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!

    Integration of odd variables II

    Abstract
    We now proceed to define integration with respect to odd variables.

    The fundamental theorem of calculus for odd variables
    Let us consider just one odd variable. This will be sufficient for our purposes for now. Following the direct analogy with integration of functions over a circle the fundamental theorem of calculus states

    \(\int D\theta \frac{\partial f(\theta)}{\partial \theta} =0\).

    We use the notation \(D\theta \) for the measure rather than \(d\theta \) as the measure cannot be associated with a one-form. We will discuss this in more detail another time.

    Definition of integration
    Recall that the general form of a function in one odd variable is

    \(f(\theta) = a + \theta b\),

    with a and b being real numbers. Thus from the fundamental theorem we have

    \(\int D\theta b =0\).

    In particular this implies

    \(\int D\theta =0\).

    Then we have

    \(\int D\theta f(\theta) = a \int D\theta + b \int D\theta \:\: \theta = b \int D\theta\:\: \theta \).

    Thus to define integration all we have to do is define the normalisation

    \(\int D\theta\:\: \theta\).

    The choice made by Berezin was to set this to unity. Other choices are also just as valid. Thus,

    \(\int D\theta f(\theta) = b\).

    Integration for several odd variables
    For the case of more than one odd variable one simply uses

    \(\int D(\theta_{1}, \theta_{2} , \cdots \theta_{n})f(\theta) = \int D\theta_{1} \int D \theta_{2} \cdots \int D\theta_{n} f(\theta)\).

    example Consider two odd variables.

    \(\int D(\overline{\theta}, \theta) \left( f_{0} + \theta \:f + \overline{\theta}\: \overline{f} + \theta \overline{\theta}F \right) = F \).

    The general rule is that (taking care with signs) the integration with respect to the measure \(D(\theta_{1}, \theta_{2} , \cdots \theta_{n})\) of a function picks out the coefficient of the \(\theta_{1}, \theta_{2} , \cdots \theta_{n}\) term.

    Integration and differentiation are the same!
    From the above we see that differentiation with respect to an odd variable is the same as integration with respect to the odd variable. This explains why we cannot associate a “top-form” with the measure. This will become more apparent when we discuss changes of variables.

    What next?
    Next we will examine how changing variables in the integration effects the measure. We will see that things look “upside down” as compared with the integration of real variables. This is anticipated by the equivalence of integration and differentiation.

    Integration of odd variables I

    Abstract
    Before we consider odd variables, let us describe how to algebraically define integration of functions over the circle.

    Functions on the circle
    Recall the Fourier expansion. It is well known that any continuous function on the circle is of the form

    \(f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right) \),

    with the a’s and b’s being constants, i.e. independent of the variable x.

    The fundamental theorem of calculus
    The fundamental theorem of calculus states that

    \(\int_{S^{1}} dx \: \frac{\partial f(x)}{\partial x } = 0 \),

    as functions on the circle are periodic.

    Integration of functions
    It turns out that integration of functions over the circle can be defined algebraically up to a choice in measure. To see this observe

    \(\int_{S^{1}} dx f(x) = \int_{S^{1}} dx \frac{a_{0}}{2} + \int_{S^{1}} dx \sum_{n=1}^{\infty}\left( a_{n} \cos(nx) + b_{n}\sin(nx) \right)\)

    Then we can write

    \(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx + \int_{S_{1}} dx \frac{\partial }{\partial x} \sum_{n=1}^{\infty} \left ( \frac{a_{n}}{n}\sin(nx) + \frac{- b_{n}}{n} \cos(nx) \right)\)

    to get via the fundamental theorem of calculus

    \(\int_{S^{1}} dx f(x) = \frac{a_{0}}{2} \int_{S^{1}} dx\).

    So we have just about defined integration completely algebraically from the fundamental theorem of calculus. All we have to do is specify the normalisation

    \(\int_{S^{1}} dx \).

    The standard choice would be

    \(\int_{S^{1}} dx = 2 \pi\),

    to get back to our usual notion of integration of periodic functions. Though it would be quite consistent to consider some other normalisation, say to unity.

    Anyway, up to a normalisation the integration of functions over the circle selects the “constant term” of the corresponding Fourier expansion.

    What next?
    So, the above construction demonstrates that integration of functions over a domain without boundaries can be defined algebraically, up to a normalisation. This served as the basis for Berezin who defined the notion of integration of odd variables.

    Recall that odd variables have no topology and no boundaries. The integration with respect to such variables cannot be in the sense of Riemann. However, thinking of functions of odd variables in analogy to periodic functions integration can be defined algebraically. We will describe this next time.

    Differential calculus of odd variables.

    Abstract
    Here we will define the notion of differentiation with respect to an odd variable and examine some basic properties.

    Definition
    Differentiation with respect to an odd variable is completely and uniquely defined via the following rules:

    1. \(\frac{ \partial \theta^{\beta} }{\partial \theta^{\alpha}} = \delta_{\alpha}^{\beta} \).
    2. Linearity:
      \(\frac{\partial}{\partial \theta }(a f(\theta)) = a \frac{\partial}{\partial \theta } f(\theta)\).
      \(\frac{\partial}{\partial \theta }( f(\theta) + g(\theta)) = \frac{\partial}{\partial \theta }f(\theta) + \frac{\partial}{\partial \theta } g(\theta)\).
    3. Leibniz rule:
      \(\frac{\partial}{\partial \theta }( f(\theta)g(\theta)) = \frac{\partial f(\theta)}{\partial \theta } + (-1)^{\widetilde{f}} f \frac{\partial g(\theta)}{\partial \theta } \).

    The operator \(\frac{\partial }{\partial \theta }\) is odd, that is it changes the parity of the function it acts on. This must be taken care of when applying Leibniz’s rule.

    Elementary properties
    It is easy to see that

    \(\frac{\partial}{\partial \theta^{\alpha}}\frac{\partial}{\partial \theta^{\beta}}+ \frac{\partial}{\partial \theta^{\beta}}\frac{\partial}{\partial \theta^{\alpha}}=0\),

    in particular

    \(\left( \frac{\partial}{\partial \theta} \right)^{2}=0\).

    Example
    \(\frac{\partial}{\partial \theta} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = b + \overline{\theta}d\).

    Example
    \(\frac{\partial}{\partial \overline{\theta}} (a + \theta b+ \overline{\theta}c + \theta \overline{\theta} d ) = c- \theta d\).

    Changes of variables
    Under changes of variable of the form \(\theta \rightarrow \theta^{\prime}\) the derivative transforms as standard

    \(\frac{\partial}{\partial \theta^{\prime}} = \frac{\partial\theta}{\partial \theta^{\prime}} \frac{\partial}{ \partial \theta}\).

    We will have a lot more to say about changes of variables (coordinates) another time.

    What next?
    We now know how to define and use the derivative with respect to an odd variable. Note that this was done algebraically with no mention of limits. As the functions in odd variables are polynomial the derivative was simple to define.

    Next we will take a look at integration with respect to an odd variable. We cannot think in terms of boundaries, limits or anything resembling the Riemann or Lebesgue notions of integration. Everything will need to be done algebraically.

    This will lead us to the Berezin integral which has the strange property that integration and differentiation with respect to an odd variable are the same.

    Random thoughts on mathematics, physics and more…