Monday, February 27, 2012

Brian Cox misunderstands locality, Pauli exclusion principle

Update: See a wonderful Twitter confrontation between Carroll and Cox, as storified by – thanks to Jon Butterworth and Twistor 59

Update: See a treatment of the two-electron double-well energy measurements

Update: See Brian Cox and lunar phases
It's very refreshing to agree with Sean Carroll (and a guy named Tom Swanson) on something.

Brian Cox – who has previously ignited both positive and negative responses on TRF – wanted to use quantum mechanics to defend the Gaia religious proposition that "everything is connected with everything else" (the last sentence of the video above). I am convinced that this main "punch line" he wanted to prove was predetermined by ideological prejudices and goals.

The vague misconception that "everything is connected with everything else" is a pillar of the broader environmentalist movement into which Cox indisputably belongs. Of course, when you say this sentence in general, it may be right or wrong: it's hard to say whether the general thesis is correct. However, when you make a more specific statement of this kind, one may usually see whether it's correct or not.

To "prove" that everything is connected with everything else, he said that when he does something to a piece of diamond, i.e. heats it by friction, the electrons in the whole Universe must immediately change their energy a little bit in order to protect the Pauli exclusions principle which, according to Cox, requires that two electrons can't have the same energy.

The statements are fundamentally wrong at many levels.

The most important misconception is that something may change immediately in the whole Universe. This is impossible because of the principle of locality whose validity is guaranteed by the special theory of relativity. You simply can't change any observable associated with a distant part of the Universe immediately; the signals can't propagate faster than light.

So what's wrong with Cox's "proof" that such a change is inevitably occurring because it's required by the Pauli exclusion principle? Well, his statement that electrons have to have different energies is wrong. Electrons can't be found in the same quantum state. But a state isn't the same thing as an energy level. That's because {energy} isn't a complete set of commuting observables: you must specify something else aside from energy to specify the state.

That's why even in a single atom, many electrons actually occupy the same energy level. We may always choose the energy eigenstates to be angular momentum eigenstates, too. That's because the angular momentum is conserved i.e. it commutes with the energy. If the total angular momentum is "J", there can be "2J+1" different electrons in the multiplet. The counterpart of "2J+1" becomes macroscopic if you consider macroscopic chunks of matter.

Of course, the discussion above is oversimplified because the angular momentum of a single electron isn't actually conserved. That's because of the interactions between different electrons. Such interactions may transfer the angular momentum from one electron to another particle – or another particle whatsoever. For a pair of distant electrons, we would probably not describe the possible states in which they can be found by the energy and the angular momentum. We would approximate the Hilbert space by the tensor product of a two-dimensional Hilbert space for two possible locations (near the nuclei) and the Hilbert space of a single atom. If the two nuclei are far from one another, the interactions are negligible and the location (here or there) may become a good quantum number aside from the spin.

If you consider a crystal such as the diamond (or metals), one may place individual electrons into states with a well-defined wave number (information about wavelength and direction of motion, given by a vector "k"). The electrons whose "k" differs by a rotation which is a symmetry of the solid inevitably contribute the same energy. Of course, for a strict crystal, the number of rotations which are "exact" symmetries are finite but in the continuum limit, there's still a two-sphere (round or squashed one) in the "k" space where all the electrons' states carry the same energy.

Once again, there is a subtlety because the angular momentum or energy of a single electron isn't a good quantum number. It's not conserved i.e. it doesn't commute with the Hamiltonian (total energy) which physically means that the interactions may transfer energy and angular momentum from one particle to another. But Cox neglected this subtlety, too. Even if this subtlety didn't exist, he would be wrong about the big points. Quantum mechanics doesn't require the energies of two particles to be different. And in fact, relativity totally prohibits something that he believes to follow from quantum mechanics – the ability of objects to instantly influence the rest of the Universe.

This video does indicate that Brian Cox is just a "rock star" who happens to promote physics. It's easier to hide your incompetency in an experimental team where all the work is a product of many people than it would be if you were a theorist, an occupation that requires more self-sufficient individuals. More generally, the video teaches us where various popular and not so popular misconceptions about physics in general and quantum mechanics in particular originate. Someone considered a physicist is just misunderstanding a technical point but he decides that this misconception could be a great story to sell to the laymen so he just sells it.

The listeners start to consider the misconception to be a piece of valid physics and they often "improve it" beyond the original point.


  1. Dr. Motl - I haven't watched the included video (work restrictions you know) but I'm confused by your assertion regarding the principle of locality. Doesn't QM allow for entangled pairs to instantaneously affect each other regardless of the distance between them? Just from reading the article, that's what I supposed Dr. Cox was referring to.

  2. Nope, Dr Lewis, entanglement doesn't imply any interaction at a distance. Entanglement is just the quantum version of correlation between the entangled subsystem, a correlation that is caused by the contact between the subsystems in the past.

    But correlation doesn't imply causation. And indeed, the correlation in the measurements of the two subsystem isn't a result of any interaction right before the measurement. Instead, it is a consequence of the correlation that has existed for a long time, since the moment when the subsystems were in mutual contact.

    Quantum field theory is a quantum theory that strictly obeys the laws of locality and doesn't allow observables in distant regions to be changed instantly. That's true in Cox's setup, in the EPR entanglement setup, and in any other situation that can occur in the history of the Universe, too. Locality is a universal law.

  3. Hi Lubos,

    I'm reluctant to ask this, but isn't unitarity a global property of the universe rather than a local one?

  4. Dear Jim, I think it's fair to say that the unitarity is a global property of the Universe, not a local one, in the sense that only if we study the whole system, we may verify that the unitarity holds.

    Why are you asking? You were the first one who used the term "unitarity" in this blog entry and thread: check it by a simple search.

    Unitarity isn't the only property that the laws of quantum field theory and others obey. They obey other laws such as the Lorentz symmetry and locality, too.


  5. Thanks, I was asking because I mistakenly understood that you were ruling out the need for a global application of QM by your emphasis on locality - ie I think that if you accept global unitarity then you would have to allow a wavefunction of the universe evolving according to Scrödinger's equation - or maybe I'm missing something - how else is global unitarity enforced?

  6. Dear Jim,

    it's very hard to understand what you're saying.

    Quantum field theory exactly obeys locality: the influence of an event such as a human decision in all spacelike separated points in spacetime is exactly equal to zero. Do you understand this simple point?

    This rule, locality, doesn't contradict unitarity in any way. These are two largely independent conditions. In fact, they're "complementary" in the sense that whenever the origin of one of them is very easy to be understood, it's harder with the other one, and vice versa. But both of them are exactly satisfied in QFT.

    If one uses Schrödinger's picture with a Hamiltonian, unitarity is equivalent to the Hermiticity of the Hamiltonian. This is satisfied by non-relativistic models of quantum mechanics but it is satisfied by the Hamiltonian of quantum field theories, too. It has to be.

    You seem to be implicitly if not explicitly using the assumption that locality has to be broken for unitarity to hold, but this assumption is, politely speaking, pure shit.


  7. You're (very reasonably) assuming a human decision is consistent with the Hamiltonian flow - in that case it has no (measurable) effect at spacelike points

    However, I agree with what you said so please don't swear

    (btw I'm regular drunk/confused poster James, google auto logged me in under this account)

  8. Dear Dr. Motl,
    Thank you for the reply. Please forgive my ignorance. I was (am?) under the impression that Bell's Theorem implied non-locality. I see that this may not be true. I've not yet read it, but found a paper by David Deutsch titled "Vindication of Quantum Locality", I'll give that a go and see if it clears up my confusion.


  9. Dear Mike,

    yours is an opinion that is widespread in the general public but it's a misconception.

    The violation of Bell's inequalities in Nature implies that either locality fails, or realism in the sense of classical physics fails in Nature, or both.

    When one looks at other types of data, it becomes totally unequivocal that locality holds while classical realism fails in Nature.

    I am not sure whether the paper by Deutsch is sensible but I am sure that there are way too many writings by the same author that are totally wrong so I couldn't recommend you discussions of conceptual questions written by this author in general.


    1. Who (or what) would you recommend? Is there a Dummies Guide to Quantum Physics? :) I don't need to see the equations or proofs, but a simplistic explanation would be nice.

      Thank you!

  10. You are just envious man. This was a lecture for 12 year-olds. That is why he is communicating science and you are not. Your imagination is taking leaps your ego cannot cash. He never said anything of what you say here. So what, instead of saying quantum state he said energy to 12 year-olds. Big deal...

  11. The correlation that gives rise to Pauli's exclusion principle has always confused me.

    Brian Cox's statement that manipulating electrons here here all electrons must be wrong, even if we generalise his statements to quantum states.

    I am having a hard time understanding this in the context of the permanent correlation among all electrons mentioned above.

    Assume an electron at the edge of the universe is described by a set of quantum numbers "A". If, at some earlier time, we manipulate an electron on earth such that it had the same set of quantum numbers "A", this would "affect" the electron at the edge of the universe such that it could never have obtained set "A" as a description,since that state is now already occupied.

    Where am I going wrong?

  12. Typo mistake: The second paragraph in my last post should have been "Brain Cox's statement that manipulating electrons here affects all electrons instantaneously must be wrong, even if we generalise his statements to quantum states."

  13. For posterity:

    I think I have resolved my issue after reading this paper and reading a few more of your blog entries.

    The probability of an observer measures any eigenvalue X is always

    P(X) = P(X | Brian rubs diamond) + P(X | Brian doesn't rub diamond)