Category Archives: Math

The Chinese Room Argument, Understanding Math and the imposter syndrome

The Chinese Room Argument

 was first published in a 1980 article by American philosopher John Searle. He imagines himself alone in a room following a computer program for responding to Chinese characters slipped under the door. Searle understands nothing of Chinese, and yet, by following the program for manipulating symbols and numerals just as a computer does, he sends appropriate strings of Chinese characters back out under the door, and this leads those outside to mistakenly suppose there is a Chinese speaker in the room.

 

So it was with me and math as an undergraduate due to a history dating back to age 10.  I hit college being very good at manipulating symbols whose meaning I was never given to understand.  I grew up 45 miles from the nearest synagogue.  My fanatically religious grandfather thought it was better not to attend services at all than to drive up there on the Sabbath.  My father was a young lawyer building a practice, and couldn’t close his office on Friday.   So my he taught me how to read Hebrew letters and reproduce how they sound, so I could read from the Torah at my Bar Mitzvah (which I did comprehending nothing).  Since I’m musical, learning the cantillations under the letters wasn’t a problem.

 

I’ve always loved math and solving problems of the plug and chug variety was no problem.  I’d become adept years earlier at this type of thing thanks to my religiously rigid grandfather.   It was the imposter syndrome writ large.  I’ve never felt like this about organic chemistry and it made a good deal of intuitive sense the first time I ran into it.  For why have a look at — https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/

 

If there is anything in math full of arcane symbols calling for lots of mechanical manipulation, it is the differential geometry and tensors needed to understand General relativity.   So I’ve plowed through a lot of it, but still don’t see what’s really going on.

 

Enter Tristan Needham’s book “Visual Differential Geometry and Forms”.  I’ve written about it several times
and Here — https://luysii.wordpress.com/2022/03/07/visual-differential-geometry-and-forms-q-take-3/

 

If you’ve studied any math, his approach will take getting used to as it’s purely visual and very UNalgebraic.  But what is curvature but a geometric concept.

 

So at present I’m about 80 pages away from completing Needham’s discussion of general relativity.  I now have an intuitive understanding of curvature, torsion, holonomy, geodesics and the Gauss map that I never had before.   It is very slow going, but very clear.  Hopefully I’ll make it to p. 333.  Wish me luck.

Visual Differential Geometry and Forms — Take 3

Visual Differential Geometry and Forms is a terrific math book about which I’ve written

here — https://luysii.wordpress.com/2021/07/12/a-premature-book-review-and-a-60-year-history-with-complex-variables-in-4-acts/

and

here — https://luysii.wordpress.com/2021/12/04/a-book-worth-buying-for-the-preface-alone-or-how-to-review-a-book-you-havent-read/

but first some neurology.

In the old days we neurologists figured out what the brain was doing by studying what was lost when parts of the brain were destroyed (usually by strokes, but sometimes by tumors or trauma).  Not terribly logical, as pulling the plug on a lamp plunges you in darkness, but the plug has nothing to do with how the lightbulb or LED produces light.  It was clear that the occipital lobe was important — destroy it on both sides and you are blind — https://en.wikipedia.org/wiki/Occipital_lobe but it’s only 10% of the gray matter of the cerebral cortex.

The information flowing into your brain from your eyes is enormous.  The optic nerve connecting the eyeball to the brain has a million fibers, and they can fire ‘up to 500 times a second.  If each firing (nerve impulse) is a bit, then that’s an information flow into your brain of a gigaBit/second.   This information is highly processed by the neurons and receptors in the 10 layers of the retina.  There are many different cell types — cells responding to color, to movement in one direction, to a light stimulus turning on, to a light stimulus turning off, etc. etc.   Over 30 cell types have been described, each responding to a different aspect of the visual stimulus.

So how does the relatively small occipital lobe deal with this? It doesn’t.At least half of your the brain responds to visual stimuli.  How do we know?   It’s complicated, but something called functional Magnetic Resonance Imaging (fMRI) picks up increased neuronal activity (primarily by the increase in blood flow it causes).

Given that half your brain is processing what you see, it makes sense to use it to ‘see’ what’s going on in Mathematics.  This is where Tristan Needham’s books come in.

If you’ve studied math at the college level with some calculus you shouldn’t have much trouble.  But you definitely need to look at Euclid as it’s used heavily throughout. Much use is made of similar triangles to derive relationships.

I’ll assume you’ve read the first two posts mentioned above.  Needham’s description of curvature and torsion of curves in 3 dimensional space is terrific.  They play a huge role in relativity, and I was able to mouth the formulas for them but they remained incomprehensible to me, as they are just symbols on the page.  Hopefully the discussion further on in the book will let me ‘see’ what they are when it comes to tensors.

He does skip about a bit passing Euclid when he tell how people living t on a 2 dimensional surface could tell it wasn’t flat (p. 19).  It involves numerically measuring the circumference of a circle.  But this involves a metric and putting numbers on lines, something Euclid never did.

Things really get interesting when he started talking about how Newton found the center of curvature of an arbitrary curve.  Typically Needham doesn’t really define curve, something obvious to a geometer, but it’s clear the curve is continuous.  Later he lets it slip that the curve is differentiable (without saying so).

So what did Newton do?   Start with a point p and find its normal line.  Then find point q near p on the curve and find its normal line and see where they intersect.  The center of curvature at p is the point of intersection of the normals as the points get closer and closer to p.

This made wonder how Newton could find the normal to an arbitrary continuous curve.  It would be easy if he knew the tangents, because Euclid very early on (Proposition 11 Book 1) tells you how to construct the perpendicular to a straight line.  It is easy for Euclid to find the tangent to a circle at point p — it’s just the perpendicular to the line formed between the center of the circle (where you put one point of the compass used by Newton) and the circle itself (the other point of the compass.

But how does Newton find the tangent to an arbitrary continuous curve?  I couldn’t find any place that he did it, but calc. 101 says that you just find the limit of secants ending at p as the other point gets closer and closer.  Clearly this is a derivative of sorts.

Finally Needham tells you that his curves in 3 dimensions are differentiable in the following oblique way.  On p. 106 he says that “each infinitesimal segment (of a curve) nevertheless lies in a plane.”  This tells you that the curve has a tangent, and a normal to it at 90 degrees (but not necessarily in the plane).  So it must be differentiable (oblique no?).   On p. 107 he differentiates the tangent in the infinitesimal plane to get the principal normal (which DOES  lie in the plane).  Shades of Bishop Berkeley (form Berkeley California is named) — differentiating the ghost of a departed object.

Addendum: 28 March ’22:  It’s was impossible for me to find a definition of surface even reading the first 164 pages.  Needham highly recommends a book I own “Elementary Differential Geometry (revised 2nd edition) by Barrett O’Neill calling it “the single most clear-eyed elegant, and (ironically) modern treatment of the subject  . . .  at the undergraduate level”.  The first edition was 1966.   In the preface to his book O’Neill says “One weakness of classical differential geometry is its lack of any adequate definition of surface”.    No wonder I had trouble.

So it’s great fun going through the book, get to “Einstein’s Curved Spacetime” Chapter 30 p. 307, “The Einstein FIeld Equation (with Matter) in Geometrical Form” complete with picture p. 326.

More Later.

A book worth buying for the preface alone (or how to review a book you haven’t read)’

Everyone in grade school writes a book report on something they haven’t completely read (or read at all). Well, I’m long past, but here’s a book worth buying for the preface alone — Visual Differential Geometry and Forms by Tristan Needham.

His earlier book Visual Complex Analysis (another book I haven’t read completely) is a masterpiece (those parts I’ve managed to read) with all sorts of complex formulas and algebraicism explained visually.  It’s best to be patient, as Needham doesn’t even get to complex differentiation until Chapter 4 (p. 188) The Amplitwist concept.

On page xviii of the preface, Needham describes the third type of calculus Newton invented, and the one he used in the great Principia Mathematica of 1687.

The first one of 1665 was basically the manipulation of power series (which I’ve never heard of.  Have you?)

The second was one we all study in school, dx dy and all that.

Newton called the third method of calculus “the synthetic method of fluxions.”

Bishop Berkeley had a field day with them.  “And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”

Laugh not at the good Bishop;   Berkeley California is named for him (as long as the woke don’t find out).

Needham gives a completely nontrivial example of the method.  He shows geometrically how the derivative of tan(theta) is 1 + [ tan(theta)]^2,.

The diagram will not be reproduced here but suffice it to say you’ll have to remember a lot of your geometry, e.g. two lines separated by a very small angle are essentially parallel, the radius of a circle is perpendicular to the tangent, what a similar triangle is.

The best thing about it is you can actually visualize the limit process taking place as a triangle shrinks and shrinks and eventually becomes ‘almost’ similar to another.  No deltas and epsilons for Newton (or Needham).  There is some algebra involving the ratios of the sides of similar triangles, but it’s trivial.

Buy the book and enjoy.   You’ll never think of differentiation the same way.  Your life will be better, and you might even meet a better class of people.

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).

 

A premature book review and a 60 year history with complex variables in 4 acts

“Visual Differential Geometry and Forms” (VDGF) by Tristan Needham is an incredible book.  Here is a premature review having only been through the first 82 pages of 464 pages of text.

Here’s why.

While mathematicians may try to tie down the visual Gulliver with Lilliputian strands of logic, there is always far more information in visual stimuli than logic can appreciate.  There is no such a thing as a pure visual percept (a la Bertrand Russell), as visual processing begins within the 10 layers of the retina and continues on from there.  Remember: half your brain is involved in processing visual information.  Which is a long winded way of saying that Needham’s visual approach to curvature and other visual constructs is an excellent idea.
Needham loves complex variables and geometry and his book is full of pictures (probably on 50% of the pages).

My history with complex variables goes back over 60 years and occurs in 4 acts.

 

Act I:  Complex variable course as an undergraduate. Time late 50s.  Instructor Raymond Smullyan a man who, while in this world, was definitely not of it.  He really wasn’t a bad instructor but he appeared to be thinking about something else most of the time.

 

Act II: Complex variable course at Rocky Mountain College, Billings Montana.  Time early 80s.  The instructor and MIT PhD was excellent.  Unfortunately I can’t remember his name.  I took complex variables again, because I’d been knocked out for probably 30 minutes the previous year and wanted to see if I could still think about the hard stuff.

 

Act III: 1999 The publication of Needham’s first book — Visual Complex Analysis.  Absolutely unique at the time, full of pictures with a glowing recommendation from Roger Penrose, Needham’s PhD advisor.  I read parts of it, but really didn’t appreciate it.

 

Act IV 2021 the publication of Needham’s second book, and the subject of this partial review.  Just what I wanted after studying differential geometry with a view to really understanding general relativity, so I could read a classmate’s book on the subject.  Just like VCA, and I got through 82 pages or so, before I realized I should go back and go through the relevant parts (several hundred pages) of VCA again, which is where I am now.  Euclid is all you need for the geometry of VCA, but any extra math you know won’t hurt.

 

I can’t recommend both strongly enough, particularly if you’ve been studying differential geometry and physics.  There really is a reason for saying “I see it” when you understand something.

 

Both books are engagingly and informally written, and I can’t recommend them enough (well at least the first 82 pages of VDGF).

 

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.

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).

The pleasures of reading Feynman on Physics – V

Feynman finally gets around to discussing tensors 376 pages into volume II in “The Feynman Lectures on Physics” and a magnificent help it is (to me at least).  Tensors must be understood to have a prayer of following the math of General Relativity (a 10 year goal, since meeting classmate Jim Hartle who wrote a book “Gravity” on the subject).

There are so many ways to describe what a tensor is (particularly by mathematicians and physicists) that it isn’t obvious that they are talking about the same thing.   I’ve written many posts about tensors, as the best way to learn something it to try to explain it to someone else (a set of links to the posts will be found at the end).

So why is Feynman so helpful to me?  After plowing through 370 pages of Callahan’s excellent book we get to something called the ‘energy-momentum tensor’, aka the stress-energy tensor.  This floored me as it appeared to have little to do with gravity, talking about flows of energy and momentum. However it is only 5 pages away from the relativistic field equations so it must be understood.

Back in the day, I started reading books about tensors such as the tensor of inertia, the stress tensor etc.  These were usually presented as if you knew why they were developed, and just given in a mathematical form which left my intuition about them empty.

Tensors were developed years before Einstein came up with special relativity (1905) or general relativity (1915).

This is where Feynman is so good.  He starts with the problem of electrical polarizability (which is familiar if you’ve plowed this far through volume II) and shows exactly why a tensor is needed to describe it, e.g. he derives  the tensor from known facts about electromagnetism.  Then on to the tensor of inertia (another derivation).  This allows you to see where all that notation comes from. That’s all very nice, but you are dealing with just matrices.  Then on to tensors over 4 vector spaces (a rank 4 tensor) not the same thing as a 4 tensor which is over a 4 dimensional vector space.

Then finally we get to the 4 tensor (a tensor over a 4 dimensional vector space) of electromagnetic momentum.  Here are the 16 components of Callahan’s energy momentum tensor, clearly explained.  The circle is finally closed.

He briefly goes into the way tensors transform under a change of coordinates, which for many authors is the most important thing about them.   So his discussion doesn’t contain the usual blizzard of superscripts and subscript.  Covariant and contravariant are blessedly absent. Here the best explanation of how they transform is in Jeevanjee “An introduction to Tensors and Group Theory for Physicists”  chapter 3 pp. 51 – 74.

Here are a few of the posts I’ve written about tensors trying to explain them to myself (and hopefully you)

https://luysii.wordpress.com/2020/02/03/the-reimann-curvature-tensor/

https://luysii.wordpress.com/2017/01/04/tensors-yet-again/

https://luysii.wordpress.com/2015/06/15/the-many-ways-the-many-tensor-notations-can-confuse-you/

https://luysii.wordpress.com/2014/12/28/how-formal-tensor-mathematics-and-the-postulates-of-quantum-mechanics-give-rise-to-entanglement/

https://luysii.wordpress.com/2014/12/08/tensors/