*Special Relativity and Classical Field Theory: The Theoretical Minimum* by Leonard Susskind and Art Friedman

The third book in Susskind’s Theoretical Minimum series (based off the video lectures here) covers special relativity and classical field theory, specifically electromagnetism (general relativity is the next planned book). It’s assumed the reader has read the first book in the series where things like the action, Lagrangian, and Hamiltonian are developed, but there’s still lots of review and hand-holding. Appendix B is a review of some vector calculus operations.

I thought this book was great, and is filled with a bunch of great nuggets even for somebody with a degree in physics. For example, one longstanding question I’ve had is why the Lagrangian only goes up to the first derivative of position (aside from the “It’s equivalent to Newton’s Second Law” cop-out). Susskind’s satisfying answer: it’s because of locality [1]. And just for fun, Appendix A is all about magnetic monopoles.

Speaking of locality, Susskind lists four fundamental principles that underlie all of physics, and then uses them directly in producing results in electromagnetism. That is he doesn’t just have an interesting philosophical discussion, but translates them into meat-and-potatoes mathematics and then uses them to solve problems. These principles are: the action principle, locality, Lorentz invariance, and gauge invariance.

This is my favorite so far of the Theoretical Minimum series. People who have only seen Maxwell’s equations developed to their vector calculus form that appears on T-shirts (for example, most engineers) will be guided to their “higher” formulation in terms of tensors. The connection between special relativity and electromagnetism is deep, and it’s with that connection that Einstein began (paragraph one!) his famous special relativity paper. With this book the reader will clearly see how once you observe that a current-carrying wire generates a magnetic field, special relativity necessarily follows.

Also, “GROUCHO!” (see p. 392)

[1] And from Coopersmith I got a satisfying answer to why the Lagrangian depends explicitly on the first derivative: it serves as an infinite set of “internal boundary conditions” to enforce continuity.

]]>*Quantum Mechanics: The Theoretical Minimum* by Leonard Susskind and Art Friedman

This book (based off of these lectures) came out a few years ago, and while I picked it up at launch I never quite finished it and it got kind of lost on the back burner. The third book on special relativity just came out, though, so I dusted this one off and finally completed it. Recall that this quantum mechanics volume is the second in Susskind’s Theoretical Minimum series, after the first book on classical mechanics.

A few years ago at my old university we had David Griffiths as an invited guest speaker. I didn’t take notes so I could be misremembering, but I recall him saying that while the physics education profession had more or less decided on the appropriate structure for a course on electromagnetism, there was still widespread disagreement about how to teach quantum mechanics. Griffiths’ own book starts with the Schrodinger equation on page one, and proceeds from simple potentials like the infinite square well up through the hydrogen atom and beyond, and then moves on to other topics like spin. Longair’s *Quantum Concepts in Physics* instead takes an historical approach.

Susskind and Friedman take a different tack: they start with spin, which doesn’t really have any classical analogue and hence is rather foreign and abstract. However, the mathematics of spin is comparatively very simple (on the order of 2×2 matrices). I’m not sure of this approach. Having take several quantum mechanics courses, it was all fairly old hat to me, but I’m not sure that a beginner wouldn’t feel a bit overwhelmed right off the start. The advantage of beginning with spin is that they are very well placed to then explore issues like entanglement (Chapters 6 and 7). They do a very thorough job developing and motivating the algebra of operators, and having started with spin there’s always ready-made and simple examples to apply the operators to.

While they never solve the infinite square well or the hydrogen atom, the last chapter is devoted to the harmonic oscillator. It’s not as thoroughly developed as in a textbook like Griffiths, since it starts off by declaring but not proving a theorem that the ground state has no nodes, and then the ground state wave function is just given to us. However, they then develop the raising and lowering operators to generate the higher-energy wave functions, whose solutions involve the Hermite polynomials.

In his video lecture series, Susskind had a second course on advanced quantum mechanics, and so I wonder if they’ll eventually publish a similar book in this series. Before that though there’s the third book on special relativity, then presumably we’ll get general relativity, cosmology, and statistical mechanics.

]]>*The Lazy Universe: An Introduction to the Principle of Least Action* by Jennifer Coopersmith

There is an interesting phenomenon of popular versions of more technical books. We have Feynman and Hibbs’ *Quantum* *Mechanics and Path Integrals* being popularized in the form of Feynman’s *QED*. David Friedman turned his *Price Theory* textbook into the wonderful *Hidden Order*. And now we have an answer to the question: but who will write a “popular” (for non-expert audiences with college-level math training) version of Cornelius Lanczos’ *The Variational Principles of Mechanics*? Jennifer Coopersmith, that’s who.

I really enjoyed this book. The principle of least action is one of those great gems that bears returning to again and again. One of my pet pursuits is finding more and better explanations for the age-old question “Why is the classical Lagrangian kinetic *minus* potential energy?” (see here for example for my attempt to understand it a bit better in the case of gravity). Coopersmith devotes Section 6.6 to that very topic.

*The Lazy Universe* is structured by beginning with some historical material and some mathematical preliminaries (advisory note: you do need to know vector calculus to really profit from this book). Then we march from the principle of virtual work, to D’Alembert’s principle, on to Lagrangian mechanics and then Hamiltonian mechanics. Unlike Susskind’s popular classical mechanics book Coopersmith doesn’t address Poisson brackets. I really like the structure and how she builds from one chapter to the next; I certainly did not sufficiently appreciate the point of virtual work and D’Alembert when I took my classical mechanics class. In the penultimate chapter there’s a survey of how the principle of least action takes its form in other branches of physics.

I’m just not sure who this book is *for*, exactly. Much of the mathematical nitty-gritty and nearly all the examples are pushed to a multitude of appendices, which makes it annoying to flip back and forth if you want the full story. This was done I think so that less mathematically-adept readers could enjoy the main thread of the argument, but that still requires knowledge of the variational calculus (for example, the Euler-Lagrange equations are not derived like in Susskind’s book). So I would recommend this to people for whom this is their second (or third, etc) pass at analytical mechanics.

2011 | 2012 | 2013 | 2014 | 2015 | 2016 | |
---|---|---|---|---|---|---|

Supercomputer (PFLOPs) | 10.51 | 17.59 | 33.86 | 33.86 | 33.86 | 93.01 |

Known exoplanets | 716 | 854 | 1055 | 1855 | 2041 | 3557 |

ATLAS integ. lumin. (fb^{-1}) |
5.46 | 28.54 | 28.54 | 28.54 | 32.89 | 71.8 |

GenBank base pairs (billions) | 135.1 | 148.4 | 156.2 | 184.9 | 203.9 | 224.9 |

Kardashev score | 0.724 | 0.725 | 0.726 | 0.727 | 0.727 | NA |

World population (billions) | 7.0 | 7.1 | 7.1 | 7.2 | 7.3 | 7.4 |

What do I usually even write here? I don’t remember. I finished 60 new books this year, only a fraction of which I blog reviewed. I successfully defended my master’s thesis and can now append MSc after my name. You know, if I was a douche. I’ve been undergoing allergy desensitization therapy the past few years, and it paid off big this summer (normally I have hard allergic conjunctivitis from the beginning of May through the end of August). No symptoms, so that’s plus one right there for modern medicine.

For people super bummed about 2016, take the large view, the long view, stop following the news so closely and read or listen to more books, meditate on how bad it could get, listen to some good tunes, and browse some dank memes.

]]>*Hidden Order: The Economics of Everyday Life* by David Friedman

*Hidden Order* is my favorite popular economics introduction, because, rather than being a sequence of interesting stories making a point, it has the logic and structure of a textbook: starting from simple beginnings, Friedman hones in on certain insights, and then gradually adds complexity to explain more and more.

This is not surprising, since before this book (published in 1997), Friedman had written an intermediate macroeconomics textbook called *Price Theory *(first edition in 1986, second edition in 1990). The chapter titles and subjects are almost word-for-word identical. The difference is a loss of mathematical detail and rigor and end-of-chapter problems; perfectly fair for a pop intro.

After an introductory Part I that introduces economics as it’s actually thought of by economists (as opposed to a naive laymen view that it’s all about money and the stock market, the far more fun stuff is things like the economics of war and the economics of driving), Part II logically constructs modern price theory in a very clean and ideal world. Part III adds a number of real-world complications, like the existence of firms (as opposed to single-person production), strategic behavior, and the uncertainties of time and chance. Part IV is subtitled “The Economist as Judge” and deals with efficiency, at least as economists define it. This is the section dealing with rent-seeking, public goods, and externalities. One simplification in this section that I wish he had done without was the assumption of constant utility of dollars (one dollar being worth the same to a rich man as a poor man) versus something more realistic like utility as a logarithmic function of dollars. The final Part V deals with some particular applications: the economics of crime, public choice theory, and the economics of dating and marriage.

In my experience (having convinced several people to read it), *Hidden Order* might need to be read “in waves”, as many paragraphs are dripping in insight and the reader might move too quickly and not be building on a solid foundation. Typically when revisiting a chapter the point sticks better having seen a bit where he’s heading. It’s definitely worth the effort though.

*Square One: The Foundations of Knowledge* by Steve Patterson

The first of a series of planned philosophy books by Steve Patterson, who runs the *Patterson in Pursuit* podcast that I listen to, *Square One* is about the ultimate grounding of human knowledge in necessary logic.

In the first chapter, Patterson lays out a tree analogy for one’s worldview, where the leaves are conclusions, the branches are premises, and the roots are the foundations. People spend most of their time on the leaves, when it’s the roots that matter most (if you get the foundations wrong, all your conclusions are in serious doubt). This is contrasted with the “web of knowledge” way of thinking where there are no foundations but instead all your beliefs consist of one interconnected layer. He argues that the roots of knowledge are the laws of logic, which are necessary and can be known for certain.

In the second chapter, Patterson addresses implausibility arguments against humans attaining such certain knowledge. Examples include the skeptical injunction to always be humble about what we can know, the evolutionary origins of the brain which emphasizes what works rather than what is necessarily true, and the implausibility of the universe needing to be sensible to humans. While these arguments have some persuasive power, they cannot establish that certain truths are unknowable. Impossibility arguments are addressed in chapter five.

Chapter 3 is the core of the book, where the laws of logic are laid down: the law of identity, the law of non-contradiction, and the law of excluded middle. Patterson hammers home that these are necessarily true and cannot be denied, for the denial of them assumes their truth. Chapter 4 extends these basic laws to some necessary consequences, such as how theory is prior to data and how we can be certain there exists the phenomena of awareness by a mind. This is not a textbook on logic, though, so he doesn’t demonstrate the formalization of propositional logic and how it can be used and manipulated (perhaps that could have made a short appendix? For people who want to explore this topic more and in particular how logical operations can be mechanized, try *The Logician and the Engineer*).

The final fifth chapter addresses “impossibility argument” counters to the claim that we can be certain about the laws of logic. Some major arguments that he addresses are the liar’s paradox (a misuse of language), quantum indeterminacy (whatever else, logic is prior to physics), and the “mere tautology” reply (mathematics can be thought of as a series of “mere tautologies”, but we sure as hell learn something new).

Short and readable, this makes for a good primer on logic and the rationalist worldview and I wouldn’t hesitate to recommend it to a logic neophyte or to somebody who claims that “we can never really know anything”. I look forward to Patterson’s subsequent books.

]]>*Stuff Matters: Exploring the Marvelous Materials that Shape Our Man-Made World* by Mark Miodownik

When I watched the BBC documentary series *Connections* (which I recommend to everybody), one of the main takeaways was how important materials science and engineering are to the modern world. No matter the impressive high-tech features of some modern product, buried in the background is likely a need for highly-pure materials unavailable, save in maybe small quantities, to our ancestors. And so I picked up Miodownik’s popular-level *Stuff Matters* to get a materials fix.

A different material is explored in each of the main chapters, and we are led on a tour of

- Steel
- Paper, of various kinds
- Concrete, including steel-reinforced concrete
- Chocolate
- Aerogels
- Plastics, particularly celluloid
- Glass
- Carbon, including graphite, diamond, carbon fiber, graphene, and nanotubes
- Ceramics, particularly porcelain
- Biomaterials, including plaster, amalgams for tooth-fillings, and titanium

There’s a concluding chapter that moves quickly between the different length scales involved in materials and tries to end pointlessly romantically.

This is not a systematic approach to understanding materials, but rather essentially a collection of popular-level essays on various materials of interest. I don’t mean that as some sort of pointed criticism; the book is what it is. Some materials I might like to have included (there’s always a question of space) are aluminum, hydrocarbons, and textiles.

When posed with the question of what one piece of knowledge would we want to pass down if all of our scientific civilization were to crumbled, the most useful might be “all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another.” I had that in the back of mind as *Stuff Matters* regularly delves into how the arrangement of atoms determines the macro-scale properties of different materials. Arrange them one way and you get diamond; arrange them in another and you get pencil graphite.

While there was a 1.0% increase in total world primary energy consumption in 2015 over 2014, that wasn’t enough to tick over the Kardashev score giving my three decimals of precision standard.

I guess I haven’t addressed issues with this score yet, so now is as good a time as any. The main problem with the Kardashev scale is the same with all other simple measures of civilization, like GDP or HDI: what you want to know might not be and probably isn’t exactly what you’re measuring. Imagine two civilizations; each produces the same amount of stuff and each happens to produce the same cultural output (same songs, same movies, etc). The only difference is that one uses twice as much energy as the other. We would conclude that the more energy-efficient civilization is the more advanced, right? But the more wasteful civilization would have a higher Kardashev score.

It’s therefore worthwhile to note while the energy consumption of Earth is still increasing, there is another trend where the energy intensity (the amount of energy needed to produce one unit of GDP) is *decreasing*.

Recent I made use of MATLAB’s ability to quickly generate animated GIFs by repeatedly writing new figures as frames. So now I can make animations like this:

This is the band structure of the Kronig-Penney model as you gradually turn on the potential (the black rectangles) and see the colored bands begin to split. The horizontal axis is in K-space, not real space, so the spacing of the barriers and wells isn’t meaningful in an absolute sense and is merely for decoration. The height of the barriers in terms of the vertical energy scale is correct, as well as the relative sizing of the barriers and wells.

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