All posts by ajb

Theories in physics

In physics the word theory is used synonymously with mathematical model or mathematical framework. The theory is a mathematical construction   that can be used to describe physical phenomena.  A theory should, at least  in principle be falsifiable, that is make predictions that can be tested.

People who are not trained in physics take theory to mean either  “hypothetical” or loosely an  “idea”.  One may hear “but it is only a theory”, which takes the physics use of the word theory out of context.

A theory, in the sense of modern physics must by definition be phrased in mathematics. We need something to mathematically manipulate and calculate things that can be tested against observation.  Without the mathematical framework it is hard to judge if an “idea” has any merit or not.

Often by theory physicists may have something  a little more specific in mind, they often mean a specified action or Lagrangian.  Most of physics can be stated in terms of actions and so it usually makes sense to start there.  Again the action or equivalently the Lagrangian are mathematical notions.

 

 

 

Astronomy Vs Astrology

Even today people confuse astronomy and astrology.  It is not hard to see why when almost every newspaper has a horoscope and  lots of adverts for astrology phone lines.  Lets set the record straight.

 

Astronomy the scientific study of  celestial bodies, for example the Sun, planets, starts, comets etc.  The science is based on observation of the  celestial bodies and the application of physical laws to such bodies.  Mathematics and physics are essential in astronomy.

 

Astrology the belief that the position of  celestial bodies influences the personality and human affairs. It is based on superstition and no physical mechanisms have been established. The superstition does not apply the scientific method and in no way follows modern scientific principles.

In short, astronomy and the closely related astrophysics and cosmology add to the human understanding of nature and our place in the Universe.  Astrology is a superstition that people exploit to make money.

It is of course true that the origins of astronomy lie in astrology. Careful observations and recording of data was necessary in order to write astrological charts.  One could equally argue that chemistry owes  a lot to alchemy.  But we have come a long way in our thinking and philosophy.  Astronomy and chemistry are sciences.

Please do not confuse the two, it is rather insulting to all astronomers!


The fundamental misunderstanding of calculus

We all know the fundamental theorems of calculus, if not check Wikipedia.  I now want to  demonstrate what has been called the fundamental misunderstanding of calculus.

Let us consider the two dimensional plane and equip it with coordinates \((x,y)\).  Associated with this choice of coordinates are  the partial derivatives

\(\left( \frac{\partial}{\partial x} , \frac{\partial}{\partial y} \right)\).

You can think about these in terms of the tangent sheaf etc. if so desired, but we will keep things quite simple.

Now let us consider a change of coordinates. We will be quite specific here for illustration purposes

\(x \rightarrow \bar{x} = x +y\),

\(y \rightarrow \bar{y} = y\).

Now think about how these effect the partial derivatives. This is really just a simple change of variables.  Let me now state  the fundamental misunderstanding of  calculus in a way suited to our example:

Misunderstanding: Despite coordinate x changing the partial derivative with respect to x remains unchanged. Despite the coordinate y remaining unchanged the partial derivative with respect to y changes.

This may seem at first counter intuitive, but is correct. Let us prove it.

Note hat we can invert the change of coordinate for x very simply

\(x = \bar{x} {-}\bar{y} \),

using the fact that y does not change. Then one needs to use the chain rule,

\(\frac{\partial}{\partial \bar{x}}  = \frac{\partial x}{\partial \bar{x}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{x}}\frac{\partial}{\partial y}   =    \frac{\partial}{\partial x}\),

\(\frac{\partial}{\partial \bar{y}}  = \frac{\partial x}{\partial \bar{y}}\frac{\partial}{\partial x}+ \frac{\partial y}{\partial \bar{y}}\frac{\partial}{\partial y}   =    \frac{\partial}{\partial y} {-} \frac{\partial}{\partial x} \).

There we are. Despite our initial gut feeling that that the partial derivative wrt y should remain unchanged we see that it is in fact the partial derivative wrt x that is unchanged.  This can course some confusion the first time you see it,  and hence the nomenclature the fundamental misunderstanding of calculus.

I apologise for forgetting who first named the misunderstanding.

 

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.

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    Update:

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

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    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!