Oct 202021
 

Earlier today, I noticed something really strange. A lamp was radiating darkness. Or so it appeared.

Of course there was a mundane explanation. Now that the Sun is lower in the sky and the linden tree in front of our kitchen lost many of its leaves already, intense sunlight was reflecting off the hardwood floor in our dining area.

Still, it was an uncanny sight.

 Posted by at 11:27 pm
Sep 302021
 

I have had it up to my eyeballs with misinformation about vaccines, mRNA vaccines in particular. People who up until 2020 could not tell the difference between acronyms like “RNA” and “WTF” suddenly became experts on molecular biology, capable of evaluating the professional literature and arriving at profound judgments, telling us that the vaccines are “fake” and such, or worse yet, they amount to “gene therapy”.

With all due respect, I first encountered the acronym “mRNA” (or its Hungarian equivalent, mRNS) not in 2020, not in 2019, but in 1980 or 81, from a Hungarian translation of Watson’s book on molecular biology of the gene.

Now granted, even if I had read that book cover-to-cover (I didn’t) it would not make me an expert on molecular biology. But I knew enough for the expression “mRNA vaccine” to make sense to me right away when it first showed up in news reports. In short, I know enough to spot the bullshit. Such as all that anti-vaccine scaremongering that has become ever so popular on the Interwebs lately.

Something similar happened 20 years ago, in the wake of 9/11. Many folks, especially Americans, who previously couldn’t tell Mohammed the prophet from Mohammed Ali, and who have never been in the same room with a textbook on comparative religion previously, suddenly became experts on Islam, making grand pronouncements about it being the religion of terror and all that. I first read a textbook on comparative religion back when I was 10 or so, from a 1927 2-volume tome on religions of the world:

This is volume one, titled “Primitive and cultural religions, Islam and Buddhism”. As with the Watson textbook, the images in this blog entry are of my own making, done just moments ago using my phone camera, of the actual books I have in my personal library.

Again, reading this book did not make me an instant expert. But it did give me enough background to spot the flood of bullshit that permeated the discussion after the 9/11 terror attacks.

Coming from a family and personal tradition that values learning, values impartial knowledge, it almost feels like physical pain, being confronted with such gross ignorance and outright lies each and every day. Enough already. Don’t listen to me, but don’t listen to the bullshit artists either. Listen to the actual experts (and not a cherry-picked subset of so-called experts who say what you want to hear). That’s what experts are for in an advanced scientific-technological society in which no human can be a master of all trades, and in which we rely on each other’s knowledge and experience.

Someone on Quora recently compared the anti-vaxxer movement to a hypothetical scenario on an airliner in distress: instead of following the crews’ instructions and donning oxygen masks, passengers stage a revolt, led by an “expert” who already knows better than the pilots how to fly the damn plane because he played with Microsoft Flight Simulator!

Groan.

 Posted by at 1:10 am
Sep 282021
 

I live in a condominium townhouse. We’ve been living here for 25 years. We like the place.

Our unit, in particular, is the middle unit in a three-unit block. The construction is reasonably sound: proper foundations, cinderblock firewalls between the units, woodframe construction within, pretty run-of-the-mill by early 1980s North American standards. We have no major complaints.

Except that… for the past several years, every so often the house wobbled a bit. Almost imperceptibly, but still. At first, I thought it was a minor earthquake (not uncommon in this region because it is still subject to isostatic rebound from the last ice age; in fact we did live through a couple of notable earthquakes since we moved in here.) But no, it was no earthquake.

I thought perhaps it was related to the downtown light rail tunnel construction? But no, the LRT tunnels are quite some ways from here and in any case, that part of the construction has been finished long ago.

But then what the bleep is it? Could I be just imagining things?

Our phones have very sensitive acceleration sensors. Not for the first time, I managed to capture one of these events. A little earlier this afternoon, I heard the woodframe audibly creak as the house began to move again. I grabbed my phone and turned on a piece of software that samples the acceleration sensor at a reasonably high rate, about 200 times a second. Here is the result of the first few seconds of sampling:

The sinusoidal signal is unmistakably there, confirmed by a quick Fourier-analysis to be a signal just above 3 Hz in frequency:

Like Sheldon Cooper in The Big Bang Theory, I can claim that no, I am not crazy, and in this case not because my mother had me tested but because my phone’s acceleration sensor confirms my perception: Something indeed wobbles the house a little, enough to register on my phone’s acceleration sensor, measuring a peak-to-peak amplitude of roughly 0.05 m/s² (the vertical axis in the first graph is in g-units.) That wobble is certainly not enough to cause damage, but it is, I admit, a bit unnerving.

So what is going on here? A neighbor engaging in some, ahem, vigorous activity? Our current neighbors are somewhat noisier than prior residents, occasionally training their respective herds of pygmy elephants to run up and down the stairs (or whatever it is that they are doing). But no, the events are just too brief in duration and too regular. Underground work, perhaps a secret hideout for the staff of the nearby Chinese embassy? Speaking of which, I admit I even thought that this ~3 Hz signal might be related to the reported cases of illness by embassy staff at several embassies around the world, but I just don’t see the connection: even if those cases are real and have an underlying common cause (as opposed to just mere random coincidences) it’s hard to see how a 3 Hz vibration can have anything to do with them.

OK, so I have a pretty good idea of what this thing isn’t, but then, what the bleepety-bleep is it?

 Posted by at 3:50 pm
Aug 142021
 

I am not happy admitting it, but it’s true: There have been a few occasions in my life when I reacted just like this XKCD cartoon character when I first encountered specific areas of research.

 Posted by at 11:48 am
Jul 302021
 

On my eighth birthday, I received a gift from a nice couple, friends of my Mom.

It was a Hungarian-language book bearing the title, “Wonders of the World,” in Hungarian, translated from the German original that was written by German-Jewish authors Artur Fürst and Alexander Moszkowski.

It was an old book, published in the 1930s. A dark green hardcover, with the etched image of a skyscraper for illustration on the cover. Its dust jacket, if it ever had one, was long gone.

But never mind that, it’s the content on these yellowed pages that matters.

It was from this book that I first learned about statistical fallacies, for instance. What is the probability that when you leave your home, the first 200 people you encounter are all males? Astronomically small, you might conclude. 2−200 ~ 6.223 × 10−61 to be a bit more precise, assuming half the population is male. A probability this small is firmly in the category of never happens. Until one morning, you step outside and the first thing you see is an all-male battalion of soldiers marching down the street…

I was reminded of this book today as I was reading about recent pronouncements of “breakthrough” infections among the vaccinated, and the reminder by experts that in a population that has a high vaccination rate, such cases are to be expected. It does not mean that the vaccine is worthless. It simply means that as the virus runs out of unvaccinated victims, to the extent it can still cause damage, increasingly it will be among the vaccinated folks. Which should make sense, except, as we well know, roughly 90% of statistical fallacies are committed by right-handed people…

Anyhow, much to my surprise, this book I love so much, from which I learned so much as a pre-teen, remains well-known in the country where it was originally published under the title Das Buch der 1000 Wunder. So well-known, in fact, that current German-language editions are readily available on Amazon, nearly a century after its initial publication. So I guess I am not the only person who finds the insights and information presented in this unassuming volume immensely valuable, especially for a child.

So let this serve as my notice of gratitude across time and space to “uncle Sandor and aunt Eva,” as they inscribed their names in the book along with their birthday wishes, for what I can now truly call a gift of a lifetime.

 Posted by at 11:32 pm
Jul 242021
 

Can you guess the author with the most physics books on what I call my “primary” bookshelf, the shelf right over my desk where I keep the books that I use the most often?

It would be Steven Weinberg. His 1972 Gravitation and Cosmology remains one of the best books ever on relativity theory, working out details in ways no other book does. His 2010 Cosmology remains a reasonably up-to-date textbook on modern cosmology. And then there is of course the 3-volume Quantum Theory of Fields.

Alas, Weinberg is no longer with us. He passed away yesterday, July 23, at the age of 88.

He will be missed.

 Posted by at 6:27 pm
Jul 142021
 

The other day, someone asked a question: Can the itensor package in Maxima calculate the Laplace-Beltrami operator applied to a scalar field in the presence of torsion?

Well, it can. But I was very happy to get this question because it allowed me to uncover some long-standing, subtle bugs in the package that prevented some essential simplifications and in some cases, even produced nonsensical results.

With these fixes, Maxima now produces a beautiful result, as evidenced by this nice newly created demo, which I am about to add to the package:

(%i1) if get('itensor,'version) = false then load(itensor)
(%i2) "First, we set up the basic properties of the system"
(%i3) imetric(g)
(%i4) "Demo is faster in 3D but works for other values of dim, too" 
(%i5) dim:3
(%i6) "We declare symmetries of the metric and other symbols"
(%i7) decsym(g,2,0,[sym(all)],[])
(%i8) decsym(g,0,2,[],[sym(all)])
(%i9) components(g([a],[b]),kdelta([a],[b]))
(%i10) decsym(levi_civita,0,dim,[],[anti(all)])
(%i11) decsym(itr,2,1,[anti(all)],[])
(%i12) "It is useful to set icounter to avoid indexing conflicts"
(%i13) icounter:100
(%i14) "We choose the appropriate convention for exterior algebra"
(%i15) igeowedge_flag:true
(%i16) "Now let us calculate the Laplacian of a scalar field and simplify"
(%i17) canform(hodge(extdiff(hodge(extdiff(f([],[]))))))
(%i18) contract(expand(lc2kdt(%)))
(%i19) ev(%,kdelta)
(%i20) D1:ishow(canform(%))
                  %1 %2  %3 %4                 %1 %2            %1 %2
(%t20)   (- f    g      g      g     ) + f    g      + f       g
             ,%4  ,%3           %1 %2     ,%2  ,%1      ,%1 %2
(%i21) "We can re-express the result using Christoffel symbols, too"
(%i22) ishow(canform(conmetderiv(D1,g)))
               %1 %4  %2 %5      %3                   %1 %2      %3
(%t22) 2 f    g      g      ichr2      g      - f    g      ichr2
          ,%5                    %1 %2  %3 %4    ,%3             %1 %2
                                              %1 %3      %2               %1 %2
                                      - f    g      ichr2      + f       g
                                         ,%3             %1 %2    ,%1 %2
(%i23) "Nice. Now let us repeat the same calculation with torsion"
(%i24) itorsion_flag:true
(%i25) canform(hodge(extdiff(hodge(extdiff(f([],[]))))))
(%i26) "Additional simplifications are now needed"
(%i27) contract(expand(lc2kdt(%th(2))))
(%i28) ev(%,kdelta)
(%i29) canform(%)
(%i30) ev(%,ichr2)
(%i31) ev(%,ikt2)
(%i32) ev(%,ikt1)
(%i33) ev(%,g)
(%i34) ev(%,ichr1)
(%i35) contract(rename(expand(canform(%))))
(%i36) flipflag:not flipflag
(%i37) D2:ishow(canform(%th(2)))
                %1 %2  %3 %4                 %1 %2    %3            %1 %2
(%t37) (- f    g      g      g     ) + f    g      itr      + f    g
           ,%1         ,%2    %3 %4     ,%1           %2 %3    ,%1  ,%2
                                                                          %1 %2
                                                               + f       g
                                                                  ,%1 %2
(%i38) "Another clean result; can also be expressed using Christoffel symbols"
(%i39) ishow(canform(conmetderiv(D2,g)))
               %1 %2  %3 %4      %5                   %1 %2    %3
(%t39) 2 f    g      g      ichr2      g      + f    g      itr
          ,%1                    %2 %3  %4 %5    ,%1           %2 %3
                     %1 %2      %3            %2 %3      %1               %1 %2
             - f    g      ichr2      - f    g      ichr2      + f       g
                ,%1             %2 %3    ,%1             %2 %3    ,%1 %2
(%i40) "Finally, we see that the two results differ only by torsion"
(%i41) ishow(canform(D2-D1))
                                   %1 %2    %3
(%t41)                       f    g      itr
                              ,%1           %2 %3
(%i42) "Last but not least, d^2 is not nilpotent in the presence of torsion"
(%i43) extdiff(extdiff(f([],[])))
(%i44) ev(%,icc2,ikt2,ikt1)
(%i45) canform(%)
(%i46) ev(%,g)
(%i47) ishow(contract(%))
                                       %3
(%t47)                         f    itr
                                ,%3    %275 %277
(%i48) "Reminder: when dim = 2n, the Laplacian is -1 times these results."

The learning curve is steep and there are many pitfalls, but itensor remains an immensely powerful package.

 Posted by at 3:51 pm
Jun 292021
 

Temperatures like this just do not exist in Canada.

When you hear that the temperature was within a hair’s breadth of 50 degrees centigrade (well over 120 F) you’d think I am talking about a spot in the Sahara Desert. Or maybe the Australian Outback. Or Death Valley.

But no, this temperature was measured earlier today in Lytton, British Columbia, Canada.

It is surreal. Scary. And deadly: apparently, dozen’s of mostly older people succumbed to this heat wave in BC.

 Posted by at 10:33 pm
May 182021
 

We have a new manuscript on arXiv. Its title might raise some eyebrows: Algebraic wave-optical description of a quadrupole gravitational lens.

Say what? Algebra? Wave optics? Yes. It means that in this particular case, namely a gravitational lens that is described as a gravitational monopole with a quadrupole correction, we were able to find a closed form description that does not rely on numerical integration, especially no numerical integration of a rapidly oscillating function.

Key to this solution is a quartic equation. Quartic equations were first solved algebraically back in the 16th century by Italian mathematicians. The formal solution is usually considered to be of little practical value, as it entails cumbersome algebra, and polynomial equations can be routinely and efficiently solved using numerical methods.

But in this case… The amazing thing is that the algebraic solution reveals so much about the physics itself!

Take this figure from our paper, for instance:

On the left is light projected by the gravitational lens, its so-called point-spread function (PSF) which tells us how light from a point source is distributed on an imaginary projection screen by the lens. On the right? Why, that’s the discriminant of the quartic equation

$$ x^4-2\eta\sin\mu \, x^3+\big(\eta^2-1\big)x^2+\eta\sin\mu \, x+{\textstyle\frac{1}{4}}\sin^2\mu=0, $$

in a plane characterized by polar coordinates \((\eta,\tfrac{1}{2}\mu)\), that is, \(\eta\) as a radial coordinate and \(\tfrac{1}{2}\mu \) as an azimuthal angle. When the discriminant is positive, the equation is expected to have four real (or four complex) roots; everywhere else, it’s a mix of real and imaginary roots. This direct connection between the algebra and the lensing phenomenon is unexpected and beautiful.

The full set of real roots of this equation can be shown in the form of an animation:

Of course one must read the paper in order for this animation to make sense, but I think it’s beautiful.

How good is this quartic solution? It is uncannily accurate. Here is a comparison of the PSF computed using the quartic solution and also using numerical integration, as well as some enlarged details from the so-called caustic boundary:

It’s only in the immediate vicinity of the caustic boundary that the quartic solution becomes less than accurate.

We can also use the quartic solution to simulate images seen through a telescope (i.e., the Einstein ring, or what survives of it, that would appear around a gravitational lens when we looked at the lens through a telescope with a point source of light situated behind the lens.) We can see again that it’s only in the vicinity of the caustic boundary that the quartic solution produces artifacts instead of accurately reproducing it when spots of light widen into arcs:

This paper was so much joy to write! Also, for the first time in my life, this paper gave us a legitimate, non-pretentious reason to cite something from the 16th century: Cardano’s 1545 treatise in which the quartic solution (as well as the cubic) are introduced, together with discussion on the meaning of taking the square root of negative numbers.

 Posted by at 5:35 pm
May 132021
 

Last fall, I received an intriguing request: I was asked to respond to an article on the topic of dark matter in an online publication that, I admit, I never heard of previously: Inference: International Review of Science.

But when I looked, I saw that the article in question was written by a scientist with impressive and impeccable credentials (Jean-Pierre Luminet, Director of Research at the CNRS Astrophysics Laboratory in Marseille and the Paris Observatory), and other contributors of the magazine included well-known personalities like Lawrence Krauss or Noam Chomsky.

More importantly, the article in question presented an opportunity to write a response that was not critical but constructive: inform the reader that the concept of modified gravity goes far beyond the so-called MOND paradigm, that it is a rich and vibrant field of theoretical research, and that until and unless dark matter is actually discovered, it remains a worthy pursuit. My goal was not self-promotion: I did not even mention my ongoing collaboration with John Moffat on his modified theory of gravity, MOG/STVG. Rather, it was simply to help dispel the prevailing myth that failures of MOND automatically translate into failures of all efforts to create a viable modified theory of gravitation.

I sent my reply and promptly forgot all about it until last month, when I received another e-mail from this publication: a thank you note letting me know that my reply would be published in the upcoming issue.

And indeed it was, as I was just informed earlier today: My Letter to the Editor, On Modified Gravity.

I am glad in particular that it was so well received by the author of the original article on dark matter.

 Posted by at 4:38 pm
Apr 192021
 

This morning, a drone took flight. It successfully took off from the ground, hovered for a few seconds, and then landed safely.

What, you ask? How is this supposed to be a big deal? There are millions of drones out there, kids playing with them and whatnot.

Oh, but this drone is special, and not only because it carries a small piece of fabric from the Wright brothers’ very first airplane.

It is special because it flew on Mars.

 Posted by at 11:42 am
Mar 272021
 

Bald eagles have a fearsome reputation as predators of the sky. They also symbolize the great United States of America.

Canada geese? Not so fearsome. They are best known for pooping a lot. (If you ever walked through an Ottawa park after it was visited by a flock of geese, you know what I am talking about.)

Yet just like the country that they are named after, these geese are not so timid after all. Here is a recent series of images (and if you search online, you see that this by no means is an exception) captured by a PEI photographer of a Canada goose, fighting off a bald eagle:

Reader’s Digest version: The eagle remained hungry that day.

 Posted by at 12:03 am
Mar 142021
 

The next in our series of papers describing the extended gravitational lens (extended, that is, in that we are no longer treating the lensing object as a gravitational monopole) is now out, on arXiv.

Here’s one of my favorite images from the paper, which superimposes the boundary of the quadrupole caustic (an astroid curve) onto a 3D plot showing the amplitude of the gravitational lens’s point-spread function.

I was having lots of fun working on this paper. It was, needless to say, a lot of work.

 Posted by at 9:18 pm
Mar 142021
 

I am reading about Heino Falcke.

Dr. Falcke is a scientist. He is the leader of the Event Horizon Telescope project, the first successful attempt to image the event horizon (actually, the shadow of the photon sphere cast on the accretion disk background) of a black hole.

One of the Event Horizon Telescope participating facilities, at Pico Veleta

Dr. Falcke also happens to be religious. A lay pastor, no less, in the Protestant Church in the Netherlands.

He represents yet another example of how faith and the sciences need not be in conflict.

I happen to be nonreligious. I even mock religion (not the religious! Never!) occasionally when I talk about “imaginary friends”, “sky daddy” or the “Flying Spaghetti Monster”. My mockery is not intended to hurt: rather, this truly is how I feel about these supernatural concepts, as surreal, outlandish flights of fancy, fairy tales, nothing more.

Yet I think I understand how faith can also give strength to people. Offer motivation. Fill their lives with meaning.

It has been invariably my experience that the company of a person of faith who is open-minded and capable of critical thinking is much preferable to that of a dogmatic atheist.

In any case, while I may not have much respect for the supernatural aspects of religion, I certainly take no issue with the basic tenets of Christianity, such as loving thy neighbor or not committing murder. If the core message of religion is to try to be a decent human being, well, I don’t need to believe in imaginary friends to accept and fully embrace these principles.

This always reminds me how the best description of Christianity I ever came across came from a devoted atheist, the late Douglas Adams (of Hitchhiker’s Guide to the Galaxy fame): “And then, one Thursday, nearly two thousand years after one man had been nailed to a tree for saying how great it would be to be nice to people for a change […]”

 Posted by at 6:38 pm
Mar 142021
 

Because I’ve been asked a lot about this lately, I thought I’d also share my own take on this calculation in my blog.

Gravitoelectromagnetism (or gravitomagnetism, even gravimagnetism) is the name given to a formalism that shows how weak gravitational fields can be viewed as analogous to electromagnetic fields and how, in particular, the motion of a test particle is governed by equations that are similar to the equations of the electromagnetic Lorentz-force, with gravitational equivalents of the electric and magnetic vector potentials.

Bottom line: no, gravitoelectromagnetism does not explain the anomalous rotation curves of spiral galaxies. The effect is several orders of magnitude too small. Nor is the concept saved by the realization that spacetime is not asymptotically flat, so the boundary conditions must change. That effect, too, is much too small, at least five orders of magnitude too small in fact to be noticeable.

To sketch the key details, the radial acceleration on a test particle due to gravitoelectromagnetism in circular orbit around a spinning body is given roughly by

$$a=-\frac{4G}{c^2}\frac{Jv}{r^3},$$

where \(r\) is the orbital speed of the test particle. When we plug in the numbers for the solar system and the Milky Way, \(r\sim 8~{\rm kpc}\) and \(J\sim 10^{67}~{\rm J}\cdot{\rm s}\), we get

$$a\sim 4\times 10^{-16}~{\rm m}{\rm s}^2.$$

This is roughly 400,000 times smaller than the centrifugal acceleration of the solar system in its orbit around the Milky Way, which is \(\sim 1.6\times 10^{-10}~{\rm m}/{\rm s}^2.\)

Taking into account that our universe is not flat, i.e., deviations from the flat spacetime metric approach unity at the comoving distance of \(\sim 15~{\rm Gpc},\) only introduces a similarly small contribution on the scale of a galaxy, of \({\cal O}(10^{-6})\) at \(\sim 15~{\rm kpc}.\)

A more detailed version of this calculation is available on my Web site.

 Posted by at 1:14 pm
Mar 012021
 

Now it is time for me to be bold and contrarian. And for a change, write about physics in my blog.

From time to time, even noted physicists express their opinion in public that we do not understand quantum physics. In the professional literature, they write about the “measurement problem”; in public, they continue to muse about the meaning of measurement, whether or not consciousness is involved, and the rest of this debate that continues unabated for more than a century already.

Whether it is my arrogance or ignorance, however, when I read such stuff, I beg to differ. I feel like the alien Narim in the television series Stargate SG-1 in a conversation with Captain (and astrophysicist) Samantha Carter about the name of a cat:

CARTER: Uh, see, there was an Earth physicist by the name of Erwin Schrödinger. He had this theoretical experiment. Put a cat in a box, add a can of poison gas, activated by the decay of a radioactive atom, and close the box.
NARIM: Sounds like a cruel man.
CARTER: It was just a theory. He never really did it. He said that if he did do it at any one instant, the cat would be both dead and alive at the same time.
NARIM: Ah! Kulivrian physics. An atom state is indeterminate until measured by an outside observer.
CARTER: We call it quantum physics. You know the theory?
NARIM: Yeah, I’ve studied it… in among other misconceptions of elementary science.
CARTER: Misconception? You telling me that you guys have licked quantum physics?

What I mean is… Yes, in 2021, we “licked” quantum physics. Things that were mysterious in the middle of the 20th century aren’t (or at least, shouldn’t be) quite as mysterious in the third decade of the 21st century.

OK, let me explain by comparing two thought experiments: Schrödinger’s cat vs. the famous two-slit experiment.

The two-slit experiment first. An electron is fired by a cathode. It encounters a screen with two slits. Past that screen, it hits a fluorescent screen where the location of its arrival is recorded. Even if we fire one electron at a time, the arrival locations, seemingly random, will form a wave-like interference pattern. The explanation offered by quantum physics is that en route, the electron had no classically determined position (no position eigenstate, as physicists would say). Its position was a combination, a so-called superposition of many possible position states, so it really did go through both slits at the same time. En route, its position operator interfered with itself, resulting in the pattern of probabilities that was then mapped by the recorded arrival locations on the fluorescent screen.

Now on to the cat: We place that poor feline into a box together with a radioactive atom and an apparatus that breaks a vial of poison gas if the atom undergoes fission. We wait until the half-life of that atom, making it a 50-50 chance that fission has occurred. At this point, the atom is in a superposition of intact vs. split, and therefore, the story goes, the cat will also be in a superposition of being dead and alive. Only by opening the box and looking inside do we “collapse the wavefunction”, determining the actual state of the cat.

Can you spot a crucial difference between these two experiments, though? Let me explain.

In the first experiment involving electrons, knowledge of the final position (where the electron arrives on the screen) does not allow us to reconstruct the classical path that the electron took. It had no classical path. It really was in a superposition of many possible locations while en route.

In the second experiment involving the cat, knowledge of its final state does permit us to reconstruct its prior state. If the cat is alive, we have no doubt that it was alive all along. If it is dead, an experienced veterinarian could determine the moment of death. (Or just leave a video camera and a clock in the box along with the cat.) The cat did have a classical state all throughout the experiment, we just didn’t know what it was until we opened the box and observed its state.

The crucial difference, then, is summed up thus: Ignorance of a classical state is not the same as the absence of a classical state. Whereas in the second experiment, we are simply ignorant of the cat’s state, in the first experiment, the electron has no classical state of position at all.

These two thought experiments, I think, tell us everything we need to know about this so-called “measurement problem”. No, it does not involve consciousness. No, it does not require any “act of observation”. And most importantly, it does not involve any collapse of the wavefunction when you really think it through. More about that later.

What we call measurement is simply interaction by the quantum system with a classical object. Of course we know that nothing really is classical. Fluorescent screens, video cameras, cats, humans are all made of a very large but finite number of quantum particles. But for all practical (measurable, observable) intents and purposes all these things are classical. That is to say, these things are (my expression) almost in an eigenstate almost all the time. Emphasis on “almost”: it is as near to certainty as you can possibly imagine, deviating from certainty only after the hundredth, the thousandth, the trillionth or whichever decimal digit.

Interacting with a classical object confines the quantum system to an eigenstate. Now this is where things really get tricky and old school at the same time. To explain, I must invoke a principle from classical, Lagrangian physics: the principle of least action. Almost all of physics (including classical mechanics, electrodynamics, even general relativity) can be derived from a so-called action principle, the idea that the system evolves from a known initial state to a known final state in a manner such that a number that characterizes the system (its “action”) is minimal.

The action principle sounds counterintuitive to many students of physics when they first encounter it, as it presupposes knowledge of the final state. But this really is simple math if you are familiar with second-order differential equations. A unique solution to such an equation can be specified in two ways. Either we specify the value of the unknown function at two different points, or we specify the value of the unknown function and its first derivative at one point. The former corresponds to Lagrangian physics; the latter, to Hamiltonian physics.

This works well in the context of classical physics. Even though we develop the equations of motion using Lagrangian physics, we do so only in principle. Then we switch over to Hamiltonian physics. Using observed values of the unknown function and its first derivative (think of these as positions and velocities) we solve the equations of motion, predicting the future state of the system.

This approach hits a snag when it comes to quantum physics: the nature of the unknown function is such that its value and its first derivative cannot both be determined as ordinary numbers at the same time. So while Lagrangian physics still works well in the quantum realm, Hamiltonian physics does not. But Lagrangian physics implies knowledge of the future, final state. This is what we mean when we pronounce that quantum physics is fundamentally nonlocal.

Oh, did I just say that Hamiltonian physics doesn’t work in the quantum realm? But then why is it that every quantum physics textbook begins, pretty much, with the Hamiltonian? Schrödinger’s famous equation, for starters, is just the quantum version of that Hamiltonian!

Aha! This is where the culprit is. With the Hamiltonian approach, we begin with presumed knowledge of initial positions and velocities (values and first derivatives of the unknown functions). Knowledge we do not have. So we evolve the system using incomplete knowledge. Then, when it comes to the measurement, we invoke our deus ex machina. Like a bad birthday party surprise, we open the magic box, pull out our “measurement apparatus” (which we pretended to not even know about up until this moment), confine the quantum system to a specific measurement value, retroactively rewrite the description of our system with the apparatus now present all along, and call this discontinuous change in the system’s description “wavefunction collapse”.

And then spend a century about its various interpretations instead of recognizing that the presumed collapse was never a physical process: rather, it amounts to us changing how we describe the system.

This is the nonsense for which I have no use, even if it makes me sound both arrogant and ignorant at the same time.


To offer a bit of a technical background to support the above (see my Web site for additional technical details): A quantum theory can be constructed starting with classical physics in a surprisingly straightforward manner. We start with the Hamiltonian (I know!), written in the following generic form:

$$H = \frac{p^2}{2m} + V({\bf q}),$$

where \({\bf p}\) are generalized momenta, \({\bf q}\) are generalized positions and \(m\) is mass.

We multiply this equation by the unit complex number \(\psi=e^{i({\bf p}\cdot{\bf q}-Ht)/\hbar}.\) We are allowed to do this trivial bit of algebra with impunity, as this factor is never zero.

Next, we notice the identities, \({\bf p}\psi=-i\hbar\nabla\psi,\) \(H\psi=i\hbar\partial_t\psi.\) Using these identities, we rewrite the equation as

$$i\hbar\partial_t\psi=\left[-\frac{\hbar^2}{2m}\nabla^2+V({\bf q})\right]\psi.$$

There you have it, the time-dependent Schrödinger equation in its full glory. Or… not quite, not yet. It is formally Schrödinger’s equation but the function \(\psi\) is not some unknown function; we constructed it from the positions and momenta. But here is the thing: If two functions, \(\psi_1\) and \(\psi_2,\) are solutions of this equation, then because the equation is linear and homogeneous in \(\psi,\) their linear combinations are also solutions. But these linear combinations make no sense in classical physics: they represent states of the system that are superpositions of classical states (i.e., the electron is now in two or more places at the same time.)

Quantum physics begins when we accept these superpositions as valid descriptions of a physical system (as indeed we must, because this is what experiment and observation dictates.)

The presence of a classical apparatus with which the system interacts at some future moment in time is not well captured by the Hamiltonian formalism. But the Lagrangian formalism makes it clear: it selects only those states of the system that are consistent with that interaction. This means indeed that a full quantum mechanical description of the system requires knowledge of the future. The apparent paradox is that this knowledge of the future does not causally influence the past, because the actual evolution of the system remains causal at all times: only the initial description of the system needs to be nonlocal in the same sense in which 19th century Lagrangian physics is nonlocal.

 Posted by at 12:48 pm
Feb 182021
 

I really cannot tell which impresses me more: The incredibly complex landing or the fact that there is now a de facto infrastructure in orbit around Mars, in the form of earlier spacecraft that provide communications relay capabilities for real-time tracking of the landing.

Or perhaps the fact that Perseverance also carries the Ingenuity helicopter. If successful, it will be the first drone to fly in the atmosphere of another planet.

The children of future settlers on Mars will be learning about these moments in school.

 Posted by at 4:06 pm
Feb 152021
 

A Dallas-Fort Worth TV station characterized Texas as the energy capital of the world as it asked the rhetorical question: How could this happen?

I have friends in Texas. One of them e-mailed me to let me know that they’ve been without power since 5 AM this morning.

Unlike the US-Canadian northeast with its interconnected power grid, Texas has its own power grid. This means, I understand, that they cannot rely on excess generating capacity in neighboring states to help with the crisis.

And the weather is bitterly cold, much colder than up here in wintry Ottawa. Right now, according to that Dallas-Fort Worth TV station it’s 8 degrees Fahrenheit but it will drop to several degrees below zero on the Fahrenheit scale overnight; that’s -20 C for us folks in metric lands.

This is not a joke. In weather like this, people can die, especially in ill-insulated homes as they struggle with unexpected secondary disasters such as bursting, frozen pipes that may very well happen.

 Posted by at 10:42 pm