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Tag Archives: QFT

As kids tend to discover rather early on, it’s fun to throw stones into water. As the stone hits the water, it gives it energy and creates ripples that move outwards in concentric circles. These ripples carry the energy outwards, and as they move further out they eventually die out and the water is smooth again (this is true even if you ignore the effect of friction). The behavior of the water is well described by the wave equation — very likely the first field equation that people are taught about in physics. In one dimension, it describes waves on a string or electric and magnetic waves. In two dimensions, it describes water waves. In three, sound waves. Waves are ubiquitous in our daily lives.

In contrast, when you kick a soccer ball, the energy you give it doesn’t spread out. It stays with the ball as it rolls away. The difference between soccer balls and water waves is that in the former case the energy density is localized, and in the latter it dissipates. So the question arises: are there any wave type phenomena with a localized, non-dissipative energy density?

Solutions of field equations that have this property are called solitary waves. We do hear of such objects from time to time – moving lumps of water that don’t die out like waves do – and usually with catastrophic consequences. These are the tsunamis. The physics of tsunamis is rather involved, but certain kinds of tsunamis involve solitary waves – those that are caused due to sharp localized impulses such as an asteroid collision.

If these solutions have the further property that they retain their shapes after collision with each other, they are known as solitons. These objects show up in theories where non-linear interactions balance out the tendency to disperse, and they give rise to lots of interesting physics!

The discovery of the solitary wave is the kind of story that physicists love to tell. It’s a example of Isaac Asimov’s principle –  “The most exciting phrase to hear in science, the one that heralds new discoveries is not ‘Eureka!’ but ‘That’s funny…'”. The story is described in this article on the history and application of solitons:

In August 1834, the naval engineer John Scott Russell was watching a horse-drawn barge on the Union Canal, Hermiston, Edinburgh, as part of his work on hull design. When the cable snapped and the barge suddenly stopped, Russel was impressed by what happened: ‘A mass of water rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some 30 feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel’.

I find it extremely cool that an idea that started off with a seemingly mundane observation of horse drawn boats can predict the existence of magnetic monopoles in certain gauge theories, or can explain flux quantization in superconductors! The important point is that rather than write off his observation as a quirk of nature, John Russell was the first person to follow through and experiment to try to understand this phenomenon.

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Nima Arkani-Hamed gave a talk on ‘Revenge of the S-Matrix: What is the simplest QFT?’

He talked about how many of the fundamental theoretical problems in physics such as determining the vacuum, the information paradox, and infinities that arise in eternal inflation come about because of problems with locality and gravity. Because of gravity, he says, locality is not an exact principle of our world. The pragmatic reason for considering such theories is that locality gives rise to a huge amount of redundancy.

He then reviewed a different way of talking about QFT which was not manifestly local, called BCFW, where you analytically continue the momenta of an incoming and an outgoing particle in some scattering process. The result was that in this formalism, N=8 SUGRA becomes the simplest QFT.

His talk is online here.

Sidney Coleman taught a legendary class on Quantum Field Theory at Harvard, the lectures of which are cherished by many of the big names in physics today. Lisa Randall calls them a cult classic. I’ve been looking for these for a while, and I was thrilled to hear from fliptomato that Harvard has generously hosted these videos online.

Here are his 1975-76 lectures on QFT: link

Since the videos quality is not that good, it’s probably essential to read along with the lecture notes, transcribed by Brian Chen at Penn.

The lectures seem to provide a pretty thorough grounding in the canonical formalism, so hopefully this aspect of QFT will be slightly less of a mystery for me if I can get through some of this. He doesn’t seem to cover the path integral approach, but I’m getting enough of that in class anyway.

“The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction.” – Sidney Coleman