What is operator calculus?
I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.
I have searched for "operator calculus" and have not found any accessible references that introduce the topic. Maybe operator calculus go under another name today, than at the time of the interview?
Can you give me a reasonably simple explanation of operator calculus, and also give some references to books on the subject?
Solution 1:
At the heart the key idea is quite simple. Namely, one views functions, endomorphisms, etc. as "numbers" and manipulates them as such (where valid). E.g. for a derivative $\rm D = d/dx\ $ we have that $\rm D^2 - c^2\ =\ (D - c)(D + c) $ just as for numbers, as long $\rm\:D\:$ commutes with $\rm\:c\:$, i.e. $\rm\:c\:$ is constant. So we can solve constant coefficient linear differential / difference equations by simply factoring their operator ("characteristic") polynomials into linear factors over $\mathbb C$. One can also similarly perform transcendental operator manipulations such as
$$\rm f(D)\ e^{t\:x}\ =\ f(t)\ e^{t\ x}$$
$$\rm e^{\:t\ D}\ f(x)\ =\ f(x+t) $$
$$\rm t^{x\ D}\ f(x)\ =\ f(t\: x) $$
and the Generalized Leibniz Rule
$$\rm g(D)\ f(x)\ = \ \sum_{n=0}^\infty\ \frac{f^{(n)}(x)\ g^{(n)}(D)}{n!}$$
For further details see Roman: Umbral Calculus and Rainville: Special Functions and especially Rota: Finite operator calculus.
Solution 2:
The operator calculus that Feynman is talking about first came to the attention of mathematicians when Freeman Dyson (learned it from Feynman and) used it to prove that the Feynman and the Schwinger formulations of QED (quantum electrodynamics) were equivalent.
The basic idea is to let time take on its natural role as a director of physical processes, so that operators acting at different times commute. To do this, one views the evolution of a physical system as a motion picture and lays out its history as on a film. This means that the mathematical convention of position on paper is replaced by position in time.
The actual mathematical foundations for this theory is of relative recent vintage. I would suggest two references. The first is more mathematical, while the second is directed to the physics inclined and includes quite a lot of history.
Feynman operator calculus: The constructive theory, Expositiones Mathematicae 29 (2011)165–203. (by T. L. Gill and W. W. Zachary)
Foundations for relativistic quantum theoryI: Feynman’s operator calculus and the Dyson conjectures, J. Math. Phys. 43(2002)69–93. (by T. L. Gill and W. W. Zachary)