Wednesday, July 30, 2014

Identical electrons in a nose you touched

Impenetrability of matter proves that generic questions about the intermediate evolution can't have sharp classical answers

A German blogger wrote a playful text
Can you touch your nose?
that partially answers this would-be deep question by philosophers.

You know, you never really touch your nose because there remains a gap between your hand and your nose. The atoms and electrons repel. The German author correctly says that the Pauli exclusion principle preventing two electrons from being in "exactly the same state" is actually more important for the "impenetrability of matter law" than the Coulomb repulsion. She also correctly discusses that the definition of "you" may be subtle. The electrons and even atoms are constantly leaving and re-entering your body. As Feynman would say, "you" is really the approximate pattern that the particles love to recreate and the approximate dance that they like to dance, not a particular collection of particles.

Also, there may be sparks. When you're tickled by static electricity, you acquire at least several microcoulombs or so which translates to trillions of electrons. Lots of electrons are moving from one surface to another.

But now. I want to discuss whether the electrons are being exchanged in the case that there is no electric spark and the surfaces are exactly electrically neutral and the number of electrons on both surfaces just stays the same.

Imagine that the electron on the surface of your hand comes really close to an electron on the surface of your nose. Your hand touches your nose. Have the electrons exchanged their places?

In classical physics, you could say either "Yes" or "No" – you just trace their trajectories.

But in quantum mechanics, electrons are mutually indistinguishable. They obey the Fermi-Dirac statistics. This is not an "extra subtlety" that you must add to the question whether you may touch your nose. It's actually the same question – the indistinguishability of electrons is actually a necessary condition for the Pauli exclusion principle to hold. It's necessary for the matter to be impenetrable!

The \(t\)-channel and \(u\)-channel diagrams for electron-electron repulsion/scattering differ by the exchange of the final (or initial) electrons.

Consider this pair of diagrams. The left picture shows the "simple" history of two electrons. One is in your hand, one is in your nose. In the initial state, they're depicted by the lines entering from the left upper corner and from the right lower corner, respectively.

They repel – they exchange a virtual photon – and you obtain the two electrons in the final state, represented by the lines in the right upper corner and the left lower corner, respectively. On the left picture, you may see that the electron that was in the hand remains in the hand. The electron that was in the nose remains in the nose.

But there is another diagram, the right part of the picture above.

On that picture, you may see that the electron that initially belonged to the hand (left upper corner) moves to the nose and becomes the nose's final electron (right upper corner). Similarly, the electron that was initially in the nose (right lower corner) moves and ends up in the hand in the final state (left lower corner). OK?

Now, in classical physics, you could ask which of these two histories took place. If the answer is the "left history", then the nose hasn't exchanged electrons with the hand. If it is the "right history", then the electrons got interchanged. In classical physics, God might in principle observe the paths and decide which of the diagrams is right.

But quantum mechanics is different and heavily restricts what God may do – and orders Him what He has to do, too (e.g. He is obliged to throw dice).

In quantum mechanics, both histories must actually be summed before you produce any physical predictions. The two parts of the picture above are known as "Feynman diagrams" and they represent different intermediate histories. If you want to calculate the probability that two electrons would be observed in two particular states in the final situation, you first have to sum two complex probability amplitudes\[

c_\text{total} = c_\text{no exchange} + c_\text{exchange} \in \CC

\] (both terms depend on the final state whose probability you calculate, but I didn't want to make the equation messy) and then you may calculate the probability as some\[

P = |c_\text{total}|^2.

\] I emphasize that we had to sum two diagrams – one without the exchange of the electrons; and one with the exchange of electrons.

In fact, if you fix e.g. the four momentum vectors of the initial and final states of the two electrons, the Fermi-Dirac statistics implies that the two contributions to the total amplitude, \(c_\text{no exchange}\) and \(c_\text{exchange}\), will have the opposite sign, at least if the final momenta of the two electrons are nearby.

If the final momenta of the two electrons (the whole state) were chosen to be equal, the two contributions would actually exactly cancel because they have the minus sign from the Fermi-Dirac statistics; and the absolute values would match. So the two diagrams would exactly cancel and you would get \(c_{\rm total}=0\).

That is the actual reason (presented in the path integral formalism) why you can't ever bring two electrons into the same state!

In other words, the cancellation between the two diagrams – that becomes perfect if the two final momentum vectors exactly agree (or if the two initial momentum vectors exactly agree) – is absolutely necessary for the "impenetrability of matter law"!

People who have a problem with the fact that quantum mechanics made it nonsensical to talk about "objective properties of the intermediate states" in an evolution – even though it was fully sensible in classical physics – think that this novelty of quantum mechanics is an optional luxury that may be questioned, obscured, "interpreted", or even denied.

But it isn't really true.

Even as obvious observations as the fact that your hand and your nose are "mutually impenetrable" existentially depends on the fact that the correct probabilities are obtained from the summation of two qualitatively different histories – in one history, the two electrons were not exchanged while in the other history, they were exchanged.

If someone (a bigot unable to say good-bye to classical physics) tells you that the questions such as "have the two electrons been interchanged during the history?" must have a clear answer, it's enough to touch your nose to prove him wrong. The fact that your hand stops when it reaches the nose and doesn't penetrate the nose is due to the fact that both conceivable histories contributed to the outcome of the experiment. In fact, the absolute values of their contributions were pretty much equal for the nearby electrons because it's the two diagrams' exact cancellation that prevents electrons from being in the same state! So not only the "electrons' history without exchange" and "electron's history with exchange" have nonzero contributions to the events; their share is 50% – 50%.

Amy made a relevant experiment at 0:50 of the video above. I recommend you to perform the same experiment with the anti-quantum bigots, too. But be careful to avoid the innocent victims!

P.S. My argument was in terms of Feynman's path-integral formulation of quantum mechanics. Someone could suggest that the need to "equally represent" the exchanging and non-exchanging histories is an artifact of Feynman's approach. But it's not. In the operator formalism, the wave functions have to be antisymmetric. The position-based basis vectors of the physical Hilbert space of two electrons are\[

\ket{\vec x_1, \vec x_2} - \ket{\vec x_2,\vec x_1}

\] which means that a sudden \(\vec x_1\leftrightarrow\vec x_2\) exchange of the two electrons changes the wave function at all (except for the overall sign or phase, perhaps). In other words, there is no way to decide whether a final state \(\ket{\vec x_{1,f}, \vec x_{2,f}}\) evolved from a particular initial state or its version with the permutation of the two electrons as long as the electrons came close to each other. Of course that the "relative weight of the two histories" is a concept that is only well-defined in Feynman's approach but the analogous wisdom that the permuted and non-permuted state vectors are equally important is valid in other pictures of quantum mechanics, too.


  1. Hello, my comment is not relevant to this topic, but I'd like to thank you for a great czech translation of Brian Greene's Elegant Universe, which I just read :)

  2. Lubos can you translate that analysis in String theoretic language where you have splitting and joining, sum over topologies and stuff like that?

  3. Thanks, Ondřeji, although it seems that you wouldn't need a translation at all. ;-)

  4. Sure, you just thicken all the Feynman diagrams to connecting tubes and pants diagrams. In models where the electrons and photons are closed strings (e.g. heterotic string theory), the diagrams replacing the QFT ones look like this one:

    Both of the Feynman diagrams become topologically the same in string theory. It's a tree-level diagram, so it's conformally equivalent to a sphere with 4 holes for the 4 external closed strings.

    In different regions of the moduli space of the Riemann surfaces, the stringy diagram reproduces the individual contributions of the QFT Feynman diagrams: closed string theory only has one Feynman diagram at each order.

    If the electrons and/or photons were open strings, the discussion would be a bit different.

    The antisymmetry with respect to two identical fermions holds identically in string theory, of course.

  5. I suggest a little fix: "But Quantum Mechanics is different and heavily restricts what god may do – and orders him what he has to do, too (e.g. he is obliged to throw Dice)."

    If there were no god, it would be necessary to invent him.
    - Voltaire

  6. Great Post, Off the topic: media is saying mass and weight are the same after all

  7. Great post, Lubos! It is interesting and instructive to picture the quantum mechanical bases of our every experience, even the most mundane ones. With enough practice, the classical picture starts to seem artificial.

  8. Lubos, have you done a post explaining how oppositely charged particles exchange a photon and have their momenta brought towards each other?

  9. Apart from saying four when I meant two tosses, There are four possible outcomes of flipping two coins all equally likely I should have said to avoid also the inclusion of cases of more than two heads, eg.
    But it makes my original point. The sleeping beauty problem is badly stated. Their can never be a probability other than 1/2 from a toss of fair coin.

  10. QM arises from conservation of angular momentum, and that from exact vacuum isotropy plus Noether's theorems. Exact vacuum isotropy toward hadrons is doubtful - parity violations, symmetry breakings, chiral anomalies, baryogenesis, Chern-Simons repair of Einstein-Hilbert action.

    Five classes of experiment can falsify exact vacuum isotropy toward hadrons, three in existing apparatus. Two of those need 24 hours. Bee being theoretically squishy is less interesting than Bee being empirically solid. Look.

    BTW, fantastic explanation. Is QM exact toward matter? It is an important question given SUSY empirical irrelevance to date,.

  11. LOL, no, I've certainly done no blog post about this point only.

    The photon that they "exchange" is a virtual photon, so it may carry any energy and momentum - even negative energies and even energies and momenta not obeying E^2 = p^2 c^2.

    So exchanging a photon with whatever momentum is needed to change the particles is just fine with the formalism.

    One may actually calculate whether two particles will attract or repel. They repel if the momenta are changed by a positive multiples of the separation. The change of the momentum is equivalent to multiplying the wave function by

    exp(i * delta p * r).

    This is the kind of change of the wave function that the Feynman diagrams cause, and because the Feynman diagram is proportional to the product of the two charges (these factors of charges come from the vertices), the value of the Feynman diagram changes the sign if you switch from opposite charges to like charges (or vice versa).

    So the Feynman diagram knows about the sign, and the change of the wave function or the average change of the momentum will be proportional to this quantity with the sign, too. There is really nothing discontinuous happening when the product of charges is zero. Negative numbers are just fine.

    At any rate, all the intuition that the change of the momentum has to be "positive" (outward) has to be based on classical physics and is therefore wrong. But to fully understand what is right, you must obviously learn the mathematics of quantum field theory.

  12. The way you drew your feynman diagrams they describe electron-positron scattering via a virtual annihilation and creation process or a virtual photon exchange in hand and nose ;)

  13. Dear Luboš,

    I think you've had the burglars in. Quite a few comments have disappeared from this Sleeping Beauty thread.

  14. Sometimes I think the world would work smoother if we all just spoke math equations. In some sense, it's what the rest of Nature is doing anyway. We need to get with the program. Thank you as well for sharing the times....Don


    arxiv shall we meet in 2 weeks?

  16. The topics of Sean's blogs are becoming progressively more annoying, much like the Peter Cook character in this skit:

  17. In the hypothetical moment when they were gone, I was a sleeping beauty.

    Didn't you overlook the "load more comments" button at the bottom? It shows up when the number of comments becomes large, and these 100+ is large.

  18. I'll stick with Ohm's law thanks.

  19. You always strike me as extra-wide awake! :)

    No, Luboš, my first assumption was that the fault was at my end so I checked thoroughly by refreshing and then reloading twice from scratch, and in all cases making sure I had exhausted the "Load more comments" feature. But it was the more recent (top) lot of comments that went AWOL.


  20. LOL, John, I trust you, you are trustworthy enough, which doesn't mean that I would bet my life that your testimony is accurate.

    Servers sometimes behave crazily. A month ago, my Sitemer broke down

    and I thought it would never count the visits again. But after weeks, it began to work again. ;-)

  21. There is nothing to show, J=σE

  22. Correct but you assume that there is a specified time direction from left to right. Lubos hasn't specifed a time direction though...

  23. Reading the sentences "That is the actual reason (presented in the path integral formalism) why you can't ever bring two electrons into the same state!" I'm a bit puzzled. It seems to me that the reason for this formalism are some observations or weaker some principle (Pauli's principle). Or the formalism is the expression of some invisible proposition in a strong scientific building. I hope it is the first one. Luboš, can you explain the logical rightness of your sentence? (Perhaps I fale to see something?)
    Like your post.

  24. Dear Holger, Feynman's path integral isn't any appendix of the Pauli's principle - do I understand you right?

    Feynman's path integral approach is a way to formulate all of quantum mechanics and every single quantum mechanical theory that has a classical limit - which includes all the quantum mechanical theories used in practice. Feynman's path integral is the right way and only way to describe any quantum mechanical theory in a way where the concept of the action (and Lagrangian) plays a role.

    The summation over different histories isn't some "added feature" to agree with the Pauli's exclusion principle. It's a completely general feature of the Lagrangian- or action-based approaches in quantum mechanics, in the most general sense. One simply has to sum over trajectories to get the right transition amplitudes, to describe the interference patterns, to explain and derive *anything* about Nature.

    So your suggestion that Feynman needed to assume Pauli's principle to derive the sum over different histories or Feynman diagrams is just completely wrong. Feynman's approach to QM is extremely robust, universal, fundamental, and vital for physics, and if you don't see it, you misunderstand something really, really badly.

    But even if Feynman's approach to QM were a spin-off of the Pauli principle, it would still be very important because the Pauli's exclusion principle is staggeringly important and fundamental, too. For example, it's necessary for the diversity of atoms which is clearly necessary for the life's existence, too, among everything else about matter.

    So I am shocked by your suggestion that any of these things fails to be deep or fundamental. They are super-deep. There exists nothing in any other science that would be equally deep.



  27. Thanks, I actually did leave a comment in that very article and it was apparently censored.