^*(\mathfrak g, \mathbf C)\otimes V^{\mathfrak g}[/tex]

The [tex]H^0[/tex] piece of this is just the [tex]V^{\mathfrak g}[/tex] that we want when we are doing BRST, but we also get quite a bit else: [tex]dim\ V^{\mathfrak g}[/tex] copies of the higher degree pieces of the Lie algebra cohomology [tex]H^*(\mathfrak g, \mathbf C)[/tex]. The Lie algebra cohomology here is quite non-trivial, but doesn’t interact in a non-trivial way with the process of identifying the invariants [tex]V^{\mathfrak g}[/tex] in [tex]V[/tex].

In the next posting I’ll turn to an example where Lie algebra cohomology interacts in a much more interesting way with the representation theory, this will be the highest-weight theory of representations, in a cohomological interpretation first studied by Bott and Kostant.

Posted in BRST | 2 Comments

Science and Science Fiction

Posted on November 18, 2008 by woit

I just set my DVR to record this evening’s broadcast on the History Channel of Parallel Universes, and noticed that the summary information about the show reads:

Some of the world’s leading physicists believe they have found evidence proving the existence of parallel universes.

One participant in the program is Clifford Johnson, who writes on his blog about how he’s gotten a bad feeling about the project after seeing a rough cut:

I’m a bit worried, if I’m honest, since this is a topic that is so easily seized upon by nutcases and sensible people alike, and is, in various forms, the fodder of so much charlatanism and mystical mumbo-jumbo. Any program in a science series on this sort of material has to be doubly careful -triply- to not give people an excuse to say that “the scientists have verified this”.

Why am I slightly worried? Well, I did not see a final cut of the show and so don’t want to go over the top here, but an early rough cut I saw did seem to potentially suffer from a problem these shows can sometimes have: A collection of practicing scientists are very carefully making comments about what is known, unknown, likely, and unlikely, and so forth, and then much of that care can be undermined by the interspersing of their remarks with clips of every physics documentary filmmaker’s favourite go-to guy who can be relied upon to say wild and wonderful things – Michio Kaku…

I also did notice in the rough cut that there were a couple of places where I’d have preferred a bit more of a reminder that string theory (a framework where some of these speculative ideas about parallel universes has recently been re-discussed in scientific -but yes, still speculative- circles) is itself an unestablished and under-developed theory that could well be cast aside one day in favour of something else. I stressed this point in the course of our shooting, but don’t know how much this got through.

One odd thing about this TV show is that it has already been done, in our universe, with the same name, featuring Michio Kaku, by the BBC back in 2001:

Everything you’re about to read here seems impossible and insane, beyond science fiction. Yet it’s all true.

Scientists now believe there may really be a parallel universe – in fact, there may be an infinite number of parallel universes, and we just happen to live in one of them. These other universes contain space, time and strange forms of exotic matter. Some of them may even contain you, in a slightly different form. Astonishingly, scientists believe that these parallel universes exist less than one millimetre away from us….

For years parallel universes were a staple of the Twilight Zone. Science fiction writers loved to speculate on the possible other universes which might exist. In one, they said, Elvis Presley might still be alive or in another the British Empire might still be going strong. Serious scientists dismissed all this speculation as absurd. But now it seems the speculation wasn’t absurd enough. Parallel universes really do exist and they are much stranger than even the science fiction writers dared to imagine.

It all started when superstring theory, hyperspace and dark matter made physicists realise that the three dimensions we thought described the Universe weren’t enough. There are actually 11 dimensions. By the time they had finished they’d come to the conclusion that our Universe is just one bubble among an infinite number of membranous bubbles which ripple as they wobble through the eleventh dimension.

In his posting, Clifford asks sensible questions about what scientists can do to keep science fiction from taking over science programs. I’ve heard that one mediagenic physicist who was offered a role in this program told them he would only participate if given the right to veto any segment involving him that misrepresented his views. He’s not in the program.

From the opposite end of the science/science-fiction issue, tomorrow in LA there will be an event to launch a new project called The Science and Entertainment Exchange. This is a program (directed by Jennifer Ouellette, who blogs about it here), aimed at improving the portrayal of science by the entertainment industry. There seems to be an increasing amount of media-interest in science-related story lines, and the goal of translating this into getting some higher-quality science out before the public is a worthy one.

One goal of this organization I guess will be to improve the science in science-fiction programs. Since, at least as far as fundamental physics goes, the battle to keep science-fiction out of science appears to have been lost, maybe there should also be an effort to improve the quality and accessibility of the fiction now spreading throughout the physics literature. Some organization could get together creative artists and other media professionals to work on this, helping out programs like “Parallel Universes” as well as popular science books and journal articles. One can’t deny that, at the moment, all of these are pretty sophomoric as creative art, as well as typically not very successful at reaching a mass audience.

There’s a lot of room for advice from visual artists about more appealing string theory vacua for use in particle physics and string cosmology. Surely a good novelist or playwright could come up with a better pre-big bang story line than “colliding branes”. As physics journals like Nuclear Physics B fill up with articles on Boltzmann Brains and the multiverse, with some help from the entertainment industry they could be marketed to a much wider audience, bringing down their cost to university libraries. A lot could be done on the marketing front: for instance it might be a good idea to include some 420 with each issue to help ensure that “mind-blowing” ideas don’t just bore people, but really do blow the mind of the target audience. The possibilities really are limitless…

Update: Just finished watching “Parallel Universes”. Wow. Almost completely free of any real scientific content, and definitely deserves an award as the most idiotic and ludicrous TV show ever made that pretends to have something to do with science. Deep into “what the bleep” territory. The problem is not just Michio Kaku. Everyone involved in the thing should be deeply ashamed of themselves.

Posted in Multiverse Mania | 5 Comments

The Complete Idiot’s Guide to String Theory

Posted on November 17, 2008 by woit

I recently acquired a copy of The Complete Idiot’s Guide to String Theory, by Scientific American’s George Musser, which has been out for a few months now. It’s a popular-level treatment of modern physics, string theory and quantum gravity, much like many other such books, but now in the “Complete Idiot’s” style of lots of cartoons, graphics, material set off in boxes, and short summaries of chapters. As such, I guess it does as good a job as any of putting this material in a form designed to sell it to as many people as possible.

Musser is an enthusiast for just about any and every speculative idea about space and time. Besides string theory, the book covers loop quantum gravity, causal dynamical triangulations, the idea that spacetime is a fluid or a giant computer, and even some ideas I’d never heard of (we live in 3 dimensions because “For the simplest particle, we can make three mutually exclusive measurements”????). The treatment is often breathless, continually going on about how “exciting” all this is. In many ways, the book reads like advertising copy, hyping the promise of ideas (with string theory getting the bulk of the attention) while mostly ignoring or minimizing their problems. For example, the chapter on symmetry contains more than two pages on the “Pros and Cons” of supersymmetry, but this turns out to be just about all “pros” until a short paragraph at the end that begins: “That said, supersymmetry raises some questions that physicists have yet to solve”.

I think I’m tempermentally allergic to this sort of discussion of science, but can see that some people like it and I realize there are arguments in its favor (get those kids and taxpayers excited about science!). Within the limits of such a genre, much of the book does a reasonable job, until the later chapters, where it starts to go off the rails.

There’s a chapter on “parallel universes” which promotes the anthropic multiverse, describing it as “the most promising scientifically” of all possible options. Despite the fact that many string theorists are extremely unhappy with seeing this kind of thing promoted as the received wisdom of their field, Musser claims that:

String theorists originally expected everything to be hard-wired but now think that almost everything is accidental

The scientific advisor for the book was Keith Dienes of the String Vacuum Project, and the list of those most prominently thanked for their help is dominated by landscape proponents Dienes, Bousso, Carroll and Tegmark.

A late chapter entitled “Ten Ways to Test String Theory” goes beyond the overly enthusiastic into the realm of the misleading and the simply untrue. According to Musser, the LHC will test string theory, GLAST will test string theory, Auger will test string theory, Planck will test string theory, LIGO will test string theory, a successor to Super-Kamiokande will test string theory, all the various dark-matter experiments will test string theory, table-top measurements of Newton’s law will test string theory, bouncing laser beams off the moon will test string theory, checking midget galaxies to see if their stars have planets will test string theory, and looking for variation of fundamental constants will test string theory. This is really egregious nonsense.

The next to last chapter is about “The String Wars”, and I appear prominently as “the most persistent and forceful critic of string theory”, paired with Lubos Motl for my “over-the-top” comments. One of the few explicit factual errors in Musser’s book is the claim that my book grew out of this blog (the book was written earlier, but took a long time to get published). The chapter is quite a bit less than even-handed in its discussion of these “wars”, and mainly devoted to shooting down the supposed arguments of critics of string theory. I come in for criticism as endlessly putting forward a “silly deadline” of less than twenty years for string theory to have succeeded in reaching its goals. This straw man argument is conclusively bested, while ignoring the real argument, which is that the huge investment in time and effort put into string theory research has just produced more and more evidence that string theory-based unification is an idea that doesn’t work. The problem is not the magnitude of the rate of progress towards understanding unification, it’s the sign. And, soon I can start going on about 25 years….

Posted in Book Reviews, This Week's Hype | 19 Comments

Notes on BRST III: Lie Algebra Cohomology

Posted on November 16, 2008 by woit

The Invariants Functor

The last posting discussed one of the simplest incarnations of BRST cohomology, in a formalism familiar to physicists. This fits into a much more abstract mathematical context, and that’s what we’ll turn to now.

Given a Lie algebra $\mathfrak g$, we’ll consider Lie algebra representations as modules over $U(\mathfrak g)$. Such modules form a category $\mathcal C_{\mathfrak g}$: what is interesting is not just the objects of the category (the equivalence classes of modules), but also the morphisms between the objects. For two representations $V_1$ and $V_2$ the set of morphisms between them is a linear space denoted $Hom_{U(\mathfrak g)}(V_1,V_2)$. This is just the set of linear maps from $V_1$ to $V_2$ that commute with the action of $\mathfrak g$:

$$Hom_{U(\mathfrak g)}(V_1,V_2)=\{\phi\in Hom_{\mathbf C}(V_1,V_2): \pi(X)\phi=\phi\pi(X)\ \forall X\in \mathfrak g\}$$

Another conventional name for this is the space of intertwining operators between the two representations.

For any representation $V$, its $\mathfrak g$-invariant subspace $V^{\mathfrak g}$ can be identified with the space $Hom_{U(\mathfrak g)}(\mathbf C, V)$, where here $\mathbf C$ is the trivial one-dimensional representation. Having a way to pick out the invariant piece of a representation also allows one to solve the more general problem of picking out the subspace that transforms like a specific irreducible $W$: just find the invariant subspace of $V\otimes W^*$.

The map $V\rightarrow V^{\mathfrak g}$ that takes a representation to its $\mathfrak g$-invariant subspace is a functor: it takes the category $\mathcal C_{\mathfrak g}$ to $\mathcal C_{\mathbf C}$, the category of vector spaces and linear maps ($\mathbf C$ – modules and $\mathbf C$ – homomorphisms). If, instead of taking

$$V\rightarrow V^{\mathfrak g}$$

one takes

$$V\rightarrow V^{\mathfrak h}$$

where [tex]\mathfrak h[/tex] is a Lie subalgebra of [tex]\mathfrak g[/tex], one again gets a functor. If [tex]\mathfrak h[/tex] is an ideal in [tex]\mathfrak g[/tex] (so that [tex]\mathfrak g/\mathfrak h[/tex] is a Lie algebra), then this functor takes [tex]\mathcal C_{\mathfrak g}[/tex] to [tex]\mathcal C_{\mathfrak g/\mathfrak h}[/tex]. This is a simple version of the situation of interest in the case of gauge theory: if [tex]V[/tex] is a state space with [tex]\mathfrak h[/tex] acting as a gauge symmetry, then [tex]V^{\mathfrak h}[/tex] will be the physical subspace, carrying an action of the algebra of operators [tex]U(\mathfrak g/\mathfrak h)[/tex].

Some Homological Algebra

It turns out that when one has a category of modules like [tex]\mathcal C_{\mathfrak g}[/tex], these can usefully be studied by considering complexes of modules, and this is the subject of homological algebra. A complex of modules is a sequence of modules and homomorphisms

$$\cdots\stackrel{\partial}\longrightarrow U\stackrel{\partial}\longrightarrow V \stackrel{\partial}\longrightarrow W\stackrel{\partial}\longrightarrow\cdots$$

such that [tex]\partial\circ\partial =0[/tex]. If the complex satisfies [tex]im\ \partial=ker\ \partial[/tex] at each module, the complex is said to be an “exact complex”.

To motivate the notion of exact complex, note that

$$0\longrightarrow V_0\longrightarrow V \longrightarrow 0$$

is exact iff [tex]V_0[/tex] is isomorphic to [tex]V[/tex], and an exact sequence

$$0\longrightarrow V_1 \longrightarrow V_0 \longrightarrow V \longrightarrow 0$$

represents the module [tex]V[/tex] as the quotient [tex]V_0/V_1[/tex]. Using longer complexes, one gets the notion of a resolution of a module [tex]V[/tex] by a sequence of n modules [tex]V_i[/tex]. This is an exact complex

$$0\longrightarrow V_n \longrightarrow\cdots\longrightarrow V_1 \longrightarrow V_0\longrightarrow V\longrightarrow 0$$

The deviation of a sequence from being exact is measured by its homology $H^*=\frac{ker\ \partial}{im\ \partial}$. Note that if one deletes [tex]V[/tex] from its resolution, the sequence

$$0\longrightarrow V_n \longrightarrow\cdots\longrightarrow V_1 \longrightarrow V_0\longrightarrow 0$$

is exact except at [tex]V_0[/tex]. Indexing the homology in the obvious way, one has [tex]H^i =0[/tex] for [tex]i>0[/tex], and [tex]H^0=V[/tex]. A sequence like this whose only homology is [tex]V[/tex] at [tex]H^0[/tex] is another manifestation of a resolution of [tex]V[/tex].

The reason this construction is useful is that, for many purposes, it allows us to replace a module whose structure we may not understand by a sequence of modules whose structure we do understand. In particular, we can replace a [tex]U(\mathfrak g)[/tex] module [tex]V[/tex] by a sequence of free modules, i.e. modules that are just sums of copies of [tex]U(\mathfrak g)[/tex] itself. This is called a free resolution, and more generally one can work with projective modules (direct summands of free modules).

A functor that takes exact complexes to exact complexes is called an exact functor. Homological invariants of modules come about in cases where one has a functor on a category of modules that is not exact. Applying such a functor to a free or projective resolution gives the homological invariants.

The Koszul Resolution and Lie Algebra Cohomology

There are many possible choices of a free resolution of a module. For the case of [tex]U(\mathfrak g)[/tex] modules, one convenient choice is known as the Koszul (or Chevalley-Eilenberg) resolution. To construct a resolution of the trivial module [tex]\mathbf C[/tex], one uses the exterior algebra on [tex]\mathfrak g[/tex] to make free modules

$$Y_k=U(\mathfrak g)\otimes_{\mathbf C}\Lambda^k(\mathfrak g)$$

and get a resolution of [tex]\mathbf C[/tex]

$$0\longrightarrow Y_{dim\ \mathfrak g}\stackrel{\partial_{dim\ \mathfrak g -1}}\longrightarrow\cdots\stackrel{\partial_1}\longrightarrow Y_1\stackrel{\partial_0}\longrightarrow Y_0\stackrel{\epsilon}\longrightarrow \mathbf C\longrightarrow 0$$

The maps are given by
$$\epsilon : u\in Y_0=U(\mathfrak g) \rightarrow \epsilon (u) = const.\ term\ of\ u$$

and
$$\partial_{k-1} (u\otimes X_1\wedge\cdots\wedge X_k)=$$
$$\sum_{i=1}^k(-1)^{i+1}(uX_i\otimes X_1\wedge\cdots\wedge\hat X_i\wedge\cdots \wedge X_k)$$
$$+\sum_{i<j} (-1)^{i+j}(u\otimes[X_i,X_j]\wedge X_1\wedge\cdots\wedge \hat X_i\wedge\cdots\wedge \hat X_j\wedge\cdots\wedge X_k)$$

To get Lie algebra cohomology, we apply the invariants functor

$$V\longrightarrow V^{\mathfrak g}=Hom_{U(\mathfrak g)}(\mathbf C, V)$$

replacing the trivial representation by its Koszul resolution. This gives us a complex with terms

$$C^k(\mathfrak g, V)=Hom_{U(\mathfrak g)}(Y_k,V)= Hom_{U(\mathfrak g)}(U(\mathfrak g)\otimes \Lambda^k(\mathfrak g),V)$$
$$=Hom_{U(\mathfrak g)}(U(\mathfrak g),Hom_{\mathbf C}(\Lambda^k(\mathfrak g),V))$$
$$=Hom_{\mathbf C}(\Lambda^k(\mathfrak g),V) =V\otimes\Lambda^k(\mathfrak g^*)$$

and induced maps $d_i$

$$0\longrightarrow C^0(\mathfrak g, V)\stackrel{d_0}\longrightarrow C^1(\mathfrak g, V)\cdots\stackrel{d_{dim\ \mathfrak g -1}}\longrightarrow C^{dim\ \mathfrak g}(\mathfrak g, V)\longrightarrow 0$$

The Lie algebra cohomology $H^*(\mathfrak g, V)$ is just the cohomology of this complex, i.e.

$$H^i(\mathfrak g, V)=\frac{ker\ d_i}{im\ d_{i-1}}|_{C^i(\mathfrak g, V)}$$

This is exactly the same definition as that of the BRST cohomology defined in physicist’s formalism in the last posting with $\mathcal H =C^*(\mathfrak g, V)$.

One has $H^0(\mathfrak g, V)=V^{\mathfrak g}$ and so gets the $\mathfrak g$-invariants as expected, but in general the cohomology will be non-zero also in other degrees.

This is all rather abstract, so in the next posting some examples will be worked out, as well as the relationship of all this to the de Rham cohomology of the group. Anthony Knapp’s book Lie Groups, Lie Algebras, and Cohomology is an excellent reference for details on Lie algebra cohomology.

Posted in BRST | 6 Comments

Notes on BRST II: Lie Algebra Cohomology, Physicist’s Version

Posted on November 15, 2008 by woit

My initial plan was to have the second part of these notes be about gauge symmetry and the problems physicists have encountered in handling it, but as I started writing it quickly became apparent that explaining this in any detail would take me into various issues that are quite interesting, but far afield from what I want to get to. So, I hope to get back to this at some point, but for now will just assume that most of my readers know what gauge symmetry is, and that the rest just need to know that:

  • The gauge group is an infinite dimensional Lie group. Locally (on space-time), it looks like a group of maps into a finite dimensional Lie group.
  • The conventional assumption is that physics is invariant under the gauge group, so the gauge group and its Lie algebra should act trivially on physical states.

    • The actual situation is quite a bit more complicated than this, but for now we’ll focus on the simplest version of the mathematical problem that comes up here, and see how the BRST formalism deals with it. This posting will begin explaining one part of this story, starting with the simplest version of BRST cohomology, in a language familiar to physicists. The next posting will deal with Lie algebra cohomology in a more general mathematical context and work out some examples. For more about the material in this posting, see, for instance, Green, Schwarz and Witten, volume I, section 3.2.1, where they go on to apply this to the Virasoro algebra, or these lecture notes from Jose O’Figueroa-Farrill .

    Physicists always begin by choosing a basis, in this case a basis [tex]X_i[/tex] of [tex]\mathfrak g[/tex] satisfying [tex][X_i,X_j]=f_{ij}^kX_k[/tex], where [tex]f_{ij}^k[/tex] are called the structure constants of [tex]\mathfrak g[/tex]. A representation [tex](\pi,V)[/tex] is then a set of linear operators [tex]K_i=\pi (X_i)[/tex] on [tex]V[/tex] satisfying [tex][K_i,K_j]=f_{ij}^kK_k[/tex]. Let [tex]\alpha^i[/tex] be a basis of the dual space [tex]\mathfrak g^*[/tex], dual to the basis [tex]X_i[/tex].

    Now, extend [tex]V[/tex] to [tex]\mathcal =V\otimes \Lambda^* (\mathfrak g^*)[/tex], where [tex]\Lambda^* (\mathfrak g^*)[/tex] is the exterior algebra on [tex]\mathfrak g^*[/tex]. On this space, define the “ghost” operator [tex]c^i[/tex] to be wedge-product with [tex]\alpha^i[/tex], and “anti-ghost” operator [tex]b_i[/tex] to be contraction (interior product) with [tex]X_i[/tex]. These operators satisfy “fermionic” anti-commutation relations

    [tex]\{c^i,c^j\}=\{b_i,b_j\}=0,\ \ \{c^i,b_j\}=\delta^i_j[/tex]

    and one can get all vectors in [tex]\mathcal H[/tex] from linear combinations of decomposable elements of [tex]\mathcal H[/tex] (those given by repeated application of the [tex]c^i[/tex] to the “vacuum vector” [tex]V\otimes \mathbf 1[/tex]).

    The ghost number operator [tex]N=c^ib_i[/tex] on [tex]\mathcal H[/tex] has eigenvectors the decomposable elements, with integer eigenvalues from 0 to dim [tex]\mathfrak g[/tex], given by the number of ghost operators needed to produce the eigenvector from a vacuum vector.

    The BRST operator is given by

    [tex]Q=c^iK_i -\frac{1}{2}f_{ij}^kc^ic^jb_k[/tex]

    which increases the ghost number by one, and has the crucial property of [tex]Q^2=0[/tex] (this comes from the fact that the [tex]f_{ij}^k[/tex] satisfy the Jacobi identity). The BRST cohomology is given by considering the space [tex]ker\ Q[/tex] of elements [tex]\chi[/tex] of [tex]\mathcal H[/tex] that are “BRST-closed”, i.e. satisfy [tex]Q\chi=0[/tex], and identifying two such elements if they are “BRST-exact”, i.e. differ by [tex]Q\lambda[/tex] for some [tex]\lambda[/tex]. So BRST cohomology is defined by

    [tex]H^*_Q(V)=\frac{ker\ Q}{im\ Q}|_{V\otimes\Lambda^*(\mathfrak g^*)[/tex]

    with [tex]H^j_Q(V)[/tex] the component of the BRST cohomology of ghost number [tex]j[/tex].

    A vector [tex]\chi=v\otimes\mathbf 1[/tex] of ghost number zero satisfies [tex]Q\chi =0[/tex] iff and only if [tex]K_iv=0[/tex] for all i, so we can identify [tex]H^0_Q(V)[/tex] with the space [tex]V^\mathfrak g[/tex] of [tex]\mathfrak g[/tex] – invariant vectors in [tex]V[/tex].

    The essence of the BRST method is to replace the problem of finding the invariant subspace [tex]V^{\mathfrak g}[/tex] of a representation [tex]V[/tex] by the problem of finding the degree zero BRST cohomology [tex]H^0_Q(V)[/tex].

    There are two different ways of putting an inner product on [tex]\Lambda^*(\mathfrak g*)[/tex] and thus getting an inner product on [tex]\mathcal H[/tex] ([tex](\pi,V)[/tex] is assumed to be unitary, so preserves a given inner product on [tex]V[/tex]).

  • Given [tex]\omega_1,\omega_2\in \Lambda^*(\mathfrak g*)[/tex], one can define

    [tex]< \omega_1,\omega_2> = \int \omega_1\omega_2\equiv coeff.\ of\ \alpha_1\wedge\cdots\wedge\alpha_{dim\ \mathfrak g}\ in\ \omega_1\wedge\omega_2[/tex]

    (this uses the “fermionic” or “Berezin” integral [tex]\int[/tex], although I have not properly dealt with signs here. ).
    This inner product is indefinite, but it makes the BRST operator [tex]Q[/tex] and ghost-operator [tex]c^i[/tex] self-adjoint.

  • Use an inner product on [tex]\mathfrak g[/tex], e.g. the Killing form for a semi-simple Lie algebra, to identify [tex] \mathfrak g[/tex] and [tex]\mathfrak g^*[/tex]. This gives a Hodge operator [tex]*_{Hodge}[/tex] on [tex]\Lambda^*(\mathfrak g*)[/tex] that takes [tex]\Lambda^i(\mathfrak g*)[/tex] to [tex]\Lambda^{dim\ \mathfrak g -i}(\mathfrak g*)[/tex], and one can define

    [tex]< \omega_1,\omega_2> = \int_G \omega_1\wedge *_{Hodge} \omega_2[/tex]

    (Note, here the integral sign is not Berezin integration, but the usual integration of differential forms over a compact manifold, in this case [tex]G[/tex])

    With this inner product [tex]Q[/tex] and [tex]c^i[/tex] are not self-adjoint on [tex]\mathcal H[/tex]. To get something self-adjoint, one can consider the operator [tex]Q + Q^\dagger[/tex] where [tex]Q^\dagger[/tex] is the adjoint of [tex]Q[/tex], but this operator does not have a definite ghost-number.

    • Posted in BRST | 15 Comments

    Short Bits

    Posted on November 11, 2008 by woit

    More about BRST is on its way, but in the meantime a lot of things have accumulated that might be of interest, so I wanted to do a quick posting about these.

    One of them does have to do with BRST. A correspondent pointed out to me that the 2009 Dannie Heineman prize for Mathematical Physics has been awarded to the four people involved in the original discovery: Carlo Becchi, Alain Rouet, Raymond Stora, and Igor Tyutin.

    Via Garrett Lisi, there’s this collection of photos of the latest Threeasfour collection. It seems that E8 is inspiring not just physicists.

    Over at the n-category cafe, John Baez has a posting about the remarkable publication record of M. S. El Naschie.

    On the experimental HEP front, it looks like the LHC will not be trying again to commission beams until next summer. Minutes of a recent meeting about LHC work are here, an outline of a schedule here.

    SLAC recently hosted an ICFA seminar, with talks available here summarizing the state of various current and proposed accelerator projects. Prospects for a photon-photon collider are discussed here.

    For the latest on the CDF anomaly, Tommaso Dorigo has started a series of detailed posting on the analysis here and here. Matt Strassler has a new paper out about this, including some discussion of possible interpretation of the results in terms of the hidden valley scenario. For more about this topic, see a recent posting at Resonaances.

    There’s a new popular book out about particle physics, Nature’s Blueprint: Supersymmetry and the Search for a Unified Theory of Matter and Force by Dan Hooper. It’s a rather breathless account of how physics is about to be revolutionized by the discovery of supersymmetry at the LHC, very much like Gordon Kane’s 2000 Supersymmetry: Unveiling the Ultimate Laws of Nature. In Kane’s version the LHC was supposed to start up in 2005 and soon discover supersymmetry, in Hooper’s the LHC start-up is moved to 2008. One change since 2000: string theory played a big role in Kane’s book, Hooper pretty much ignores it.

    The December issue of Discover Magazine is out, with Hawking on the cover for a story about the “50 Best Brains in Science”. Terry Tao and Edward Witten are on the list, and the magazine includes a nice appreciation of Witten by John Schwarz, who writes about his experience co-authoring a book on string theory with Witten, explaining that:

    Witten is both deep and fast: After thinkings through the ideas, he can compose an essentially error-free 100 page manuscript, often describing breakthrough original research, on his computer in a day. His papers and lectures set a new standard for clarity of exposition. And he shows no signs of slowing down.

    This year, Witten is working at CERN, and there’s a talk by him scheduled in the string theory seminar there next week, topic TBA. Maybe Jester will report on this.

    In other Discover-related news, Cosmic Variance has announced that they have “sold out to the man”, and will now be going corporate, signing up with Discover to be one of their blogs.

    Also in the new Discover Magazine is a long article promoting the multiverse entitled Science’s Alternative to an Intelligent Creator: the Multiverse Theory. The author’s take on the story is that we really only have two choices: believe in God and intelligent design, or believe in the Landscape. He seems to have gotten this from Susskind:

    The physicist Leonard Susskind once told me that without a multiverse theory, there may be no other explanation for life other than intelligent design.

    The author’s note reports that the article came about through Templeton funding:

    For this issue, he [Tim Folger] traveled to Cambridge, England, as a Templeton-Cambridge Journalism Fellow in Science and Religion to learn what physicists have to say about how the universe seems custom-tailored to favor life.

    In keeping with his theme, Folger quotes many proponents of the multiverse, and only one critic: John Polkinghorne, an ex-physicist and current Anglican priest who has motive to want to keep a role for God.

    There’s some rather out-there stuff at the end from Andrei Linde:

    As for Linde, he is especially interested in the mystery of consciousness and has speculated that consciousness may be a fundamental component of the universe, much like space and time. He wonders whether the physical universe, its laws, and conscious observers might form an integrated whole. A complete description of reality, he says, could require all three of those components, which he posits emerged simultaneously. “Without someone observing the universe,” he says, “the universe is actually dead.”

    The History Channel is running a series on The Universe. Next week the multiverse is being promoted, in an episode Parallel Universes. Here’s the summary:

    Some of the world’s leading physicists believe they have found startling new evidence showing the existence of universes other than our own. One possibility is that the universe is so vast that an exact replica of our Solar System, our planet and ourselves exists many times over. These Doppelganger Universes exist within our own Universe; in what scientist now call “The Multiverse.” Today, trailblazing experiments by state of the art particle colliders are looking for evidence of higher dimensions and Parallel Universes. If proof is found, it will change our lives, our minds, our planet, our science and our universe.

    I learned about this from Clifford Johnson’s blog. He’ll be one of the physicists featured in the episode, as well as in the following one, entitled Light Speed. The next episode, Sex in Space, which will explore the “physiological, psychological and cultural challenges of sex in space” presumably will not be starring any theoretical physicists.

    Update: It seems that selling pseudo-science with the argument “it’s either this or religion” works.

    Update: The links above to the LHC Performance Committee’s site have now been closed to outside access. For the last few years the web-sites of the groups responsible for getting the LHC working have been open to the public, but it looks like there now has been a change of policy. The tentative schedule now inaccessible to the public showed that it is repairs to sector 34 that will determine when they can get going again. The process of getting damaged magnets out of the tunnel, making repairs, getting replacements installed, then testing everything, is what may delay everything into next summer.

    Update: For some commentary on the Strassler paper, see Tommaso Dorigo here. Slashdot features the Discover article, promoting the idea that the string theory landscape is “Science’s Alternative To an Intelligent Creator”.

    Posted in BRST, Experimental HEP News, Multiverse Mania | 52 Comments