### Dirac, Heaviside, Physics, and Electrical Engineering

Wednesday, March 27th, 2013Over at Copasetic Flow, I wondered briefly last week whether electrical engineering notation and quantum mechanics notation had shared a common background and if so, which was the progenitor[1]. A few days ago I came across a helpful article by J. D. Jackson, If you’re on the physics side of things, yes, that Jackson, (careful, language in the subtitles).

Dr. Jackson points out[5] that what most of us call the Dirac delta function was introduced in its modern form 35 years before Dirac’s definition[2] by Oliver Heaviside, first in The Electrician, and then in his Electromagnetic Theory volume I[4]. So, as early as 1895, (before quantum mechanics was invented), the two fields were already intertwined.

If you’ve never read Heaviside’s book, you should, (go ahead, it’s free on Google Books). He introduces a sizable amount of the notation that we use in EE. First, there’s the Heaviside step function. Then, there’s the entire field of operational calculus.

If you vaguely remember a linear systems course in EE where derivatives suddenly went from being denoted as d/dt to being denoted as multiplication by p, that’s operational calculus and Heaviside invented it. I remember being awestruck that I could now do differential calculus simply by multiplying by p, (taking the derivative), or dividing by p, (taking the integral). Consequntly, this is one of the notations that seems to be similar between EE and quantum mechanics. Reading Heaviside, it seems to have its origin in physics. p in physics was and is the symbol for momentum which is the time derivative, (d/dt) of position. Heaviside doesn’t elucidate the point, but on page 232 he starts using the p notation, (“for convenience”), and apparently never goes back. Later on he’ll realize that Fourier analysis is also kind of fun and start using omega, or frequency, as a derivative and one over omega as an integral. Both of these really nice notations have never been adopted by the physics texts. Interestingly though, quantum mechanics will pick up the whole Fourier habit and denote p, (momentum), as the Fourier conjugate of position space.

I only really have one gripe with Heaviside, great hero of engineering though he may be. He killed the quaternions Man! I mean dead, for decades. The text I’ve referenced here is where he did it. He replaced them with the easier to work with, (he thought), vectors. They started turning back up in the 1960′s, 70′s, and 80′s when people began to realize that they were very nice for doing efficient rotations, (think computer graphics and games). It also became apparent that they could be used to provide geometric explanations of quantum mechanics, (more on that later). On the fringe side of things, they’re of course the reason the Russians got to the moon before us and also how they got ahead in teleportation technology, but I digress.

So, in summary, quantum mechanics and EE do seem to be intertwined, the p notation in operational calculus, and the ‘Dirac’ delta point towards a common heritage, but there’s no smoking gun yet as far as locating a common link between the two in a single person.

**References:**

1. Copasetic Flow about the history of EE and QM

http://copaseticflow.blogspot.com/2013/03/separataed-at-birth-quantum-mechanics.html

2. Dirac’s Introduction of what will be called the Dirac Delta Function

http://rspa.royalsocietypublishing.org/content/180/980/1.full.pdf+html

3. Heaviside’s introduction of the same function, (the derivative of what will be called the Heaviside step function), 35 years earlier

http://books.google.com/books?id=8GBNAAAAYAAJ&dq=editions%3A21BCEIltjgcC&pg=PA600#v=onepage&q&f=true

4. Heaviside’s Electromagnetic Theory

http://books.google.com/books?id=9ukEAAAAYAAJ&printsec=frontcover&dq=oliver+heaviside&hl=en&sa=X&ei=lhlTUcn_CJTryAH6ioGQCQ&ved=0CDwQ6AEwAg

5. Jackson on Dirac and Heaviside

http://ajp.aapt.org/resource/1/ajpias/v76/i8/p704_s1?ver=pdfcov