Monday, March 12, 2012

Brian Cox and lunar phases

Prof Brian Cox is a physicist who was the main presenter of the one-hour-long 2011 BBC special, "Guide to the Moon", among many other programs. You may expect him to know something about the Moon; you may also be wrong. How does it work in practice? Steve Davis pointed out the following cute story; see the Student Room:
Trailer for a radio programme on BBC Radio Wales this morning.

It's a phone-in item for programme guest Prof. Brian Cox.

Little kid (sounded 5 or 6 years old) asks:
Why is the moon sometime round and sometimes looks like a banana? (As near as I can remember the exact question)

Prof Cox: "That's the shadow of the earth." :-(

Now that was either some really bad editing of the programme for the trailer (I sincerely hope so) or some really bad physics. (I sincerely hope not.)
I will explain where the phases of the Moon come from later although most readers know the reason and most of those who don't have understood it from the picture above. But at this moment, let me mention that the unexpected answer wasn't due to bad editing. He did say it and the January 28th, 2012 trailer above actually wasn't the first time when he did. See a discussion at Swanson's blog.

In his comment, Brian Cox wrote lots of wrong things about locality in quantum mechanics and the Pauli principle but he also mentioned the lunar story:
But here is the point I want to make. Everybody goofs, as Sean said in his initial tweet when this debate arose. I goof – as Stevie C points out in the comments here, I apparently said in a [January 19th, 2012 BBC Radio Wales] radio interview that the phases of the Moon are caused by the Earth’s shadow, which is clearly bollocks! Unless I was talking about a lunar eclipse, I can’t understand what I must have been thinking. Probably the end of a long day.
He continues to say that everyone goofs etc. Well, I am not sure whether everyone goofs about the origin of the lunar phases or similarly elementary questions, especially among physics professors who have claimed to be experts on the subject. But I do appreciate that Cox confessed his sins and I do agree that people, including smart ones, may sometimes err and the probability for a smart person to err doesn't really go down when we talk about elementary topics.

Instead, I am pointing this example out because many people are capable of understanding that Cox's answer is wrong. Cox's comments about the Pauli principle and a hypothetical non-locality in quantum mechanics are wrong to an exactly the same extent except that, unfortunately, the percentage of people who understand the Pauli principle and the reason why locality is respected by various quantum systems is much lower.

Because too many people misunderstood my technical explanation, the comment about the lunar phases is the simplest argument I see to argue that Brian Cox often says things that are bollocks. This argument isn't perfect but unlike the technical one, it may be more accessible to the laymen.

Lunar phases

The lunar phases have nothing to do with shadows, dead kir and dear Brian Cox. The Earth could only leave a shadow on the Moon if all these three celestial bodies, the Sun, the Earth, and the Moon, are located on a single straight line and the Earth is in the middle. But this occurs very rarely. When it does, there's indeed a shadow of the Earth on the Moon and it's called lunar eclipse. (The solar eclipse occurs when the three bodies are on the line but the Moon is in the middle and prevents us from seeing the whole Sun.)

But the eclipses are very rare. They don't even occur once per month because the plane in which the Earth orbits the Sun and the plane of Moon's orbit around the Earth don't coincide; they're tilted relatively to one another so the celestial bodies which is why they usually miss the "conjunction".

Most of the time, the three bodies are not located at a line at all. Instead, they're vertices of a triangle. The Sun acts like a light bulb that can make an otherwise dark Moon look bright. If this light bulb is located somewhere on the left side from the Moon (imagine that you are looking at the Moon from the Earth), the left half of the moon will be illuminated and the right side will remain dark; the Moon will look like a "C". If the Sun is on the right side, the left half will remain dark while the right side is illuminated and the Sun will look like a "D".

When the Sun is approximately behind the Moon, it only illuminates the half of the Moon that we can't see anyway (because it has turned away from us) so the half of the Moon on our side remains dark; that's the new moon. However, when the Sun is located in the same direction from us as the Moon (however, the Sun is much further), it's usually a day over here and the direct sunlight beats the light reflected from the Moon so we don't see any Moon, anyway. If the new moon is "exact", which rarely takes place, we get the solar eclipse.

When the Sun is approximately on the opposite side from the Earth than the Moon and if we can see the Moon, we can see "our side" of the Moon (it's night right now because the Sun is beneath the horizon) which is the same side that is illuminated by the Sun, too. We see the full moon; the whole Moon is a bright disk. As the Moon keeps on orbiting around the Earth (and the Earth plus Moon revolve around the Sun, but the latter motion is slower than the lunar motion), the lunar phases alternate. So we may also see something in between a "C" and a new moon, i.e. a thin banana, and so on.

Back to quantum mechanics

This was a silly discussion, wasn't it? My point is that there exists an understandable answer and it dramatically differs from Brian Cox's answer.

Most of the discussion on Swanson's blog is dedicated to the original disagreements concerning quantum mechanics. Many people say many wrong things and many irrelevant things.

For example, Brian Cox mentions that the Feynman propagator is nonzero even if the two points are space-like-separated, i.e. \[

G_F(x,y)\neq 0\text{ for } (x-y)^\mu (x-y)_\mu \lt 0.

\] He thinks that it's an argument in favor of his original statement that by playing with a diamond, he instantly changes the energies of all electrons in the Universe because such a change is demanded by the Pauli exclusion principle. However, the Feynman propagator proves nothing of the sort. Instead, the nonzero value of the Feynman propagator at space-like separations means that there's an entanglement, i.e. correlation, between the fields at the two points. Why? Note that the Feynman propagator is \[

G_F(x-y) = i \bra 0 T [\Phi(x)\Phi(y)] \ket 0.

\] It's the expectation value of the product of fields at these two points \(x,y\). Well, these two operators should be time-ordered by \(T\); it's a technicality saying that the field at a lower time – i.e. lower value of \(x^0\) or \(y^0\) – should appear on the right side. But if \(x,y\) are space-like-separated, the ordering doesn't really matter because the operators commute at space-like separations. In fact, this vanishing of the commutator at space-like separations is the key fact that prohibits "instant communication" envisioned by Cox in his diamond story.

If we don't discuss the commutator but the separate (time-ordered) product, as seen in the Feynman propagator, it tells us about a correlation. How does it work? The point is that \[

\bra 0 \Phi(x) \Phi(y) \ket 0

\] is positive for space-like-separated points \(x,y\). It means that the values of the fields at these points, \(\Phi(x)\) and \(\Phi(y)\), tend to have the same sign. Why? The vacuum is full of random oscillations of fields which is the reason why our measurements (of the values of the fields) tend to have somewhat random outcomes; it's quantum mechanics, stupid. Note that the fields and their time derivatives can't be both strictly zero because that would violate the uncertainty principle.

However, these "random waves in the vacuum" also occur for wavelengths that are long. And if there is a wave in the field \(\Phi(x)\) whose wavelength is long, it means that the value of the field at two nearby points – that are (much) shorter than the wavelength – will be very close to each other. If they're nonzero, they will have the same sign. So there's this correlation between the values of quantum fields in the vacuum. I can make the statement that the correlation exists even though I can't say (in advance) what the shape of the random wave exactly is; in fact, the point of the term "correlation" is that it is used in the statistical context when we can't predict the individual outcomes.

Because this correlation is studied in a quantum theory, we should really call it "entanglement", not just "correlation", but it's really meant to be exactly the same thing. Quantum mechanics is a different theory than classical physics so it says different things about the correlations of different quantities, and how correlations between some properties may co-exist with correlations in others and how strong these correlations may be. However, at the end, the interpretation is still the same: entanglement is a correlation between properties of two parts of a larger system.

However, this nonzero value of the Feynman propagator, while proving correlation, doesn't mean that any faster-than-light communication is possible. The correlation is nonzero not because the regions would be communicating "right now". Instead, it is nonzero because the fields at the two points \(x,y\) have evolved from fields at earlier points and one may find "common ancestors", i.e. points \(z\) that belong to the past light cones of \(x\) as well as \(y\). As the quantum field was trying to minimize the energy – by sending all excitations and excessive local disturbances away (this is just a different way of saying that particles in the region were escaping) – it was establishing a particular correlation between different points that is characteristic for the ground state of the quantum field, the vacuum.

So the nonzero Feynman propagator at space-like separations isn't a consequence of some "action at a distance"; instead, it is a result of a mutual contact of the two "regions" in the past. In the same way, the nonzero value of the quantum field isn't a prerequisite of a nonlocal behavior, either. Assume that \(x^0\lt y^0\) and the measurement of \(\Phi(x)\) makes some impact on the properties of the field around the other point, \(\Phi(y)\), which is space-like-separated.

It would mean that the probabilistic distribution for \(\Phi(y)\) would have to depend on whether or not we made the the measurement of \(\Phi(x)\). However, it doesn't depend because \[

\Phi(x)\Phi(y) = \Phi(y) \Phi(x)

\] as I mentioned earlier. The commutator of space-like-separated operators vanishes. On the left-hand side, the operator \(\Phi(y)\) is on the right side of the product so it acts on ket vectors before \(\Phi(x)\) which is the factor on the left side. However, the product is the same, regardless of the order, which means that the values of \(\Phi(y)\) don't depend on whether we did a measurement of \(\Phi(x)\) before or not.

Of course, this independence is required by the special theory of relativity. If \(x,y\) are space-like-separated points in your spacetime, the question "which of them is earlier" depends on the inertial system. In some reference frames, we may have \(x^0 \lt y^0\). In others, we will have \(x'^0 \gt y'^0\). So the physical results mustn't depend on questions how we order \(\Phi(x)\) and \(\Phi(y)\) in the products: there's no God-given ordering and quantum field theory must be able to yield some expectation values of the product in a given spacetime without any reference to a particular inertial system.

(I should also discuss that that the Hamiltonian, the operator dictating how the system evolves in time, contains pieces governing the evolution of the regions around \(x\) and around \(y\) and they also commute with one another which guarantees the independent evolution of both subsystems or regions in time.)

This wasn't necessarily the clearest or most explicit presentation of the reason and one could do better. Be sure that the conclusions won't change. The quantum fields at space-like separations commute with each other. And this vanishing commutator (or anticommutator in the case of pairs of fermionic operators) is a necessary condition for the independent evolution of fields at space-like separations. This absence of an "action at a distance" is required by the special theory of relativity because if such superluminal influences existed, they would be equivalent – by switching to a different reference frame – to actions affecting the past. That would break causality and allow us logically contradictory "closed time-like curves" in a Minkowski spacetime. (One could find a way to kill your grandfather before he had his first sex with your grandmother, you know the story.)

I also need to emphasize that there is no "inevitable" contradiction between quantum mechanics and special relativity. In fact, quantum field theory – discussed above – is the simplest class of theories or framework (string theory is the only other framework with this property that we know) that perfectly respects the principles of special relativity and perfectly respects the postulates of quantum mechanics, too.

Some people's idea that quantum mechanics actually "sends signals faster than light" or "instantly influences space-like separated regions or electrons" is just flawed. It is typically a result of these people's trying to imagine a "classical model" that "emulates" quantum mechanics. Such a classical model may indeed require nonlocal interactions and they would violate the principles of special relativity.

However, the real world isn't a classical model of quantum mechanics. The real world is an intrinsically, genuinely quantum system. It's not a classical caricature, it's the real quantum thing, stupid. It doesn't need (and it doesn't allow) any superluminal communication whatsoever. The contradictions with special relativity only exist if we demand that the world isn't quantum; it's a classical model of a quantum world. But the world isn't a classical model and that's why these contradictions evaporate. For this reason, special relativity and quantum mechanics are "on the same side of a barricade". Special relativity is another argument why we must accept that quantum mechanics with its positivist, probabilistic rules is the only right description while any attempt to "emulate" quantum mechanics by something that isn't truly quantum is just wrong.

And that's the memo.

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