This and That

Posted on December 17, 2008 by woit

This week’s Nature has a nice cover story on Lyn Evans, who has been leading the construction of the LHC. The story mentions one of the problems of his high-profile job:

Evans has found himself the subject of more than one ad hominem attack in physics chat rooms and blogs; he knows because he Googles to find out.

While beam commissioning won’t start up again at the LHC until at least next July, at the Tevatron things have been going extremely well. Last week they set a new luminosity record, accumulating 74 pb-1. For more about this, there’s a posting at Symmetry Breaking.

The Boston Globe has an interview with Lisa Randall, who is writing the libretto for an opera to be entitled “Hypermusic Prologue: A projective opera in seven planes”.

Lieven le Bruyn has a posting about David Mumford and the so-called “Red Book”, the notes for his course on algebraic geometry. This includes a reproduction of Mumford’s picture of Spec Z[x], together with explanations of what all the squiggles mean. From this posting I also learned about a wonderful book on the topic of “Five Centuries of French Mathematics”, available here.

Taking a look at the Theoretical Particle Physics Jobs Rumor Mill, things are looking quite bad for tenure-track jobs in string theory or, more generally, any formal work on quantum field theory. It seems that what US physics departments most want now are cosmologists and “astro-particle physicists”. One place that plans to do a lot of hiring in this area is Arizona State, which is advertising for 8-10 new faculty appointments in these areas, and a similar number of postdocs, to be hired over the next 5 years. All of a sudden the field of “string cosmology” starts to make a lot more sense.

One organization that may need a lot of string theory instructors is the Maharishi Central University which will offer “Unified Field Based Education”:

The groundbreaking curriculum of Maharishi Central University is based upon the most advanced scientific knowledge of our age: the discovery of the Unified Field. During the past quarter century, modern physics has explored progressively more fundamental levels of nature’s functioning at the atomic, nuclear and sub-nuclear scales, culminating in the recent discovery of the Unified Field—a single, universal field of nature’s intelligence at the foundation of the universe.

This Unified Field, or “E8xE8 superstring field,” is the crowning achievement of fifty years of advanced research in quantum gravity theory, and is expressed most concisely in the following, compact Lagrangian, or “super-formula,” presented, for simplicity, in the super-conformal gauge…

The summary of the curriculum goes on to explain how the superstring field “provides the long-sought, mathematically rigorous, interdisciplinary foundation for all the sciences, and for the whole field of academic study,” and that “Without such knowledge, the entire field of education is essentially baseless.”

The plan seems to be to build 50 universities, one in each state, with a construction cost of $16 million each. They’re looking for investors, who are told that each university will enroll 200 students who will pay $45,000/year, generating an income of $9 million per year, so “This will render financing completely risk free.” This money-raising effort is related to the one discussed here.

The first such university is being built at the “exact geographic center” of the US, a point about 12 miles northeast of Smith Center, Kansas. The news from Raja Robert Wynne, Mayor of Maharishi Vedic City and Raja of Invincible New Zealand, Armenia, Kenya, Pakistan, Iraq, Vanuatu, Liberia, and Burundi for the Global Country of World Peace, is that there are 10 buildings now under construction. From an AP article about this, according to founding president John Hagelin

“The ultimate vision is 40,000 students. We’re probably not interested in something smaller than 10,000 students”… He said it would take more than $100 million to start up the university – which he had wanted to have open two years after construction began – and that kind of money isn’t easy to find amid a national banking crisis. Because of that, he said, a more reasonable estimate would be that the university will open in five to 10 years.

The locals seem to not be very happy about all this, worried by the presence of a Mexican construction company with Mexican workers at the site. One such Kansan is the Rev. Dennis Lambert, whose church is nearby, who says “We consider them to be a cult”. The AP article explains that

Lambert was among a small group of people who in 2006 dug up what they believe to be a Hindu idol on a rural property that meditators had once owned about 10 years ago. The figure, a hollow metal animal, contained fake jewels symbolic of the nine planetary gods, he said.

“The fake jewels were crushed and the metal deal was destroyed with heat,” Lambert said. “It was believed to have demonic influence and that’s the way we dealt with it.”

Posted in Experimental HEP News, Uncategorized | 27 Comments

Notes on BRST VII: The Harish-Chandra Homomorphism

Posted on December 15, 2008 by woit

The Casimir element discussed in the last posting of this series is a distinguished quadratic element of the center [tex]Z(\mathfrak g)=U(\mathfrak g)^\mathfrak g[/tex] (note, here [tex]\mathfrak g[/tex] is a complex semi-simple Lie algebra), but there are others, all of which will act as scalars on irreducible representations. The information about an irreducible representation V contained in these scalars can be packaged as the so-called infinitesimal character of [tex]V[/tex], a homomorphism

[tex]\chi_V: Z(\mathfrak g)\rightarrow \mathbf C[/tex]

defined by [tex]zv=\chi_V(z)v[/tex] for any [tex]z\in Z(\mathfrak g)[/tex], [tex]v\in V[/tex]. Just as was done for the Casimir, this can be computed by studying the action of [tex]Z(\mathfrak g)[/tex] on a highest-weight vector.

Note: this is not the same thing as the usual (or global) character of a representation, which is a conjugation-invariant function on the group [tex]G[/tex] with Lie algebra [tex]\mathfrak g[/tex], given by taking the trace of a matrix representation. For infinite dimensional representations [tex]V[/tex], the character is not a function on [tex]G[/tex], but a distribution [tex]\Theta_V[/tex]. The link between the global and infinitesimal characters is given by

[tex]\Theta_V(zf)=\chi_V(z)\Theta_V(f)[/tex]

i.e. [tex]\Theta_V[/tex] is a conjugation-invariant eigendistribution on [tex]G[/tex], with eigenvalues for the action of [tex]Z(\mathfrak g)[/tex] given by the infinitesimal character. Knowing the infinitesimal character gives differential equations for the global character.

The Harish-Chandra Homomorphism

The Poincare-Birkhoff-Witt theorem implies that for a simple complex Lie algebra [tex]\mathfrak g[/tex] one can use the decomposition (here the Cartan subalgebra is [tex]\mathfrak h=\mathfrak t_{\mathbf C}[/tex])

[tex]\mathfrak g=\mathfrak h \oplus \mathfrak n^+ \oplus \mathfrak n^-[/tex]

to decompose [tex]U(\mathfrak g)[/tex] as

[tex]U(\mathfrak g) =U(\mathfrak h) \oplus (U(\mathfrak g)\mathfrak n^+ + \mathfrak n^-U(\mathfrak g))[/tex]

and show that If [tex]z\in Z(\mathfrak g)[/tex], then the projection of z onto the second factor is in [tex]U(\mathfrak g)\mathfrak n^+\cap\mathfrak n^-U(\mathfrak g)[/tex]. This will give zero acting on a highest-weight vector. Defining [tex]\gamma^\prime: Z(\mathfrak g)\rightarrow Z(\mathfrak h)[/tex] to be the projection onto the first factor, the infinitesimal character can be computed by seeing how [tex]\gamma^\prime(z)[/tex] acts on a highest-weight vector.

Remarkably, it turns out that one gets something much simpler if one composes [tex]\gamma^\prime[/tex] with a translation operator

[tex]t_\rho: U(\mathfrak h)\rightarrow U(\mathfrak h)[/tex]

corresponding to the mysterious [tex]\rho\in \mathfrak h^*[/tex], half the sum of the positive roots. To define this, note that since [tex]\mathfrak h[/tex] is commutative, [tex]U(\mathfrak h)=S(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex], the symmetric algebra on [tex]\mathfrak h[/tex], which is isomorphic to the polynomial algebra on [tex]\mathfrak h^*[/tex]. Then one can define

[tex]t_\rho (\phi(\lambda))=\phi(\lambda -\rho)[/tex]

where [tex]\phi\in \mathbf C[\mathfrak h^*][/tex] is a polynomial on [tex]\mathfrak h^*[/tex], and [tex]\lambda\in\mathfrak h^*[/tex].

The composition map

[tex]\gamma=t_\rho\circ\gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex]

is a homomorphism, known as the Harish-Chandra homomorphism. One can show that the image is invariant under the action of the Weyl group, and the map is actually an isomorphism

[tex]\gamma: Z(\mathfrak g)\rightarrow \mathbf C[\mathfrak h^*]^W[/tex]

It turns out that the ring [tex]\mathbf C[\mathfrak h^*]^W[/tex] is generated by [tex]dim\ \mathfrak h[/tex] independent homogeneous polynomials. For [tex]\mathfrak g=\mathfrak{sl}(n,\mathbf C)[/tex] these are of degree [tex]2, 3,\cdots,n[/tex] (where the first is the Casimir).

To see how things work in the case of [tex]\mathfrak g=\mathfrak{sl}(2,\mathbf C)[/tex], where there is one generator, the Casimir [tex]\Omega[/tex], recall that

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)[/tex]

so one has
[tex]\gamma^\prime(\Omega)= \frac{1}{4}(h +\frac{1}{2}h^2)[/tex]

Here [tex]t_\rho(h)=h-1[/tex], so

[tex]\gamma(\Omega)=\frac{1}{4}((h-1)+\frac{1}{2}(h-1)^2)=\frac{1}{8}(h^2-1)[/tex]

which is invariant under the Weyl group action [tex]h\rightarrow -h[/tex].

Once one has the Harish-Chandra homomorphism [tex]\gamma[/tex], for each[tex] \lambda\in\mathfrak h^*[/tex] one has a homomorphism

[tex]\chi_{\lambda}: z\in Z(\mathfrak g)\rightarrow \chi_\lambda(z)=\gamma(z)(\lambda)\in \mathbf C[/tex]

and the infinitesimal character of an irreducible representation of highest weight [tex]\lambda[/tex] is [tex]\chi_{\lambda + \rho}[/tex].

The Casselman-Osborne Lemma

We have computed the infinitesimal character of a representation of highest weight [tex]\lambda[/tex] by looking at how [tex]Z(\mathfrak g)[/tex] acts on [tex]V^{\mathfrak n^+}=H^0(\mathfrak n^+,V)[/tex]. On [tex]V^{\mathfrak n^+}, z\in Z(\mathfrak g)[/tex] acts by

[tex]z\cdot v = \chi_V(z)v[/tex]

This space has weight [tex]\lambda[/tex], so [tex]U(\mathfrak h)=\mathbf C[\mathfrak h^*][/tex] acts by evaluation at [tex]\lambda[/tex]

[tex]\phi\cdot v=\phi(\lambda)v[/tex]

These two actions are related by the map [tex]\gamma^\prime: Z(\mathfrak g)\rightarrow U(\mathfrak h)[/tex] and we have

[tex]\chi_V(z)=(\gamma^\prime(z))(\lambda)=(\gamma(z))(\lambda + \rho)[/tex]

It turns out that one can consider the same question, but for the higher cohomology groups [tex]H^k(\mathfrak n^+,V)[/tex]. Here one again has an action of [tex]Z(\mathfrak g)[/tex] and an action of [tex]U(\mathfrak h)[/tex]. [tex]Z(\mathfrak g)[/tex] acts on k-cochains [tex]C^k(\mathfrak n^+,V)= Hom_{\mathbf C}(\Lambda^k\mathfrak n^+,V)[/tex] just by acting on [tex]V[/tex], and this action commutes with [tex]d[/tex] so is an action on cohomology. [tex]U(\mathfrak h)[/tex] acts simultaneously on [tex]\mathfrak n^+[/tex] and on [tex]V[/tex], again in a way that descends to cohomology. The content of the Casselman-Osborne lemma is that these two actions are again related in the same way by the Harish-Chandra homomorphism. If [tex]\mu[/tex] is a weight for the [tex]\mathfrak h[/tex] action on [tex]H^k(\mathfrak n^+,V)[/tex], then

[tex]\chi_V(z)=(\gamma^\prime(z))(\mu)=(\gamma(z))(\mu + \rho)[/tex]

Since [tex]\chi_V(z)=(\gamma(z))(\lambda + \rho)[/tex], one can use this equality to show that the weights occurring in [tex]H^k(\mathfrak n^+,V)[/tex] must satisfy

[tex](\mu +\rho)=w(\lambda + \rho)[/tex]

and thus

[tex]\mu=w(\lambda + \rho)-\rho[/tex]

for some element [tex]w\in W[/tex]. Non zero elements of [tex]H^k(\mathfrak n^+,V)[/tex] can be constructed with these weights, and the Casselman-Osborne lemma used to show that these are the only possible weights. This gives the computation of [tex]H^k(\mathfrak n^+,V)[/tex] as an [tex]\mathfrak h[/tex] – module referred to earlier in these notes, which is known as Kostant’s theorem (the algebraic proof was due to Kostant, an earlier one using geometry and sheaf cohomology was due to Bott).

For more details about this and a proof of the Casselman-Osborne lemma, see Knapp’s Lie Groups, Lie Algebras and Cohomology, where things are worked out for the case of [tex]\mathfrak g=\mathfrak{gl}(n,\mathbf C)[/tex] in chapter VI.

Generalizations

So far we have been considering the case of a Cartan subalgebra [tex]\mathfrak h\subset \mathfrak g[/tex], and its orthogonal complement with a choice of splitting into two conjugate subalgebras, [tex]\mathfrak n^+ \oplus \mathfrak n^-[/tex]. Equivalently, we have a choice of Borel subalgebra [tex]\mathfrak b\subset \mathfrak g[/tex], where [tex]\mathfrak b =\mathfrak h \oplus \mathfrak n^+[/tex]. At the group level, this corresponds to a choice of Borel subgroup [tex]B\subset G[/tex], with the space [tex]G/B[/tex] a complex projective variety known as a flag manifold. More generally, much of the same structure appears if we choose larger subgroups [tex]P \subset G[/tex] containing [tex]B[/tex] such that [tex]G/P[/tex] is a complex projective variety of lower dimension. In these cases [tex]Lie\ P=\mathfrak l \oplus \mathfrak u^+[/tex], with [tex]\mathfrak l[/tex] (the Levi subalgebra) a reductive algebra playing the role of the Cartan subalgebra, and [tex]\mathfrak u^+[/tex] playing the role of [tex]\mathfrak n^+[/tex].

In this more general setting, there is a generalization of the Harish-Chandra homomorphism, now taking [tex]Z(\mathfrak g)[/tex] to [tex]Z(\mathfrak l)[/tex]. This acts on the cohomology groups [tex]H^k(\mathfrak u^+,V)[/tex], with a generalization of the Casselman-Osborne lemma determining what representations of [tex]\mathfrak l[/tex] occur in this cohomology. The Dirac cohomology formalism to be discussed later generalizes this even more, to cases of a reductive subalgebra [tex]\mathfrak r[/tex] with orthogonal complement that cannot be given a complex structure and split into conjugate subalgebras. It also provides a compelling explanation for the continual appearance of [tex]\rho[/tex], as the highest weight of the spin representation.

Posted in BRST | 2 Comments

Status of Superstring and M-theory

Posted on December 11, 2008 by woit

A write-up by John Schwarz of his Erice lectures from this past summer has now appeared on the arXiv, with the title Status of Superstring and M-theory. In his second lecture, Schwarz provides a good review of the various attempts to do “string phenomenology” by trying to find a “string background” that doesn’t conflict with known particle physics. He devotes particular attention to the newest of these backgrounds, so-called “F-theory local models”, providing a summary of the rather complicated constructions involved. Schwarz doesn’t describe any experimental predictions of such models, just noting:

It will be very interesting to see what predictions can be made before the experimental results pour in and whether they turn out to be correct.

For more discussion of these models and the question of whether they predict anything, see here.

Schwarz begins with an account of his interactions with Sidney Coleman at Aspen and elsewhere:

I recall him once saying that there are three things that he does not like, all of which are becoming popular: supersymmetry, strings, and extra dimensions. Obviously, my views are quite different, but this did not lessen my regard for him, nor did it harm our personal relationship. In fact, I respected his honesty, especially as he did not try to impose his prejudices on the community.

About the anthropic landscape issue, he has this to say:

Perhaps the absurdly large number of flux vacua that typically arise in flux compactifications has discouraged people from trying to construct viable particle physics models. In fact, this large number of vacua has motivated the suggestion that various parameters of Nature (such as the cosmological constant) should be studied statistically on the landscape. I don’t really understand the logic of doing this, since this approach seems to assume implicitly that Nature corresponds to a more or less random vacuum. This in turn is motivated by some vague idea about how Universes are spawned in the Multiverse in a process of eternal inflation. Then the story gets even more entangled when the anthropic principle is brought into the discussion. Some people are enthusiastic about this approach, but I find it fundamentally defeatist. It is not the way I like to think about particle physics.

Meanwhile, public promotion of the Multiverse continues, with the opinion pages of Britain’s The Independent today featuring a piece by Bernard Carr entitled Fifth dimensions, space bubbles and other facets of the multiverse. Carr describes the “growing popularity” of the multiverse proposal, ending with:

But is the “multiverse” a proper scientific proposal or just philosophy? Despite the growing popularity of the proposal, the idea is speculative and currently untestable – and it may always remain so. Astronomers may never be able to observe the other universes with their telescopes and particle physicists may never be able to detect the extra dimensions with their accelerators. So, although some physicists favour the multiverse because it may do away with the need for a creator, others regard the idea as equally metaphysical. What is really at stake is the nature of science itself.

Carr characterizes some multiverse proponents as atheists favoring something that doesn’t seem to fit into the conventional scientific method because it gives an answer to the argument from design for a deity. For more about this all-too-common argument for the multiverse, being promoted by Susskind and others, see here. In answer to such claims about religion being promoted by physicists, New Scientist this week is running a sensible piece by Amanda Gefter entitled Why it’s not as simple as God vs the multiverse. It makes the obvious point about the multiverse-God dichotomy:

Science never boils down to a choice between two alternative explanations. It is always plausible that both are wrong and a third or fourth or fifth will turn out to be correct.

Update: For more multiverse mania, see today’s colloquium at Perimeter here. The intense promotion of this pseudo-science continues, but I don’t think it’s getting any traction.

Update: Yet more media attention to the God vs. Multiverse debate, now from the Guardian.

Posted in Multiverse Mania, Uncategorized | 10 Comments

Educational Malpractice

Posted on December 9, 2008 by woit

According to the New York Times, Scarsdale High School has decided to get rid of their Advanced Placement classes, including AP Physics, replacing them with a new curriculum that cost “$40,000 to bring in 25 professors from Harvard, Yale, New York University and other top colleges.”

“We have the luxury of being able to move beyond the A.P.,” John Klemme, Scarsdale’s principal, said in a recent interview. “If people called it a gold curriculum in the past, I refer to this version as the platinum curriculum.”

What’s the change in this new “platinum curriculum” as far as physics is concerned?

Physics students now study string theory — a hot topic in some college courses that is absent from the Advanced Placement exam.

Posted in Uncategorized | 55 Comments

Notes on BRST VI: Casimir Operators

Posted on December 3, 2008 by woit

For the case of [tex]G=SU(2)[/tex], it is well-known from the discussion of angular momentum in any quantum mechanics textbook that irreducible representations can be labeled either by j, the highest weight (here, highest eigenvalue of [tex]J_3[/tex] ), or by [tex]j(j+1)[/tex], the eigenvalue of [tex]\mathbf{J\cdot J}[/tex]. The first of these requires making a choice (the z-axis) and looking at a specific vector in the representation, the second doesn’t. It was a physicist (Hendrik Casimir), who first recognized the existence of an analog of [tex]\mathbf{J\cdot J}[/tex] for general semi-simple Lie algebras, and the important role that this plays in representation theory.

The Casimir Operator

Recall that for a semi-simple Lie algebra [tex]\mathfrak g[/tex] one has a non-degenerate, invariant, symmetric bi-linear form [tex](\cdot,\cdot)[/tex], the Killing form, given by

[tex](X,Y)= tr(ad(X)ad(Y))[/tex]

If one starts with [tex]\mathfrak g[/tex] the Lie algebra of a compact group, this bilinear form is defined on [tex]\mathfrak g_{\mathbf C}[/tex], and negative-definite on [tex]\mathfrak g[/tex]. For a simple Lie algebra, taking the trace in a different representation gives the same bilinear form up to a constant. As an example, for the case [tex]\mathfrak g_{\mathbf C}={\mathfrak{sl}(n,\mathbf C)}[/tex], one can show that

([tex]X,Y)=2n\ tr(XY)[/tex]

here taking the trace in the fundamental representation as [tex]n[/tex] by [tex]n[/tex] complex matrices.
One can use the Killing form to define a distinguished quadratic element [tex]\Omega[/tex] of [tex]U(\mathfrak g)[/tex], the Casimir element

[tex]\Omega=\sum_iX_iX^i[/tex]

where [tex]X_i[/tex] is an orthonormal basis with respect to the Killing form and [tex]X^i[/tex] is the dual basis. On any representation [tex]V[/tex], this gives a Casimir operator

[tex]\Omega_V=\sum_i\pi(X_i)\pi(X^i)[/tex]

Note that, taking the representation [tex]V[/tex] to be the space of functions [tex]C^\infty(G)[/tex] on the compact Lie group G, [tex]\Omega_V[/tex] is an invariant second-order differential operator, (minus) the Laplacian.

[tex]\Omega[/tex] is independent of the choice of basis, and belongs to [tex]U(\mathfrak g)^{\mathfrak g}[/tex], the subalgebra of [tex]U(\mathfrak g)[/tex] invariant under the adjoint action. It turns out that [tex]U(\mathfrak g)^{\mathfrak g}=Z(\mathfrak g)[/tex], the center of [tex]U(\mathfrak g)[/tex]. By Schur’s lemma, anything in the center [tex]Z(\mathfrak g)[/tex] must act on an irreducible representation by a scalar. One can compute the scalar for an irreducible representation [tex](\pi,V)[/tex] as follows:

Choose a basis [tex](H_i, X_{\alpha},X_{-\alpha})[/tex] of [tex]\mathfrak g_{\mathbf C}[/tex] with [tex]H_i[/tex] an orthonormal basis of the Cartan subalgebra [tex]\mathfrak t_{\mathbf C}[/tex], and [tex]X_{\pm\alpha}[/tex] elements of [tex]\mathfrak n^{\pm}[/tex] in the [tex]\pm\alpha[/tex] root-spaces of [tex]\mathfrak g_{\mathbf C}[/tex], orthonormal in the sense of satisfying

[tex](X_{\alpha},X_{-\alpha})=1[/tex]

Then one has the following expression for [tex]\Omega[/tex]:

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots} (X_{\alpha} X_{-\alpha} +X_{-\alpha}X_{\alpha})[/tex]

To compute the scalar eigenvalue of this on an irreducible representation [tex](\pi,V_{\lambda})[/tex] of highest weight [tex]\lambda[/tex], one can just act on a highest weight vector [tex]v\in V^{\lambda}=V^{\mathfrak n^+}[/tex]. On this vector the raising operators [tex]\pi(X_{\alpha})[/tex] act trivially, and using the commutation relation

[tex][X_{\alpha},X_{-\alpha}]=H_{\alpha}[/tex]

([tex]H_{\alpha}[/tex] is the element of [tex]\mathfrak t_{\mathbf C} [/tex] satisfying [tex](H,H_{\alpha})=\alpha(H)[/tex]) one finds

[tex]\Omega=\sum_i H_i^2 + \sum_{+\ roots}H_{\alpha}= \sum_i H_i^2 +2H_{\rho}[/tex]

where [tex]\rho[/tex] is half the sum of the positive roots, a quantity which keeps appearing in this story. Acting on [tex]v\in V^{\lambda}[/tex] one finds

[tex]\Omega_{V_{\lambda}}v=(\sum_i\lambda(H_i)^2+2\lambda (H_{\rho}))v[/tex]

Using the inner-product [tex]< \cdot,\cdot>[/tex] induced on [tex]\mathfrak t^*[/tex] by the Killing form, this eigenvalue can be written as:

[tex]< \lambda,\lambda>+2< \lambda,\rho>=||\lambda+\rho||^2- ||\rho||^2[/tex]

In the special case [tex]\mathfrak g = \mathfrak {su}(2),\ \mathfrak g_{\mathbf C}=\mathfrak sl(2,\mathbf C)[/tex], there is just one positive root, and one can take

[tex]H_1=h=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\ X_{\alpha}=e=\begin{pmatrix}0&1\\0&0\end{pmatrix},\ X_{-\alpha}=f=\begin{pmatrix}0&0\\1&0\end{pmatrix}[/tex]

Computing the Killing form, one finds

[tex](h,h)=8,\ (e,f)=4[/tex]

and

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}(ef +fe)=\frac{1}{8}h^2 + \frac{1}{4}(h +2fe)[/tex]

On a highest weight vector [tex]\Omega[/tex] acts as

[tex]\Omega=\frac{1}{8}h^2 + \frac{1}{4}h=\frac{1}{8}h(h+2)=\frac{1}{2}(\frac{h}{2}(\frac{h}{2} +1))[/tex]

This is 1/2 times the physicist’s operator [tex]\mathbf{J\cdot J}[/tex], and in the irreducible representation [tex]V_n[/tex] of spin [tex] j=n/2[/tex], it acts with eigenvalue [tex]\frac{1}{2}j(j+1)[/tex].

In the next posting in this series I’ll discuss the Harish-Chandra homomorphism, and the question of how the Casimir acts not just on [tex]V^{\mathfrak n^+}=H^0(\mathfrak n^+,V)[/tex], but on all of the cohomology [tex]H^*(\mathfrak n^+,V)[/tex]. After that, taking note that the Casimir is in some sense a Laplacian, we’ll follow Dirac and introduce Clifford algebras and spinors in order to take its square root.

Posted in BRST | 8 Comments

BRST News

Posted on December 2, 2008 by woit

I should finish writing the next installment of the Notes on BRST series soon, but thought I’d post here about two pieces of BRST-related news, concerning the “B” and the “T”.

  • The “T” in BRST is I.V. Tyutin, whose Lebedev preprint N. 39 from 1975 is considered to be one of the first uses of what was later to become known as BRST symmetry. This paper was never published and the preprint has not been widely available (in particular, I’ve never seen a copy of the original). This evening one of the new preprints in the arXiv hep-th section is a copy of the 1975 preprint, making it now available on the web.
  • The “B” in BRST is Carlo Becchi, who together with Camillo Imbimbo has written an article on BRST for Scholarpedia entitled Becchi-Rouet-Stora-Tyutin symmetry. Scholarpedia has the interesting feature of making available (here) the discussion between reviewers and authors of the article, which can be enlightening.
    • Posted in BRST | 4 Comments

    From Quarks to Strings

    Posted on December 2, 2008 by woit

    There’s a new preprint on the arXiv from Polyakov, entitled From Quarks to Strings, in which he tells the story of his involvement with string theory over the years. He begins:

    In the sixties I was not much interested in string theory. The main reason for that was my conviction that the world of elementary particles should allow field theoretic description and that this description must be closely analogous to the conformal bootstrap of critical phenomena. At the time such views were very far from the mainstream. I remember talking to one outstanding physicist. When I said that the boiling water may have something to do with the deep inelastic scattering, I received a very strange look. I shall add in the parenthesis that this was a beginning of the long series of ”strange looks” which I keep receiving to this day.

    Another reason for the lack of interest was actually the lack of abilities. I could not follow a very complicated algebra of the early works on string theory and didn’t have any secret weapon to struggle with it.

    After the asymptotic freedom breakthrough, Polyakov quickly saw that a non-perturbative understanding of gauge theory was needed. He (independently from Wilson) developed lattice gauge theory, but acknowledges that, unlike Wilson, he did not have the Wilson loop criterion for confinement. In 3d, he worked out the dual-superconductor picture, where monopoles are responsible for confinement, and he has interesting comments about efforts over the years to use instantons in 4d, something that works for the N=2 supersymmetric case (Seiberg-Witten, and Nekrasov), but not for QCD itself.

    The strong-coupling lattice expansion was one thing that encouraged him to look for a gauge/string duality as a way to solve QCD. One idea was to write dynamical equations for the Wilson loop (later called Migdal-Makeenko equations) and find a solution to these as a path integral over surfaces, thus a string theory. About this he writes:

    This action is called now the Polyakov action, demonstrating the Arnold theorem,stating that things are never called after their true inventors.

    He derived the critical dimension for the string, but was much more interested in trying to understand non-critical strings, especially the four-dimensional string as a tool for studying gauge theories, and the 3d string as a tool to solve the 3d Ising model. Later he realized that it was natural to think of the Liouville mode as a fifth dimension, and by 1996 was studying the idea of using warped 5 dimensions to get gauge/string duality. Here’s how he describes what he was doing in the lead-up to the breakthrough by Maldacena which led to the AdS/CFT conjecture:

    At this point I was certain that I have found the right language for the gauge/ strings duality. I attended various conferences, telling people thatit is possible to describe gauge theories by solving Einstein-like equations (coming from the conformal symmetry on the world sheet) in five dimensions. The impact of my talks was close to zero. That was not unusual and didn’t bother me much. What really caused me to delay the publication ([12]) for a couple of years was my inability to derive the asymptotic freedom from my equations. At this point I should have noticed the paper of Igor Klebanov [13] in which he related D3 branes described by the supersymmetric Yang Mills theory to the same object described by supergravity. Unfortunately I wrongly thought that the paper is related to matrix theory and I was skeptical about this subject. As a result I have missed this paper which would provide me with a nice special case of my program. This special case was presented little later in full generality by Juan Maldacena [14] and his work opened the flood gates.

    He goes on to make some intriguing comments about the questions of integrability, and that of how to truly understand and derive gauge/string duality from first principles:

    The problem of reproducing gauge perturbation theory from the string theory side remains unsolved (and extremely important).

    Why should we care about the derivation from the first principles ? After all, in physics we value not so much the proved theorems but correct and powerful statements. However, in this case the lack of the derivation really impedes progress. We do not know how far the gauge /string duality can be extended and generalized. The enormous accumulation of special cases has been useful but not sufficient for deeper understanding. This is why I think that establishing the foundations is one of the most important problems in the field.

    Polyakov seems to have always been a skeptic about the idea of using the 10 dimensional superstring to construct a unified theory, instead hoping that some understanding of the 4d non-critical string might lead somewhere. Like most string theorists, he takes the attitude that we need to wait for the results from the LHC and hope that a new clue will emerge and tell us how to make progress along these lines:

    As for the problem of string unification, it seems to me that non-critical strings may have some future. However, it may be wise to wait for some more information about Nature (specifically about supersymmetry) which we expect to get from the LHC.

    Posted in Uncategorized | 3 Comments

    Latest From the LHC

    Posted on November 25, 2008 by woit

    A talk at CERN today by Jorg Wenninger gives an update on the problems at Sector 34 and more information about what the prospects are for restarting the machine next year.

    The cause of the accident has been identified as excessive resistance in a busbar interconnection between two magnets. Looking at logged data from before the accident, evidence for this excessive resistance was seen. Checking all the other sectors, a hint of a similar problem was found in one other cell, and that dipole will be replaced.

    50 magnets are in the process of being removed from Sector 34, all to be out by Christmas. To avoid future similar accidents, the quench protection system is being upgraded, and the commissioning procedures will include a systematic search for excessive resistance problems. These measures can be implemented before next summer. There is also a plan is to add pressure release valves on every dipole cryostat, but this is highly problematic since it will require warming up all the sectors and likely would not allow the LHC to run with beam during 2009. The summary for 2009 plans reads:

    Plan A:

  • Restart in (late) summer of 2009 with beam.
  • Beam intensity and energy limited to minimize any risk.

    • Plan B:

  • No beam before a complete ‘upgrade’ of the pressure relief system is implemented on all sectors.
  • Excludes beam in 2009.

    • Final decision in February?

    On a more cheerful note, tonight PBS will be broadcasting a documentary about the search for the Higgs at Fermilab called The Atom Smashers. It looks like this program should be about 10^(10^5) times better than a recent one featuring theorists. One of the filmmakers has a blog here. With the LHC out of commission for a while, the Higgs search at the Tevatron is where the action is, and the experimenters there may be the ones to find the Higgs or rule it out.

    Update: Two more recent presentations with information (including pictures!) about the LHC accident, repairs and plans for the future are here and here. For now, the plan is for the machine to be cold again next July.

    Posted in Experimental HEP News | 18 Comments

    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