Quantum Field Theory as Simply as Possible

5.0 out of 5 stars Tired of popularizations? Buy this book.

The following is my review published in Amazon — the other 25 reviews are interesting and somewhat divergent– here’s a link to them all — https://www.amazon.com/Quantum-Field-Theory-Simply-Possible/dp/0691174296#customerReviews.

 

 

I’ve put in two further thoughts I left out to keep the Amazon review to a reasonable length (Addendums 1 and 2).

 

Disclaimer: I have probably spent more time with Quantum Field Theory As Simply As Possible (QFTASAP) than most as I was a lay reader providing commentary to Dr. Zee as he was writing it. I came to know Dr. Zee after he responded to some questions I had while going through another of his books, Group Theory in a Nutshell for Physicists. We’ve corresponded for years about math, physics, medicine, biology and life both turning out to be grumpy old Princeton Alums (1960 and 1966). If this makes me a friend of his despite never having met and living on opposite coasts — and if this precludes this review appearing on Amazon, so be it.

Should you buy the book? It depends on two things: (1) your ability and (2) your background. Dr. Zee spends a lot of time in the preface describing the rather diverse collection of people he is writing for. “I am particularly solicitous of the young, the future physicists of the world. .. delighted beyond words if some college students, or even a few high school students are inspired by this book .. “

Addendum 1  In retirement, I met one such high school student whil auditing an abstract algebra course at a local college.  He was simultaneously doing his German homework while listening to the lecture with one ear.  He did not go on in physics but did get a double summa in math and physics at an Ivy League university (not Princeton despite my attempts to get him to go there).

 

I fall into another class of reader for QFTASAP which the author mentions, “scientists, engineers, medical doctors, lawyers and other professionals . … Quite a few are brave enough to tackle my textbooks. I applaud these older readers, and address them as I write.”

So you need to know the background I bring to the book to put what I say in perspective. I am a retired neurologist. I did two years of graduate work in chemistry ’60 – ’62 before going to medical school. All grad students in chemistry back then took quantum mechanics, solving the Schrodinger equation to see where atomic (and molecular orbitals) come from.

At my 50th reunion, I met a classmate I didn’t know as an undergraduate, Jim Hartle, a world class relativist still writing papers with Hawking, so I decided to try and learn relativity so I’d have something intelligent to say to him if we met at another reunion. I studied his book on Gravitation (and Dr. Zee’s). Unfortunately COVID19 has stopped my attendance at reunions. To even begin to understand gravitation (which is what general relativity is about), you must first study special relativity. So my background was perfect for QFTASAP, as quantum field theory (QFT) tries to merge special (not general) relativity and quantum mechanics.

Do not despair if you have neither background as Dr. Zee starts the book explaining both in the first 99 pages or so. His style is very informal, with jokes, historical asides, and blinding clarity. As a retired MD I can’t speak to how accurate any of it is, but the publisher notes that his textbooks have been used at MIT and Cal Tech, which is good enough for me.

So quantum mechanics and special relativity gets you to base camp for the intellectual ascent to QFT — which takes the rest of the book (to page 342). If this sounds daunting, remember thAt physics majors and physics grad student’s QFT courses last a year (according to Dr. Zee). So read a little bit at a time.

The big leap (for me) was essentially abandoning the idea of force and thinking about action (which was totally unfamiliar). Much of the math, as a result is rather unfamiliar, and even if you took calculus, the integrals you will meet look like nothing you’ve ever seen.
e.g. d^4x e psibar (x) gamma^mu psi(x) A_mu(x) all under the integral sign
Fortunately, on p. 154 Dr. Zee says “I have to pause to teach you how to read this hieroglyphic.” This is very typical of his informal and friendly teaching style.

QFTASAP contains all sorts of gems which deepened my understanding of stuff I’d studied before. For example, Dr. Zee shows how special relativity demolishes the notion of simultaneity, then he goes even farther and explains how this implies the existence of antiparticles. Once you get integrals like the above under your belt, he gives a coherent explanation of where and how the idea of the expanding universe comes from and how it looks mathematically.

There is much, much more: gauge theory, Yang Mills, the standard model of particle physics etc. etc.

To a Princetonian, some of the asides are fascinating. One in particular tells you why you or your kid should want to go there (spoiler alert — not to meet the scion of a wealthy family, or an heiress, not to form connections which will help you in your career). He mentions that there was an evening seminar for physics majors given by a young faculty member (33) about recent discoveries in physics. In 1964 the same young prof (James Cronin) said that he had discovered something exiting — in 1980 he got the Nobel for it. For my part, it was John Wheeler (he of the black hole, wormholes etc) teaching premeds and engineers (not future physicists) freshman physics and bringing in Neils Bohr to talk to us. So go to Princeton for the incredible education you will get, and the way Princeton exposes their undergraduates to their very best faculty.

Addendum #2 — As a Princeton chemistry major, my undergraduate adviser was Paul  Schleyer , Princeton ’52, Harvard PhD ’56. We spent a lot of time together in his lab, and would sometimes go out for pizza after finishing up in the lab of an evening. For what working with him was like please see — https://luysii.wordpress.com/2014/12/15/paul-schleyer-1930-2014-a-remembrance/ and https://luysii.wordpress.com/2014/12/14/paul-schleyer-1930-2014-r-i-p/

Contrast this with Harvard where I did chemistry graduate work from ’60 – ’62.   None of the 7 people who were in the department back then who later won the Nobel prize later (Woodward, Corey, Hoffmann, +4 more) did any undergraduate teaching.  I did most of the personal teaching the Harvard undergrads got — 6 hours a week as a teaching assistant in the organic chemistry lab.  I may have been good, but I was nowhere as good as I would be if I stayed in the field for 8 more years.   I thought the Harvard students were basically cheated. 

I guess every review should have a quibble, and I do; but it’s with the publisher, not Dr. Zee. The whole book is one mass of related concepts and is filled with forward and backward references to text, figures and diagrams. Having a page to go to instead of Chapter III, 1 or figure IV.3.2 would make reading much easier. Only the publisher could do this once the entire text has been laid out.

One further point. QFTASAP clarified for me the differences between the (substantial) difficulties of medicine and the (substantial) difficulties of theoretical physics. When learning medicine you are exposed to thousands of unrelated (because we don’t understand what lies behind them) facts. That’s OK because you don’t need to remember all of them. Ask the smartest internist you know to name the 12 cranial nerves or the 8 bones of the wrist. The facts of theoretical physics are far fewer, but you must remember, internalize and use them — that’s why QFTASAP contains all these forward and backward references.

There is a ton more to say about the book and I plan to write more as I go through the book again. If interested, just Google Chemiotics II now and then. QFTASAP is definitely worth reading more than once.

 

Helpful

Orwell does quantum mechanics

I can find no evidence that George Orwell knew or cared about quantum mechanics when he wrote 1984 in 1948.

He invented the term doublethink to describe the thought control of the totalitarian society in the novel.

“Doublethink means the power of holding two contradictory beliefs in one’s mind simultaneously, and accepting both of them.”

For chemists to think about the quantum mechanics of the atom, doublethink is necessary as this old post will show.

Here are a few examples from the Novel

War is Peace

Love is Hate

Ignorance is Strength

Freedom is Slavery

Here are a few doublethinks from Quantum Mechanics as applied to the atoms loved by chemists everywhere

Atomic orbitals in atoms have angular momentum as do curved trajectories. Trajectories don’t exist in atoms (or anywhere in the quantum world)

Relativistic corrections must be made for rapidly moving electrons in heavy elements close to the nucleus.  They may move but they don’t have trajectories.

The two s atomic orbital has a node where electrons are never found.

Electrons are found on both sides of the node.  How do they get there?

A moving charge creates a magnetic field, and a magnetic field affects a moving charge, so placing a moving charge in a magnetic field should alter its energy. This accounts for the Zeeman effect (the splitting of spectral lines in a magnetic field). Trajectories help you understand this (even if they can’t really exist in the confines of the atom).

Here is an old post on the subject.

Be warned, it’s technical

Doublethink and angular momentum — why chemists must be adept at it

Chemists really should know lots and lots about angular momentum which is intimately involved in 3 of the 4 quantum numbers needed to describe atomic electronic structure. Despite this, I never really understood what was going until taking the QM course, and digging into chapters 10 and 11 of Giancoli’s physics book (pp. 248 -310 4th Edition).

Quick, what is the angular momemtum of a single particle (say a planet) moving around a central object (say the sun)? Well, its magnitude is the current speed of the particle times its mass, but what is its direction? There must be a direction since angular momentum is a vector. The (unintuitive to me) answer is that the angular momemtum vector points upward (resp. downward) from the plane of motion of the planet around the center of mass of the sun planet system, if the planet is moving counterclockwise (resp. clockwise) according to the right hand rule. On the other hand, the momentum of a particle moving in a straight line is just its mass times its velocity vector (e.g. in the same direction).

Why the difference? This unintuitive answer makes sense if, instead of a single point mass, you consider the rotation of a solid (e.g. rigid) object around an axis. All the velocity vectors of the object at a given time either point in different directions, or if they point in the same direction have different magnitudes. Since the object is solid, points farther away from the axis are moving faster. The only sensible thing to do is point the angular momentum vector along the axis of rotation (it’s the only thing which has a constant direction).

Mathematically, this is fairly simple to do (but only in 3 dimensions). The vector from the axis of rotation to the planet (call it r), and the vector of instantaneous linear velocity of the planet (call it v) do not point in the same direction, so they define a plane (if they do point in the same direction the planet is either hurtling into the sun or speeding directly away, hence not rotating). In 3 dimensions, there is a unique direction at 90 degrees to the plane. The vector cross product of r and v gives a vector pointing in this direction (to get a unique vector, you must use the right or the left hand rule). Nicely, the larger r and v, the larger the angular momentum vector (which makes sense). In more than 3 dimensions there isn’t a unique direction away from a plane, which is why the cross product doesn’t work there (although there are mathematical analogies to it).

This also explains why I never understood momentum (angular or otherwise) till now. It’s very easy to conflate linear momentum with force and I did. Get hit by a speeding bullet and you feel a force in the same direction as the bullet — actually the force you feel is what you’ve done to the bullet to change its momentum (force is basically defined as anything that changes momentum).

So the angular momentum of an object is never in the direction of its instantaneous linear velocity. But why should chemists care about angular momentum? Solid state physicists, particle physicists etc. etc. get along just fine without it pretty much, although quantum mechanics is just as crucial for them. The answer is simply because the electrons in a stable atom hang around the nucleus and do not wander off to infinity. This means that their trajectories must continually bend around the nucleus, giving each trajectory an angular momentum.

Did I say trajectory? This is where the doublethink comes in. Trajectory is a notion of the classical world we experience. Consider any atomic orbital containing a node (e.g. everything but a 1 s orbital). Zeno would have had a field day with them. Nodes are surfaces in space where the electron is never to be found. They separate the various lobes of the orbital from each other. How does the electron get from one lobe to the other by a trajectory? We do know that the electron is in all the lobes because a series of measurements will find the electron in each lobe of the orbital (but only in one lobe per measurement). The electron can’t make the trip, because there is no trip possible. Goodbye to the classical notion of trajectory, and with it the classical notion of angular momentum.

But the classical notions of trajectory and angular momentum still help you think about what’s going on (assuming anything IS in fact going on down there between measurements). We know quite a lot about angular momentum in atoms. Why? Because the angular momentum operators of QM commute with the Hamiltonian operator of QM, meaning that they have a common set of eigenfunctions, hence a common set of eigenvalues (e.g. energies). We can measure these energies (really the differences between them — that’s what a spectrum really is) and quantum mechanics predicts this better than anything else.

Tomorrow:  Orwell does Stanford

Yesterday: Orwell does China  — https://luysii.wordpress.com/2022/12/26/orwell-does-china/

Why there’s more to chemistry than quantum mechanics

As juniors entering the Princeton Chemistry Department as majors in 1958 we were told to read “The Logic Of Modern Physics” by P. W. Bridgeman — https://en.wikipedia.org/wiki/The_Logic_of_Modern_Physics.   I don’t remember whether we ever got together to discuss the book with faculty, but I do remember that I found the book intensely irritating.  It was written in 1927, in early hay day of quantum mechanics.  It  said that all you could know was measurements (numbers on a dial if you wish) without any understanding of what went on in between them.

I thought chemists knew a lot more than that.  Here’s Henry Eyring — https://en.wikipedia.org/wiki/Henry_Eyring_(chemist)https://en.wikipedi developing transition state theory a few years later in 1935 in the department.  It was pure ideation based on thermodynamics, which was developed long before quantum mechanics and is still pretty much a quantum mechanics free zone of physics (although people are busy at work on the interface).

Henry would have loved a recent paper [ Proc. Natl. Acad. Sci. vol. 118 e2102006118 ’21 ] where the passage of a molecule back and forth across the free energy maximum was measured again and again.

A polyNucleotide hairpin of DNA  was connected to double stranded DNA handles in optical traps where it could fluctuate between folded (hairpin) and unfolded (no hairpin) states.  They could measure just how far apart the handles were and in the hairpin state the length appears to be 100 Angstroms (10 nanoMeters) shorter than the unfolded state.

So they could follow the length vs. time and measure the 50 microSeconds or so it took to make the journey across the free energy maximum (e.g. the transition state). A mere 323,495 different transition paths were studied.  You can find much more about the work here — https://luysii.wordpress.com/2022/02/15/transition-state-theory/

Does Bridgeman have the last laugh — remember all that is being measured are numbers (lengths) on a dial.

Here’s another recent paper Eyring would have loved — [ Proc. Natl. Acad. Sci. vol. 119 e2112372118 ’22  — ] https://www.pnas.org/doi/epdf/10.1073/pnas.2112382119  ]

The paper studied Barnase, a 110 amino acid protein which degrades RNA (so much like the original protein Anfinsen studied years ago).  Barnase is highly soluble and very stable making it one of the E. Coli’s of protein folding studies.

The new wrinkle of the paper is that they were able to study the folding and unfolding and the transition state of single molecules of Barnase at different temperatures (an experiment which would have been unlikely for Eyring to even think about doing in 1935 when he developed transition state theory, and yet this is exactly the sort of thing what he was thinking about but not about proteins whose structure was unknown back then).

This allowed them to determine not just the change in free energy (deltaG)  between the unfolded (U) and the transition state (TS) and the native state (N) of Barnase, but also the changes in enthalpy (delta H) and entropy (delta S) between U and TS and between N and TS.

Remember delta G = Delta H – T delta S.  A process will occur if deltaG is negative, which is why an increase in entropy is favorable, and why the decrease in entropy between U and TS is unfavorable.   You can find out more about this work here — https://luysii.wordpress.com/2022/03/25/new-light-on-protein-folding/

So the purely mental ideas of Eyring are being confirmed once again (but by numbers on a dial).  I doubt that Eyring would have thought such an experiment possible back in 1935.

Chemists know so much more than quantum mechanics says we can know.  But much of what we do know would be impossible without quantum mechanics.

However, Eyring certainly wasn’t averse to quantum mechanics, having written a text book Quantum Chemistry with Walter and Kimball on the very subject in 1944.

Fiber bundles at last

As an  undergraduate, I loved looking at math books in the U-store.  They had a wall of them back then, now it’s mostly swag.  The title of one book by a local prof threw me — The Topology of Fiber Bundles.

Decades later I found that to understand serious physics you had to understand fiber bundles.

It was easy enough to memorize the definition, but I had no concept what they really were until I got to page 387 of Roger Penrose’s marvelous book “The Road to Reality”.  It’s certainly not a book to learn physics from for the first time.  But if you have some background (say just from reading physics popularizations), it will make things much clearer, and will (usually) give you a different an deeper perspective on it.

Consider a long picket fence.  Each fencepost is just like every other, but different, because each has its own place.  The pickets are the fibers and the line in the ground on which they sit is something called the base space.

What does that have to do with our 3 dimensional world and its time?

Everything.

So you’re sitting at your computer looking at this post.  Nothing changes position as you do so.  The space between you and the screen  is the same.

But the 3 dimensional space you’re sitting in is different at every moment, just as the pickets are different at every position on the fence line.

Why?  Because you’re siting on earth.  The earth is rotating, the solar system is rotating about the galactic center, which is itself moving toward the center of the local galactic cluster.

Penrose shows that this is exactly the type of space implied by Galilean relativity. (Yes Galileo conceived of relativity long before Einstein).   Best to let him speak for himself.   It’s a long quote but worth reading.

“Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship’s motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.”

I’d read this many times, but Penrose’s discussion draws out what Galileo is implying. “Clearly we should take Galileo seriously.  There is no meaning to be attached to notion that any particular point in space a minute from now is to be judged as the same point in space that I have chosen. In Galilean dynamics we do not have just one Euclidean 3-space as an arena for the actions of the physical world evolving with time, we have a different E^3 for each moment in time, with no natural identification between these various E^3 ‘s.”

Although it was obvious to us that the points of our space retain their identity from one moment to the next, they don’t.

Penrose’s book is full of wonderful stuff like this.  However, all is not perfect.  Physics Nobelist Frank Wilczek in his review of the book [ Science vol. 307 pp. 852 – 853 notes that “The worst parts of the book are the chapters on high energy physics and quantum field theory, which in spite of their brevity contain several serious blunders.”

However, all the math is fine, and Wilczek says “the discussions of the conformal geometry of special relativity and of spinors are real gems.”

Since he doesn’t even get to quantum mechanics until p. 493 (of 1049) there is a lot to chew on (without worrying about anything other than the capability of your intellect).

 

More homework assignments

Homework assignment #1:  design a sequence of 10 amino acids which binds to the same sequence in the reverse order forming a plane 4.8 Angstroms thick.

Homework assignment #2 design a sequence of 60 amino acids which forms a similar plane 4.8 Angstroms thick, such that two 60 amino acid monomers bind to each other.

Feel free to use any computational or theoretical devices currently at our disposal, density functional theory, force fields, rosetta etc. etc.

Answers to follow shortly

Hint:  hundreds to thousands of planes can stack on top of each other.

Also I’ve written about phase changes in the past — https://luysii.wordpress.com/2020/12/20/neuroscience-can-no-longer-ignore-phase-separation/

A superb review of the subject is available if you have a subscription to Neuron [ Neuron vol. 109 pp. 2663 – 2681 ’21 ]

Mathematics and the periodic table

It isn’t surprising that math is involved in the periodic table. Decades before the existence of atoms was shown for sure (Einstein in 1905 on Brownian motion — https://physicsworld.com/a/einsteins-random-walk/) Mendeleev arranged the known elements in a table according to their chemical properties. Math is great at studying and describing structure, and the periodic table is full of it. 

What is surprising, is how periodic table structure arises from math that ostensibly has absolutely nothing to do with chemistry.  Here are 3 examples.

The first occurred exactly 60 years ago to the month in grad school.  The instructor was taking a class of budding chemists through the solution of the Schrodinger equation for the hydrogen atom. 

Recursion relations are no stranger to the differential equations course, where you learn to (tediously) find them for a polynomial series solution for the differential equation at hand. I never really understood them, but I could use them (like far too much math that I took back then).

So it wasn’t a shock when the QM instructor back then got to them in the course of solving the hydrogen atom (with it’s radially symmetric potential). First the equation had to be expressed in spherical coordinates (r, theta and phi) which made the Laplacian look rather fierce. Then the equation was split into 3, each involving one of r, theta or phi. The easiest to solve was the one involving phi which involved only a complex exponential. But periodic nature of the solution made the magnetic quantum number fall out. Pretty good, but nothing earthshaking.

Recursion relations made their appearance with the solution of the radial and the theta equations. So it was plug and chug time with series solutions and recursion relations so things wouldn’t blow up (or as Dr. Gouterman put it, the electron has to be somewhere, so the wavefunction must be zero at infinity). MEGO (My Eyes Glazed Over) until all of a sudden there were the main quantum number (n) and the azimuthal quantum number (l) coming directly out of the recursions.

When I first realized what was going on, it really hit me. I can still see the room and the people in it (just as people can remember exactly where they were and what they were doing when they heard about 9/11 or (for the oldsters among you) when Kennedy was shot — I was cutting a physiology class in med school). The realization that what I had considered mathematical diddle, in some way was giving us the quantum numbers and the periodic table, and the shape of orbitals, was a glimpse of incredible and unseen power. For me it was like seeing the face of God.

The second and third examples occurred this year as I was going through Tony Zee’s book “Group Theory in a Nutshell for Physicists”

The second example occurs with the rotation group in 3 dimensions, which is a 3 x 3 invertible matrix, such that multiplying it by its transpose gives the identity, and such that is determinant is +1.  It is called SO(3)

Then he tensors 2 rotation matrices together to get a 9 x 9 matrix.  Zee than looks for the irreducible matrices of which it is composed and finds that there is a 3×3, a 1×1 and a 5×5.  The 5×5 matrix is both traceless and symmetric.  Note that 5 = 2(2) + 1.  If you tensor 3 of them together you get (among other things 3(2) + 1)   = 7;   a 7 x 7 matrix.

If you’re a chemist this is beginning to look like the famous 2 L + 1 formula for the number of the number of magnetic quantum numbers given an orbital quantum number of L.   The application of a magnetic field to an atom causes the orbital momentum L to split in 2L + 1 magnetic eigenvalues.    And you get this from the dimension of a particular irreducible representation from a group.  Incredible.  How did abstract math know this.  

The third example also occurs a bit farther along in Zee’s book, starting with the basis vectors (Jx, Jy, Jz) of the Lie algebra of the rotation group SO(3).   These are then combined to form J+ and J-, which raise and lower the eigenvalues of Jz.  A fairly long way from chemistry you might think.  

All state vectors in quantum mechanics have absolute value +1 in Hilbert space, this means the eigenvectors must be normalized to one using complex constants.  Simply by assuming that the number of eigenvalues is finite, there must be a highest one (call it j) . This leads to a recursion relation for the normalization constants, and you wind up with the fact that they are all complex integers.  You get the simple equation s = 2j where s is a positive integer.  The 2j + 1 formula arises again, but that isn’t what is so marvelous. 

j doesn’t have to be an integer.  It could be 1/2, purely by the math.  The 1/2 gives 2 (1/2) + 1 e.g two numbers.  These turn out to be the spin quantum numbers for the electron.  Something completely out of left field, and yet purely mathematical in origin. It wasn’t introduced until 1924 by Pauli — long after the math had been worked out.  

Incredible.  

Tensors — again, again, again

“A tensor is something that transforms like a tensor” — and a duck is something that quacks like a duck. If you find this sort of thing less than illuminating, I’ve got the book for you — “An Introduction to Tensors and Group Theory for Physicists” by Nadir Jeevanjee.

He notes that many physics books trying to teach tensors start this way, without telling you what a tensor actually is.  

Not so Jeevanjee — right on the first page of text (p. 3) he says “a tensor is a function which eats a certain number of vectors (known as the rank r of the tensor) and produces a number.  He doesn’t say what that number is, but later we are told that it is either C or R.

Then comes  the crucial fact that tensors are multilinear functions. From that all else flows (and quickly).

This means that you know everything you need to know about a tensor if you know what it does to its basis vectors.  

He could be a little faster about what these basis vectors actually are, but on p. 7 you are given an example explicitly showing them.

To keep things (relatively) simple the vector space is good old 3 dimensional space with basis vectors x, y and z.

His rank 2 tensor takes two vectors from this space (u and v) and produces a number.  There are 9 basis vectors not 6 as you might think — x®x, x®y, x®z, y®x, y®y, y®z, z®x, z®y, and z®z.    ® should be read as x inside a circle

Tensor components are the (real) numbers the tensor assigns to the 9 — these are written T(x®x) , T(x®y) T( x®z), T(y®x), T(y®y), T(y®z), T(z®x), T(z®y), and T(z®z)– note that there is no reason that T(x®y) should equal T(y®x) any more than a function R^2 –> R should give the same values for (1, 2) and (2, 1).

One more complication — where do the components of u and v fit in?  u is really (u^1, u^2, u^3) and v is really (v^1, v^2, v^3)

They multiply each other and the T’s  — so the first term of the tensor (sometimes confusingly called a tensor component)

is u^1 * v^1 * T(x®x)  and the last is u^3 * v^3 T(z®z).  Then the 9 tensor terms/components are summed giving a number. 

Then on pp. 7 and 8 he shows how a change of basis matrix (a 3 x 3 matrix written A^rs where rs, is one of 1, 2, 3) with nonZero determinant) gives the (usually incomprehensible) formula 

 

T^i’j’ = A^ik * A^jl T * (k, l)  where i, j, k, l are one of x, y, and z (or 1, 2, 3 as usually written)

 

So now you have a handle on the cryptic algebraic expression for tensors and what happens to them on a change of basis (e.g. how they transform).  Not bad for 5 pages of work — certainly not everything, but enough to make you comfortable with what follows — dual vectors, invariance, symmetric etc. etc.

 

Just knowing the multilinearity of tensors and just 2 postulates of quantum mechanics is all you need to understand entanglement — yes truly.  Yes, and you don’t need the Schrodinger equation, or differential equations at all, just linear algebra. 

 

Here is an old post to show you exactly how this works

 

How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement

Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.

If you’re a bit foggy on just what entanglement is — have a look at https://luysii.wordpress.com/2010/12/13/bells-inequality-entanglement-and-the-demise-of-local-reality-i/. Even better; read the book by Zeilinger referred to in the link (if you have the time).

Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.

Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)

Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ but there are several more.

Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at https://luysii.wordpress.com/2014/12/08/tensors/

Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.

So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.

Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.

Multilinearity of the tensor product is what you’d think

(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)

= v ® w + v ® w’ + v’ ® w + v’ ® w’

You get all 4 tensor products in this case.

This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)

Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”

http://planetmath.org/simpletensor

Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.

What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).

So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.

v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.

Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.

But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.

Hydrogen bonding — again, again

I’ve been thinking about hydrogen bonding ever since my senior thesis in 1959. Although its’ role in the protein alpha helix had been known since ’51 and in the DNA double helix since ’53, little did we realize at the time just how important it would be for the workings of the cell. So I was lucky Dr. Schleyer put me at an IR spectrometer and had me make a bunch of compounds, to look for hydrogen bonding of OH, NH and SH to the pi electrons of the benzene ring. I had to make a few of them, which involved getting a (CH2)n chain between the benzene ring and the hydrogen donor. Just imagine the benzene as the body of a scorpion and the (CH2) groups as the length of the tail.  The SH compounds were particularly nasty, and people would look at their shoes when I’d walk into the eating club. Naturally the college yearbook screwed things up and titled my thesis “Studies in Hydrogen Bombing”, to which my parents’ friends would say — he looks like such a nice young man, why was he doing that?

At any rate I’m going to talk about a recent paper [ Science vol. 371 pp. 160 – 164 ’21 ] on the nature of the bond in the F H F – anion.  It’s going to be pretty hard core stuff with relatively little explanatory material. You’ve either been previously exposed to this stuff or you haven’t.  So this post is for the cognoscenti.  Hold on, it’s going to be wild ride.

In conventional hydrogen bonds, the donor (D) atom is separated from the Acceptor atom (A) by 2.7 Angstroms or more, and the hydrogen nucleus is found closer to A where the potential energy minimum is found.

So it looks like this D – H . .. A

The D-H bond isn’t normal, but is stretched  and weakened.  This means that it takes less energy to stretch it meaning that it absorbs infrared radiation at a lower frequency (higher wavelength) — red shift if you will. 

Such is what we were looking for and we found it comparing 

Benzene (CH2)n OH vibrations to butanol, pentanol, hexanol, etc etc. cyclohexane (CH2)n OH.

As the D – A distance shrinks there is ultimately a flat bottomed single well potential, where H becomes a confined particle (but still delocalized) betwen D and A.

The vibrations of protons in hydrogen bonds deviate markedly from the classic quantum harmonic oscillator beloved by physicists.  Here the energy levels on solving the classic H psi = E psi equation of quantum mechanics are evenly spaced (see Lancaster & Blundell “Quantum Field Theory” p. 20.)

However in real molecules, as you ascend the vibrational ladder, conventional hydrogen bonds show a decrease in the difference between energy levels (positive anharmonicity).  By contrast, when proton confinement dictates the potential shape in short hydrogen bonds (when D and A are close together, mimicking the particle in a box model in quantum mechanics) the spacing between states increases (negative anharmonicity).

The present work shows that in FHF- the proton motion is superharmonic — https://en.wikipedia.org/wiki/Subharmonic_function — which they don’t describe very well. 

When the F F distance gets below 2.4 Angstroms, covalent bonding starts to become a notable contributor to the short hydrogen bond, and the authors actually have evidence that there is overlap in FHF- between the 3s orbital of H and the 2 Pz orbitals of the donor and the acceptor atoms, yielding a stabilization of the resulting molecular orbital. 

Is that cool or what.  The bond sits right on the borderland between a covalent bond and a hydrogen bond, taking on aspects of both. 

 

The Representation of group G on vector space V is really a left action of the group on the vector space

Say what? What does this have to do with quantum mechanics? Quite a bit. Practically everything in fact. Most chemists learn quantum mechanics because they want to see where atomic orbitals come from. So they stagger through the solution of the Schrodinger equation where the quantum numbers appear as solution of recursion equations for power series solutions of the Schrodinger equation.

Forget the Schrodinger equation (for now), quantum mechanics is really written in the language of linear algebra. Feynman warned us not to consider ‘how it can be like that’, but at least you can understand the ‘that’ — e.g. linear algebra. In fact, the instructor in a graduate course in abstract algebra I audited opened the linear algebra section with the remark that the only functions mathematicians really understand are the linear ones.

The definitions used (vector space, inner product, matrix multiplication, Hermitian operator) are obscure and strange. You can memorize them and mumble them as incantations when needed, or you can understand why they are the way they are and where they come from. So if you are a bit rusty on your linear algebra I’ve written a series of 9 posts on the subject — here’s a link to the first https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/– just follow the links after that.

Just to whet your appetite, all of quantum mechanics consists of manipulation of a particular vector space called Hilbert space. Yes all of it.

Representations are a combination of abstract algebra and linear algebra, and are crucial in elementary particle physics. In fact elementary particles are representations of abstract symmetry groups.

So in what follows, I’ll assume you know what vector spaces, linear transformations of them, their matrix representation. I’m not going to explain what a group is, but it isn’t terribly complicated. So if you don’t know about them quit. The Wiki article is too detailed for what you need to know.

The title of the post really threw me, and understanding requires significant unpacking of the definitions, but you need to know this if you want to proceed further in physics.

So we’ll start with a Group G, its operation * and e, its identity element.

Next we have a set called X — just that a bunch of elements (called x, y, . . .), with no further structure imposed — you can’t add elements, you can’t mutiply them by real numbers. If you could with a few more details you’d have a vector space (see the survival guide)

Definition of Left Action (LA) of G on set X

LA : G x X –> X

LA : ( g, x ) |–> (g . x)

Such that the following two properties hold

l. For all x in X LA : (e, x) |–> (e.x) = x

2. For all g1 and g2 in G LA ( g1 * g2), x ) |–> ( g1 . (g2 . x )

Given vector space V define GL(V) the set of invertible linear transformations (LTs) of vector space. GL(V) becomes a group if you let composition of linear transformations become its operation (it’s all in the survival guide.

Now for the definition of representation of Group G on vector space V

It is a function

rho: G –> GL(V)

rho: g |–> LTg : V –> V linear ; LTg == Linear Transformation labeled by group element g

The representation rho defines a left group action on V

LA : (g, v) |–> LTg (V) — this satisfies the two properties above of a left action given above — think about it.

Now you’re ready for some serious study of quantum mechanics. When you read that the representation is acting on some vector space, you’ll know what they are talking about.

Math can be hard even for very smart people

50 McCosh Hall an autumn evening in 1956. The place was packed. Chen Ning Yang was speaking about parity violation. Most of the people there had little idea (including me) of what he did, but wanted to be eyewitnesses to history.. But we knew that what he did was important and likely to win him the Nobel (which happened the following year).

That’s not why Yang is remembered today (even though he’s apparently still alive at 98). Before that he and Robert Mills were trying to generalize Maxwell’s equations of electromagnetism so they would work in quantum mechanics and particle physics. Eventually this led Yang and Mills to develop the theory of nonAbelian gauge fields which pervade physics today.

Yang and James Simons (later the founder of Renaissance technologies and already a world class mathematician — Chern Simons theory) later wound up at Stony Brook. Simons, told him that gauge theory must be related to connections on fiber bundles and pointed him to Steenrod’s The Topology of Fibre Bundles. So he tried to read it and “learned nothing. The language of modern mathematics is too cold and abstract for a physicist.”

Another Yang quote “There are only two kinds of math books: Those you cannot read beyond the first sentence, and those you cannot read beyond the first page.”

So here we have a brilliant man who invented significant mathematics (gauge theory) along with Mills, unable to understand a math book written about the exact same subject (connections on fiber bundles).