The Complete Idiot’s Guide to String Theory

Posted on November 17, 2008 by woit

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

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

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

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

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

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

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

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

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

Notes on BRST III: Lie Algebra Cohomology

Posted on November 16, 2008 by woit

The Invariants Functor

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

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

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

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

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

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

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

one takes

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

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

Some Homological Algebra

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

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

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

To motivate the notion of exact complex, note that

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

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

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

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

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

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

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

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

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

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

The Koszul Resolution and Lie Algebra Cohomology

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

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

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

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

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

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

To get Lie algebra cohomology, we apply the invariants functor

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

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

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

and induced maps $d_i$

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

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

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

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

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

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

Posted in BRST | 6 Comments

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

Posted on November 15, 2008 by woit

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

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

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

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

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

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

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

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

    The BRST operator is given by

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    • Posted in BRST | 15 Comments

    Short Bits

    Posted on November 11, 2008 by woit

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    Notes on BRST I: Representation Theory and Quantum Mechanics

    Posted on November 5, 2008 by woit

    This is the first posting of a planned series that will discuss the BRST method for handling gauge symmetries and related mathematical topics. I’ve been writing a more formal paper about this, but given the substantial amount of not-well-known background material involved, it seems like a good idea to first put together a few expository accounts of some of these topics. And what better place for this than a blog?

    Many readers who are used to my usual attempts to be newsworthy and entertaining, often by scandal-mongering or stirring up trouble of one kind or another, may be very disappointed in these posts. They’re quite technical, hard to follow, and of low-to-negative entertainment value. You probably would do best to skip them and wait for more of the usual fare, which should continue to appear from time to time.

    Quantum Mechanics and Representation Theory

    A quantum mechanical physical system is given by the following mathematical structure:

  • A Hilbert space [tex]\mathcal H[/tex], the “space of states”. A state of the physical system is determined by a vector [tex]|\psi\rangle\in \mathcal H[/tex], with unit norm (i.e. [tex]||\psi||^2=\langle \psi|\psi\rangle=1[/tex]).
  • An algebra [tex]\mathcal O[/tex] that acts on [tex]\mathcal H[/tex]. To each physical observable corresponds a self-adjoint operator [tex]O\in\mathcal O[/tex]. Eigenvectors in [tex]\mathcal H[/tex] of this operator correspond to states where the observable has a well-defined value, which is the eigenvalue.

    • If a physical system has a symmetry group [tex]G[/tex], there is a unitary representation [tex](\Pi, \mathcal H)[/tex] of [tex]G[/tex] on [tex]\mathcal H[/tex]. This means that for each [tex]g\in G[/tex] we get a unitary operator [tex]\Pi(g)[/tex] satisfying

    [tex]\begin{displaymath}\Pi(g_3)=\Pi(g_2)\Pi(g_1)\ \text{if}\ g_3=g_1g_2\end{displaymath}[/tex]

    i.e. the map [tex]\Pi[/tex] from group elements to unitary operators is a homomorphism. The [tex]\Pi(g)[/tex] act on [tex]\mathcal O[/tex] by taking an operator [tex]O[/tex] to its conjugate [tex]\Pi(g)O(\Pi(g))^{-1}[/tex].

    When [tex]G[/tex] is a Lie group with Lie algebra [tex](\mathfrak g, [\cdot,\cdot])[/tex], differentiating [tex]\Pi[/tex] gives a unitary representation [tex](\pi, \mathcal H)[/tex] of [tex]\mathfrak g[/tex] on [tex]\mathcal H[/tex]. This means that for each [tex]X\in \mathfrak g[/tex] we get a skew-Hermitian operator [tex]\pi(X)[/tex] on [tex]\mathcal H[/tex], satisfying

    [tex]\pi(X_3)=[\pi(X_1),\pi(X_2)]\ \text{if}\ X_3=[X_1,X_2][/tex]

    i.e. the map [tex]\pi[/tex] taking Lie algebra elements [tex]X[/tex] (with the Lie bracket in [tex]\mathfrak g[/tex]) to skew-Hermitian operators (with commutator of operators) is a homomorphism. On [tex]\mathcal O[/tex], [tex]\mathfrak g[/tex] acts by the differential of the conjugation action of [tex]G[/tex], this action is just that of taking the commutator with [tex]\pi(X)[/tex].

    The Lie bracket is not associative, but to any Lie algebra [tex]\mathfrak g[/tex], one can construct an associative algebra [tex]U(\mathfrak g)[/tex] called the universal enveloping algebra for [tex]\mathfrak g[/tex]. If one identifies [tex]X\in \mathfrak g[/tex] with left-invariant vector fields on [tex]G[/tex], which are first-order differential operators on functions on [tex]G[/tex], then [tex]U(\mathfrak g)[/tex] is the algebra of left-invariant differential operators on [tex]G[/tex] of all orders, with product the composition of differential operators. A Lie algebra representation is precisely a module over [tex]U(\mathfrak g)[/tex], i.e. a vector space with an action of [tex]U(\mathfrak g)[/tex].

    So, the state space [tex]\mathcal H[/tex] of a quantum system with symmetry group [tex]G[/tex] carries not only a unitary representation of [tex]G[/tex], but also a unitary representation of [tex]\mathfrak g[/tex], or equivalently, an action of the algebra [tex]U(\mathfrak g)[/tex]. [tex]X\in \mathfrak g[/tex] acts by the operator [tex]\pi(X)[/tex]. In this way a representation [tex]\pi[/tex] gives a sub-algebra of the algebra [tex]\mathcal O[/tex] of observables. Most of the important observables that show up in practice come from a symmetry in this way. An interesting philosophical question is whether the quantum system that governs the real world is purely determined by symmetry, i.e. such that ALL its observables come from symmetries in this manner.

    Some Examples

    Much of the structure of common quantum mechanical systems is governed by the fact that they carry space-time symmetries. In our 3-space, 1-time dimensional world, these include:

  • Translations in space: [tex]G=\mathbf R^3, \mathfrak g =\mathbf R^3[/tex], Lie Bracket is trivial.
    For each basis element [tex]e_j\in \mathfrak g[/tex] one gets a momentum operator [tex]\pi(e_j)=iP_j[/tex]
  • Translations in time: [tex]G=\mathbf R, \mathfrak g =\mathbf R[/tex]. If [tex]e_0[/tex] is a basis of [tex]\mathfrak g[/tex], [tex]i\pi(e_0)=H[/tex], the Hamiltonian operator. The fact that this operator generates time-translations is just Schrodinger’s equation.
  • Rotations in 3-space: [tex]G=SO(3)[/tex], or its double cover [tex]G=Spin(3)=SU(2), \mathfrak g = \mathbf R^3[/tex], with bracket given by the vector product. For each basis element [tex]e_j\in \mathfrak g[/tex] one gets an angular momentum operator [tex]\pi(e_j)=iJ_j[/tex]. These operators do not commute, so cannot be simultaneously diagonalized.

    • Another example is the symmetry of phase transformations of the state space [tex]\mathcal H[/tex]. Here [tex]G=U(1), \mathfrak g=R[/tex], and one gets an operator [tex]Q_e[/tex] that can be normalized to have integral eigenvalues.

    This last example also comes in a local version, where we make independent phase transformations at different points in space-time. This is an example of a “gauge symmetry”, and the question of how it gets represented on the space of states is what will lead us into the BRST story. Next posting in the series will be about gauge symmetry, then on to BRST.

    If you want some idea of where this is headed, you can take a look at slides from a colloquium talk I gave recently at the Dartmouth math department. They’re very sketchy, the postings in this series should add some detail.

    Posted in BRST | 63 Comments

    The Circus Begins

    Posted on November 2, 2008 by woit

    Friday’s arXiv posting of the paper by CDF about the multi-muon anomaly they are seeing has already generated three different conjectural explanations of what physics might be responsible for this. Undoubtedly many, many more are on the way.

  • Some members of the CDF collaboration have posted a paper entitled Phenomenological interpretation of the multi-muon events reported by the CDF collaboration. This explains the large numbers of muons with a rather baroque mechanism, conjecturing the production of a 300 GeV heavy particle decaying through a chain of 3 lighter particles, the last of which is supposed to be the long-lived (20 picosecond) one. This interpretation was part of the original draft PRL from last June/July. The CDF collaboration as a whole seems to have decided not to support the draft PRL and this interpretation, instead releasing just a PRD paper that describes the anomaly without trying to interpret its significance. It also seems that only two-thirds of the collaboration put their names on the PRD paper, and the interpretation paper was put out just by a small group. The whole story is somewhat reminiscent of the “Superjet” affair (see Tommaso Dorigo’s multi-part discussion here), which also involved a PRD publication about an anomaly signed by the collaboration, and an interpretation (in terms of squarks) signed by a much smaller group led by Paolo Giromini.
  • An hour or so after the Giromini et. al. paper came in on Friday, a group of string theorists had posted the 40-page Towards Realistic String Vacua on hep-th claiming to explain the CDF results with a class of string vacua:

    We also describe model-independent physical implications of this scenario. These include the masses of anomalous and non-anomalous U(1)’s and the generic existence of a new hyperweak force under which leptons and/or quarks could be charged. We propose that such a gauge boson could be responsible for the ghost muon anomaly recently found at the Tevatron’s CDF detector.

    If the Giromini et. al. explanation invoking 4 new particles is baroque, it’s hard to know what the right word is for the far more complicated constructions that are described in this paper.

  • There’s a new version this evening of the 3-week old Arkani-Hamed/Weiner LHC Signals for a SuperUnified Theory of Dark Matter, in which they claim to have a new signature for supersymmetry, with a large fraction of all SUSY events looking exactly like what CDF described. Oddly enough, the changes to the paper don’t include a mention of the CDF result. This paper also invokes a rather baroque mechanism, involving both the supersymmetric extension of the standard model and a whole new complicated dark sector.

    • This last paper is also supposed to explain the PAMELA data, and papers with other explanations of this are starting to flood hep-ph.

    So far, all the explanations of the anomaly seen by CDF look suspiciously complicated, which may be one reason that many members of CDF are so skeptical about the whole thing that they were unwilling to sign on to the PRD submission. But I’m sure that many more proposals for how to explain the anomaly are being drafted at this very moment, and maybe one of them will be more convincing.

    Update: Over at Tommaso Dorigo’s blog there’s a short posting about Giromini et. al., and an exchange with Nima Arkani-Hamed, who claims to have had no inside knowledge of the CDF “lepton jets” when he wrote his paper with Weiner predicting them. He also explains how the exact mechanism discussed in that paper is unlikely to explain the CDF result since their observed rate is too high for this.

    Update: New Scientist has the story, emphasizing the possible relation to the work of Arkani-Hamed and Weiner:

    So what could it be? As it happens, Weiner and Nima Arkani-Hamed of the Institute for Advanced Study in Princeton, New Jersey, and colleagues have developed a theory of dark matter – the enigmatic stuff thought to make up a large proportion of the universe – to explain recent observations of radiation and anti-particles from the Milky Way.

    Their model posits dark matter particles that interact among themselves by exchanging “force-carrying” particles with a mass of about 1 gigaelectronvolts.
    The CDF muons appear to have come from the decay of a particle with a mass of about 1 GeV. So could they be a signature of dark matter? “We are trying to figure that out,” says Weiner. “But I would be excited by the CDF data regardless.”

    CDF spokesperson Jacobo Konigsberg is quoted as saying:

    we haven’t ruled out a mundane explanation for this, and I want to make that very clear

    Update: Then there’s Slashdot, where the hypothetical CDF particle is advertised as accounting for the Arkani-Hamed et. al. theory of dark matter.

    Update: Another story, at Physics World, which has more from various people at CDF. Again, that Arkani-Hamed/Weiner “predicted a CDF–like signal”, although the problem with the rate being too low is mentioned.

    Also Nature, where one learns:

    Theorists are already coming up with ideas about what might be producing the excess muons. One possibility is that they stem from the decay of a heavier, yet-to-be-discovered particle — perhaps related to dark matter, an unseen material that is believed to make up some 85% of matter in the Universe.

    Another idea from string theory evokes seven-dimensional ‘branes’ — theoretical surfaces that are inhabited by exotic particles manifested as strings. These higher-dimensional branes might be home to force-carrying particles that interact weakly with our three-dimensional world and create a faint, but traceable, signal in the data.

    But Adam Falkowski, a theorist at CERN, Europe’s particle accelerator laboratory near Geneva, Switzerland, says that the explanations need some work, and cautions against attempting to force the data to fit into particular theories.

    Update: More press stories here and here.

    Posted in Experimental HEP News | 24 Comments

    Discovery of a New Particle?

    Posted on October 30, 2008 by woit

    Except for the excitement surrounding first beams in the LHC, particle physics has been an all-too-quiet subject recently. It looks like that may be about to change, with a dramatic new result announced by the CDF experiment this evening, in a preprint entitled Study of multi-muon events produced in p-pbar collisions at sqrt(s)=1.96 TeV.

    The CDF result originates in studies designed to determine the b-bbar cross-section by looking for events where a b-bbar pair is produced, each component of the pair decaying into a muon. The b-quark lifetime is of order a picosecond, so b-quarks travel a millimeter or so before decaying. The tracks from these decays can be reconstructed using the inner silicon detectors surrounding the beam-pipe, which has a radius of 1.5 cm. They can be characterized by their “impact parameter”, the closest distance between the extrapolated track and the primary interaction vertex, in the plane transverse to the beam.

    If one looks at events where the b-quark vertices are directly reconstructed, fitting a secondary vertex, the cross-section for b-bbar production comes out about as expected. On the other hand, if one just tries to identify b-quarks by their semi-leptonic decays, one gets a value for the b-bbar cross-section that is too large by a factor of two. In the second case, presumably there is some background being misidentified as b-bbar production.

    The new result is based on a study of this background using a sample of events containing two muons, varying the tightness of the requirements on observed tracks in the layers of the silicon detector. The background being searched for should appear as the requirements are loosened. It turns out that such events seem to contain an anomalous component with unexpected properties that disagree with those of the known possible sources of background. The number of these anomalous events is large (tens of thousands), so this cannot just be a statistical fluctuation.

    One of the anomalous properties of these events is that they contain tracks with large impact parameters, of order a centimeter rather than the hundreds of microns characteristic of b-quark decays. Fitting this tail by an exponential, one gets what one would expect to see from the decay of a new, unknown particle with a lifetime of about 20 picoseconds. These events have further unusual properties, including an anomalously high number of additional muons in small angular cones about the primary ones.

    The exciting possibility here is that a new, relatively long-lived particle has been observed, one that decays in some way that leads to a lot more muons than one gets from Standard Model states. It should be remembered though that this is an extraordinary claim requiring extraordinary evidence, and the possibility remains that this is some sort of background or detector effect that the CDF physicists have missed. It should also be made clear that this paper is not a claim by CDF to have discovered a new particle, rather it is written up as a description of the anomalies they have found, leaving open the possibility that these come from some standard model processes or detector characteristics that they do not yet understand.

    The overwhelming success of the Standard Model during the past 30 years has meant that essentially all claims from accelerator experiments to see some new, non-SM physics have turned out to be mistaken. As a result, collaborations like CDF are now extremely careful about making such claims and will only do so after the most rigorous possible review. It’s a remarkable event that this one has gotten out, signed off on by the entire collaboration (although from what I understand, people can drop their names from the publication list of a specific paper if they disagree with it, maybe one should check this author list carefully…).

    What would really be convincing would be a confirmation of this from D0, the other Tevatron detector. The D0 collaboration would not only be working with a detector that has somewhat different characteristics, but would also have some motivation to find a problem with the result from their competition. If they also see it, that would be pretty extraordinary evidence. Another sort of extraordinary evidence would be to see evidence for the same kind of new particle in other channels.

    This will undoubtedly unleash a flood of papers from theorists promoting models that extend the Standard Model in ways that would produce something with the observed experimental signature. This is not a signature characteristic of supersymmetry or any of the other known heavily-studied classes of models. If real, as far as I’m aware it’s something genuinely unexpected. Perhaps phenomenology experts can point to some less well-known models with this kind of signature. The only such thing I’m aware of is a very recent paper from three weeks ago by Arkani-Hamed and Weiner, entitled LHC Signals for a SuperUnified Theory of Dark Matter. They discuss a theory of dark matter involving a new hidden gauge symmetry, broken near the GeV scale, saying that this is “motivated directly by striking Data from the PAMELA and ATIC collaborations”. In these models there can be Gev-scale Higgs and gauge particles decaying to an anomalously large number of leptons. They discuss the question of whether the parameters of such models can be adjusted to give large decay lengths, and predict the observation of events that “contain at least two “lepton jets”: collections of n > 2 leptons, with small angular separations and GeV scale invariant masses”, pretty much just what CDF sees . Since the CDF paper undoubtedly has been the topic of intense discussion among the 450 or so physicists in the collaboration for many months now, the most likely explanation for the appearance of a new theory paper a few weeks ago discussing exactly the signatures in question is that news of what’s in the paper got out to some theorists early. Even if this particular result goes away, this gives some indication of what sorts of things are likely to happen once LHC data starts being collected and analyzed.

    The bottom line though is that for the first time in quite a while, there’s some very exciting and potentially revolutionary news in particle physics. It’s coming not out of the LHC, which is still a hope for the future, but from a currently functioning machine which is producing more data every day. If this result holds up, this data contains a wealth of information about some new physics which will likely revolutionize our understanding of elementary particle physics. Particle physics may already have started to move out of its doldrums.

    Update: This evening I came across an unexpected source of information about this, one which might explain why news of this result may have leaked out a while ago. More about this later.

    Update: The results from PAMELA mentioned here, and which are listed as motivation for the Arkani-Hamed/Weiner paper, are now officially out. For discussion of this from someone much better informed than me, see this posting at Resonaances.

    Update: There’s an excellent detailed posting about the paper from CDF’s own Tommaso Dorigo. If you’re interested in understanding exactly what is going on here, that’s where you should start.

    Update: For entertainment, there’s always Lubos.

    Update: I’m now free to explain what I was alluding to when I earlier mentioned an “unexpected source of information”. Yesterday evening while I was trying to find out more about the CDF result, its relation to previously published experimental results, and to possible phenomenological models, I was running some Google searches on relevant terms. One of these turned up something very surprising: the first result was a summary of CDF’s internal review of drafts of PRL and PRD papers on the subject, the second was the PRL draft. Both of these documents were from early July, and part of a publicly accessible directory containing all the materials from a review of the draft at that time. Investigating further, it became clear that the CDF web-server was seriously misconfigured, allowing directory listings and public access to a wide array of their work materials.

    I wrote an e-mail to people at CDF warning them about the problem, heard back quickly, and have just checked that they have fixed it, so I don’t think I’ll cause them a problem other than embarassment by telling this story here. It seems likely that these materials have been publicly accessible and indexed by Google probably for several months now.

    I confess to reading these documents, figuring that anything really interesting that is Googleable is fair game. Since they clearly were not intended for public consumption I won’t disseminate information about them beyond this story about their existence and the fact that the PRL draft contained tentative material interpreting the data in terms of new physics, the sort of thing the released paper avoids. One thing I can say is that it is very impressive to see the amount of effort and very serious scientific work behind a review of this kind. A lot went into this, and presumably a lot more has gone into it since last July. I understand why CDF does not make this kind of thing public, but it actually would be a wonderful example for the public of how science is done to do so.

    Update: See Resonaances for a discussion of the dark sector model building.

    Update: Another CDF blogger heard from: John Conway.

    Posted in Experimental HEP News | 42 Comments

    Job Posting

    Posted on October 25, 2008 by woit

    This is the time of year during which a large number of physicists and mathematicians must turn their attention to the problem of finding employment for next fall. Physics Today has a jobs site here, which has a new posting that may interest some of my readers. Thanks to Michael Williams for pointing this out to me.

    If applicants need a PD18 form, it’s available here.

    Posted in Uncategorized | 17 Comments

    Quantum Physicist Offers Solution To Global Market Meltdown

    Posted on October 16, 2008 by woit

    I briefly met John Hagelin when I was an undergraduate. At the time he was a Harvard particle theory graduate student, soon to get his degree and start a quite respectable career in the subject. His biography on his web-site claims that:

    His scientific contributions in the fields of electroweak unification, grand unification, super-symmetry and cosmology include some of the most cited references in the physical sciences. He is also responsible for the development of a highly successful Grand Unified Field Theory based on the Superstring. Dr. Hagelin is therefore at the pinnacle of achievement among the elite cadre of physicists who have fulfilled Einstein’s dream of a “theory of everything” through their mathematical formulation of the Unified Field—the most advanced scientific knowledge of our time.

    Hagelin moved on from doing physics research, first to help run the Maharishi International University (now the Maharishi University of Management) in Fairfield, Iowa, then to run for president as the candidate of the Natural Law Party (it seems that “Joe the Plumber” was a member of this). Nowadays, the Maharishi has passed away, and Hagelin has moved to New York, where he is Executive Director of the Global Financial Capital of New York.

    Soon after Hagelin moved to New York, his organization bought an impressive building down near Wall Street (chosen because “It’s one of the very few buildings in all of New York City that’s oriented due east”), with the goal “to inspire financiers to come forth to support the creation of Heaven on Earth”. They also are buying up land near Princeton “where, with the support of the township, a university for 5,000 Yogic Flyers is to be established.” Part of the plan seems to be to raise $3 billion or so from the financial industry, by offering 10% interest on 15 year loans.

    Last year they held a news conference to explain how the a “surging stock market” was one

    of the concrete signs of the success of the Invincible America Assembly in Iowa—the largest-ever scientific demonstration project to document the effects of large group meditations on the economic and social trends of the nation, according to Dr. John Hagelin, world-renowned quantum physicist, executive director of the International Center for Invincible Defense, and President of the Global Union of Scientists for Peace, who is leading the Assembly.

    This year the story is a bit different, with a recent press release entitled Quantum Physicist Offers Solution to Global Market Meltdown explaining that:

    a group of nearly 2,000 advanced experts is now in place at Maharishi University of Management in Fairfield, Iowa. He said the influence of coherence generated by this group is helping to calm the nation in the midst of the global crisis, but a larger, more powerful group of 8,000 experts (the square root of one percent of the world’s population) is needed to neutralize worldwide fears and re-establish confidence in the global markets.

    The cost to establish this group on a permanent basis, Dr. Hagelin said, would be negligible compared to what has been lost in a single hour during the current financial crisis.

    In a recent Open Letter to the Yogic Flyers of the Nation, Hagelin, the Raja of Invincible America, urges them on as follows:

    I am writing you from Wall Street, where I am living and working with members of my national team at the Global Financial Capital of New York, one block from the New York Stock Exchange. We are starting to make remarkably good progress in our efforts to bring Maharishi’s knowledge of enlightenment and invincibility to leaders here whose thinking and actions vitally impact the whole world. However, ultimately, our success is dependent on you. We will be successful when the leaders are receptive to our message. But their openness is 100% dependent upon the numbers in the Domes. Why? Because a rise in national consciousness directly translates into a rise in openness among leaders of business, health, education, defense, government, etc.

    At this critical time I urge everyone to fly together in large groups

    Maharishi said we need 2500 experts flying together to guarantee invincibility for America. This requirement is because the turbulence in the collective consciousness of one country can easily spread like a wildfire to create a similar turbulence in another country. And this is why, at this critical time, with the economic stability of the world hanging in the balance, I urge everyone in Iowa and everyone in the country to recommit to fly together in large groups.

    Update: Today was a beautiful day, and on a bike ride downtown I stopped to take a look at 70 Broad Street, the headquarters of Global Financial Capital of New York. Looking in the windows of the below ground level and the lobby level, the place looked pretty much abandoned. The building has some floors upstairs, maybe there is some activity up there.

    They did just put out this press release, announcing a press conference to be held in the lobby at 11am on Tuesday and webcast. Hagelin will be announcing a $1 billion plan to fund 1000 experts in New York and 8000 experts in Iowa who will “create coherence for the whole world—the basis of an invincible, prosperous global economy.” No word in the press release about where the $1 billion was going to come from.

    Update: Video of the press conference is here. The plan is simple: just give Hagelin and his followers $1 billion, and they’ll have this financial crisis all sorted out soon, by means of experts using the unified field.

    Posted in Uncategorized | 87 Comments

    Princeton Companion to Mathematics

    Posted on October 15, 2008 by woit

    I just recently got my hands on a copy of the new Princeton Companion to Mathematics, and I fear that this is likely to seriously impact my ability to get things done for a while, as I devote too much time to happily reading many of its more than 1000 pages.

    The book is an amazing document (and physically, a beautiful, if weighty object), unlike anything else I know of. Its coverage of mathematics and mathematical culture is very wide and sometimes deep, but it makes no attempt to be comprehensive. Thus, the accurate title “Companion to” rather than “Encyclopedia of”. The most remarkable aspect of the book is the extremely high quality of the contributions from a large number of different authors. It includes many wonderful long expository articles, mostly at a level that a good undergraduate math student could hope to appreciate, with much of the book accessible to an even wider audience. The articles are often written by some of the best researchers and expositors around. For example, one can find Barry Mazur writing on Algebraic Numbers, Janos Kollar on Algebraic Geometry, Cliff Taubes on Differential Topology, Ingrid Daubechies on Wavelets, Persi Diaconis on Mathematical Statistics, and many, many others of similar quality. The table of contents is available here.

    The book also includes extensive articles on historical topics in mathematics and short biographies of a large number of mathematicians, as well as coverage of applications and a section largely devoted to describing the art of problem-solving and how mathematics really gets created. This section includes a beautiful set of five essays called “Advice to a Young Mathematician”, which give five different equally fascinating perspectives from some of the best in the subject about how they achieved what they did, as well as what they have learned from years of helping students become researchers. The authors of these pieces are Michael Atiyah, Bela Bollobas, Alain Connes, Dusa McDuff, and Peter Sarnak. Luckily for all young (and old) mathematicians, this chapter is freely available here.

    The person most responsible for this is clearly the editor (and author of some of the pieces), Fields Medalist Timothy Gowers, who had help from many others, including fellow Fields Medalist Terry Tao. Gowers has a weblog, and he has written about the book in these entries (and there’s a podcast interviewing him on the book web-site at PUP). Terry Tao has a posting about the book here.

    If you’re looking for a gift for someone with a serious interest in mathematics, no matter what their background, you won’t do any better than this.

    Posted in Uncategorized | 23 Comments