Contact structures and supersymmetric mechanics

Contact structures and supersymmetric mechanics

Andrew James Bruce

We establish a relation between contact structures on supermanifolds and supersymmetric mechanics in the superspace formulation. This allows one to use the language of contact geometry when dealing with supersymmetric mechanics.

arXiv:1108.5291v1 [math-ph]

In the preprint above I show that aspects of  d=1, N=2 supersymmetric quasi-classical mechanics in the superspace formulation can be understood in terms of  a contact structure on the supermanifold \(R^{1|2}\).


In particular if we pick local coordinates \((t, \theta, \bar{\theta})\) then the super contact structure is given by


\(\alpha = dt + i \left(  d \bar{\theta}\theta + \bar{\theta} d \theta  \right)\),
which is a Grassmann odd one form. One could motivate the study of such a one form as a “superisation” of the contact form on \(R^{3}\).


Associated with any odd one form that is nowhere vanishing is a hyperplane distribution of codimension (1|0). That is we have a subspace of the tangent bundle that contains one less even vector field in its (local) basis as compared to the  tangent bundle.  This is why we should refer to the above structure as an even (pre-)contact structure.


The hyperplane distribution associated with the super contact structure is spanned by two odd vector fields. These odd vector fields are exactly the SUSY covariant derivatives. More over we do have a genuine contact structure as the exterior derivative of the super contact form is non-degenerate on the hyperplane distribution. For more details see the preprint.


Generalising contact structures  on manifolds to  supermanifolds appears fairly straight forward. We have the non-classical case of odd contact structures to also handle, here the hyperplane distribution is of corank (0|1), i.e. one less odd vector field. There is also a subtly when defining kernels and contactomorphisms as we will have to take care with nilpotent objects.

Comments on the preprint will be very much appreciated.



Update A third revised version has now been submitted. 08/02/2012

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.