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Hiding Behind Abstractions and Terminology

As mathematical formalisms become more and more beautiful, it is increasingly easy to be trapped by the formalism and to become a ‘slave’ to the formalism. We used to be ‘slaves’ to Newton’s laws when we regarded everything as a collection of particles. After the discovery of quantum theory,4 we become ‘slaves’ to quantum field theory. At the moment, we want to use quantum field theory to explain everything and our education does not encourage us to look beyond quantum field theory. However, to make revolutionary advances in physics, we cannot allow our imagination to be trapped by the formalism. We cannot allow the formalism to define the boundary of our imagination. The mathematical formalism is simply a tool or a language that allows us to describe and communicate our imagination. Sometimes, when you have a new idea or a new thought, you might find that you cannot say anything. Whatever you say is wrong because the proper mathematics or the proper language with which to describe the new idea or the new thought have yet to be invented. Indeed, really new physical ideas usually require a new mathematical formalism with which to describe them. This reminds me of a story about a tribe. The tribe only has four words for counting: one, two, three, and many-many. Imagine that a tribe member has an idea about two apples plus two apples and three apples plus three apples. He will have a hard time explaining his theory to other tribe members. This should be your feeling when you have a truly new idea. Although this book is entitled Quantum field theory of many-body systems, I hope that after reading the book the reader will see that quantum field theory is not everything. Nature’s richness is not bounded by quantum field theory

Xiao-Gang Wen

A review for the Americal mathematical Society offered this deep statement in praise of the book: “It is often deeper to know why something is true rather than to have a proof that it is true.” (Indeed, a Fields Medalist once told me that top mathematicians secretly think like physicists and after they work out the broad outline of a proof they then dress it up with epsilons and deltas. I have no idea if this is true only for one, for many, or for all Fields Medalists. I suspect that it is true for many.)

Quantum Field Theory in a Nutshell by A. Zee

It has been my experience, by the way, that such simple derivations are much more useful for scientific thinking than more formal ones; so it’s unfortunate that textbooks (and academic papers) are almost always dominated by the latter. I am always pleasantly surprised by how much easier it is to talk science one-on-one with someone than it is read their papers. That’s because in a one-on-one conversation a scientist will talk to you in the language that s/he uses to think about the problem, whereas when writing a paper everyone gets paranoid that they’ll say something incorrect and be called out for it. But as my undergraduate advisor used to say, “what’s a factor of \pi between friends?”Brian Skinner

On the other hand, a prominent blogger once offered the advice that it’s dangerous to blog without tenure. There is a caricatured image of grad students as working 200% of the time. While this is clearly not true in practice, it still doesn’t look great when you hit a ‘rough patch’ in your research but you still manages to make regular blog posts. Further, no matter how many insightful posts you write, you’re always a single bone-headed statement away from offending someone senior with a lot of power over your future. (So when the grown-ups are having blogo-wars with one another, Junior would be wise enough to stay out of it.) [I will note, however, that I’ve heard a few people say that blogging has *helped* their early careers.]

I haven’t complete closed the door to future blogging. Maybe somewhere down the line I’d be interested in joining a group blog of young scientists, but this very-hypothetical situation wouldn’t happen in the near future and would only occur after a long talk with my adviser.

Flip Tanedo

The Problem with Most Textbooks and Papers

Most textbooks and papers are not written to help you, but instead to prove how smart the author is.

Many people write things that are hard to understand on purpose. This is done, because if no one understands what you write, you don't get criticized. In addition, you appear extremely smart. This was already noted in 1966 by Cornelius Lanczos in his book “The Variational Principles of Mechanics”:

Many of the scientific treatises of today are formulated in a half-mystical language, as though to impress the reader with the uncomfortable feeling that he is in the permanent presence of a superman.

In a similar spirit V. I. Arnold noted:

It is almost impossible for me to read contemporary mathematicians who, instead of saying “Petya washed his hands,” write simply: There is a $t_1 <0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto \text{Petya}(t_1)$ belongs the set of dirty hands, and a $t_{2}, t_{1}<t_{2}\le 0,$, such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.


“Thus in elemental silicon, where there are many electrons locked up in the chemical bonds, it is possible to pull an electron out of a chemical bond to make a hole. This hole is then mobile, and acts in every way like an extra electron added to the silicon, except that its electric charge is backward. This is the antimatter effect. Unfortunately, the hole idea makes no sense in the absence of something physically analogous to a solid's bond length, since this length fixes the density of electrons one is ripping out. Without it, the background electron density would have to be infinity. However, such a length conflicts fundamentally with the principle of relativity, which forbids space from having any preferred scales. No solution to this dilemma has ever been found. Instead, physicists have developed clever semantic techniques for papering it over. Thus instead of holes one speaks of antiparticles. Instead of a bond length one speaks of an abstraction called the ultraviolet cutoff ,a tiny length scale introduced into the problem to regulate it-which is to say, to cause it to make sense. Below this scale one simply aborts one's calculations, as though the equations were becoming invalid at this scale anyway because it is, well, the bond length. One carries the ultraviolet cutoff through all calculations and then argues at the end that it is too small to be measured and therefore does not exist. Much of quantum electrodynamics, the mathematical description of how light communicates with the ocean of electrons ostensibly pervading the universe, boils down to demonstrating the unmeasurableness of the ultraviolet cutoff. This communication,which is large, has the fascinating implication that real light involves motion of something occupying the vacuum of space, namely all those electrons (and other things as well), although the extent of this motion depends sensitively on the value of the ultraviolet cutoff which is not known. There are endless arguments about what kinds of regularization are best, whether the cutoff is real or fictitious,whether relativity should be sacrificed, and who is too myopic to see the truth. It is just dreadful. The potential of overcoming the ultraviolet problem is also the deeper reason for the allure of string theory,a microscopic model for the vacuum that has failed to account for any measured thing.” from a Different Universe, by R. Laughlin

“The similarity between sound and light requires explanation, for there is no obvious reason for their quantum mechanics to be the same. In the case of sound-quantization may be deduced from the underlying laws of quantum mechanics obeyed by the atoms. In the case of light it must be postulated. This logical loose end is enormously embarrassing, and is something we physicists prefer to disguise in formal language. Thus we say that light and sound obey the Planck law by virtue of canonical quantization and the bosonic nature of the underlying degrees of freedom.But this is no explanation at all, for the reasoning is circular. Stripped of its complexity, “canonical quantization” simply boils down to requiring light to have properties modeled after those of sound.from a Different Universe, by R. Laughlin

Further Reading

causes/hiding_behind_abstraction.1509700975.txt.gz · Last modified: 2017/11/03 10:22 (external edit)