Yang Mills


To define the Yang-Mills Lagrangian, we need to define the ‘Trace’ of an End(E) valued form. Recall that the Trace of a matrix is the sum of its diagonal enTries. The Trace is independent of the choice of basis – an invariant notion that is independent of the choice of basis. A definition of the Trace that mkes this clear is as follows. Consider {End(V) \simeq V \otimes V^*} – an isomorphism that does not depend on any choice of basis – so the pairing between V and {V^*} defines a linear map Tr: End(V) \rightarrow \mathbb{R} v \otimes f \mapsto f(v) To see that this v is really a Trace, pick {e_i} of V and let {\epsilon^j} be a dual basis of {V^*}. Writing {T \;\in\; End(V)} as T = T^i_j e_i \otimes \epsilon^j We have Tr(T) = T^i_je_i(\epsilon^j) = T^i_j \delta_i^j = T^i_i which is of course the sum of the diagonal enTries. \newline This implies that if we have a section T of {End(E),} we can define a funciton Tr(T) on the base manifold M whose value at {p \in M} is the Trace of the endomorphism T(p) of the fiber {E_p}: Tr(T)(p) = Tr(T(p)) If {T \in \Gamma(End(E))} and {\omega \in \Omega^p(M)} we define

\displaystyle Tr(T \otimes \omega) = Tr(T)\omega

Now we can write down the Yang-Mills Lagrangian: If D is a connection on E, this is the n-form given by

\displaystyle \mathcal{L}_{YM} = \frac{1}{2} Tr(F \wedge \textasteriskcentered F) \ \ \ \ \ (1)

where F is the curvature of D. Note that by the defintion of the hodge star operator (also in this collection of notes), we can write this in local co-ordinates as

\displaystyle \mathcal{L}_{YM} = \frac{1}{4} Tr (F_{\mu \nu}F^{\mu \nu})vol \ \ \ \ \ (2)

If we integrate {\mathcal{L}_{YM}} over M we get the Yang-Mills action

\displaystyle S_YM = \frac{1}{2} \int_M Tr (F \wedge \textasteriskcentered F) \ \ \ \ \ (3)

This needs some elaboration. So let us explain these formulas better. We choose the physics convention {F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu -ig[A_\mu,A_\nu]} where the generators of the Lie algebra are Hermitian.

\displaystyle \mathcal{L}_{YM} = \mathcal{I} = - \int Tr(F \wedge \textasteriskcentered F) \ \ \ \ \ (4)

by another convention (there is a lot of ambiguity of signs in this subject). The first thing to note is that F has vector and Lie algebra indices. The Trace is over the Lie algebra, not over the vector indices. The vector indices are just those of the field sTrength in QED. In Yang-Mills the curvature form is Lie Algebra valued. \newline In this case {F_{\mu \nu} = F^{a}_{\mu \nu}T^a} where the summation convention is used,and where {T^a} are the generators of {\mathfrak{su}(n)}. To be explicit, F has not only tensor components but maTrix components (F_{\mu \nu})_{ij} = F^{a}_{\mu \nu}T^a T^a_{ij} The inner product of F with itself {<F,F> = \int F \wedge \textasteriskcentered F} where \textasteriskcentered is the Hodge \textasteriskcentered – operator. Thus we are calculating {\mathcal{I} = -Tr<F,F>}. It is a standard exercise to find the exterior product of two r-forms. We find F \wedge \textasteriskcentered F = \frac{1}{2!}F_{\mu \nu}F^{\mu \nu} dx^1 \wedge \cdots \wedge dx^4 Note that the differential forms don’t ‘know’ the Lie algebra. The algebra hasn’t come into the calcuation yet. {Tr(F_{\mu \nu}F^{\mu \nu})=Tr(F^{a}_{\mu \nu}T^aF^{\mu \nu b}T^b} {= Tr(T^aT^b)F^{a}_{\mu \nu}F^{\mu \nu b}} {= \frac{1}{2}\delta^{ab}F^{a}_{\mu \nu}F^{\mu \nu b}} {=\frac{1}{2}F^{a}_{\mu \nu}F^{\mu \nu a}} where e have used the standard normalization convention for the {T^a}, {Tr T^a T^b = \frac{1}{2}\delta^{ab}}. (This comes from the fact we want {\mathfrak{su}(2)} to live in {\mathfrak{su}(n)}, and the generators of {\mathfrak{su}(2)} are taken to be {T^a = \frac{\sigma^a}{2}} where {\sigma^a} are the Pauli matrices.) Thus, we find \mathcal{I} = - Tr (F \wedge \textasteriskcentered F) which can be written as \frac{1}{4} \int d^4 x F^{a}_{\mu \nu}F^{a \mu \nu}

Tony Judt on Society


I posted the following on my Facebook account last night ‎

‘Thinking “economistically,” as we have done now for thirty years, is not intrinsic to humans. There was a time when we ordered our lives differently.’

– Tony Judt
A friend of mine challenged me to provide some analysis, so here I provide some.
Tony Judt was writing a book aimed at young people who en masse seem to have lost a political activism that previous generations have. There are serious problems caused by rampant neo-liberalism and the domination of policy making by economic concerns. Someone quipped to me that the religion of our age is ‘pop culture, economics, business and money making’

Tony was thinking of the fact that policy considerations are largely dominated by economic concerns. Not to mention the economic dogma of the Chicago School – .Empirically the ‘efficient market hypothesis’ is false. We can have an argument about how false it is, some other time. Also I think Judt makes a very clear point (see any of his articles on this on the NYRB on this set of topics) that economic considerations dominating political discussion, aren’t a natural occurrence they are a matter of taste. What about ‘is it right’. I don’t think notions of ‘fairness’ or ‘morality’ should be neglected in political discourse. And this is very important for some defence of the social democratic model.
What is pertinent about this article is that economic thinking which dominates our policy concerns is not necessarily a natural way to think about this. A lot of these problems in policy – for instance bringing morality into it – which I personally think is one reason we should try to conquer the problems of poverty – are discursive. We have forgotten how to talk in different ways. The fields of human endeavour are large and it is dangerous to marry policy and economics )and bastardised economic thinking at that) too closely.
My friend Sam posted this “In the modern age of policy, economic analysis supersedes other decision-making criteria that most of us use every day such as ethics, morality, and principles such as robustness and precaution. As a result we have entered in to a political paradigm which totally relies on models to justify a government’s supposedly utilitarian agenda. The choice and subsequent blame becomes not that of the elected decision maker but put squarely on the models and the limits of human ability in building such a model. Blame becomes diffuse, as does responsibility. This makes for bad policy making.” on his blog a few years ago. It seemed profound then and it seems profound now.

On Fubini-Study Metrics


1. A Little Complex Analysis

We want to introduce the notion of a ‘Fubini-Study’ metric which is important in Complex Manifold Theory and Differential Geometry (and the associated theories such as Mathematical Physics). But first we need to introduce a little Complex Analysis. The source is of course Griffiths and Harris. Let M be a complex manifold, {p \in M} any point, and {z=(z_{1},\cdots,z_{n})} a holomorophic co-ordinate system around p. There are three different notions of a tangent space to M at p,which we now describe:

  • {T_{\mathbb{R},p}(M)} is the usual real tangent space to M at p,when we consider M a real manifold of dimension 2n. {T_{\mathbb{R},p}(M)} can be realized as the space of {\mathbb{R}-}linear derivations on the ring of real-valued {C^{\infty}}-functions in a neighbourhood of p; if we write {z_i = x_i + iy_i}, {T_{\mathbb{R},p}(M) = \mathbb{R}(\frac{\partial}{\partial x_{i}}, \frac{\partial}{\partial y_i}}.
  • {T_{\mathbb{C},p}(M) = T_{\mathbb{R},p}(M)\otimes_{\mathbb{R}} \mathbb{C}} is called the complexified tangent space to M at p. It can be realized as the space of {\mathbb{C} -} linear derivations in the ring of complex valued {C^{\infty}}-functions on M around p. We can write {T_{\mathbb{C},p}(M) = \mathbb{C}{\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial y_i}}}
    ={\mathbb{C}{\frac{\partial}{\partial z_{i}},\frac{\partial}{\partial \bar{z}_i}}}

  • {T'_p(M)= \mathbb{C}{\frac{\partial}{\partial z_{i}}}\subset T_{\mathbb{C}, p}(M)} is called the holomorphic tangent space to M at p. It can be realized as the subspace of {T_{\mathbb{C},p}(M)} consisting of derivations that vanish on antiholomorphic functions (i.e. F such that T is holomorphic), and so is independent of the holomorphic co-ordinate system chosen. The subspace {T''_p(M)= \mathbb{C}{\frac{\partial}{\partial \bar{z}_{i}}}} is called the antiholomorphic tangent space to M at p; clearly {T_{\mathbb{C},p}(M) = T'_p(M) \oplus T''_p(M)}

Now we consider some Calculus on Complex Manifolds. Let M be a complex manifold of dimension n. A hermitian metric on M is given by a positive definite hermitian inner product {(,)_z: T'_z(M) \otimes T'_z(M)       \rightarrow \mathbb{C}} on the holomorphic tangent space at z for each {z \in M},
depending smootly on z – that is, such that for local co-ordinates z on M the function
{h_ij(z) = (\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_j})_z} are {C^{\infty}}

Writing {(,)_z} in terms of the basis {{dz_i \otimes d\bar{z}_j}} for {(T'_z(M) \otimes \bar{T'_z(M)}^{\textasteriskcentered} = T^{\textasteriskcentered\textquoteright}_z(M) \otimes T^{* \textquotedblright}_{z}(M)}, the hermitian metric is given by {ds^{2} = \sum_{i,j} h_{ij}(z) dz_i \otimes d \bar{z}_j} So let us describe the Fubini-Study Metric Let {z_0,\cdots,z_n} be co-ordinates on {\mathbb{C}^{n+1}} and denote by {\pi:\mathbb{C}^{n+1} -{0}    \rightarrow \mathbb{P}^n} the standard projection map. Let {U \subset \mathbb{P}^{n}} be an open set and {Z: U \rightarrow \mathbb{C}^{n-1} - {0}} a lifting of U, i.e. a holomorphic map with {\pi \circ z = id}; consider the differential form
{\omega = \dfrac{i}{2\pi}\partial \bar{\partial}log\|z\|^{2}} If {Z':U   \rightarrow \mathbb{C}^{n-1} - {0}} is another lifting, then {Z' = f.Z} with f a nonzero holomorphic function, so that
{\dfrac{i}{2\pi}\partial \bar{\partial}log\|z\|^{2} = \frac{i}{2 \pi}\partial \bar{\partial} (log\|z\|^{2} + log f + log \tilde{f})}
{= \omega + \dfrac{i}{2\pi}(\partial \bar{\partial}log f - \bar{\partial} \partial log \tilde{f})} = {\omega} Therefore {\omega}is independent of the lifting chosen; since liftings always exist locally, {\omega} is a globally defined differential form in {\mathbb{P}^{n}}. (By the sheaf properties of differential forms) Clearly {\omega} is of type (1,1). To see that {\omega} is positive, first note that the unitary group {U(n+1)} acts transitively on {\mathbb{P}^{n}} and leaves the form {\omega} positive everywhere if it is positive at one point. Now let {{w_i = z_i/z_0}} be co-oridnates on the open set {U_{0} = (z_0 \neq 0)}in {\mathbb{P}^{n}} and use the lifting {Z = (1,w_1,\cdots,w_n)} on {U_0} ; we have (after some substitutions

{\omega = \dfrac{i}{2 \pi} [\frac{\sum dw_i \wedge d\bar{w}_i}{1 + \sum w_i \bar{w}_i} - \frac{(\sum \bar{w}_i dw_i \wedge \sum w_i d\bar{w}_i)}{(1 + \sum w_i \bar{w}_i)^{2}}]} At the point {[1,0,\cdots,0]}, \\ {\omega = \frac{i}{2\pi} \sum dw_i \wedge d \bar{w}_i > 0} Thus {\omega} defines a particular hermitian metric on the projective complex space called the Fubini-Study metric. That was the aim of the article!

On Academic talks


On Academic talks

Cosma Shalizi, has an excellent talk on Academic talks.

I suggest one reads it.

I merely quote my favourite part:

  1. The point of the talk is not to please you, by reminding yourself of what a badass you are, but to tell your audience something useful and interesting. (Note to graduate students: It is important that you internalize that you are, in fact, a badass, but it is also important that then you move on. Needing to have your ego stroked by random academics listening to talks is a sign that you have not yet reached this stage.) Unless something matters to your actual message, it really doesn’t belong in the main body of the talk.
  2. You can stick an arbitrary amount of detail in the “I’m glad you asked that” slides, which go after the one which says “Thank you for your attention! Any questions?”.
  3. You also can and should put all these details in your paper, and the people who really care, to whom it really matters, will go read your paper. Once again, think of an academic talk as an extended oral abstract.

Internalise that you are in fact a bad ass. I wish more Professors gave advice like that.

The notion of home


The notion of home

Ribbon farm has an interesting and well written account of the changing notion of home.

It is an interesting concept, for some of us, especially as we have movable offices, or ‘third spaces’ one can easily survive in cities above a certain size. Hiking trails, a park, a few coffee houses, etc being enough for some people who are for instance writers, or software engineers.

     A very interesting discussion of home, I myself however am an Irish national who is living in Luxembourg. I still end up in an Irish bar of course every so often, and am part of some expat community. Phones and facebook of course act as a sort of pacifier.