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A must read for all you Guinness drinkers

W. T. Lee and M. G. Devereux [1] review the bubble formation processes in carbonated drinks,  like fizzy pop or champagne and compare this with heavy stout drinks.  Stout beers have lots of dissolved nitrogen and this makes the physics slightly different to the carbonated drinks.  Specifically, although the same mechanisms apply the time scales are very different.  Stouts will not spontaneously form a head of foam. This means that a widget or similar needs to be added to cans in order to  aid the nucleation of gas bubbles.

In the paper both the mathematical and experimental issues are discussed.  I recommend you use the paper as a conversation starter next time you are wasted on stout at your local ale house.

Bottoms up!

References

[1] W. T. Lee and M. G. Devereux. Foaming in stout beers.  arXiv:1105.2263v1 [physics.chem-ph]

Higher Lie-Schouten brackets

I thought it would be interesting to point out a geometric construction related to  \(L_{\infty}\)-algebras.  (See earlier post here) Recall that given a Lie algebra \((\mathfrak{g}, [,] )\) one can associate on the dual vector space a linear Poisson structure known as the Lie-Poisson bracket.  So, as a  manifold \((\mathfrak{g}^{*}, \{, \}) \) is a Poisson manifold.  It is convenient to  replace the “classical” language of linear and replace this with a graded condition. That is, if we associate weight one to the coordinates on \(\mathfrak{g}^{*} \) then the Lie-Poisson bracket is of weight minus one.

The Lie Poisson bracket is very important in deformation quantisation (both formal and C*-algebraic). There are some nice theorems and results that I should point to at some later date.

Now, it is also known that one has an odd version of this known as the Lie-Schouten brackets on \(\Pi \mathfrak{g}^{*}\). The key difference is the shift in the Grassmann parity of the “linear” coordinates.  Note that this all carries over to Lie super algebras with no problem.  I will drop the prefix super from now on…

 

So, let us look at the situation for \(L_{\infty}\)-algebras. We understand these either as a series of higher order brackets on a vector space  \(U\) that satisfies a higher order generalsiation of the Jacobi identities or more conveniently we can understand all this in terms of a homological vector field on the formal manifold \(\Pi U\).

Definition An \(L_{\infty}\)-algebra is a vector space \(V = \Pi U\) together with a homological vector field \(Q = (Q^{\delta} + \xi^{\alpha} Q_{\alpha}^{\delta} + \frac{1}{2!} \xi^{\alpha} \xi^{\beta} Q_{\beta \alpha}^{\delta} + \frac{1}{3!} \xi^{\alpha} \xi^{\beta} \xi^{\gamma} Q_{\gamma \beta \alpha}^{\delta} + \cdots) \frac{\partial}{\partial \xi^{\delta}}\),

where we have picked coordinates on \(\Pi U\)  \(\{  \xi^{\alpha}\}\). Note that these coordinates are odd as compared to the coordinates on \(U\). Thus we assign the Grassmann parity \(\widetilde{\xi^{\alpha}} = \widetilde{\alpha} + 1\)  Note that \(Q\) is odd and that if we restrict to the quadratic part then we are back to Lie algebras.

I will simply state the result, rather than derive it.

Proposition Let \((\Pi U, Q)\) be an \(L_{\infty}\)-algebra. Then the formal manifold \(\Pi U^{*}\) has a homotopy Schouten algebra structure.

Let us pick local coordinates \(\{ \eta_{\alpha}\}\) on \(\Pi U^{*}\). Furthermore, we consider this as a graded manifold and attach a weight of one to each coordinate.  A general function,  a  “multivector” has the form

\(X = \stackrel{0}{X} + X^{\alpha} \eta_{\alpha} + \frac{1}{2!}X^{\alpha \beta}\eta_{\beta} \eta_{\alpha} + \cdots \)

The higher Lie-Schouten brackets are given by

\((X_{1}, X_{2}, \cdots, X_{r}) = \pm Q_{\alpha_{r}\cdots  \alpha_{1} }^{\beta}\eta_{\beta}\frac{\partial X_{1}}{\partial \eta_{\alpha_{1}}} \cdots   \frac{\partial X_{1}}{\partial \eta_{\alpha_{r}}}\),

being slack with an overall sign.  Note that with respect to the natural weight the n-bracket has weight (1-n). Thus not unexpectedly, restricting to n=2 gives an odd bracket of weight minus one: up to conventions this is the Lie-Schouten bracket of a Lie algebra.

The above collection of brackets forms an \(L_{\infty}\)-algebra in the “odd super” conventions that satisfies a derivation rule of the product of “multivectors”. Thus the nomenclature homotopy Schouten algebra and higher Lie-Schouten bracket.

A similar statement holds in terms of a  homotopy Poisson algebra on \(U^{*}\). Here the brackets as skewsymmetric and of  even/odd Grassmann parity for even/odd number of arguments.  (I rather the odd conventions overall).

Now this is quite a new construction and the technical exploration of this nice geometric construction awaits to be explored. How much of the geometric theory associated with Lie algebras and Lie groups carries over to \(L_{\infty}\)-algebras and \(\infty\)-groups is an open question.

Details can be found in Andrew James Bruce ” From \(L_{\infty}\)-algebroids to higher Schouten/Poisson structures”, Reports on Mathematical Physics Vol. 67, (2011), No. 2  (also on the arXiv).

 

Also see earlier post here on Lie infinity algebroids.

What is mathematical physics?

This is a question that naturally arises as I consider myself to be a mathematical physicist, so I do mathematical physics. But what is mathematical physics?

I don’t think there is any fully agreed on definition of mathematical physics and like any branch of mathematics and physics it evolves and grows. That said, there is roughly two common themes:

  • Doing physics like it is mathematics
  • That is trying to apply mathematical rigour in the constructions and calculations of physics. This is often very hard as physics often requires lots of simplifications and approximations. A lot of physical interpretation and intuition can enter into the work. Physics for the most part is not mathematics and lots of results in theoretical physics lack the rigour required by mathematicians.

  • Studying the mathematical structures required in physics and their generalisations
  • Mathematics is the framework in which one constructs physical theories of nature. As such mathematics, as mathematics is fundamental in developing our understanding of the world around us. This part mathematical physics is about studying the basic structures behind physics, often with little or no direct reference to a specific physical systems. This can lead to natural generalisations of the mathematical structures encountered and give a wider framework to understand physics.

    We see that mathematical physics is often closer to mathematics than physics. I see it as physically motivated mathematics , though this motivation is often very technical.

    Of course this overlaps to some extent with theoretical physics. However, the motivation for theoretical physics is to create and explore physical models, hopefully linking them with reality. Mathematical physics is more concerned with the mathematical structures. Both I think are important and feed of each other a lot. Without development in mathematical physics, theoretical physics would have less mathematical structure and without theoretical physics, mathematical physics would lack inspiration.

    29th North British Mathematical Physics Seminar

    I have been invited to give a talk at the 29th North British Mathematical Physics Seminar (NBMPS) in Edinburgh on the 16th February 2011.

    The NBMPS has been running since 2001 and is a forum for mathematical physicists in North Britain to meet up. They organise four one-day meetings that are held in rotation every year in Durham, Edinburgh, York and Nottingham.

    The 29th meeting is in Edinburgh and I am listed a the first speaker!

    The topic of my talk will be my preprint on Odd Jacobi manifolds and classical BV-gauge systems. This paper is also discussed on my blog here.

    I will place a link to the slides in due course.

    I will also place an update of the event at some later date.

    ************************************************
    Update:

    The talk I felt went well. I had a few questions and comments, but nothing off putting or massively critical.

    *************************************************

    Update: The slides for the talk can be found here.

    What is geometry?

    This is a question I am not really sure how to answer. So I put it to Sir Michael Atiyah after his Frontiers talk in Cardiff. In essence he told me that geometry is any mathematics that you can imagine as pictures in your head.

    To me this is in fact a very satisfactory answer. Geometry a word that literally means “Earth Measurement” has developed far beyond its roots of measuring distances, examining solid shapes and the axioms of Euclid.

    Another definition of geometry would be the study of spaces. Then we are left with the question of what is a space?

    Classically, one thinks of spaces, say topological or vector spaces as sets of points with some other properties put upon them. The notion of a point seems deeply tied into the definition on a space.

    This is actually not the case. For example all the information of a topological space is contained in the continuous functions on that space. Similar statements hold differentiable manifolds for example. Everything here is encoded in the smooth functions on a manifold.

    This all started with the Gelfand representation theorem of C*-algebras, which states that “commutative C*-algebras are dual to locally compact Hausdorff spaces”. I won’t say anything about C*-algebras right now.

    In short one can instead of studying the space itself one can study the functions on that space. More than this, one can take the attitude that the functions define the space. In this way you can think of the points as being a derived notion and not a fundamental one.

    This then opens up the possibility of non-commutative geometries by thinking of non-commutative algebras as “if they were” the algebra of functions on some non-commutative space.

    Also, there are other constructions found in algebraic geometry that are not set-theoretical. Ringed spaces and schemes for example.

    So, back to the opening question. Geometry seems more like a way of thinking about problems and constructions in mathematics rather than a “stand-alone” topic. Though the way I would rather put it that all mathematics is really geometry!

    Should you beleive everything on the arXiv?

    For those of you who do not know, the arXiv is an online repository of reprints in physics, mathematics, nonlinear science, computer science, qualitative biology, qualitative finance and statistics. In essence it is a place that scientists can share their work and work in progress, but note that it is not peer reviewed. The arXiv is owned and operated by Cornell University and all submissions should be in line with their academic standards.

    So, can you believe everything on the arXiv?

    In my opinion overall the arXiv is contains good material and is a vital resource for scientists to call upon. Many new works can be made public this way, before being published in a scientific journal. Indeed, most of the published papers I have had call to use have versions on the arXiv. Moreover, the service is free and requires no subscription.

    However, there can be errors and mistakes in the preprints, both “editorial” but more importantly scientifically. Interestingly, overall the arXiv is not full of crackpot ideas despite it being quite open. There is a system of endorsement in place meaning that an established scientist should say that the first preprint you place on the arXiv is of general interest to the community. This stops the very eccentric quacks in their tracks.

    There has been some widely publicised examples of preprints on the arXiv that have cursed a stir within the scientific community. Two well-known examples include

    A. Garrett Lisi, An Exceptionally Simple Theory of Everything arXiv:0711.0770v1 [hep-th],

    and more recently

    V.G.Gurzadyan and R.Penrose, Concentric circles in WMAP data may provide evidence of violent pre-Big-Bang activity arXiv:1011.3706v1 [astro-ph.CO],

    both of which have received a lot of negative criticism. Neither has to date been published in a scientific journal.

    Minor errors and editing artefacts can be corrected in updated versions of the preprints. Should preprints on the arXiv be found to be in grave error, the author can withdraw the preprint.

    With that in mind, the arXiv can be a great place to generate feedback on your work. I have done this quite successfully in the past. This allowed me to get some useful comments and suggestion on work, errors and all.

    My advice is to view all papers and preprints with some scepticism, even full peer review can not rule out errors. Though, always be more confident with published papers and arXiv preprints that have gone under some revision. Note that generally people who place preprints on the arXiv are not trying to con or trick anyone, all errors will be genuine mistakes.

    Odd Jacobi structures and BV-gauge systems

    Abstract
    In this paper we define Grassmann odd analogues of Jacobi structures on supermanifolds. We then examine their potential use in the Batalin-Vilkovisky formalism of classical gauge theories.

    arXiv:1101.1844v1 [math-ph]

    In my latest preprint I construct a Grassmann odd analogue of Jacobi structures on supermanifolds.

    Without any details (being slack with signs) an odd Jacobi structure on a supermanifold is an ” almost Schouten structure”, \(S\) that is an odd function on the total space of the cotangent bundle of the supermanifold quadratic in fibre coordinates and a homological vector field \(Q\) on the supermanifold together with the compatibility conditions

    \(L_{Q}S = \{\mathcal{Q}, S \} = 0\),
    \( \{S, S \} = 2 S \mathcal{Q}\),

    where \(\mathcal{Q} \in C^{\infty}(T^{*}M)\) is the “Hamiltonian” or principle symbol of the Homological vector field. The brackets here are the canonical Poisson brackets on the cotangent bundle.

    An odd Lie bracket can then be constructed on \(C^{\infty}(M)\)

    \([f,g] = \pm \{ \{ S,f \},g\} \pm \{ \mathcal{Q},fg \} \).

    So, this odd bracket satisfies all the properties of a Schouten bracket i.e. symmetry and the Jacobi identity, but the Leibniz rule is not identically satisfied. There is an “anomaly” to the Leibniz rule of the form

    \( [f,gh] = \pm [f,g]h \pm g [f,h] \pm [f,1] gh\)

    In the preprint I examine the basic properties of odd Jacobi manifolds. The definition and study of odd Jacobi manifolds appears to be missing from the previous literature despite the wide interest in Schouten manifolds and Q-manifolds in mathematical physics.

    One should note that for classical or even Jacobi structures (if you know what these are) the Reeb vector field has no constrain on it like being homological. For odd structures the homological condition is essential.

    I also consider if the classical BV-antifield formalism can be generalised to odd Jacobi manifolds. In short, does one require the antibracket to be a Schouten bracket or can one weaken the Leibniz rule? I show that it looks possible to extend the BV formalism, classically anyway to odd Jacobi manifolds with the extra condition that the extended classical action not just be a Maurer-Cartan element,

    \([s,s] = 0\),

    but in addition should be Q-closed,

    \(Qs =0\).

    Much work needs to be done to generalise the BV formalism to odd Jacobi manifolds including adding the required gradings of ghost number, antifield number etc as well as understanding the quantum aspects.

    UPDATE: 22 March 2011. I have found a mistake in one of the examples I suggest. This is corrected and an updated version of the preprint will appear in due course. The mistake does not really effect the rest of the preprint.

    Integration of odd variables III

    Abstract
    We will proceed to describe how changes of variables effects the integration measure for odd variables. We will do this via a simple example rather than in full generality.

    Integration measure with two odd variables
    Let us consider the integration with respect to two odd variables, \(\{ \theta, \overline{\theta} \}\). Let us consider a change in variables of the form

    \(\theta^{\prime} = a \theta + b \overline{\theta}\),
    \( \overline{\theta}^{\prime} = c \theta + d \overline{\theta}\),

    where a,b,c,d are real numbers (or complex if you wish).

    Now, one of the basic properties of integration is that it should not depend on how you parametrise things. In other worlds we get the same result whatever variables we chose to employ. For the example at hand we have

    \( \int D(\overline{\theta}^{\prime}, \theta^{\prime}) \theta^{\prime} \overline{\theta}^{\prime} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

    Thus, we have

    \(\int D(\overline{\theta}^{\prime}, \theta^{\prime}) (ad-bc)\theta \overline{\theta} = \int D(\overline{\theta}, \theta) \theta \overline{\theta}\).

    In order to be invariant we must have

    \(\int D(\overline{\theta}^{\prime}, \theta^{\prime})= \frac{1}{(ad-bc) }D(\overline{\theta}, \theta) \).

    Note that the factor (ad-bc) is the determinant of a 2×2 matrix. However, note that we divide by this factor and not multiply in the above law. This is a general feature of integration with respect to odd variables, one divides by the determinant of the transformation matrix rather than multiply. This generalises to non-linear transformations that mix even and odd coordinates on a supermanifold. This is the famous Berezinian. A detailed discussion is outside the remit of this introduction.

    Furthermore, note that the transformation law for the measure is really the same as the transformation law for derivatives. Thus, the Berezin measure is really a mixture of algebraic and differential ideas.

    What next?
    I think this should end our discussion of the elementary properties of analysis with odd variables. I hope it has been useful to someone!