6 Feb

Maps, Lattices and Linear Spaces

Posted by hishamuddinz in Mathematics, Sets.

This site has been left barren for a long time since I had too much to do. As the name suggests, this is a catch-up blog of learning and re-learning things. Much of the entries here will be directed to my students and beginners to what may be called as standard materials for theoretical physics. Experts are welcome to give comments, constructive criticisms and of course corrections. As in the earlier version of Ketchup Spills, I will base the entries on Nirmala Prakash’s book “Mathematical Perspectives on Theoretical Physics” coverage of topics.

This post will start with Section 0.1 on some basic definitions. It begins with maps f: E\rightarrow F from set E to set F and perhaps the most familiar example is that of a function f: R \rightarrow R. Of course, one can generalize both domain and range of the map to include all possible things that you want to use to describe physics, e.g. coordinate charts of a manifold, representations. It is interesting to note that normally we write f(x) as representing the function (map) but strictly speaking this is just the image of the function; subtly hidden is the idea of the map includes both the domain and codomain. Slightly archaic notation is the indicial notation f_x but this can be confused with another use of such symbol i.e. the derivative of f with respect to x. Also the calling of E being the index set of f  is perhaps due to the ‘enumeration’ of the image f(x) by the members of E.

Consider A \subset E, then the inclusion map \iota: A \hookrightarrow E maps x \in A to the same corresponding element in E. Prakash seems to use the term canonical map – not sure where this comes from.

A useful notion is the idea of a set X \subset E  being stable under a map f: E \rightarrow E  which means f(X) \subset X. This is perhaps important in describing fixed points and also important say in some geometrical information of dynamical systems. Maybe the experts have something to say.

Lattice structure is another important structure to be defined – it is an partially ordered set E whose finite non-empty subsets have both least upper bound and greatest lower bound. The familiar example would be finite intervals of the real line. But it goes more primitive than that and plays an important role in logic.

An often used structure in physics is linear spaces or more commonly vector spaces*. This is a set E equipped with two operations namely linear addition among its elements (called vectors) and scalar multiplication of elements from a field on the vectors. Note that linear spaces are already quite rich in structure. One probably need the following ‘primitive’ structures before arriving to the concept of linear spaces:

  • Group: a set G equipped with a product with a unit element and is associative and admits inverses.
  • Ring: an additive group X with a product structure and is distributive over the addition.
  • Field: a ring K such that its multiplication is a group structure for the subset K^*\equiv K\backslash\{0\}

*Note in addition: Some prefer to use vector spaces for affine spaces i.e. linear spaces without the ‘origin’ (intuitively space of directed line segments).

For linear spaces X,\ Y over field K, a linear mapping f: X\rightarrow Y is a mapping such that for x,\ x'\in X and \alpha,\ \beta\in K, it obeys f(\alpha x + \beta x') = \alpha f(x) + \beta f(x'). There are plenty of examples of this from quantum mechanics i.e. from linear operators.

I leave this post with a good reference for discussing these primitive structures in orderly fashion namely, Ian R. Porteous, “Topological Geometry“.

26 Sep

Resurrected

Posted by hishamuddinz in Uncategorized. Tagged: .

I’m resurrecting my Ketchup Spills blog – now done on wordpress. The reason I’m doing this is simply because that it supposed to have LaTeX capabilities. So here goes two examples:

i\hbar\cfrac{\partial}{\partial t}\left|\Psi(t)\right>=H\left|\Psi(t)\right>

G_{\mu\nu}=\kappa T_{\mu\nu}

This will be my learning out aloud blog. As such, please read at your own risk of misunderstanding. Friendly criticisms, comments and advice from experts are welcome!!