Locality of QFTs and Bell's Inequality

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Lubos Motl

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Jun 4, 2001, 1:34:46 PM6/4/01
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Hello!

I found the newsgroups using lynx and I am trying to reply via pine for
the first time. :-) BTW have you seen John Baez's paper on octonions? I
hope you find it lovely, too.

> It is important to understand that this is a matter of philosophical opinion
> and not scientific fact. Quantum mechanics violates classical definitions of
> locality. Bell proved it and Eberhard proved it without reference to

This is not the way in which physicists think about the question of
locality. Quantum mechanics does not satisfy the axioms of a classical
theory and therefore the classical definitions of locality are not
relevant.

In order to judge whether a theory is local or not, one must use the
correct physical definitions - i.e. quantum mechanical ones. And by these
definitions, quantum field theories such as the Standard Model are "local
quantum field theories" even though they are fully quantum mechanical
theories that violate Bell's inequalities, for example. Everyone calls
them this way and we know why.

Just formulate a quantum field theory - Klein-Gordon field theory for the
sake of simplicity - in the Heisenberg picture. You have something like
(Box + m^2) Phi = 0. Right, Phi is a quantum mechanical field-operator
satisfying some commutation relations and one can make probabilistic
predictions only (and calculate expectation values of various
observables): the rules of quantum mechanics cannot be broken.

But nevertheless the equation is the same equation as the equation
satisfied by the *classical* Klein-Gordon field - for example it is
Lorentz invariant. Therefore the value of the operator at some point in
spacetime (and various average values, probabilities etc. that one can
construct out of it) is determined purely by its values in the point's
past light cone. Because of this and because the commutators such as
[Phi(x),Phi(y)] vanish for x,y spacelike-separated, all the predictions
what happens at point X in spacetime (probabilities, average values of
something etc.) depend only on the inputs in the past light cone of X.
This is why we say that Quantum Field Theories such as QED are local. This
is also why information cannot be transmitted by superluminal speeds.

> hidden variables. ("Bell's Theorem without Hidden Variables",
> _Nuovo Cimento_, V 30 B, pp 75-89, 1977). Quantum mechanics is

Einstein never accepted the probabilistic nature of Quantum Mechanics.
When I was 17, I was his faithful follower. There have been obviously very
many people like that, many of them much older. John Bell, another
follower, wanted to prove that the question whether the probabilistic
interpretation of QM is plausible or whether we must return to the
"complete" (classical) description of reality can be decided
scientifically (people used to say that such a question could never be
tested). And therefore he finally discovered the inequalities saying that
(if we assume basics of classical logic) in every experiment, the
correlations between certain quantities must be smaller than his bound.
Because QM was predicting generically higher correlations (as was known
already to Einstein, in fact), Bell believed that when such experiments
are performed, they must inevitably give results incompatible with QM and
hidden variables would win the war.

Unfortunately for Bell, the experiments have been done (some of the most
obvious ones have been done recently in the context of "quantum
teleportation") and they showed clearly that predictions of QM worked very
well and Bell's inequalities are really violated in Nature. Bell was very
unhappy till the end of his life that instead of killing QM, he offered a
powerful tool to rule out the theories of hidden variables.

Bell's inequalities are violated in Nature, so what are the wrong
assumptions of Bell's proof? Well, Bell assumed the "classical logic",
just like Einstein. If two photons from a decaying positronium are far
from each other, each of them must be in some "real" state, independent of
the state of his friend. But QM dictates otherwise. The maximum we can say
about the photons is their wave function that can contain correlations
between the two photons, they can be in the state |RR>+|LL>, for example
(both right-handed or both left-handed). Although it is not clear which
polarization of the first photon will be measured (the likelihood is
50:50 for L:R), we can be sure that the result will be the same as for the
second photon.

Such "remote correlations" in the wave function can sound "nonlocal" but
they do not allow to send information faster than light - simply because
none is able to command our photon in which state it should be measured.
If this were possible, because of the correlation we could also command
the remote photon to be measured left-handed (or right-handed) and we
could send information almost immediately. But in the real world, it's not
us who decides about the results of the probabilistic experiments but
rather (undefined) God throwing dice (and ask about his Majesty is beyond
the scope of science). Locality is saved. If we ignore "our" photon and
study only the remote one, the probability for it to be left-handed or
right-handed will be 50:50 regardless of what we do. Forget the
superluminal Internet. In the world of hidden variables this would be
possible, (maybe violating relativity and maybe causality?), in the real
world it is not possible.

Such "remote correlations" exist also in classical physics, in fact. Bell
talks about the Bertlmann's socks. One of them is red, one of them is
blue, and when Mr. Bertlmann sees a red sock on his left foot, he can be
sure that the right foot has a blue sock on it (and vice versa). If
something chaotic happens in the morning, we can still describe the state
of Bertlmann's socks by the classical probabilistic distribution
"50% for left-red and right-blue and 50% for left-blue and right-red".

The only difference in Quantum Mechanics is that the probabilistic
distribution must be derived from a complex wave function (that allows for
the interference, higher correlations etc.) and that there does not exist
a more accurate "classical" description of reality. In fact, the
observation that Bell's inequalities are violated together with requiring
Lorentz invariance IMPLY that the world must be probabilistic as I have
indicated.

The rules of Quantum Mechanics are weird. If thousands of philosophers
tried to invent the strangest thing possible, they would have bever
discovered a thing as strange as Quantum Mechanics (Sidney Coleman).
Nevertheless the current facts are too obvious and the proponents of
hidden variables etc. should finally give up. Dirac, Feynman, von Neumann,
Bohr, Born etc. were sure which interpretation was correct and they did
not need so impressing experimental data which we have today.

Furthermore the Copenhagen interpretation was modernized - we can talk
about the "consistent histories", there is already no conceptual
difference between the micro- and macro- objects. One can derive special
properties of macro-objects from the first principles (decoherence etc.),
see Omnes, Hartle, Gell-Mann, Zurek, Griffiths etc. For me personally, the
question whether we can return to the age of "classical logic" is
certainly closed for many years. And I am sad that such a guru of physics
in the 70s, Gerardus 't Hooft, is spending his time with a similar
deterministic religion.

Another topic is locality in quantum gravity and string theory etc. I
believe that QG inevitably violates some rules of locality, otherwise one
can derive Hawking's loss of information in black holes etc. But this is a
completely different level of thinking which can hardly restore the
paradigms of classical physics, especially because the exotic features of
QG are irrelevant in the realm of atomic physics where the laws of QM work
very well.

Best wishes
Lubos
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E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
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Superstring/M-theory is the language in which God wrote the world.

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