Tag Archives: Quantum mechanics

Book Review — The Universe Speaks in Numbers

Let’s say that back in the day, as a budding grad student in chemistry you had to take quantum mechanics to see where those atomic orbitals came from.   Say further, that as the years passed you knew enough to read News and Views in Nature and Perspectives in Science concerning physics as they appeared. So you’ve heard various terms like J/Psi, Virasoro algebra, Yang Mills gauge symmetry, Twisters, gluons, the Standard Model, instantons, string theory, the Atiyah Singer theorem etc. etc.  But you have no coherent framework in which to place them.

Then “The Universe Speaks in Numbers” by Graham Farmelo is the book for you.  It will provide a clear and chronological narrative of fundamental physics up to the present.  That isn’t the main point of the book, which is an argument about the role of beauty and mathematics in physics, something quite contentious presently.  Farmelo writes well and has a PhD in particle physics (1977) giving him a ringside seat for the twists and turns of  the physics he describes.  People disagree with his thesis (http://www.math.columbia.edu/~woit/wordpress/?p=11012) , but nowhere have I seen anyone infer that any of Farmelo’s treatment of the physics described in the book is incorrect.

40 years ago, something called the Standard Model of Particle physics was developed.  Physicists don’t like it because it seems like a kludge with 19 arbitrary fixed parameters.  But it works, and no experiment and no accelerator of any size has found anything inconsistent with it.  Even the recent discovery of the Higgs, was just part of the model.

You won’t find much of the debate about what physics should go from here in the book.  Farmelo says just study more math.  Others strongly disagree — Google Peter Woit, Sabine Hossenfelder.

The phenomena String theory predicts would require an accelerator the size of the Milky Way or larger to produce particles energetic enough to probe it.  So it’s theory divorced from any experiment possible today, and some claim that String Theory has become another form of theology.

It’s sad to see this.  The smartest people I know are physicists.  Contrast the life sciences, where experiments are easily done, and new data to explain arrives weekly.

 

 

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Off to band camp for adults 2018

No posts for a while, as I’ll be at a chamber music camp for adult amateurs (or what a friend’s granddaughter calls — band camp for adults).  In a week or two if you see a beat up old Honda Pilot heading west on the north shore of Lake Superior, honk and wave.

I expect the usual denizens to be there — mathematicians, physicists, computer programmers, MDs, touchy-feely types who are afraid of chemicals etc. etc. We all get along but occasionally the two cultures do clash, and a polymer chemist friend is driven to distraction by a gentle soul who is quite certain that “chemicals” are a very bad thing. For the most part, everyone gets along. Despite the very different mindsets, all of us became very interested in music early on, long before any academic or life choices were made.

So, are the analytic types soulless automatons producing mechanically perfect music which is emotionally dead? Are the touchy-feely types sloppy technically and histrionic musically? A double-blind study would be possible, but I think both groups play pretty much the same (less well than we’d all like, but with the same spirit and love of music).

A few years ago I had the pleasure of playing Beethoven with Heisenberg —   along with an excellent violinist I’ve played with for years, the three of us read Beethoven’s second piano trio (Opus 1, #2) with Heisenberg’s son Jochem (who, interestingly enough, is a retired physics professor).  He is an excellent cellist who knows the literature cold.  The violinist and I later agreed that we have rarely played worse.  Oh well. Heisenberg, of course, was a gentleman throughout.

Later that evening, several of us had the pleasure of discussing quantum mechanics with him. He didn’t disagree with my idea that the node in the 2S orbital (where no electron is ever found) despite finding the electron on either side of the node, forces us to give up the idea of electron trajectory (aromatic ring currents be damned).   He pretty much seemed to agree with the Copenhagen interpretation — macroscopic concepts just don’t apply to the quantum world, and language trips us up.

One rather dark point about the Heisenberg came up in an excellent book about the various interpretations of what Quantum Mechanics actually means: “What Is Real?” by Adam Becker.  I have no idea if the following summary is actually true, but here it is.   Heisenberg was head of the German nuclear program to develop an atomic bomb.  Nuclear fission was well known in Germany, having been discovered there.  An old girl friend wrote a book about Lise Meitner, one of the discoverers and how she didn’t get the credit she was due.

At the end of the war there was an entire operation to capture German physicists who had worked on nuclear development (operation Alsos).  Those captured (Heisenberg, Hahn, von Laue and others) were taken to Farm Hall, an English manor house which had been converted into a military intelligence center.  It was supplied with chalkboards, sporting equipment, a radio, good food and secretly bugged to high heaven.  The physicists were told that they were being held “at His Majesty’s pleasure.”.  Later they told the American’s had dropped the atomic bomb.  They didn’t believe it as their own work during the war led them to think it was impossible.

All their discussions were recorded, unknown to Heisenberg.  It was clear that the Germans had no idea how to build a bomb even though they tried.  However  Heisenberg  and von Weizsacker constructed a totally false narrative, that they had never tried to build a bomb, but rather a nuclear reactor.  According to Becker, Heisenberg was never caught out on this because the Farm Hall transcripts were classified.  It isn’t clear to me from reading Becker’s book, when they were UNclassified, but apparently Heisenberg got away with it until his death in 1978.

Amazing stuff if true

 

A creation myth

Sigmund Freud may have been wrong about penis envy, but most lower forms of scientific life (chemists, biologists) do have physics envy — myself included.  Most graduate chemists have taken a quantum mechanics course, if only to see where atomic and molecular orbitals come from.  Anyone doing physical chemistry has likely studied statistical mechanics. I was fortunate enough to audit one such course given by E. Bright Wilson (of Pauling and Wilson).

Although we no longer study physics per se, most of us read books about physics.  Two excellent such books have come out in the past year.  One is “What is Real?” — https://www.basicbooks.com/titles/adam-becker/what-is-real/9780465096053/, the other is “Lost in Math” by Sabine Hossenfelder whose blog on physics is always worth reading, both for herself and the heavies who comment on what she writes — http://backreaction.blogspot.com

Both books deserve a long discursive review here. But that’s for another time.  Briefly, Hossenfelder thinks that physics for the past 30 years has become so fascinated with elegant mathematical descriptions of nature, that theories are judged by their mathematical elegance and beauty, rather than agreement with experiment.  She acknowledges that the experiments are both difficult and expensive, and notes that it took a century for one such prediction (gravitational waves) to be confirmed.

The mathematics of physics can certainly be seductive, and even a lowly chemist such as myself has been bowled over by it.  Here is how it hit me

Budding chemists start out by learning that electrons like to be in filled shells. The first shell has 2 elements, the next 2 + 6 elements etc. etc. It allows the neophyte to make some sense of the periodic table (as long as they deal with low atomic numbers — why the 4s electrons are of lower energy than the 3d electons still seems quite ad hoc to me). Later on we were told that this is because of quantum numbers n, l, m and s. Then we learn that atomic orbitals have shapes, in some wierd way determined by the quantum numbers, etc. etc.

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 in college).

So it wasn’t a shock when the QM instructor back in 1961 got to them in the course of solving the Schrodinger equation for 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 variables, 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, the instructor, 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 recursion relations.

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.

But what interested me the most about “Lost in Math” was Hossenfelder’s discussion of the different physical laws appearing at different physical scales (e.g. effective laws), emergent properties and reductionism (pp. 44 –> ).  Although things at larger scales (atoms) can be understood in terms of the physics of smaller scales (protons, neutrons, electrons), the details of elementary particle interactions (quarks, gluons, leptons etc.) don’t matter much to the chemist.  The orbits of planets don’t depend on planetary structure, etc. etc.  She notes that reduction of events at one scale to those at a smaller one is not an optional philosophical position to hold, it’s just the way nature is as revealed by experiment.  She notes that you could ‘in principle, derive the theory for large scales from the theory for small scales’ (although I’ve never seen it done) and then she moves on

But the different structures and different laws at different scales is what has always fascinated me about the world in which we exist.  Do we have a model for a world structured this way?

Of course we do.  It’s the computer.

 

Neurologists have always been interested in computers, and computer people have always been interested in the brain — von Neumann wrote “The Computer and the Brain” shortly before his death in 1958.

Back in med school in the 60s people were just figuring out how neurons talked to each other where they met at the synapse.  It was with a certain degree of excitement that we found that information appeared to flow just one way across the synapse (from the PREsynaptic neuron to the POST synaptic neuron).  E.g. just like the vacuum tubes of the earliest computers.  Current (and information) could flow just one way.

The microprocessors based on transistors that a normal person could play with came out in the 70s.  I was naturally interested, as having taken QM I thought I could understand how transistors work.  I knew about energy gaps in atomic spectra, but how in the world a crystal with zillions of atoms and electrons floating around could produce one seemed like a mystery to me, and still does.  It’s an example of ’emergence’ about which more later.

But forgetting all that, it’s fairly easy to see how electrons could flow from a semiconductor with an abundance of them (due to doping) to a semiconductor with a deficit — and have a hard time flowing back.  Again a one way valve, just like our concept of the synapses.

Now of course, we know information can flow the other way in the synapse from POST synaptic to PREsynaptic neuron, some of the main carriers of which are the endogenous marihuana-like substances in your brain — anandamide etc. etc.  — the endocannabinoids.

In 1968 my wife learned how to do assembly language coding with punch cards ones and zeros, the whole bit.  Why?  Because I was scheduled for two years of active duty as an Army doc, a time in which we had half a million men in Vietnam.  She was preparing to be a widow with 2 infants, as the Army sent me a form asking for my preferences in assignment, a form so out of date, that it offered the option of taking my family with me to Vietnam if I’d extend my tour over there to 4 years.  So I sat around drinking Scotch and reading Faulkner waiting to go in.

So when computers became something the general populace could have, I tried to build a mental one using and or and not logical gates and 1s and 0s for high and low voltages. Since I could see how to build the three using transistors (reductionism), I just went one plane higher.  Note, although the gates can be easily reduced to transistors, and transistors to p and n type semiconductors, there is nothing in the laws of semiconductor physics that implies putting them together to form logic gates.  So the higher plane of logic gates is essentially an act of creation.  They do not necessarily arise from transistors.

What I was really interested in was hooking the gates together to form an ALU (arithmetic and logic unit).  I eventually did it, but doing so showed me the necessity of other components of the chip (the clock and in particular the microcode which lies below assembly language instructions).

The next level up, is what my wife was doing — sending assembly language instructions of 1’s and 0’s to the computer, and watching how gates were opened and shut, registers filled and emptied, transforming the 1’s and 0’s in the process.  Again note that there is nothing necessary in the way the gates are hooked together to make them do anything.  The program is at yet another higher level.

Above that are the higher level programs, Basic, C and on up.  Above that hooking computers together to form networks and then the internet with TCP/IP  etc.

While they all can be reduced, there is nothing inherent in the things that they are reduced to which implies their existence.  Their existence was essentially created by humanity’s collective mind.

Could something be going on in the levels of the world seen in physics.  Here’s what Nobel laureate Robert Laughlin (he of the fractional quantum Hall effect) has to say about it — http://www.pnas.org/content/97/1/28.  Note that this was written before people began taking quantum computers seriously.

“However, it is obvious glancing through this list that the Theory of Everything is not even remotely a theory of every thing (2). We know this equation is correct because it has been solved accurately for small numbers of particles (isolated atoms and small molecules) and found to agree in minute detail with experiment (35). However, it cannot be solved accurately when the number of particles exceeds about 10. No computer existing, or that will ever exist, can break this barrier because it is a catastrophe of dimension. If the amount of computer memory required to represent the quantum wavefunction of one particle is Nthen the amount required to represent the wavefunction of k particles is Nk. It is possible to perform approximate calculations for larger systems, and it is through such calculations that we have learned why atoms have the size they do, why chemical bonds have the length and strength they do, why solid matter has the elastic properties it does, why some things are transparent while others reflect or absorb light (6). With a little more experimental input for guidance it is even possible to predict atomic conformations of small molecules, simple chemical reaction rates, structural phase transitions, ferromagnetism, and sometimes even superconducting transition temperatures (7). But the schemes for approximating are not first-principles deductions but are rather art keyed to experiment, and thus tend to be the least reliable precisely when reliability is most needed, i.e., when experimental information is scarce, the physical behavior has no precedent, and the key questions have not yet been identified. There are many notorious failures of alleged ab initio computation methods, including the phase diagram of liquid 3He and the entire phenomenonology of high-temperature superconductors (810). Predicting protein functionality or the behavior of the human brain from these equations is patently absurd. So the triumph of the reductionism of the Greeks is a pyrrhic victory: We have succeeded in reducing all of ordinary physical behavior to a simple, correct Theory of Everything only to discover that it has revealed exactly nothing about many things of great importance.”

So reductionism doesn’t explain the laws we have at various levels.  They are regularities to be sure, and they describe what is happening, but a description is NOT an explanation, in the same way that Newton’s gravitational law predicts zillions of observations about the real world.     But even  Newton famously said Hypotheses non fingo (Latin for “I feign no hypotheses”) when discussing the action at a distance which his theory of gravity entailed. Actually he thought the idea was crazy. “That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro’ a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it”

So are the various physical laws things that are imposed from without, by God only knows what?  The computer with its various levels of phenomena certainly was consciously constructed.

Is what I’ve just written a creation myth or is there something to it?

Relativity becomes less comprehensible

“To get Hawking radiation we have to give up on the idea that spacetime always had 3 space dimensions and one time dimension to get a quantum theory of the big bang.”  I’ve been studying relativity for some years now in the hopes of saying something intelligent to the author (Jim Hartle), if we’re both lucky enough to make it to our 60th college reunion in 2 years.  Hartle majored in physics under John Wheeler who essentially revived relativity from obscurity during the years when quantum mechanics was all the rage. Jim worked with Hawking for years, spoke at his funeral and wrote this in an appreciation of Hawking’s work [ Proc.Natl. Acad. Sci. vol. 115 pp. 5309 – 5310 ’18 ].

I find the above incomprehensible.  Could anyone out there enlighten me?  Just write a comment.  I’m not going to bother Hartle

Addendum 25 May

From a retired math professor friend —

I’ve never studied this stuff, but here is one way to get more actual dimensions without increasing the number of apparent dimensions:
Start with a 1-dimensional line, R^1 and now consider a 2-dimensional cylinder S^1 x R^1.  (S^1 is the circle, of course.)  If the radius of the circle is small, then the cylinder looks like a narrow tube.  Make the radius even smaller–lsay, ess than the radius of an atomic nucleus.  Then the actual 2-dimensional cylinder appears to be a 1-dimensional line.
The next step is to rethink S^1 as a line interval with ends identified (but not actually glued together.  Then S^1 x R^1 looks like a long ribbon with its two edges identified.  If the width of the ribbon–the length of the line interval–is less, say, than the radius of an atom, the actual 2-dimensional “ribbon with edges identified” appears to be just a 1-dimensional line.
Okay, now we can carry all these notions to R^2.  Take S^1 X R^2, and treat S^1 as a line interval with ends identified.  Then S^1 x R^2 looks like a (3-dimensional) stack of planes with the top plane identified, point by point, with the bottom plane.  (This is the analog of the ribbon.)  If the length of the line interval is less, say, than the radius of an atom, then the actual 3-dimensional s! x R^2 appears to be a 2-dimensional plane.
That’s it.  In general, the actual n+1-dimensional S^1 x R^n appears to be just n-space R^n when the radius of S^1 is sufficiently small.
All this can be done with a sphere S^2, S^3, … of any dimension, so that the actual k+n-dimensional manifold S^k x R^n appears to be just the n-space R^n when the radius of S^k is sufficiently small.  Moreover, if M^k is any compact manifold whose physical size is sufficiently small, then the actual k+n-dimensional manifold M^k x R^n appears to be just the n-plane R^n.
That’s one way to get “hidden” dimensions, I think. “

NonAlgorithmic Intelligence

Penrose was right. Human intelligence is nonAlgorithmic. But that doesn’t mean that our physical brains produce consciousness and intelligence using quantum mechanics (although all matter is what it is because of quantum mechanics). The parts (even small ones like neurotubules) contain so much mass that their associated wavefunction is too small to exhibit quantum mechanical effects. Here Penrose got roped in by Kauffman thinking that neurotubules were the carriers of the quantum mechanical indeterminacy. They aren’t, they are just too big. The dimer of alpha and beta tubulin contains 900 amino acids — a mass of around 90,000 Daltons (or 90,000 hydrogen atoms — which are small enough to show quantum mechanical effects).

So why was Penrose right? Because neural nets which are inherently nonAlgorithmic are showing intelligent behavior. AlphaGo which beat the world champion is the most recent example, but others include facial recognition and image classification [ Nature vol. 529 pp. 484 – 489 ’16 ].

Nets are trained on real world images and told whether they are right or wrong. I suppose this is programming of a sort, but it is certainly nonAlgorithmic. As the net learns from experience it adjusts the strength of the connections between its neurons (synapses if you will).

So it should be a simple matter to find out just how AlphaGo did it — just get a list of the neurons it contains, and the number and strengths of the synapses between them. I can’t find out just how many neurons and connections there are, but I do know that thousands of CPUs and graphics processors were used. I doubt that there were 80 billion neurons or a trillion connections between them (which is what our brains are currently thought to have).

Just print out the above list (assuming you have enough paper) and look at it. Will you understand how AlphaGo won? I seriously doubt it. You will understand it less well than looking at a list of the positions and momenta of 80 billion gas molecules will tell you its pressure and temperature. Why? Because in statistical mechanics you assume that the particles making up an ideal gas are featureless, identical and do not interact with each other. This isn’t true for neural nets.

It also isn’t true for the brain. Efforts are underway to find a wiring diagram of a small area of the cerebral cortex. The following will get you started — https://www.quantamagazine.org/20160406-brain-maps-micron-program-iarpa/

Here’s a quote from the article to whet your appetite.

“By the end of the five-year IARPA project, dubbed Machine Intelligence from Cortical Networks (Microns), researchers aim to map a cubic millimeter of cortex. That tiny portion houses about 100,000 neurons, 3 to 15 million neuronal connections, or synapses, and enough neural wiring to span the width of Manhattan, were it all untangled and laid end-to-end.”

I don’t think this will help us understand how the brain works any more than the above list of neurons and connections from AlphaGo. There are even more problems with such a list. Connections (synapses) between neurons come and go (and they increase and decrease in strength as in the neural net). Some connections turn on the receiving neuron, some turn it off. I don’t think there is a good way to tell what a given connection is doing just by looking a a slice of it under the electron microscope. Lastly, some of our most human attributes (emotion) are due not to connections between neurons but due to release of neurotransmitters generally into the brain, not at the very localized synapse, so it won’t show up on a wiring diagram. This is called volume neurotransmission, and the transmitters are serotonin, norepinephrine and dopamine. Not convinced? Among agents modifying volume neurotransmission are cocaine, amphetamine, antidepressants, antipsychotics. Fairly important.

So I don’t think we’ll ever truly understand how the neural net inside our head does what it does.

A book recommendation, not a review

My first encounter with a topology textbook was not a happy one. I was in grad school knowing I’d leave in a few months to start med school and with plenty of time on my hands and enough money to do what I wanted. I’d always liked math and had taken calculus, including advanced and differential equations in college. Grad school and quantum mechanics meant more differential equations, series solutions of same, matrices, eigenvectors and eigenvalues, etc. etc. I liked the stuff. So I’d heard topology was cool — Mobius strips, Klein bottles, wormholes (from John Wheeler) etc. etc.

So I opened a topology book to find on page 1

A topology is a set with certain selected subsets called open sets satisfying two conditions
l. The union of any number of open sets is an open set
2. The intersection of a finite number of open sets is an open set

Say what?

In an effort to help, on page two the book provided another definition

A topology is a set with certain selected subsets called closed sets satisfying two conditions
l. The union of a finite number number of closed sets is a closed set
2. The intersection of any number of closed sets is a closed set

Ghastly. No motivation. No idea where the definitions came from or how they could be applied.

Which brings me to ‘An Introduction to Algebraic Topology” by Andrew H. Wallace. I recommend it highly, even though algebraic topology is just a branch of topology and fairly specialized at that.

Why?

Because in a wonderful, leisurely and discursive fashion, he starts out with the intuitive concept of nearness, applying it to to classic analytic geometry of the plane. He then moves on to continuous functions from one plane to another explaining why they must preserve nearness. Then he abstracts what nearness must mean in terms of the classic pythagorean distance function. Topological spaces are first defined in terms of nearness and neighborhoods, and only after 18 pages does he define open sets in terms of neighborhoods. It’s a wonderful exposition, explaining why open sets must have the properties they have. He doesn’t even get to algebraic topology until p. 62, explaining point set topological notions such as connectedness, compactness, homeomorphisms etc. etc. along the way.

This is a recommendation not a review because, I’ve not read the whole thing. But it’s a great explanation for why the definitions in topology must be the way they are.

It won’t set you back much — I paid. $12.95 for the Dover edition (not sure when).

More Quantum Weirdness

If you put 3 pigeons in two pigeonholes, two pigeons must be in one of them. Not so says quantum mechanics in a new paper [ Proc. Natl. Acad. Sci. vol. 113 pp. 532 – 535 ’16 ]. I can’t claim to understand the paper, despite auditing a course in QM in the past decade, but at least I do understand the terms they are throwing about. I plan to print it out and really give it a workout

The paper does involve something called a beam splitter, which splits photon waves into two parts. I’ve never understood how this works on a mechanistic level. Perhaps no such understanding is possible. Another thing I don’t understand is what happens when a photon wave (or particle) is reflected from a mirror. Perhaps no such understanding is possible. Another thing I don’t understand is how a photon is diffracted when it passes through a slit. It must in some way sense the edge of the slit. I’ve certainly watched waves being diffracted by a breakwater — just go up to the bar at the top of the Sears Tower in Chicago, get an estimate on a beer and look out at lake Michigan.

Can anyone out there help?

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.

Urysohn’s Lemma

“Now we come to the first deep theorem of the book,. a theorem that is commonly called the “Urysohn lemma”.  . . .  It is the crucial tool used in proving a number of important theorems. . . .  Why do we call the Urysohn lemma a ‘deep’ theorem?  Because its proof involves a really original idea, which the previous proofs did not.  Perhaps we can explain what we mean this way:  By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student ought to be able to go through the book and work out the proofs independently.  (It would take a good deal of time and effort, of course, and one would not expect the student to handle the trickier examples.)  But the Uyrsohn lemma is on a different level.  It would take considerably more originality than most of us possess to prove this lemma.”

The above quote is  from  one of the standard topology texts for undergraduates (or perhaps the standard text) by James R. Munkres of MIT. It appears on  page 207 of 514 pages of text.  Lee’s text book on Topological Manifolds gets to it on p. 112 (of 405).  For why I’m reading Lee see https://luysii.wordpress.com/2012/09/11/why-math-is-hard-for-me-and-organic-chemistry-is-easy/.

Well it is a great theorem, and the proof is ingenious, and understanding it gives you a sense of triumph that you actually did it, and a sense of awe about Urysohn, a Russian mathematician who died at 26.   Understanding Urysohn is an esthetic experience, like a Dvorak trio or a clever organic synthesis [ Nature vol. 489 pp. 278 – 281 ’12 ].

Clearly, you have to have a fair amount of topology under your belt before you can even tackle it, but I’m not even going to state or prove the theorem.  It does bring up some general philosophical points about math and its relation to reality (e.g. the physical world we live in and what we currently know about it).

I’ve talked about the large number of extremely precise definitions to be found in math (particularly topology).  Actually what topology is about, is space, and what it means for objects to be near each other in space.  Well, physics does that too, but it uses numbers — topology tries to get beyond numbers, and although precise, the 202 definitions I’ve written down as I’ve gone through Lee to this point don’t mention them for the most part.

Essentially topology reasons about our concept of space qualitatively, rather than quantitatively.  In this, it resembles philosophy which uses a similar sort of qualitative reasoning to get at what are basically rather nebulous concepts — knowledge, truth, reality.   As a neurologist, I can tell you that half the cranial nerves, and probably half our brains are involved with vision, so we automatically have a concept of space (and a very sophisticated one at that).  Topologists are mental Lilliputians trying to tack down the giant Gulliver which is our conception of space with definitions, theorems, lemmas etc. etc.

Well one form of space anyway.  Urysohn talks about normal spaces.  Just think of a closed set as a Russian Doll with a bright shiny surface.  Remove the surface, and you have a rather beat up Russian doll — this is an open set.  When you open a Russian doll, there’s another one inside (smaller but still a Russian doll).  What a normal space permits you to do (by its very definition), is insert a complete Russian doll of intermediate size, between any two Dolls.

This all sounds quite innocent until you realize that between any two Russian dolls an infinite number of concentric Russian dolls can be inserted.  Where did they get a weird idea like this?  From the number system of course.  Between any two distinct rational numbers p/q and r/s where p, q, r and s are whole numbers, you can  always insert a new one halfway between.  This is where the infinite regress comes from.

For mathematics (and particularly for calculus) even this isn’t enough.  The square root of two isn’t a rational number (one of the great Euclid proofs), but you can get as close to it as you wish using rational numbers.  So there are an infinite number of non-rational numbers between any two rational numbers.  In fact that’s how non-rational numbers (aka real numbers) are defined — essentially by fiat, that any series of real numbers bounded above has a greatest number (think 1, 1.4, 1.41, 1.414, defining the square root of 2).

What does this skullduggery have to do with space?  It says essentially that space is infinitely divisible, and that you can always slice and dice it as finely as you wish.  This is the calculus of Newton and the relativity of Einstein.  It clearly is right, or we wouldn’t have GPS systems (which actually require a relativistic correction).

But it’s clearly wrong as any chemist knows. Matter isn’t infinitely divisible, Just go down 10 orders of magnitude from the visible and you get the hydrogen atom, which can’t be split into smaller and smaller hydrogen atoms (although it can be split).

It’s also clearly wrong as far as quantum mechanics goes — while space might not be quantized, there is no reasonable way to keep chopping it up once you get down to the elementary particle level.  You can’t know where they are and where they are going exactly at the same time.

This is exactly one of the great unsolved problems of physics — bringing relativity, with it’s infinitely divisible space together with quantum mechanics, where the very meaning of space becomes somewhat blurry (if you can’t know exactly where anything is).

Interesting isn’t it?