Tag Archives: Quantum Mechanics

Getting to the Bottom of Things Physics-Wise

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One of the things that made physicist Richard Feynman famous was his invention of the Feynman diagram. Feynman diagrams provide a method for understanding the interactions between sub-atomic particles. Here, for example, is a diagram that represents the collision of an electron and a positron, resulting in the creation of two photons:

640px-Feynman_EP_Annihilation.svg

Using Feynman diagrams to describe certain interactions turns out to be extremely challenging, however. The collision of two gluons, resulting in the creation of four less energetic gluons, requires 220 diagrams. Some calculations based on this methodology cannot be done without the aid of a powerful computer, since they require thousands, millions and even billions of mathematical terms.

Over the years, various physicists and mathematicians have found ways to simplify these calculations. Recently, in fact, physicists have discovered a new geometrical object, the use of which simplifies the calculations to an amazing degree. They call the new structure an “amplituhedron”:

Interactions that were previously calculated with mathematical formulas thousands of terms long can now be described by computing the volume of the corresponding jewel-like “amplituhedron,” which yields an equivalent one-term expression.

“The degree of efficiency is mind-boggling,” said Jacob Bourjaily, … one of the researchers who developed the new idea. “You can easily do, on paper, computations that weren’t feasible even with a computer before.”

For example, calculating the volume of the amplituhedron in this diagram, which represents the interaction of 8 gluons, provides the same result as 500 pages of algebra based on Feynman diagrams:

amplituhedron-drawing_web

I learned about this latest development in an article called “A Jewel at the Heart of Physics”, which can be found here:

https://www.simonsfoundation.org/quanta/20130917-a-jewel-at-the-heart-of-quantum-physics/

The article suggests that the discovery of the amplituhedron might one day lead to major consequences for our understanding of the universe: “a theory of quantum gravity that would seamlessly connect the large- and small-scale pictures of the universe”:

Attempts thus far to incorporate gravity into the laws of physics at the quantum scale have run up against nonsensical infinities and deep paradoxes. …The amplituhedron is not built out of space-time and probabilities; these properties merely arise as consequences of the jewel’s geometry. The usual picture of space and time, and particles moving around in them, is a construct.

If two methods give the same results, the simpler method is clearly preferable from a pragmatic point of view. In fact, it’s likely in such a case that the simpler method better reflects the way the universe works. Perhaps the simplicity of this new method indicates that fundamental reality is simpler than previously believed:

Beyond making calculations easier or possibly leading the way to quantum gravity, the discovery of the amplituhedron could cause an even more profound shift, [physicist Nima] Arkani-Hamed said. That is, giving up space and time as fundamental constituents of nature and figuring out how the Big Bang and cosmological evolution of the universe arose out of pure geometry.

“In a sense, we would see that change arises from the structure of the object,” he said. “But it’s not from the object changing. The object is basically timeless.”

None of this means that there are tiny amplituhedrons underlying the universe, floating around outside space and time (whatever that would mean). It’s not even clear (to me anyway) what “pure geometry” is, since the geometry of an object, whether real or imagined, usually refers to its spatial characteristics.

Nevertheless, this latest labor-saving device may help physicists get closer to the bottom of things, assuming there is a bottom to get to.

The Uncertainty Principle and Us

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It’s difficult to discuss physics if you aren’t a physicist or don’t understand the math involved. Nevertheless, what physicists tell us about the world is so strange that it’s hard not to discuss it sometimes, whether we understand it or not. (The brilliant physicist and all-around cool guy Richard Feynman once said that nobody understands quantum mechanics, but some understand it better than others.)

There are philosophers who specialize in the philosophy of physics and aren’t shy about discussing physics at all, among themselves and with physicists. One of these philosophers, Craig Callender, has recently written two interesting articles for the New York Times. In these articles, Callender argues that Werner Heisenberg’s uncertainty principle, probably the best-known part of quantum mechanics, shouldn’t be as famous as it is. 

Heisenberg was one of the founders of quantum mechanics. He published the uncertainty principle in 1927. If you look up “uncertainty principle” now, you’ll find statements like this: “The position and momentum of a particle cannot be simultaneously measured with arbitrarily high precision” and “The uncertainty principle is at the foundation of quantum mechanics: you can measure a particle’s position or its velocity, but not both.”

Well, here is Callender on quantum mechanics:

[Quantum mechanics is] a complex theory, but its basic structure is simple. It represents physical systems – particles, cats, planets – with abstract quantum states. These quantum states provide the chances for various things happening. Think of quantum mechanics as an oddsmaker. You consult the theory, and it provides the odds of something definite happening….

The quantum oddsmaker can answer … questions for every conceivable property of the system. Sometimes it really narrows down what might happen: for instance, “There is a 100 percent chance the particle is located here, and zero percent chance elsewhere.” Other times it spreads out its chances to varying degrees: “There is a 1 percent chance the particle is located here, a 2 percent change it is located there, a 1 percent chance over there and so on.”

According to Callender:

The uncertainty principle simply says that for some pairs of questions to the oddsmaker, the answers may be interrelated. Famously, the answer to the question of a particle’s position is constrained by the answer to the question of its velocity, and vice versa. In particular, if we have a huge ensemble of systems each prepared in the same quantum state, the more the position is narrowed down, the less the velocity is, and vice versa. In other words, the oddsmaker is stingy: it won’t give us good odds on both position and velocity at once.

Callender then points out that he hasn’t said anything about measurement or observation:

The principle is about quantum states and what odds follow from these states. To add the notion of measurement is to import extra content. And as the great physicist John S. Bell has said, formulations of quantum mechanics invoking measurement as basic are “unprofessionally vague and ambiguous.” After all, why is a concept as fuzzy as measurement part of a fundamental theory?

Callender later shares another quote from J. S. Bell (considered by some to be the greatest physicist of the second half of the 20th century):

What exactly qualifies some physical systems to play the role of “measurer”? Was the wavefunction [the quantum state] of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system … with a Ph.D.? If the theory is to apply to anything but highly idealized laboratory operations, are we not obliged to admit that more or less “measurement-like” processes are going on more or less all the time, more or less everywhere?

When physicists use their instruments to measure a subatomic particle’s position or momentum, the instruments affect the particle. It’s the interaction at the subatomic level between the instrument and the particle that’s important, not the fact that the interaction has something to do with measurement, observation, mental energy or human consciousness. We aren’t that important in the vast scheme of things.

Viewing the theory of quantum mechanics as a cosmic oddsmaker may seem unhelpful. We want to know what’s going on at the subatomic level that results in the theory calculating certain odds. Heisenberg thought physicists shouldn’t even think about an underlying reality — they should simply focus on the results of their observations. But some (many?) physicists working today believe that quantum mechanics is an incomplete theory that will eventually be replaced by a more fundamental theory, possibly one that explains away the apparent randomness that exists at the subatomic level (that’s what Einstein thought too). Their hope is that uncertainty will one day be replaced by certainty, or something closer to it.

If you do a Google search for “uncertainty principle consciousness”, you’ll probably get more than 8 million results. If you search for “uncertainty principle measurement”, you can get more than 32 million. Professor Callender thinks those numbers should be much, much smaller.

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This is Callender’s first article in the Times:

http://opinionator.blogs.nytimes.com/2013/07/21/nothing-to-see-here-demoting-the-uncertainty-principle/

Here he responds to questions and criticisms from readers:

http://opinionator.blogs.nytimes.com/2013/07/25/return-of-the-stingy-oddsmaker-a-response/

Those Crazy, Mixed Up Photons

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On the website of the American Association for the Advancement of Science, a physicist recently wrote:

Suppose you have a quantum particle of light, or photon. It can be polarized so that it wriggles either vertically or horizontally. The quantum realm is … hazed over with unavoidable uncertainty, and thanks to such quantum uncertainty, a photon can … be polarized vertically and horizontally at the same time. If you then measure the photon, however, you will find it either horizontally polarized or vertically polarized, as the two-ways-at-once state randomly ‘collapses’ one way or the other.

This two-ways-at-once state is called “superposition”. The idea is that something can be in more than one state (or “position”) at one time, i.e. a super-position.

However, saying that a photon can be polarized vertically and horizontally at the same time, or that it can be in a “two-ways-at-once” state, looks extremely suspicious. It’s hard to know what such a statement means, if anything. After all, language is based on logic (it wouldn’t work otherwise) and logic is based on the law of contradiction: proposition P cannot be both true and false, assuming that P has a single, precise meaning.

The proposition that photon p is polarized vertically at time t has a single, precise meaning. So does the proposition that photon p is polarized horizontally at time t. Yet these statements certainly look contradictory. It looks as if we have to give up the law of contradiction in order to accept them both.

To avoid the contradiction, however, it might be preferable to say that a photon can be in an indeterminate state, in which its polarization is neither vertical nor horizontal. It’s potentially in either state, but it’s not in either one until its state is measured (or otherwise affected), at which point the photon randomly ends up in one state or the other.

Viewed in probabilistic terms, the fate of Schrödinger’s cat doesn’t seem to be a problem (to me anyway). It was alive when it was put in the box and presumably remained alive unless it was poisoned as the result of a random sub-atomic event. We don’t have to say that the cat is now both dead and alive (or in some twilight state). It’s just a cat that may have died and there is a certain probability that it did.

But then there is the famous double-split experiment. This experiment shows that photons don’t behave like cats (or dogs) or, in the philosopher J. L. Austin’s phrase, “medium-sized dry goods”. A single photon travels through two slits and creates a wave-pattern on the other side, even though common sense tells us that the photon can only travel through one slit or the other. The bizarre but reasonable conclusion is that the photon actually takes every possible path through the two openings, not just in theory, but in fact.

Fortunately, there isn’t any contradiction in saying that the photon goes through slit 1 and slit 2 at the same time, since saying that it goes through slit 2 doesn’t conflict with saying that it also goes through slit 1. In similar fashion, photons can be polarized horizontally and vertically at the same time, because that’s the kind of thing that can happen to the crazy little bastards (i.e. sub-atomic particles).

We are used to saying things like “a person can’t be in two places at the same time” (many episodes of Law and Order are based on that premise). Logic tells us that if the number 5 is odd, it can’t be even. Logic and experience tell us that if Miss Scarlet was in the billiard room, she wasn’t in the conservatory. That’s how numbers and people work. Photons don’t work that way. It’s extremely strange, but not incomprehensible and not contradictory.

http://news.sciencemag.org/sciencenow/2013/05/physicists-create-quantum-link-b.html