Dyson on Birds and Frogs

Posted on January 7, 2009 by woit

Next month’s Notices of the AMS has an essay by Freeman Dyson entitled Frogs and Birds, which was written for his planned Einstein Public Lecture. In it, he divides mathematicians up into two species: birds, who “fly high in the air and survey broad vistas” (i.e. seek abstraction, unification and generalization), and frogs, who “see only the flowers that grow nearby” (i.e. study the details of specific examples).

Dyson himself is resolutely a frog, but writes that “many of my best friends are birds”, and argues that both birds and frogs are needed to do justice to the breadth and depth of the subject of mathematics. Frog that he is, his essay covers a variety of quite different special topics that have drawn his attention, linked together only weakly by the bird/frog theme. These include a discussion of the roles of complex numbers and linearity in quantum mechanics, a proposed idea about how to attack the Riemann hypothesis (try and enumerate 1d-quasicrystals, since the zeros of the zeta function have this structure), and a collection of profiles and anecdotes about various mathematicians and physicists (Besicovitch, Weyl, Yang, Manin, von Neumann).

Personally I suppose I fit well into Dyson’s bird category, but among the best mathematicians that I know, the frog/bird distinction is often unclear. Many of them make their reputation by proving rather abstract and general theorems, but these proofs are often the result of a huge amount of detailed investigation of examples. I agree with Dyson that both points of view are needed, and see the most successful cases of progress in mathematics coming from mathematicians who avoid the temptation to fly too high into arid abstraction, or sink too deep into irrelevant detail.

Dyson includes a long section on string theory, which I’ll include here:

I would like to say a few words about string theory. Few words, because I know very little about string theory. I never took the trouble to learn the subject or to work on it myself. But when I am at home at the Institute for Advanced Study in Princeton, I am surrounded by string theorists, and I sometimes listen to their conversations. Occasionally I understand a little of what they are saying. Three things are clear. First, what they are doing is first-rate mathematics. The leading pure mathematicians, people like Michael Atiyah and Isadore Singer, love it. It has opened up a whole new branch of mathematics, with new ideas and new problems. Most remarkably, it gave the mathematicians new methods to solve old problems that were previously unsolvable. Second, the string theorists think of themselves as physicists rather than mathematicians. They believe that their theory describes something real in the physical world. And third, there is not yet any proof that the theory is relevant to physics. The theory is not yet testable by experiment. The theory remains in a world of its own, detached from the rest of physics. String theorists make strenuous efforts to deduce consequences of the theory that might be testable in the real world, so far without success.

My colleagues Ed Witten and Juan Maldacena and others who created string theory are birds, flying high and seeing grand visions of distant ranges of mountains. The thousands of humbler practitioners of string theory in universities around the world are frogs, exploring fine details of the mathematical structures that birds first saw on the horizon. My anxieties about string theory are sociological rather than scientific. It is a glorious thing to be one of the first thousand string theorists, discovering new connections and pioneering new methods. It is not so glorious to be one of the second thousand or one of the tenth thousand. There are now about ten thousand string theorists scattered around the world. This is a dangerous situation for the tenth thousand and perhaps also for the second thousand. It may happen unpredictably that the fashion changes and string theory becomes unfashionable. Then it could happen that nine thousand string theorists lose their jobs. They have been trained in a narrow specialty, and they may be unemployable in other fields of science.

Why are so many young people attracted to string theory? The attraction is partly intellectual. String theory is daring and mathematically elegant. But the attraction is also sociological. String theory is attractive because it offers jobs. And why are so many jobs offered in string theory? Because string theory is cheap. If you are the chairperson of a physics department in a remote place without much money, you cannot afford to build a modern laboratory to do experimental physics, but you can afford to hire a couple of string theorists. So you offer a couple of jobs in string theory, and you have a modern physics department. The temptations are strong for the chairperson to offer such jobs and for the young people to accept them. This is a hazardous situation for the young people and also for the future of science. I am not saying that we should discourage young people from working in string theory if they find it exciting. I am saying that we should offer them alternatives, so that they are not pushed into string theory by economic necessity.

Finally, I give you my own guess for the future of string theory. My guess is probably wrong. I have no illusion that I can predict the future. I tell you my guess, just to give you something to think about. I consider it unlikely that string theory will turn out to be either totally successful or totally useless. By totally successful I mean that it is a complete theory of physics, explaining all the details of particles and their interactions. By totally useless I mean that it remains a beautiful piece of pure mathematics. My guess is that string theory will end somewhere between complete success and failure. I guess that it will be like the theory of Lie groups, which Sophus Lie created in the nineteenth century as a mathematical framework for classical physics. So long as physics remained classical, Lie groups remained a failure. They were a solution looking for a problem. But then, fifty years later, the quantum revolution transformed physics, and Lie algebras found their proper place. They became the key to understanding the central role of symmetries in the quantum world. I expect that fifty or a hundred years from now another revolution in physics will happen, introducing new concepts of which we now have no inkling, and the new concepts will give string theory a new meaning. After that, string theory will suddenly find its proper place in the universe, making testable statements about the real world. I warn you that this guess about the future is probably wrong. It has the virtue of being falsifiable, which according to Karl Popper is the hallmark of a scientific statement. It may be demolished tomorrow by some discovery coming out of the Large Hadron Collider in Geneva.

I don’t know where Dyson got the estimate of ten thousand string theorists; my own estimate would be more like one to two thousand (with the number strongly dependent on how you decide who is a “string theorist”). The large yearly Strings200X conferences that bring together a sizable fraction of active string theory community tend to draw roughly 500 people.

The Princeton-centric assumption that there are lots of string theory jobs embedded in his question “And why are so many jobs offered in string theory?” is quite problematic, as any young string theorist on the job market could explain to him. There actually aren’t a lot of string theory jobs out there, and a lot of Ph.D.s in the subject being produced, leading to a lot of ex-string theorists now working in the financial industry and elsewhere. These days, if you are going to choose your field based on where the jobs are, you become an LHC phenomenologist or a cosmologist. If you want to be a string theorist, you better be a string phenomenologist or a string cosmologist. Also rather unrealistic is Dyson’s “it could happen that nine thousand string theorists lose their jobs”, due to tenure in the academic system. Even if a consensus develops over the next few years that string theory was all a big mistake, twenty years from now there will still be a cadre of (older) people working in the field.

Dyson’s idea, that 50-100 years from now, a new revolution in physics will show how string theory fits in may be right. It also may be that this has already happened, as much of the field has moved into the study of gauge-string dualities, where string theory provides a useful approximation for strongly coupled systems, and the idea that it unifies particle physics is falling by the wayside.

Posted in Uncategorized | 14 Comments

Best Job in the US

Posted on January 6, 2009 by woit

According to the latest JobsRated listing released today, the best job in the US is that of mathematician. Pay is good, stress is low, and you don’t have to get your hands dirty, but can sit in front of a computer monitor all day. Nice work if you can get it. The job of physicist is significantly less desirable: down at number 13, not quite as good as working as a philosopher (number 12), but a bit better than being a parole officer (number 14).

Posted in Uncategorized | 17 Comments

New Year Links

Posted on January 5, 2009 by woit

There’s a conference going on in Jerusalem now on the topic of Particle Physics in the Age of the LHC. Some slides and other talk materials are here, video may start appearing here. Not clear when the “Age of the LHC” is; unfortunately we’re still a ways away from first collisions, even farther from new physics. Next year, starting in May, the KITP will be running a program on The First Year of the LHC, which may also be jumping the gun a bit, at least to the extent that the topic is LHC physics results. Last year’s LHC program, Physics of the Large Hadron Collider, has a web-site that still begins with the counter-factual “The Large Hadron Collider (LHC) will begin operation by the end of 2007.”

Also next year, the KITP will be running another supposedly LHC-related program, entitled Strings at the LHC and in the Early Universe. I wonder what the KITP director thinks of this, since he’s on record as thinking it unlikely that the LHC will have anything to say about string theory. A much less dubious KITP program about string theory is the one starting today, with the title Fundamental Aspects of String Theory. This program focuses on the current lack of understanding of what string theory really is:

Over the last decade, string theory has seen important conceptual and technical advances on a host of long-standing problems involving non-pertur-bative and strongly-coupled physics. However, the fundamental ingredients of superstring theory and M-theory are still not well understood, and this five month program will be directed at these open questions.

The first week will be devoted to introductory talks about string field theory and the pure spinor formalism, two quite different attempts to give a new and different formulation of string theory.

Also starting today is the big annual meeting of the AMS, held this year in Washington, DC. One of the important features of this meeting is that many institutions, especially smaller ones, do their initial interviews for next year’s jobs at the meeting. This coming hiring season promises to be an exceptionally brutal one for job candidates, with financial problems leading to freezes and reduced hiring at many places. One resource for young mathematicians on the job market is the web-site of the Young Mathematicians Network.

I wrote about Witten’s talk on quantum Yang-Mills theory at the Yau birthday conference here. A write-up of the talk is now available as a preprint here.

There’s a new book coming out this month that I’m looking forward to reading, Graham Farmelo’s biography of Dirac, entitled The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius. Nature Physics has a review here.

This week’s Science Saturday featured John Horgan and George Johnson discussing the state of science journalism and what it has to do with blogging. As science journalists, they take exception to the point of view common among scientists that their job is just to try and accurately transmit to the public the claims being made by scientists.

Posted in Uncategorized | 8 Comments

Dis-moi qui tu aimes (je te dirai qui tu hais)

Posted on January 2, 2009 by woit

A colleague has very helpfully provided me with a copy of the murder mystery set at the IHES that I wrote about recently here, and I’ve just finished reading it. Since I’m not much of an afficionado of this genre of fiction, I can’t really evaluate how good a murder mystery it is. But as a memoir of the IHES during the 1980s, it is excellent. A claim at the beginning of the book that “any resemblance to real persons is just coincidence” seems to be one of the few things in it (besides the murder) that is fiction. As far as I can tell, the descriptions of all characters correspond precisely to someone at the IHES during that period, with only the names changed. I’m guessing that all or most of the anecdotes about these characters also correspond to reality.

It’s a roman a clef, so here’s the key for the major characters:

Andre Grusin = Leon Motchane
Henrik Dekker = Nicolaas Kuiper
Charles Bouleaux = Marcel Berger
Antoine Fleuret = Alain Connes
Jacob Zuram = Barry Mazur
Boris Grekov = Mikhael Gromov
Jacques Chevalier = Pierre Deligne

Among the minor characters, I suspect

Joe Bub = Dennis Sullivan
David Amir = Ofer Gabber
Albert Toudy = Adrien Douady

I don’t think I’ll be giving away too much of the plot to mention that, since the novel was written nearly twenty years ago, back when string theory was a hot topic, one of the plot twists involves string theory. There’s a discovery that “superstring theory is renormalizable and predicts that gluonic interactions are colorless”.

Posted in Book Reviews | 7 Comments

Notes on BRST IX: Clifford Algebras and Lie Algebras

Posted on December 29, 2008 by woit

Note: I’ve started putting together the material from these postings into a proper document, available here, which will be getting updated as time goes on. I’ll be making changes and additions to the text there, not on the blog postings. For most purposes, that will be what people interested in this subject will want to take a look at.

When a Lie group with Lie algebra [tex]\mathfrak g[/tex] acts on a manifold [tex]M[/tex], one gets two sorts of actions of [tex]\mathfrak g[/tex] on the differential forms [tex]\Omega^*(M[/tex]). For each [tex]X\in \mathfrak g [/tex] one has operators:

  • [tex]\mathcal L}_X: \Omega^k(M)\rightarrow\Omega^k(M),[/tex] the Lie derivative along the vector field on [tex]M[/tex] corresponding to [tex]X[/tex]

    • and

  • [tex]i_X:\Omega^k(M)\rightarrow\Omega^{k-1}(M)[/tex], contraction by the vector field on [tex]M[/tex] corresponding to [tex]X[/tex]

    • These operators satisfy the relation

    [tex]di_X+i_Xd={\mathcal L}_X[/tex]

    where [tex]d[/tex] is the de Rham differential [tex]d:\Omega^k(M)\rightarrow \Omega^{k+1}(M)[/tex], and the operators [tex]d, i_X, \mathcal L_X[/tex] are (super)-derivations. In general, an algebra carrying an action by operators satisfying the same relations satisfied by [tex]d, i_X, \mathcal L_X[/tex] will be called a [tex]\mathfrak g[/tex]-differential algebra. It will turn out that the Clifford algebra [tex]Cliff(\mathfrak g)[/tex] of a semi-simple Lie algebra [tex]\mathfrak g[/tex] carries not just the Clifford algebra structure, but the additional structure of a [tex]\mathfrak g[/tex]-differential algebra, in this case with [tex]\mathbf Z_2[/tex], not [tex]\mathbf Z[/tex] grading.

    Note that in this section the commutator symbol will be the supercommutator in the Clifford algebra (commutator or anti-commutator, depending on the [tex]\mathbf Z_2[/tex] grading). When the Lie bracket is needed, it will be denoted [tex][\cdot,\cdot]_{\mathfrak g}[/tex].

    To get a [tex]\mathfrak g[/tex]-differential algebra on [tex]Cliff(\mathfrak g)[/tex] we need to construct super-derivations [tex]i_X^{Cl}[/tex], [tex]{\mathcal L}_X^{Cl}[/tex], and [tex]d^{Cl}[/tex] satisfying the appropriate relations. For the first of these we don’t need the fact that this is the Clifford algebra of a Lie algebra, and can just define

    [tex]i_X^{Cl}(\cdot)=[-\frac{1}{2}X,\cdot][/tex]

    For [tex]{\mathcal L}_X^{Cl}[/tex], we need to use the fact that since the adjoint representation preserves the inner product, it gives a homomorphism

    [tex]\widetilde{ad}:\mathfrak g \rightarrow \mathfrak{spin}(\mathfrak g)[/tex]

    where [tex]\mathfrak{spin}(\mathfrak g)[/tex] is the Lie algebra of the group [tex]Spin(\mathfrak g)[/tex] (the spin group for the inner product space [tex]\mathfrak g[/tex]), which can be identified with quadratic elements of [tex]Cliff(\mathfrak g)[/tex], taking the commutator as Lie bracket. Explicitly, if [tex]X_a[/tex] is a basis of [tex]\mathfrak g[/tex], [tex]X_a^* [/tex] the dual basis, then

    [tex]\widetilde{ad}(X)=\frac{1}{4}\sum_a X_a^*[X,X_a]_{\mathfrak g}[/tex]

    and we get operators acting on [tex]Cliff(\mathfrak g)[/tex]

    [tex]{\mathcal L}_X^{Cl}(\cdot)=[\widetilde{ad}(X),\cdot][/tex]

    Remarkably, an appropriate [tex]d^{Cl}[/tex] can be constructed using a cubic element of [tex]Cliff(\mathfrak g)[/tex]. Let

    [tex]\gamma= \frac{1}{24}\sum_{a,b}X^*_aX^*_b[X_a,X_b]_{\mathfrak g}[/tex]

    then

    [tex]d^{Cl}(\cdot)=[\gamma, \cdot][/tex]

    [tex]d^{Cl}\circ d^{Cl}=0[/tex] since [tex]\gamma^2[/tex] is a scalar which can be computed to be [tex]-\frac{1}{48}tr\Omega_{\mathfrak g}[/tex], where [tex]\Omega_{\mathfrak g}[/tex] is the Casimir operator in the adjoint representation.

    The above constructions give [tex]Cliff(\mathfrak g)[/tex] the structure of a filtered [tex]\mathfrak g[/tex]-differential algebra, with associated graded algebra [tex]\Lambda^*(\mathfrak g)[/tex]. This gives [tex]\Lambda^*(\mathfrak g)[/tex] the structure of a [tex]\mathfrak g[/tex]-differential algebra, with operators [tex]i_X, \mathcal L_X, d[/tex]. The cohomology of this differential algebra is just the Lie algebra cohomology [tex]H^*(\mathfrak g, \mathbf C)[/tex].

    [tex]Cliff(\mathfrak g)[/tex] can be thought of as an algebra of operators corresponding to the quantization of an anti-commuting phase space [tex]\mathfrak g[/tex]. Classical observables are anti-commuting functions, elements of [tex]\Lambda^*(\mathfrak g^*)[/tex]. Corresponding to [tex]i_X, \mathcal L_X, d[/tex] one has both elements of [tex]\Lambda^*(\mathfrak g^*)[/tex] and their quantizations, the operators in [tex]Cliff(\mathfrak g)[/tex] constructed above.

    For more details about the above, see the following references

  • A. Alekseev and E. Meinrenken, The non-commutative Weil algebra, Invent. Math 139, 135-172 (2000), or arXiv:math/9903052
  • E. Meinrenken, Clifford algebras and Lie groups, 2005 Toronto lecture notes
  • G. Landweber, Multiplets of representations and Kostant’s Dirac operator for equal rank loop groups, Duke Mathematical Journal 110, 121-160 (2001), or arXiv:math/0005057
  • B. Kostant and S. Sternberg, Symplectic reduction , BRS cohomology and infinite-dimensional Clifford algebras, Ann. Physics 176, 49-113 (1987)
    • Posted in BRST | 4 Comments

    Physics Murder Mystery

    Posted on December 27, 2008 by woit

    After the recent news that Lisa Randall is writing the libretto for an opera, there’s further evidence that particle theorists in Cambridge are moving in the direction of creative writing. Today’s Wall Street Journal has a feature article about various people’s plans for 2009. One of these is Frank Wilczek, who writes:

    I’m writing a physics murder mystery. The idea is that two men and two women from Harvard and MIT collaborate and discover dark matter. It’s clear that they should win a Nobel Prize, but according to the rules of the prize, only three people at most can share.

    This is an entertaining idea for a plot, and perhaps it has some personal resonance with Wilczek. For much of his career, he was well-known to be one of the people responsible for a definitely Nobel-prize caliber discovery, but he did not have a Nobel prize. By some counts, there were four people (Gross, Politzer, ‘t Hooft, Wilczek) who had a hand in the discovery of asymptotic freedom back in 1973. It was only with the award of a Nobel for related work to ‘t Hooft and Veltman in 1999 that the numerical obstruction to an asymptotic freedom award was removed, with the award going to the other three in 2004. Over this quarter century or so, surely it did not occur to any of the four that it might not be an entirely bad thing if one of them didn’t live to a ripe old age….

    Next month I’ll be spending a week or so in Paris, partly for vacation, partly to attend a conference about Grothendieck’s mathematical legacy, to be held at the IHES, a place I’ve never before visited. There’s a murder mystery about the IHES that I’ve heard about but haven’t yet read, so I hope to get a copy in France. The author is Nicole Gaume, who worked for the IHES director, and was forced out when a new director (Marcel Berger) came on the scene. Under the pen-name Margot Bruyère she wrote a roman à clef featuring the mathematicians of the IHES and the murder of a new director. The book first came out under the title Dis-moi qui tu aimes (je te dirai qui tu hais), but was republished in 2002 under the new title Maths à mort. For more information about the book, see here.

    Update: I just noticed that Wilczek has posted on his web-site an essay about Hermann Weyl’s Philosophy of Mathematics and Natural Science that will be the introduction to a new edition of the book appearing next year.

    Posted in Uncategorized | 13 Comments

    Notes on BRST VIII: Clifford Algebras

    Posted on December 21, 2008 by woit

    Clifford Algebras

    Clifford algebras are well-known to physicists, in the guise of matrix algebras generated by the [tex]\gamma[/tex] -matrices first used in the Dirac equation. They also have a more abstract formulation, which will be the topic of this posting. One way to think about Clifford algebras is as a “quantization” of the exterior algebra, associated with a symmetric bilinear form.

    Given a vector space [tex]V[/tex] with a symmetric bilinear form [tex](\cdot,\cdot)[/tex], the associated Clifford algebra [tex]Cliff (V,(\cdot,\cdot))[/tex] can be defined by starting with the tensor algebra [tex]T^*(V)[/tex] ([tex]T^k(V)[/tex] is the k-th tensor power of [tex]V[/tex]), and imposing the relations

    [tex]v\otimes w + w\otimes v = -2(v,w)1[/tex]

    where [tex]v,w\in V=T^1(V),\ 1\in T^0(V)[/tex]. Note that many authors use a plus instead of a minus sign in this relation. The case of most interest in physics is [tex]V=\mathbf R^4, (\cdot,\cdot)[/tex] the Minkowski inner product of signature (3,1). The theory of Clifford algebras for real vector spaces [tex]V[/tex] is rather complicated. Here we’ll stick to complex vector spaces [tex]V[/tex], where the theory is much simpler, partially because over [tex]\mathbf C[/tex] there is, up to equivalence, only one non-degenerate symmetric bilinear form. We will suppress mention of the bilinear form in the notation, writing [tex]Cliff(V)[/tex] for [tex]Cliff(V,(\cdot,\cdot)).[/tex]

    For a more concrete definition, one can choose an orthonormal basis [tex]e_i[/tex] of [tex]V[/tex]. Then [tex]Cliff(V)[/tex] is the algebra generated by the [tex]e_i[/tex], with multiplication satisfying the relations

    [tex]e_i^2=-1,\ \ e_ie_j=-e_je_i\ \ (i\neq j)[/tex]

    One can show that these complex Clifford algebras are isomorphic to matrix algebras, more precisely

    [tex]Cliff(\mathbf C^{2n})\simeq M(\mathbf C, 2^n),\ \ \ Cliff(\mathbf C^{2n+1})\simeq M(\mathbf C, 2^n)\oplus M(\mathbf C, 2^n)[/tex]

    Clifford Algebras and Exterior Algebras

    The exterior algebra [tex]\Lambda^*(V)[/tex] is the algebra of anti-symmetric tensors, with product the wedge product [tex]\wedge[/tex]. This is also exactly what one gets if one takes the Clifford algebra [tex]Cliff(V)[/tex], with zero bilinear form. Multiplying a non-degenerate symmetric bilinear form [tex](\cdot,\cdot)[/tex] by a parameter [tex]t[/tex] gives for non-zero [tex]t[/tex] a Clifford algebra [tex]Cliff(V, t(\cdot,\cdot))[/tex] that can be thought of as a deformation of the exterior algebra [tex]\Lambda^*(V)[/tex]. Thinking of the exterior algebra on [tex]V[/tex] of dimension n as the algebra of functions on n anticommuting coordinates, the Clifford algebra can be thought of as a “quantization” of this, taking functions (elements of [tex]\Lambda^*(V)[/tex]) to operators (elements of [tex]Cliff(V)[/tex], matrices in this case).

    While [tex]\Lambda^*(V)[/tex] is a [tex]\mathbf Z[/tex] graded algebra, [tex]Cliff(V)=Cliff^{even}(V)\oplus Cliff^{odd}(V)[/tex] is only [tex]\mathbf Z_2[/tex]-graded, since the Clifford product does not preserve degree but can change it by two when multiplying generators. The Clifford algebra is filtered by a [tex]\mathbf Z[/tex] degree, taking [tex]F_p(Cliff(V))\subset Cliff(V)[/tex] to be the subspace of elements that can be written as sums of [tex]\leq p[/tex] generators. The exterior algebra is naturally isomorphic to the associated graded algebra for this filtration

    [tex]\Lambda^p(V)\simeq F_p(Cliff(V))/F_{p-1}(Cliff(V))[/tex]

    [tex]\Lambda^*(V)[/tex] and [tex]Cliff(V)[/tex] are isomorphic as vector spaces. One choice of such an isomorphism is given by composing the skew-symmetrization map

    [tex]v_1\wedge v_2\wedge\cdots\wedge v_p=\frac{1}{p!}\sum_{s\in S_p}sgn(s)v_{s(1)}\otimes v_{s(2)}\otimes\cdots\otimes v_{s(p)}[/tex]

    with the projection [tex]T^*(V)\rightarrow Cliff(V)[/tex]. Denoting this map by q, it is sometimes called the “quantization map”. Using an orthonormal basis [tex]e_i[/tex], [tex]q[/tex] acts as

    [tex]q(e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_p})=e_{i_1}e_{i_2}\cdots e_{i_p}[/tex]

    The inverse [tex]\sigma=q^{-1}:Cliff(V)\rightarrow \Lambda^*(V)[/tex] is sometime called the “symbol map”.

    This identification as vector spaces is known as the “Chevalley identification”. Using it, one can think of the Clifford algebra as just an exterior algebra with a different product.

    Clifford Modules and Spinors

    Given a Clifford algebra, one would like to classify the modules over such an algebra, the Clifford modules. Such a module is given by a vector space [tex]M[/tex] and an algebra homomorphism

    [tex]\pi: Cliff(V)\rightarrow End(M)[/tex]

    To specify [tex]\pi[/tex], we just need to know it on generators, and see that it satisfies

    [tex]\pi(v)\pi(w) +\pi(w)\pi(v)= -2(v,w)Id[/tex]

    One such Clifford module is [tex]M=\Lambda^*V[/tex], with

    [tex]\pi(v)\omega=v\wedge\omega – i_v\omega[/tex]

    where [tex]i_v[/tex] is contraction by [tex]v[/tex]. This gives the inverse to the quantization map (the symbol map [tex]\sigma[/tex]) as

    [tex]\sigma: a\in Cliff(V)\rightarrow \pi(a)1\in \Lambda^*(V)[/tex]

    [tex]\Lambda^*(V)[/tex] is not an irreducible Clifford module, and we would like to decompose it into irreducibles. For [tex]dim_{\mathbf C}V =2n[/tex] even, there will be a single such irreducible [tex]S[/tex], of dimension [tex]2^n[/tex], and the module map [tex]\pi:Cliff(V)\rightarrow End(S)[/tex] is an isomorphism. In the rest of this posting we’ll stick to the this case, for the odd dimensional case see the references mentioned at the end.

    To pick out an irreducible module [tex]S\subset \Lambda^*(V)[/tex], one can begin by choosing a linear map [tex]J:V\rightarrow V[/tex] such that [tex]J^2=-1[/tex] and [tex]J[/tex] is orthogonal [tex]((Jv,Jw)=(v,w))[/tex]. Then let [tex]W_J\subset V[/tex] be the subspace on which [tex]J[/tex] acts by [tex]+i[/tex], [tex]\overline W_J[/tex] be the subspace on which [tex]J[/tex] acts by [tex]-i[/tex]. Note that [tex]V[/tex] is a complex vector space, and now has two linear maps on it that square to [tex]-1[/tex], multiplication by [tex]i[/tex], and multiplication by [tex]J[/tex]. [tex]W_J[/tex] is an isotropic subspace of [tex]V[/tex], since

    [tex](v_1,v_2)=(Jv_1,Jv_2)=(iv_1,iv_2)=-(v_1,v_2)[/tex]

    for any [tex]v_1,v_2\in W_J[/tex]. We now have a decomposition [tex]V=W_j\oplus \overline W_J[/tex] into two isotropic subspaces. Since the bilinear form is zero on these subspaces, we get two subalgebras of the Clifford algebra, [tex]\Lambda^*(W_J)[/tex] and [tex]\Lambda^*(\overline{W_J})[/tex]. It turns out that one can choose [tex]S\simeq \Lambda^*(W_J)[/tex].

    One can make this construction very explicit by picking a particular [tex]J[/tex], for instance the one that acts on the element of an orthonormal basis by [tex]Je_{2j-1}=e_{2j},\ Je_{2j}=-e_{2j-1}[/tex] for [tex]j=1,\cdots n[/tex]. Letting [tex]w_j=e_{2j-1}+ie_{2j}[/tex] we get a basis of [tex]W_J[/tex]. To get an explicit representation of [tex]S[/tex] as a [tex]Cliff(V)[/tex] module isomorphic to [tex]\Lambda^*(\mathbf C^n)[/tex], we will use the formalism of fermionic annihilation and creation operators. These are the operators on an exterior algebra one gets from wedging by or contracting by an orthonormal vector, operators [tex]a_i^+[/tex] and [tex]a_i[/tex] for [tex]i=1,\cdots,n[/tex] satisfying

    [tex]\{a_i,a_j\}=\{a^+_i,a^+_j\}=0[/tex]

    [tex]\{a_i,a^+_j\}=\delta_{ij}[/tex]

    In terms of these operators on [tex]\Lambda^*(\mathbf C^n)[/tex], [tex]Cliff(n)[/tex] acts by

    [tex]e_{2j-1}=a_j^+-a_j[/tex]

    [tex]e_{2j}=-i(a^+_j+a_j)[/tex]

    The Spin Representation

    The group that preserves [tex](\cdot,\cdot)[/tex] is [tex]O(n,\mathbf C)[/tex], and its connected component of the identity [tex]SO(n,\mathbf C)[/tex] has compact real form [tex]SO(n)[/tex]. [tex]SO(n)[/tex] has a non-trivial double cover, the group [tex]Spin(n)[/tex]. One can construct [tex]Spin(n)[/tex] explicitly as invertible elements in [tex]Cliff(V)[/tex] for [tex]V=\mathbf R^n[/tex], and its Lie algebra using quadratic elements of [tex]Cliff(V)[/tex], with the Lie bracket given by the commutator in the Clifford algebra.

    For the even case, a basis for the Cartan subalgebra of [tex]Lie\ Spin(2n)[/tex] is given by the elements

    [tex]\frac{1}{2}e_{2j-1}e_{2j}[/tex]

    These act on the spinor module [tex]S\simeq\Lambda^*(\mathbf C^n)[/tex] as

    [tex]\frac{1}{2}e_{2j-1}e_{2j}=-i\frac{1}{2}(a_j^+-a_j)(a_j^++a_j)=i\frac{1}{2}[a_j,a_j^+][/tex]

    with eigenvalues [tex](\pm\frac{1}{2},\cdots,\pm\frac{1}{2})[/tex]. [tex]S[/tex] is not irreducible as a representation of [tex]Spin(2n)[/tex], but decomposes as [tex]S=S^+\oplus S^-[/tex] into two irreducible half-spin representations, corresponding to the even and odd degree elements of [tex]\Lambda^*(\mathbf C^n)[/tex].

    With a standard choice of positive roots, the highest weight of [tex]S^+[/tex] is

    [tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},+\frac{1}{2})[/tex]

    and that of [tex]S^-[/tex] is

    [tex](+\frac{1}{2},+\frac{1}{2}\cdots,+\frac{1}{2},-\frac{1}{2})[/tex]

    Note that the spinor representation is not a representation of [tex]SO(2n)[/tex], just of [tex]Spin(2n)[/tex]. However, if one restricts to the [tex]U(n)\subset SO(2n)[/tex] preserving [tex]J[/tex], then the [tex]\Lambda^*(W_J)[/tex] are the fundamental representations of this [tex]U(n)[/tex]. These representations have weights that are 0 or 1, shifted by [tex]+\frac{1}{2}[/tex] from those of the spin representation. One can’t restrict from [tex]Spin(2n)[/tex] to [tex]U(n)[/tex], but one can restrict to [tex]\tilde U(n)[/tex], a double cover of [tex]U(n)[/tex]. On this double cover the notion of [tex]\Lambda^n(\mathbf C^n)^{\frac{1}{2}[/tex] makes sense and one has, as [tex]\tilde U(n)[/tex] representations

    [tex]S\otimes \Lambda^n(\mathbf C^n)^{\frac{1}{2}}\simeq\Lambda^*(\mathbf C^n)[/tex]

    So, projectively, the spin representation is just [tex]\Lambda^*(\mathbf C^n)[/tex], but the projective factor is a crucial part of the story.

    The above has been a rather quick sketch of a long story. For more details, a good reference is the book Spin Geometry by Lawson and Michelsohn. Chapter 12 of Segal and Pressley’s Loop Groups contains a very geometric version of the above material, in a form suitable for generalization to infinite dimensions. My notes for my graduate class also have a bit more detail, see here.

    In the next posting we’ll see what happens when one chooses [tex]V=\mathfrak g[/tex], and studies the Clifford algebra [tex]Cliff(\mathfrak g)[/tex]

    Posted in BRST | 6 Comments

    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