# 26

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"

# 26

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The Feynman path integral (PI) and Schrödinger's equation (SE) are completely equivalent formulations of QM in the sense that they give the same time evolution of an initial state. They have exactly the same information content. It's true that you can derive SE from the PI, while the reverse derivation isn't very natural. On the other hand, the PI is mathematically completely non-rigorous (roughly, the space of paths is too large) while SE evolution can be made precise.

Practically, the PI cannot be used to solve almost anything except the harmonic oscillator. This is a serious handicap in QM, since SE can be used to solve many problems exactly. But in quantum field theory, all the calculations are perturbations around harmonic oscillators, so the PI can be very useful.

Many physicists would agree that the PI is more "fundamental" because it's gives insight into QFT and theoretical physics. But the distinction is largely a matter of taste.

The way Feynman expresses the flow of amplitude to a certain point given a prior configuration is as a weighted sum over space of sums over path weights. The sum over space is simply weighted by the amplitude distribution of the given configuration and the weight of each path is but itself a sum over time of a quantity called Lagrangian (more precisely the complex exponential of this quantity but whatever) along said path.

Since this quantity is the difference between kinetic and potential energy, it normally should only depends on the position and time derivatives along the path. In that sense the path integral formalism for a finite number of particles is independent of the derivative of the amplitude distribution itself and thus of Schrödinger equation.

If one now goes to a situation with an infinite number of degrees of freedom, that is a field, and tries to implement there also a path integral formalism, then the equation changes slightly. Amplitude doesn't flow from one point to the other but rather between field configurations. In that case the second sum is not over all possible paths in between two points but over all possible field configurations in between two field configurations. Doing so, the quantity used to weight configurations now depends on the amplitude and space derivatives of the field everywhere.

And if one fancies a Schrödinger equation for a quantum field, then in the interacting and non-relativistic case this equation turns out to be nonlocal and nonlinear.

Feynman paths would basically correspond to points in spacetime configuration space, ie, histories, rather than points on plain ole position configuration space, wouldn't it?

(Actually, summing over histories is basically one way of explaining WHY in GR things follow geodesics. Think of metric as being analogous to refractive index, affecting the "optical" path length, so you end up getting the same idea as principle of least action.)

Ever since I read QED a few years ago, I've wanted to write a Quantum Ray-Tracing package that would use a discrete version of this summation over arrows to render scenes composed of a 3D grid of particles. It would have the advantage that certain classical ray-tracing problems having to do with questions of what, exactly, is a surface and its normal would go away. It would also correctly render diffraction gratings, butterfly wings and oil slicks, just given their physical arrangements.

On the negative side, it would require some serious R&D into rendering algorithms to get the computation times down to acceptable levels. Alas, I've never had the leisure to spend that kind of time on the problem.

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.

Eliezer, one thing that is confusing me is that you are trying to show that the billiard ball and the "particles have identities" analogy is wrong. At the same time you keep speaking from "the photon". In the quotation the impression I get is that "the photon" splits up into the different paths it travels. Why does it split up in the first place? Again the "splitting up" assumes that there is a particle(alias small billiard ball) but your writing seems to imply this.

Btw, I have posted a question to your last entry "The quantum arena" which unfortunately wasn't answered and has to do with this confusion.

Thanks, Roland

PS: I'm no physicist and from reading the other comments I have the impression that most who are following this are physicists or at least have quite an advanced knowledge of QM. Please don't subestimate the inferential distance for those of us who don't have all that knowledge.

Very late response:

I think that the splitting of the photon's path is pretty much entirely a human construction - the smaller the components it is split into, the more accurate the calculation, and each partition is itself an approximation that can be refined by splitting it up further in exactly the same manner. Essentially, it's a shortcut to doing a path integral over the entire range down to the planck level. Maybe... I'm not sure!

As Jess says, Schrödinger and Feynman are formally equivalent: either can be derived from the other. So if the question of which is more "fundamental" can be answered at all, it will have to be from other considerations. My own favorite way to think about the difference between the two pictures is in terms of computational complexity. The Schrödinger equation can be seen as telling us that quantum computers can be simulated by classical computers in exponential time: just write out the whole amplitude vector to reasonable precision, which takes exponentially many floating-point numbers, then update it step by step. The Feynman path integral can be seen as telling us that quantum computers can be simulated by classical computers in polynomial space: just add up the amplitudes of all paths leading to the quantum computer accepting, reusing the same memory from one path to another. Since polynomial space is contained in exponential time, the Feynman picture yields the better simulation -- and on that basis, one could argue that it's the more "fundamental" of the two representations.

Since Scott Aaronson has chimed in, it is worth pointing to this discussion on his blog in which Greg Kuperberg explains the Hilbert space issues from the previous thread.

What Roland's PS said :)

On the subject of 3D rendering:

Treating light as a classical wave can also produce pretty good experimental results on the scale of everyday life. Ray tracing algorithms ignore the properties light shares with classical waves, such as diffraction. I suspect that you don't need "quantum amplitude tracing" algorithm for more accurate 3D rendering, just a "classical wave tracing" algorithm. (Ordinary ray tracing is already rather computationally expensive anyway...)

You know, I enjoyed this post when I first read it, but now upon further thought it doesn't make any sense at all.

We're talking about the fundamental nature of reality, right? Photons are a fundamental thing? Taking all paths from S to P is fundamental? Little rotating arrows corresponding to wavelength is fundamental? OK, fine.

But what the heck is this "mirror" thing you then introduced? I'm supposed to assume that a mirror is a fundamental component of reality too?

No, obviously a mirror is just made up of atoms, which is just a pile of subatomic particles, also. You don't explain how a photon interacts with even a single other particle, but we're supposed to know how it interacts with a mirror? Especially a "flat" mirror, when we know that real physical mirrors must be very bumpy at a subatomic level. And a lot of them are silver; what's so special about silver atoms? Why is a mirror different from a (not very) flat rock?

See, here's the problem: you spend all this time telling us how our macroscopic intuitions are wrong, and we can't trust them, and that QM is the reality of how the universe works. And then, in the explanation of QM, you slip in a "mirror", and rely on our naive pre-QM common-sense understanding of mirrors to complete the example. But you've just told us that those intuitions are false!

I think you need to at least give some QM explanation of what a mirror "is", before using it in this example.

I'm not a physicist so my question may be really old hat, but whatever.

I can think of two situations in which one ends up with a diffusion equation but in which the underlying physics is quite different.

First, the flow of heat in a solid. Here there is a continuous 'heat flows down a temperature gradient' picture that is mathematically equivalent to a picture in which individual particles follow Brownian motions. Physically, the former is just a sort of averaged version of the latter - some accounting short cuts - while the latter is some way closer to reality; the particles are really diffusing.

Second, the flow of water in an aquifer. Here the Darcian flow is proportional to the pressure gradient. For the sake of argument, imagine a medium that is a perfectly regular and homogenous 3D network of tiny tubes or something. In this case, there is no 'diffusion' of the fluid particles; they flow in a completely deterministic (indeed reversible) way through the network. But of course, one could presumably 'solve' the aquifer equation with a Monte Carlo similation of a diffusion process, if one really wanted to or if that was handy.

So to my question: Does the Feynman path integral purport to represent what's actually going on in any sense? Or is it more in the nature of a device for solving the problem? Or is this one of those things that is not answerable?

Sorry I asked that wrong. I don't mean heat flow in the first case, there are no diffusing particles there. Say concentration of tracer in fluid suspension or something.

"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."

Which would seem to make it a ruler as well as a clock. But then, since general relativity made time an axis like space, I have sometimes wondered why we don't measure time in meters or distance in seconds.

To expand on that point, we also measure energy in hertz, and temperatures in Joules, and ultimately everything in pure numbers. :)

We do.

The speed of light is used to define not only the lightyear, but also the common metre.

I know this is an old post, but I'm hoping someone will see this. I read this a long time ago and have been thinking about QM questions (non-professionally) for a while. Recently, I started to wonder about a specific question regarding this post. Specifically, I'm thinking about the idea that we are summing paths leading to "identical configurations". While the various paths the photon takes in this problem do appear to lead to the same configuration, it seems to me that this is only true if you are just looking at the configuration of the photon and the mirror. The path A takes much more time to be completed than the path G, and it seems to me that during that time, the configuration of the rest of the universe would change as well, so the two configurations aren't the same.

I think this understanding is probably wrong, but I have about twenty guesses as to mistakes I could be making, and no clue which ones are genuine. Can anyone who has studied QM more help me out?

You don't need to add them ALL up at the same time, just notice that as you get further and further from the middle, each part begins canceling with nearer and nearer neighbors. To be more concrete: at some point, you start sending your pulse. The shortest path/specular reflection gets the signal there first; other paths begin contributing later. After a short time, the time offset to get to the destination is large enough that the beginning of the pulse from one angle is cancelling with the middle of the pulse from a neighboring angle. Beyond that point, unless the packet had some special structure, there's not much in the way of reflection.

To be perfectly frank, the mirror isn't necessary for this problem to work - all it really needs to do is justify Huygens' principle.

This also goes a way towards addressing DonGeddis's question - pretend the mirror isn't there, and reflect the upward rays down. The mirror no longer exists, and this now becomes the question of why light doesn't spontaneously turn angles for no reason at all. Is that better?

That's a pretty good way of explaining it. I actually read QED last summer, after posting this, and (I believe in chapter 3) Feynman covers this topic briefly. EY just didn't describe it. Thanks for posting the clarification!

Okay, so where did those arrows come from? I see how the graph second from the top corresponds to the amount of time a particle, were particles to exist, would take if it bounced, if it could bounce, because it's not actually a particle, off of a specific point on the mirror. But how does one pull the arrows out of that graph?

Feynman talks about this between 59:33 and 60:32 of part one of his 1979 Douglas Robb lectures.

Between 29:41 and 36:27 of part two, he draws the "arrows" diagram on the chalkboard.

If you find this topic interesting, you'll enjoy all four parts of the lecture series. See also 63:26 to 63:35 of part one, which is relevant to your other question.

Edit: To explicitly answer your question, the angle of each arrow is proportional to the height of the graph above that arrow. Note that different heights on the graph can correspond to identical angles, since (for example) 0 radians, 2pi radians, and 4pi radians are all the same angle.

Thank you very much.

When we sum over all paths some paths are longer than others. The argument says that the phase arrow will move further round because the time is longer. If the time is longer the the path won't end at the destination at the right time to coincide with the other paths. So how can this work?

The amplitudes don't coincide at the end. In fact some are pointing oppositely to each other and so cancel out. The final amplitude for a photon at P is the sum of the configurations coming into P. The amplitudes don't equal each other, but they can be added together to yield the amplitude for a photon at P.

(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.)

Feynman really does give you the amplitude for going from one point distribution to another point distribution. The formula for the path integral doesn't involve any derivatives of the amplitude distribution. But your fundamental point is still correct. Nature can't be viewed as classical just by thinking only in terms of point distributions. This is because the point distribution evolves into a non-point distribution. So even if you start out thinking in terms of point distributions you are immediately forced to consider other distributions.

(You might be worried that the point distribution has infinite second derivative, and so can't be evolved using the Schrodinger equation. But if you turn down your rigour dial you can find the solution:

phi = exp[i x^2 / (4t) ]/sqrt[4 pi i t]

(This is the solution for a free particle in one dimension where I've picked the mass hbar/2 for convenience.) One can sort of see how this becomes a point distribution as t tends to zero. The amplitude becomes very oscillatory everywhere except zero, and at zero all those oscillations cancel out. Meanwhile the magnitude increases like 1/sqrt(t) as t tends to zero, so at zero it has the correct value of sqrt(infintiy).)

what does the path integral actually look like?

Do you understand ordinary integration?