Tag Archives: physics

Thank you, Dr. Majumder

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Dr. Jaydeep Majumder (07 June 1972 – 22 July 2009)

Recently I finished the first part of my master’s thesis related to (complex) algebraic geometry. There are not many (useful) books available on this topic, and most of them are very costly. In fact, my college library couldn’t buy enough copies of books in this topic. However, fortunately,  Dr. Jaydeep Majumder‘s books were donated to the library and they will make my thesis possible:

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Principles of Algebraic Geometry by Joseph Harris and Phillip Griffiths

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Algebraic Curves and Riemann Surfaces by Rick Miranda

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Hodge Theory ans Complex Algebraic Geometry – I by Claire Voisin

While reading the books, I assumed that that these books were donated after the death of some old geometer. But I was wrong. He was a young physicist, who barely spent a month at NISER. A heart breaking reason for the books essential for my thesis to exist in the college library.

Dr. Majumder was a theoretical high energy physicist who did research in String Theory. He obtained his Ph.D. under the supervision of Prof. Ashoke Sen at HRI. He joined NISER as Reader-F in June 2009, and was palnning to teach quantum mechanics during the coming semester. Unfortunately, on 22 July 2009 at the young age of 37 he suffered an untimely death due to brain tumor.

I just wanted to say that Dr. Majumder has been of great help even after his death. The knowldege and good deeds never die. I really wish he was still alive and we could discuss the amazing mathematics written in these books.

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Imaginary Angles

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You would have heard about imaginary numbers and most famous of them is i=\sqrt{-1}. I personally don’t like this name because all of mathematics is man/woman made, hence all mathematical objects are imaginary (there is no perfect circle in nature…) and lack physical meaning. Moreover, these numbers are very useful in physics (a.k.a. the study of nature using mathematics). For example, “time-dependent Schrödinger equation

\displaystyle{i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat H \Psi(\mathbf{r},t)}

But, as described here:

Complex numbers are a tool for describing a theory, not a property of the theory itself. Which is to say that they can not be the fundamental difference between classical and quantum mechanics (QM). The real origin of the difference is the non-commutative nature of measurement in QM. Now this is a property that can be captured by all kinds of beasts — even real-valued matrices. [Physics.SE]

For more of such interpretation see: Volume 1, Chapter 22 of “The Feynman Lectures in Physics”. And also this discussion about Hawking’s wave function.

All these facts may not have fascinated you, but the following fact from Einstein’s Special Relativity should fascinate you:

In 1908 Hermann Minkowski explained how the Lorentz transformation could be seen as simply a hyperbolic rotation of the spacetime coordinates, i.e., a rotation through an imaginary angle. [Wiki: Rapidity]

Irrespective of the fact that you do/don’t understand Einstein’s relativity, the concept of imaginary angle appears bizarre. But, mathematically its just another consequence of non-euclidean geometry which can be interpreted as Hyperbolic law of cosines etc. For example:

\displaystyle{\cos (\alpha+i\beta) = \cos (\alpha) \cosh (\beta) - i \sin (\alpha) \sinh (\beta)}

\displaystyle{\sin (\alpha+i\beta) = \sin (\alpha) \cosh (\beta) + i \cos (\alpha) \sinh (\beta)}

Let’s try to understand what is meant by “imaginary angle” by following the article “A geometric view of complex trigonometric functions” by Richard Hammack. Consider the complex unit circle  U=\{z,w\in \mathbb{C} \ :  \  z^2+w^2=1\} of \mathbb{C}^2, in a manner exactly analogous to the definition of the standard unit circle in \mathbb{R}^2. Apparently U is some sort of surface in \mathbb{C}^2, but it can’t be drawn as simply as the usual unit circle, owing to the four-dimensional character of \mathbb{C}^2. But we can examine its lower dimensional cross sections. For example, if  z=x+iy and w=u+iv then by setting y = 0 we get the circle x^2+u^2=1 in x-u plane for v=0 and the hyperbola x^2-v^2 = 1 in x-vi plane for u=0.

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The cross-section of complex unit circle (defined by z^2+w^2=1 for complex numbers z and w) with the x-u-vi coordinate space (where z=x+iy and w=u+iv) © 2007 Mathematical Association of America

These two curves (circle and hyperbola) touch at the points ±o, where o=(1,0) in \mathbb{C}^2, as illustrated above. The symbol o is used by Richard Hammack because this point will turn out to be the origin of complex radian measure.

Let’s define complex distance between points \mathbf{a} =(z_1,w_1) and \mathbf{b}=(z_2,w_2) in \mathbb{C}^2 as

\displaystyle{d(\mathbf{a},\mathbf{b})=\sqrt{(z_1-z_2)^2+(w_1-w_2)^2}}

where square root is the half-plane H of \mathbb{C} consisting of the non-negative imaginary axis and the numbers with a positive real part. Therefore, the complex distance between two points in \mathbb{C}^2 is a complex number (with non-negative real part).

Starting at the point o in the figure above, one can move either along the circle or along the right-hand branch of the hyperbola. On investigating these two choices, we conclude that they involve traversing either a real or an imaginary distance. Generalizing the idea of real radian measure, we define imaginary radian measure to be the oriented arclength from o to a point p on the hyperbola, as

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(a) Real radian measure (b) Imaginary radian measure. © 2007 Mathematical Association of America

If p is above the x axis, its radian measure is \beta i with \beta >0, while if it is below the x axis, its radian measure is \beta i with \beta <0. As in the real case, we define \cos (\beta i) and \sin (\beta i) to be the z and w coordinates of p. According to above figure (b), this gives

\displaystyle{\cos (\beta i) = \cosh (\beta); \qquad \sin (\beta i) = i \sinh (\beta)}

\displaystyle{\cos (\pi + \beta i) = -\cosh (\beta); \qquad \sin (\pi + \beta i) = -i \sinh (\beta)}

Notice that both these relations hold for both positive and negative values of \beta, and are in agreement with the expansions of  \cos (\alpha+i\beta)  and \sin (\alpha+i\beta)  stated earlier.

But, to “see” what a complex angle looks like we will have to examine the complex versions of lines and rays. Despite the four dimensional flavour, \mathbb{C}^2 is a two-dimensional vector space over the field \mathbb{C}, just like \mathbb{R}^2 over \mathbb{R}.

Since a line (through the origin) in \mathbb{R}^2 is the span of a nonzero vector, we define a complex line in \mathbb{C}^2 analogously. For a nonzero vector u in \mathbb{C}^2, the complex line \Lambda through u is span(u), which is isomorphic to the complex plane.

In \mathbb{R}^2, the ray \overline{\mathbf{u}} passing through a nonzero vector u can be defined as the set of all nonnegative real multiples of u. Extending this to \mathbb{C}^2 seems problematic, for the word “nonnegative” has no meaning in \mathbb{C}. Using the half-plane H (where complex square root is defined) seems a reasonable alternative. If u is a nonzero vector in \mathbb{C}, then the complex ray through u is the set \overline{\mathbf{u}} = \{\lambda u \ : \  \lambda\in H\}.

Finally, we define a complex angle is the union of two complex rays \overline{\mathbf{u}_1} and \overline{\mathbf{u}_2} .

I will end my post by quoting an application of imaginary angles in optics from here:

… in optics, when a light ray hits a surface such as glass, Snell’s law tells you the angle of the refracted beam, Fresnel’s equations tell you the amplitudes of reflected and transmitted waves at an interface in terms of that angle. If the incidence angle is very oblique when travelling from glass into air, there will be no refracted beam: the phenomenon is called total internal reflection. However, if you try to solve for the angle using Snell’s law, you will get an imaginary angle. Plugging this into the Fresnel equations gives you the 100% reflectance observed in practice, along with an exponentially decaying “beam” that travels a slight distance into the air. This is called the evanescent wave and is important for various applications in optics. [Mathematics.SE]

Mathematical Relations

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In this post I will share my perception of relation of mathematics with other academic disciplines. All this is based on my very limited knowledge of various disciplines.

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Shape doesn’t signify anything.

Mathematics deals with study of properties of numbers (or the symbols representing them) and geometric objects (not in classical sense, it can mean manifolds also). In my opinion, there is no partition of mathematics into “applied” or “pure”, but intersections with other subjects. The term applied Mathematics doesn’t make any sense to me. Mathematics is somehow applicable in various places. For me, mathematics is what people call “pure” mathematics (what about “impure” Mathematics??).  Also now I agree with the vastly established belief that art and mathematics are similar, since both involve abstract ideas motivated but physical situations (at some point).

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Truth Lies Deception and Coverups – Democracy Under Fire (Source: http://goo.gl/yUHi93)

All experimental sciences (physics, chemistry, biology, economics) are based on statistics. Since statistics is a young discipline (only a couple of centuries old) many times we get wrong interpretation of results. As far as real life is concerned, study of statistics gives us a powerful tool for predicting future and Probability Theory acts as the connecting link between statistics and mathematics. Understanding of statistics affects us on daily basis since (effective) government policies are framed keeping statistical analysis in mind. Unfortunately, most of universities don’t have separate department for statistics.

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P vs NP Problem in Relationships (http://ctp200.com/comic/6; CC BY-NC 4.0)

Study of algorithms is one of the most important aspect of computer science (I am not talking about software industry…). What surprises me is that Euclid’s division algorithm is  one of the most efficient division algorithm even for computers! The neglected subject of Logic, which is supposed to be foundations of mathematics, flourishes in computer science. P vs NP is another “millennium open problem“.

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Convincing (http://xkcd.com/833/ ; CC BY-NC 2.5)

For me, Economics like Statistics is full of imperfections due to real life complications (so many dependencies to account for). Game Theory appears to be the connecting link between mathematics and economics.

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We all know that the needs of physicists are responsible for development of calculus and study of differential equations. On the other hand, theoretical physics (quantum mechanics, string theory) depends heavily on the developments in algebra.