Feynman Paths


Previously in seriesThe Quantum Arena

At this point I would like to introduce another key idea in quantum mechanics.  Unfortunately, this idea was introduced so well in chapter 2 of QED: The Strange Theory of Light and Matter by Richard Feynman, that my mind goes blank when trying to imagine how to introduce it any other way.  As a compromise with just stealing his entire book, I stole one diagram—a diagram of how a mirror really works.


In elementary school, you learn that the angle of incidence equals the angle of reflection.  But actually, saith Feynman, each part of the mirror reflects at all angles.

So why is it that, way up at the human level, the mirror seems to reflect with the angle of incidence equal to the angle of reflection?

Because in quantum mechanics, amplitude that flows to identical configurations (particles of the same species in the same places) is added together, regardless of how the amplitude got there.

To find the amplitude for a photon to go from S to P, you've got to add up the amplitudes for all the different ways the photon could get there—by bouncing off the mirror at A, bouncing off the mirror at B...

The rule of the Feynman "path integral" is that each of the paths from S to P contributes an amplitude of constant magnitude but varying phase, and the phase varies with the total time along the path.  It's as if the photon is a tiny spinning clock—the hand of the clock stays the same length, but it turns around at a constant rate for each unit of time.

Feynman graphs the time for the photon to go from S to P via A, B, C, ...  Observe: the total time changes less between "the path via F" and "the path via G", then the total time changes between "the path via A" and "the path via B".  So the phase of the complex amplitude changes less, too.

And when you add up all the ways the photon can go from S to P, you find that most of the amplitude comes from the middle part of the mirror—the contributions from other parts of the mirror tend to mostly cancel each other out, as shown at the bottom of Feynman's figure.

There is no answer to the question "Which part of the mirror did the photon really come from?"  Amplitude is flowing from all of these configurations.  But if we were to ignore all the parts of the mirror except the middle, we would calculate essentially the same amount of total amplitude.

This means that a photon, which can get from S to P by striking any part of the mirror, will behave pretty much as if only a tiny part of the mirror exists—the part where the photon's angle of incidence equals the angle of reflection.

Unless you start playing clever tricks using your knowledge of quantum physics.

For example, you can scrape away parts of the mirror at regular intervals, deleting some little arrows and leaving others.  Keep A and its little arrow; scrape away B so that it has no little arrow (at least no little arrow going to P).  Then a distant part of the mirror can contribute amplitudes that add up with each other to a big final amplitude, because you've removed the amplitudes that were out of phase.

In which case you can make a mirror that reflects with the angle of incidence not equal to the angle of reflection.  It's called a diffraction grating.  But it reflects different wavelengths of light at different angles, so a diffraction grating is not quite a "mirror" in the sense you might imagine; it produces little rainbows of color, like a droplet of oil on the surface of water.

How fast does the little arrow rotate?  As fast as the photon's wavelength—that's what a photon's wavelength is.  The wavelength of yellow light is ~570 nanometers:  If yellow light travels an extra 570 nanometers, its little arrow will turn all the way around and end up back where it started.

So either Feynman's picture is of a very tiny mirror, or he is talking about some very big photons, when you look at how fast the little arrows seem to be rotating.  Relative to the wavelength of visible light, a human being is a lot bigger than the level at which you can see quantum effects.

You'll recall that the first key to recovering the classical hallucination from the reality of quantum physics, was the possibility of approximate independence in the amplitude distribution.  (Where the distribution roughly factorizes, it can look like a subsystem of particles is evolving on its own, without being entangled with every other particle in the universe.)

The second key to re-deriving the classical hallucination, is the kind of behavior that we see in this mirror.  Most of the possible paths cancel each other out, and only a small group of neighboring paths add up.  Most of the amplitude comes from a small neighborhood of histories—the sort of history where, for example, the photon's angle of incidence is equal to its angle of reflection.  And so too with many other things you are pleased to regard as "normal".

My first posts on QM showed amplitude flowing in crude chunks from discrete situation to discrete situation.  In real life there are continuous amplitude flows between continuous configurations, like we saw with Feynman's mirror.  But by the time you climb all the way up from a few hundred nanometers to the size scale of human beings, most of the amplitude contributions have canceled out except for a narrow neighborhood around one path through history.

Mind you, this is not the reason why a photon only seems to be in one place at a time.  That's a different story, which we won't get to today.

The more massive things are—actually the more energetic they are, mass being a form of energy—the faster the little arrows rotate. Shorter wavelengths of light having more energy is a special case of this.  Compound objects, like a neutron made of three quarks, can be treated as having a collective amplitude that is the multiplicative product of the component amplitudes—at least to the extent that the amplitude distribution factorizes, so that you can treat the neutron as an individual.

Thus the relation between energy and wavelength holds for more than photons and electrons; atoms, molecules, and human beings can be regarded as having a wavelength.

But by the time you move up to a human being—or even a single biological cell—the mass-energy is really, really large relative to a yellow photon.  So the clock is rotating really, really fast.  The wavelength is really, really short.  Which means that the neighborhood of paths where things don't cancel out is really, really narrow.

By and large, a human experiences what seems like a single path through configuration space—the classical hallucination.

This is not how Schrödinger's Cat works, but it is how a regular cat works.

Just remember that this business of single paths through time is not fundamentally true.  It's merely a good approximation for modeling a sofa.  The classical hallucination breaks down completely by the time you get to the atomic level.  It can't handle quantum computers at all.  It would fail you even if you wanted a sufficiently precise prediction of a brick.  A billiard ball taking a single path through time is not how the universe really, really works—it is just what human beings have evolved to easily visualize, for the sake of throwing rocks.

(PS:  I'm given to understand that the Feynman path integral may be more fundamental than the Schrödinger equation: that is, you can derive Schrödinger from Feynman.  But as far as I can tell from examining the equations, Feynman is still differentiating the amplitude distribution, and so reality doesn't yet break down into point amplitude flows between point configurations.  Some physicist please correct me if I'm wrong about this, because it is a matter on which I am quite curious.)


Part of The Quantum Physics Sequence

Next post: "No Individual Particles"

Previous post: "The Quantum Arena"