Śrīdhara Brāhmaṇa: Quantum Field Theory

309 1

As I had mentioned in an earlier post, I have decided to get to the bottom of Quantum Field Theory (QFT) as a leisure activity during these times of COVID-19.

I am not just interested in knowing what QFT is, but also on how it came to be.

Its “origin story” so to speak.

I am also additionally very interested in being able to communicate QFT without mathematical (or, more pragmatically, with as little as possible) notation, in everyday vernacular, the raison d’etre of my Śrīdhara Brāhmaṇa series to begin with. 

To this last point, I was ecstatic that Michael Faraday wrote (in 1857) to James Clerk Maxwell (who in 1855, at the age of 24, had published the first of his three landmark papers, On Faraday’s Lines of Force):

There is one thing I would be glad to ask you. When a mathematician engaged in investigating physical actions and results has arrived at his conclusions, may they not be expressed in common language as fully, clearly and definitely as in mathematical formulae? If so, would it not be great boon to such as I to express them so?—translating them out of the hieroglyphics that we also might work upon them by experiment. I think it must be so, because I have always found that you could convey to me a perfectly clear idea of your conclusions.

It gets better.

It was fascinating to read about the great commercial undertaking (in 1861) by Great Eastern (with William Thomson, aka Lord Kelvin, as lead scientific consultant, having earlier entered University of Glasgow as a student at the age of 10, having been home-tutored by his father who was a professor of Mathematics there) to lay a cable across the Atlantic, and that it, serendipitously, converged with Maxwell’s theoretical project:

the actual determination of the fundamental electromagnetic units.

Why?

Contracts must specify standards of performance, while performance must be certified by measurements with reference to officially recognized standards.

In 1862, Maxwell joined the committee established (in 1861, the year when he had published his second paper, On Physical Lines of Force) by the British Academy for the Advancement of Science to set the needed electrical standards. The only example available at that time to build from was from Germany, work initiated by Gauss and carried on by Weber and Kohlrausch.

The first task of the committee was to provide a sure standard for electrical resistance. So we got Ohm.

Once the Ohm was established, it was left up to Maxwell (who was also an ingenious experimenter in addition to being a gifted mathematician, and who borrowed much of the mathematics that was previously laid down by Thomson) to put to test his own audaciously imaginative proposition that

light is made of electricity and magnetism.

Like Maxwell, Thomson’s genius was not only in theory (thermodynamics, for instance), but he also excelled in the design and perfection of instruments both for the laboratory and the commercial world: he was the triumphant engineer of the Atlantic cable.

Lord Kelvin was a 19th Century Academic Capitalist!😊

All of this culminated in Maxwell’s third paper, a masterpiece, published in 1865:

 A Dynamical Theory of the Electromagnetic Field.

What is a field?

The field is a new fundamental mode of thought in which the whole, as a fully connected, coherent system, is prior, while the parts derive their significance through their membership in that whole….a system in a mathematically configured state, sensitive throughout as a whole in which a change at any one point is necessarily propagated to every other.

And what mathematics did Maxwell use?

Maxwell was introduced to the work of Lagrange, Traité Analytique, by his life-long friend Peter Guthrie Tait (and Thomson).

The power of Lagrange’s methods lies in their generality and their abstraction that is purely algebraic with no concern to anything that is anything actually physical.

So, can I explain this stuff without mathematics? Let me try.

I will begin near the end of the nursery rhyme, with

One Little Monkey Jumping on a Bed.

This bed is a mattress of springs.

Suppose a monkey is jumping up and down on a horizontal mattress (made of identical interconnected springs on a square grid that is infinite on both axis), at one of the points that connects any four springs.

How does this monkey bobbing (physics terminology: harmonic oscillation) cascade to the other interconnected points (of four springs) in the square grid, over time, as the various springs are pulled (stretched) and they, in return, because they are springs, pull back?

It is a messy math problem.

If we make some simplifications (“continuum limit” pioneered by Cauchy), we obtain something tractable (partial differential equation with a simple enough structure, wave equation) that captures the dynamics of propagation of the monkey-induced bobbing across the mattress. 

Based on the spring constant (“how springy a spring is”) and how energetically the monkey is jumping, a sinusoidal formula with an amplitude and frequency is the solution of how the effect of monkey bobbing is propagating through the mattress.

That is:

If (x,y) is the point (on the mattress) from where the monkey is jumping, and t is the time since the monkey started to jump, this sinusoidal formula tells us the amplitude (and how it is changing) at this location, and how the wave – the bobbing effect – is traveling across the mattress.

Indeed, one can, abstractly, view this time-dependent sinusoidal solution on the 2-dimensional mattress as a (2+1) dimensional field: 2 for the x-y plane (representing space), and the +1 to represent time. 

The period of oscillation and the amplitude are important characteristics of this field as is the speed of wave propagation. 

Welcome to classical field theory. It is just a “continuum approximation” of a monkey jumping on a bed.

The intellectual heritage of QFT – Fields and Lagrangian formalism — can be traced back to James Clerk Maxwell and development of what is now called classical electromagnetism.

To this monkey bobbing, add Einstein‘s restrictions that no wave can travel faster than light, and that all observers, regardless of how fast they themselves are moving, will view the speed of the light to be the same, something I have discussed previously in Śrīdhara Brāhmaṇa: Pythagoras Theorem and Time Dilation. (Special Relativity.)

This restricts the types of Lagrangians that we can use.

Of course, we need to be in a (3+1) dimensional field and not just looking at wave propagation due to monkey bobbing on a flat mattress.

We are almost there. We need to quantize the fields.

What this means is that we need to view the fields as operators that satisfy certain commutative relations.

This is done differently than in (non-relativistic) Quantum Mechanics, that I had discussed in an earlier post Newton v Schrodinger (and Heisenberg), where we had wave functions.

QFT is therefore sometimes called Second Quantization.

And, like in the case of a wave function, the field is not in our (3+1) physical space but in the abstract conceptual space (with imaginary numbers) as I have discussed in an earlier post on Imaginary Numbers and Electron Spin.

Putting it all together:

QFT is a Lagrangian field theory, the Lagrangians restricted to satisfy Special Relativity (“Lorentz invariant”), and the fields are quantized.

You must think that I have simplified QFT too much, and that there must be more to it than worrying about the effects of monkey bobbing.

Let me close with a plea from Tony Zee, Quantum Field Theory in a Nutshell (2003):

Even after 75 years, the whole subject of QFT remains rooted in this harmonic paradigm. We have not been able to get away from the basic notions of oscillations and wave packets. Indeed, string theory, the heir to QFT, is still firmly founded on this harmonic paradigm. Surely, a brilliant young physicist, perhaps a reader of this book, will take us beyond.

Said differently:

No more monkeys jumping on the bed!😏

1 comment

  1. Interesting simplified way of explaining a complicated concept.
    Although I have a background in EE but more with machines and energy conversion and never got introduced or had the opportunity to use Maxwell’s equations, I find the article very interesting to give an uninitiated and long-time retiree a vague idea by the author. Thanks and keep up your contribution.

Leave a Reply