Working Miracles with the Differential Operator - I
26th January 2014
[This article is excerpted from the FEM Lectures by Rahul Bhattacharya. This is the first of the three part lecture on the topic of Operational Calculus.
To many, who have no formal education in physics, quantum mechanics - one of the foundations for explaining the origin and evolution of our universe - may
seem utterly esoteric and complicated. Even the most brilliant of physicists consider this branch of physics extremely challenging, and at times totally
counter-intuitive. But to Roger Penrose, the famed mathematical physicist it is nothing short of a miracle. Here's a quote from his recent masterpiece, The
Road to Reality,
||...but we should not turn down a miracle when it is presented to us! What miracle is this? The miracle is the fact that these seemingly gross
absurdities of experimental fact - that waves are particles and that particles are waves - can all be accommodated within a beautiful mathematical formalism,
a formalism in which momentum is indeed with 'differentiation with respect to position', and energy with 'differentiation with respect to time'.
To me, the miracle is that we get a first glimpse of this brilliant act of God, when we take our first lessons in differential calculus in high school. When
we start to study about the notion of or we get
a hint of how our Creator thinks and how beautifully - and in the most simple manner - He has woven the fabric of our universe.
One of the most fundamental mathematical concepts - something that, as I said earlier, we actually learn in high school - is the notion of differential
operator. This is what high school kids around the world learn in their first lessons on differential calculus.
When we study the mathematical notion of differentiation we learn of things like , which is explained
to us as the first mathematical derivative of the variable with respect to the variable . Of course, we are seldom told that the symbol "" in our
differentiation problems can work magic with physics. We are also not told that this ' something by something' is in fact, an operator and that it can be treated exactly like an algebraic variable. (The
notion of operator, of course, is not an easy one and it is only when we study mathematical physics in college - or, those electrical engineering students,
who take a first course on Operational Calculus - that we truly begin to understand the essence of this idea.) But let's not judge our high school teacher
too harshly. After all, he or she did a good job in teaching differential calculus to us.
All this begs a very important question, one that has prompted us to write this article. Time and again students in our CFE classes and tutorials ask this
question: does the time step matter? In other words, since Monte Carlo simulation is studying the temporal evolution of a system does it matter how we slice
time - in millionth of a second, seconds, minutes, hours, days or weeks - to study the evolution of the system. As stated in the beginning of this article in
Monte Carlo method we study systems which are governed by differential equations and these differential equations contain terms such as where is the differential operator with respect to time. If we have
to experimentally understand such equations on a computer we need to figure out how small should "" be,
ideally. Or, more importantly, does the size of "" - which is the time step - matter.
A differential operator, denoted by , refers to the computation of the first mathematical derivative
with respect to a variable . That is, is the
first mathematical derivative of the variable with respect to the variable .
Another way to look at this is to write as
Which implies that the operator "acts on the variable ". More generally speaking, if is a function of the variable then when we can write
This signifies that the operator "acts on the function ". To simply things - hopefully, this isn't an oversimplification - we can understand the phrase "acts on the function " as meaning being "multiplied with the function . In other
words, we can also think of "" as an algebraic variable, just like the variables and ; this is where things start to become at once very simple and very
esoteric, though not necessarily complex.
Isn't it absurd to think of as an algebraic variable? Right from the day when we were introduced to
the discipline of differential and integral calculus we have understood the term as an operation which
signifies that it is the limit (where, in the limit of ) of the term which, in turn, implies the rate of change of with respect to as we keep making the changes in smaller and smaller.
How is it then that suddenly we start talking about "" as an algebraic variable, instead of as a
limiting case of the change of change of two variables. Does it mean that the variable is subject to
the same laws of addition, subtraction and commutation, like other algebraic variables do?
Let us denote the variable - it is in fact, an "operator", though we have not quite properly defined this operator - by the capital letter, .
Thus, if is an algebraic variable then does it follow the same rules as other algebraic variables? If
we have two variables, and then we have the
following commutation relationship between them:
Can the same be said for the variables, and ?
The answer is no. When the s and the s get mixed
up, we have something very weird - I use this word for the want of a better word - happening in nature. And, this is where the miracle starts to take shape.
Even though can be treated as an algebraic variable, it does not commute when multiplied with . The commutation rule does not hold true.
It would seem bizarre - to me it seems like an act of God - to many if I were to say that the above inequality in indeed - within a certain framework of
analysis and under a proper perspective - the basis for the famous Heisenberg's uncertainty equation in quantum mechanics.
Even a cursory visual inspection of the above inequality raises a bigger, and seemingly a far more fundamental, question, the answer to which can only be
appreciated within the framework of mathematical physics that deals with the laws of quantum mechanics. Very briefly stated, what is exactly meant by the
second term in the above equation? The notion seems totally counter-intuitive. What possible
mathematical significance can this have given the fact that there is nothing to right of so the
conventional interpretation of as that of being a way of finding the first derivative of a function
(or a variable) does not hold true, because there is no variable or a function on the right of .
This would be a wonderful journey where one can unlock some of the greatest secrets of physics - and our universe - by studying this seemingly absurd looking
variable or operator, .
- Heaviside's Operational Calculus, Ernst Julius Berg, McGraw Hill, 1936
- Electrical Circuit Theory and the Theory of Operational Calculus, Lectures by John R. Carson, Moore School of Electrical Engineering, University of
- A treatise on differential equations, George Boole, McMillan, 1859;
- The Road to Reality, Roger Penrose, Vintage Books, 2005
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