If only one thing were enough

 kw: book reviews, nonfiction, science, explanations, interdisciplinary

For Marcus Chown, explaining things isn't just a "man thing," it's a lifelong passion. He bites off a very big chunk to chew, to explain 21 science ideas most people find hard to comprehend, in The One Thing You Need to Know: The Simple Way to Understand the Most Important Ideas in Science. Dr. Chown seems to be used to juggling a passel of sciences.

The best I can do with a book this comprehensive is to limn a sampling:

• The Second Law of Thermodynamics – For the sake of background, the First Law of Thermodynamics is Conservation of Energy, or in cosmological terms, Conservation of Mass-Energy. The Second Law can be stated a few ways: "Work requires a flow of energy" and "Entropy must always increase" are two easy ones. Implied in these two statements is the prefixed caveat that "In a closed system…" Thus, building a house decreases entropy (a measure of disorder), but it does so only locally. In total, there is a great increase in overall entropy. Think of a huge pile of sawdust and other wastes… For you or I to grow from a fertilized egg cell into a baby, then to an adult, decreases entropy within our body, but increases entropy even more in the Universe as a whole. There's a curious statement on p61: "…the energy of a photon is proportional to its temperature…" The author is making the case that for each photon the Earth receives from the Sun, it emits 200 photons of lower energy. Photons don't really have a temperature, but in a thermal regime, it takes a hot object to emit photons of higher energy, so the statement is useful shorthand. The "average" photon from the Sun conveys a greenish color to our eyes and has a wavelength near 550 nm, and an energy of about 4.4 eV (look up electron-Volts). Of course, the Sun emits photons with a very wide range of wavelengths and thus energies. The average temperature of the Earth is 15°C (59°F), so it radiates infrared photons into space with and "average" wavelength of about 10,000 nm or 10 µ, and an energy of 0.124 eV. The ratio 4.4/0.124 = 35.5. That's rather different from 200, but it would take a much more elaborate analysis to produce a more definitive value, and that's not what this book (or this review) is all about.
• Atoms – This is a simpler concept, as long as we stay with pre-quantum-mechanical explanations. The word "atom" comes from the Greek word atomos, meaning "un-cuttable" or "indivisible". The philosopher Democritus 2,400 years ago asked, "If I cut this piece of pottery in half, and then do it again, and again…could I go on forever?" He declared, "No." But there was no way to prove it. Now we have abundant proof and demonstrations, including a microscope called AFM, for "atomic force microscope", that can produce an image, magnified several million times, of the atoms on a surface. X-ray methods allow us to visualize the arrangement of atoms in a crystal. But they are not longer "un-cuttable". When I was a physics student, "atom smashers" of a few types, including a synchrotron at my college, routinely banged ions against one another, "splitting" atoms into smaller pieces. Now, we think that electrons and quarks are the truly un-cuttable entities. Probably, but stay tuned…
• The Standard Model – I got out of physics because I was a college senior during the heyday of the "particle zoo", when the number of "-ons" and "resonances" and other items showering out of atom smashers had grown to a list of 100 or more. A few years later physicists proposed the "Eightfold Way", which was tweaked and modified and added to, until now we can make this diagram:

Ordinary matter is entirely composed of the leftmost column of 4 "leptons" and the 5 "bosons". There is a caution, though: all the leptons have anti-matter "twins", such as the positron, which is the anti-electron. The gluon, photon, Z, and higgs have no anti-bosons, or one can say they are each their own antiparticle. The W has an anti-W.

Thus, the particle zoo is smaller now, with "only" 30 "fundamental" particles, rather than a hundred or so.

This is a great synthesis, but it is still incomplete. We don't know if gravity is quantized, or what to do with a "graviton" if such a critter exists. I presume it would be a boson.

The introduction to this chapter (#15) includes the quote, "People want to know about what's going on with what's in the universe, what are particles like, what are the basic rules of nature. There's a lot of curiosity out there." by Sheldon Lee Glashow. I'd say, for "people" he really meant "scientists" or even "cosmologists." For the rest of the human race, the curiosity is mainly directed to "What's my next meal?" and "Where can I sleep safely?" and "Can I get laid tonight?"

• Quantum Computers – The hype about these is like entropy; it is ever-increasing. And so we find it here. One useful point is made, and this must be the "one thing" for this chapter: Quantum mechanical math only applies to an isolated thing, whether an electron, an atom, a buckyball, or anything else, in a very low-temperature vacuum chamber (i.e., isolated from the Universe; I guess gravity doesn't count). Constructs larger than single particles need to maintain "coherence," such as that seen in Bose-Einstein condensates. Anything at all from the outside that interacts with the "thing" will cause it to "decohere" and enter a fixed state that is described by classical mechanics, not quantum mechanics. That "anything at all" includes photons with extremely low energies, which is why Bose-Einstein condensates can only be created in an extremely rarefied vacuum at a temperature less than one degree above absolute zero. Apparently, thermal photons emitted by the walls of a chamber at such a low temperature are either too sparse to disrupt the condensate, or too low in energy to do so. Anyway, the math of quantum multiplicity shows that adding a single qubit to an array of qubits doubles the number of final states it can use, thus doubling the complexity of the problems it can solve. The trouble is, a quantum computer can only produce a single output, so it is best suited to doing something like cracking a single password. Like second-grade math teachers, for a quantum computer there is "only one right answer". I guess matrix math is out of reach. If you know how passwords are cracked with current equipment, you know that they cannot be tackled one at a time; a hacker typically gathers the "hashes" from thousands to millions of passwords and cross-matches them against a "universal hash generator". If a hacker can extract a few or a few hundred passwords that way, he can make a ton of money exploiting just those, and the uncracked ones can be left for a later, more rigorous attempt. I really haven't seen another problem that quantum computers are suited for, and the author doesn't suggest any either. But along the way he makes a wonderful statement about the current state of science: "Something physicists never like to admit is that they have only ever solved one problem exactly: the two-body problem." That's the orbit of two objects about one another under the force of gravity only. He is right! Everything else is approximated. Science has some distance yet to go.
• The Big Bang – If you begin with the current state of the Universe, and the observation that all the galaxies are separating from one another at a rate that varies primarily with their distance, you can "extrapolate to zero" and wind the Universe back to the initial state of zero volume and infinite temperature that "must" have begun everything. This was determined before 1930. More detailed observations and analyses since then have found three "hangups":
1. The background "temperature" is too uniform; it should express more of the initial turmoil unless there was time for the temperature to equalize. It is posited that a slightly slower start, during the first trillionth of a trillionth of a trillionth of a second, was followed by a very brief period of enormous expansion, dubbed Inflation, for about a billionth of a trillionth of a trillionth of a second, at which time the Universe was the size of a softball, and then continued expanding at a more "sedate" rate comparable to what we see today. This is kind of like blowing up a weather balloon with C-4.
2. The gravity of all visible matter is too small for galaxies to have formed in the calculated time (13.8 billion years) since time-zero, and the gravity of all visible matter in a galaxy is too small to hold the stars in their measurable orbits. It is posited that the actual mass of gravitating "stuff" is about seven times as great as what we can see; the extra "stuff" is called Dark Matter. So far, we can only know it from its gravity.
3. Observations of distant Type 1a supernovae seem anomalous; calculations based on their brightness indicate that universal expansion is speeding up. It is posited that a kind of negative gravity extracted from "vacuum energy", dubbed Dark Energy, is responsible. I personally think that we don't yet know enough about how Type 1a supernovae behaved in the first billion years or so, when the "metals" content (everything except hydrogen and helium) of the Universe was very, very small.

The author concludes this chapter (#21) by saying, "…there is a strong suspicion that there is a deeper, more fundamental cosmological theory to be found, which will merge inflation, dark matter and dark energy into a more appealing, seamless entity." I would think that a proper theory would make all three superfluous. Time will tell

It's a very enjoyable book. I'd have preferred each chapter to begin with an introductory blurb, stating "the concept to be grasped" and "the one thing that'll help you grasp it". I don't really see any "one thing" in any of the chapters. But it's cool anyway.

Today's buzzword: nonrenormalizable

 kw: science, philosophy of science, explanations

In a post last week I reviewed The Grand Biocentric Design: How Life Creates Reality, by Robert Lanza, MD and Matej Pavšič, PhD, with Bob Berman. Much of the post expressed my objections to the theory that "life creates reality", a theory based on a dramatic over-extension of the Copenhagen Interpretation of quantum theory (I disagree also with Neils Bohr, who concocted CI). 

I decided now to add a brief discussion of a point made several times in the book, that the version of quantum gravity theory these authors espouse is "non-renormalizable". The word is never explained. Perhaps I can at least give the public some understanding of "normalization" and "renormalization".

A problem arose in the early 1900's when scientists began to solidify the mathematics of quantum electrodynamics (i.e., the dynamics of charged particles in the quantum realm). If equations for the behavior of an electron (for example) were to be solved, partway through the solution one had to contend with expressions that divided by zero, leading to "infinities". Eventually, mathematical methods were developed that "renormalized" these equations, so that the term(s) leading to division by zero could be removed before one actually had to calculate the result.

To understand renormalization, it helps to first understand normalization, which is used to derive the basic equations of differential calculus. The term "derivative" means a function, derived from another function, that expresses the slope at any point. It is the basis for a large family of functions that are needed to optimize functions and, used "in reverse", to calculate areas and volumes (among many other useful summation operations). 

I will illustrate by deriving the "first derivative" of two functions. The first is a basic parabolic function, y = 5x². This states that, in an x-y coordinate system, the value of y is found by squaring x and multiplying by 5. Thus, for x = 3, y = 5*3*3 = 45. The second is a simple quartic, y = x⁴. For this one, when x = 3, y = 3*3*3*3 = 81. Both functions are very useful in mechanics: the position of an object falling in a uniform gravitational field is expressed by a parabolic function, and the energy radiated by a heated body is related to the temperature by a quartic function.

This illustration shows both functions graphed in the domain x = [-1,2].

NOTE: There are several notations used in calculus, because of the complicated history of its discovery three-plus centuries ago. Here I use notation based on that of Gottfried Liebniz (d. 1716).

The mathematical slope of either of these curves is dy/dx, and one can approximate it by calculating the functions at x and x+dx, for very small dx. But the exact value can only be found when dx = 0!

Below is the derivation for the parabolic equation as I learned it in high school.

"Lim" means the limit of the expression in brackets, as the term in parentheses is satisfied. We do the derivation using dx and dy as algebraic variables.

The three lines in the middle expand the function. The following line is the "normalization": dividing by dx removes dx from one of the terms in the equation.

Then, as shown in the last line, when we set dx to zero, the term that contains it vanishes, leaving us with a function in x only. This function is the first derivative of function y1.

Let's drive the point home by performing the same operation on the quartic function:

This time, multiplying out the function results in a larger number of terms. As before, the calculations are shown in the three-line cluster in the middle. The next step, normalization, divides by dx to yield one term that doesn't include dx, and three others that include it. Setting dx to zero yields the first derivative of function y2.

To summarize: Normalization is a mathematical process of deriving expressions that would contain a division by zero if carried out with numbers, but then dividing out the term that will become zero, so that a perfectly calculable function remains.

We can do a simple verification to see how these derivatives perform. Here is a table of the two functions, calculated from the values of x as given. The graph above was produced from these numbers.

In the graph above the green lines are anchored to x = 1.5. Let's start with dx = 0.1 and calculate y1 and y2 at x = 1.6, subtracting values found in this table at 1.5:

y1(1.6) = 5*1.6*1.6 = 12.8; dy1 = 12.8 - 11.25 = 1.55; dy1/dx = 15.5.

y2(1.6) = (1.6)⁴ = 6.5536; dy2 = 6.5536 - 5.0625 = 1.4911; dy2/dx = 14.911.

We can shrink dx to 0.01, so we calculate at 1.51. The results are then 

dy1 = 11.4005 - 11.25 = 0.1505; dy1/dx = 15.05.

dy2 = 5.19885601 - 5.0625 = 0.13635601; dy2/dx = 13.635601.

These numbers seem to be closing in on 15 and something more than 13.

Now let's find exact slopes using the two derivatives:

Slope #1 = 10*1.5 = 15.0

Slope #2 = 4*(1.5)³ = 13.5

It's possible to use smaller and smaller values of dx to get a converging series, from which we can get pretty accurate results. However, the mathematics of the derivative ensure that the values calculated by them are exact.

Based on all that, what is Renormalization? Here I must get conceptual because the functional expressions are enormous, and people got Nobel prizes for figuring them out. Calculus, which is already based on Normalization, was used to determine the functions needed by quantum electrodynamics. The equations were labored over until a version of each derivation could be performed that "divided out" the expressions (analogous to dx but much more elaborate) that caused trouble. This second dividing-out process is renormalization.

In one of his books, Richard Feynman tells of being asked, at the Nobel Prize ceremony, "What did you do?" He answered, "Buddy, if I could tell you that in one minute, it wouldn't be worth a Nobel Prize." Renormalization was part of it.

So what is Non-renormalizable? The authors of The Grand Biocentric Design make much of their contention that a quantum theory of gravity cannot be renormalized. No such theory has yet been fully developed, as the authors admit. Since we don't know what form such a theory will take, we (and they) don't have a way to determine if their contention is anywhere close to being correct.

On a plane ride several years ago I sat with a young man who, it developed, is a cosmologist. I said, "I have a question I've been hoping to ask a cosmologist, if you're willing to give it a crack." He agreed. I asked, "If gravity is a quantum phenomenon, that means there is a gravitational quantum, a 'graviton'. The most powerful sources of these gravitons are black holes. What is it, for a black hole or for any other massive body, that emits the gravitons? If they start 'inside', where the mass is, how do they get out?" He said, "That's two questions, but it's worth some thought." After a half hour of silence he said, "By analogy to photons, which are emitted when charged particles such as electrons are accelerated or make a quantum jump from one orbital to another: The motion of an electron may cover at most a nanometer or two, particularly in an orbital transition. The photon that is emitted has a wavelength of a few hundred nanometers or more. We don't yet know what is the wavelength of a typical graviton. But I think gravitons are emitted over an area larger than the typical black hole, not from inside it. That's the best I can say on short notice." That was pretty good! I reckon it will be a while before a better answer is in the offing.