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Time crystals would be a perpetuum mobile

One of the widely shared recent articles at was

Time crystals—how scientists created a new state of matter
three days ago. The text claims that Frank Wilczek's 2012 idea about quantum time crystals has been experimentally proven to be right. Not bad. Time crystals have previously attracted some funding from Microsoft, too. Great. Frank Wilczek is playful, smart, and cool but I find this whole industry to be nothing else than a children's game meant to fool themselves. What has been seen is completely trivial while Wilczek's claims that were actually new and provoking are demonstrably impossible.

What's going on?

A normal crystal may have atoms or molecules at regularly spaced places (a lattice)\[

(x,y,z) \in \ZZ^3.

\] In some units or a coordinate system, the three coordinates are integer-valued. This setup breaks the group of spatial translations from the continuous group \(\RR^3\) to the discrete subgroup \(\ZZ^3\), assuming that the crystal is infinite. Now, Wilczek's general idea is that he wants the same "symmetry breaking" to be applied to the translations in time, too. Effectively, his new "material", the quantum time crystal, is doing something special – or reaching the maximum value of some observable – at moments \(t\in \ZZ\) in some units, too.

Already at this point, you may see a misconception that leads to Wilczek's impossible proposals. There's a very general implicit problem in his usage of the word "material" or "object" for something that has some properties at different moments of time. Sorry but a "material" or an "object" is fully described by some information at a single moment of time, e.g. \(t=0\). If you need to talk about the values of observables at many or all values of time \(t\), then you are not talking about a "material" or "object" but rather a process. The misguided analogies between "crystals" and hypothetical "time crystals" may be said to result from this confusion mixing objects and processes. By the way, there were lots of the very same confusion in the literature about S-branes (Strominger or spacelike branes).

But let us look at the problem from a slightly different, although basically equivalent, angle.

More conceptually, Wilczek's time crystal is defined as an object that has the property that in its ground state (the state of lowest energy), an observable that we may call \(x(t)\) is a non-constant or periodic function of time. Something keeps on spinning indefinitely.

There exists a name for a hypothetical object that is oscillating indefinitely. The name is perpetuum mobile, or a perpetual motion machine of the first kind. In his original paper, Wilczek is aware of this point and acknowledges that his gadget therefore perilously close to fitting the definition of a perpetual motion machine.
Well, he's far too modest here. His gadget wouldn't be just close to a perpetuum mobile; it would be one. Wilczek just rebranded the perpetual motion machines just like cold fusion was rebranded as "low energy nuclear reactions" and creationism was rebranded as "the intelligent design". The main difference between the millions of "inventors" of perpetuum mobile in the past and Frank Wilczek is, we are led to believe, that Frank Wilczek is really smart and a Nobel prize winner and so on, so unlike the numerous losers and/or crackpots before him, he actually succeeded.

Design by a predecessor of Wilczek's.

The new papers don't show anything of the sort, I am confident although I haven't read them in their entirety. They just present some atomic physics systems that respond periodically when they're stimulated by some periodic pulses of laser light or something like that. What a shock that sustained, periodic external influences lead to sustained, periodic responses. As far as I can see, these "insights" are absolutely trivial and absolutely don't justify the statement that Wilczek's claims were shown true. I don't have enough motivation to read these papers because I find it obvious that they're just papers by clueless experimenters who observe something, they don't understand what they're actually seeing, and they say that it agrees with some theorist's wrong paper.

Shortly after Wilczek published his 2012 paper, exchanges between Wilczek and Patrick Bruno, a critic who indeed said that Wilczek's objects are impossible because they're perpetual motion machines, began. I guess that this 2013 paper with a no-go theorem was the last salvo by Bruno in his battles. Watanabe and Oshikawa added another no-go-theorem in their 2014 paper.

Instead of discussing any specifics of the experiments that just ignore all these results, let me say what I think is the deep theoretical misconception that led Wilczek to say all these things.

He clearly thinks that the spatial translations and temporal translations are analogous – after all, space and time are linked by the Lorentz symmetry in special relativity and they are naturally unified to spacetime translations – and because it's possible to spontaneously break the spatial translations (by creating crystals), it must be possible to do the same with temporal translations (by creating time crystals), so the only remaining task is to decide how to do it nicely.

But this "complete democracy" between space and time is wrong for a simple reason.

The reason is that things in the spacetime evolve and the observables (e.g. fields) aren't quite independent in every spacetime point. At most, you may determine the initial conditions e.g. at a spacelike hypersurface \(t=0\). Once you determine the fields (and their derivatives or canonical momenta) at \(t=0\), their values in the whole spacetime are determined by the field equations of motion. The surface where you can pick the initial conditions is referred to as the Cauchy surface and even in general relativity where many things are flexible, there exist very good reasons why this surface should better be space-like and contain no timelike vectors in it.

In quantum field theory, the reason why timelike Cauchy slices would be no good is simple: the field equations guarantee that timelike-separated fields almost always have nonzero commutators. So you simply can't determine these values independently because of the uncertainty principle!

Because the Cauchy surface is spacelike and not timelike, the "complete democracy" between space and time is broken. The equations of motion and commutators etc. are still Lorentz-covariant but the required spacelike signature of the Cauchy surfaces implies that you have much less freedom to determine how things depend on time than the freedom to decide how things depend on the spatial coordinates. And that's probably the key point that Wilczek is overlooking.

So if you choose the most general object that may exist in the spacetime, you may always determine it by some information at a Cauchy surface which is morally equivalent to the three-dimensional spacelike \(t=0\) hypersurface. And whether or not its evolution in time will be constant or non-constant, periodic or aperiodic, and damped or not damped, is completely determined by the dynamical laws of your physical theory. You simply cannot prescribe these things.

That differs from the case of ordinary crystals where you have the freedom to distribute the atoms to the lattice sites in the three-dimensional space. But you simply don't have the freedom to dictate whether some peaks reappear periodically after every period \(\Delta t\). The question what happens is dictated by the dynamical laws of physics.

Another pre-Wilczek model of a classical time crystal. Such concepts remind me of the big-government leftists. It's spectacularly clear that the more redistribution or more moving parts you add, the more energy (or money) the process will cost, waste, or demand for the machinery to run. But they always ignore or understate some energy cost, driven by the unchangeable belief that the perpetuum mobile or the big government is a great idea

OK, do the dynamical laws of physics allow the ground state of an object to indefinitely oscillate, to have an observable \(x(t)\) that is non-constant, like \(x_0\cos\omega t\)? Let's use the elementary rules of quantum mechanics to say something about this question. Well, it's easy. If the state \(\ket\psi\) is a ground state, it really means that it is an eigenstate of the energy operator\[

H\ket\psi = E_0 \ket\psi

\] where \(E_0\) is the lowest eigenvalue in the whole spectrum. But we don't even need to know that it's the lowest one – although this was the statement that Bruno – focusing on particular systems proposed by Wilczek – was proving incorrect. I only need that the state is an eigenstate. As undergraduate students learn in the first lectures of quantum mechanics – when they are taught about the time-independent and time-dependent Schrödinger equation – the evolution of the energy eigenstate in time is unavoidably stationary,\[

\ket{\psi(t)} = \exp(Ht / i\hbar) \ket{\psi(0)}.

\] Only the overall phase of the state is changing with time. That phase has no physical consequences (at an isolated moment of time) and it cancels in all the expectation values etc. which are therefore constant:\[

\bra{\psi(t)} x(t) \ket{\psi(t)} = {\rm const}.

\] So the ground states are simply stationary and no observable that may be measured in them may oscillate. Period. That's it. There are no quantum time crystals.

If some expectation value in a state depends on time, the state must unavoidably be a superposition of many energy eigenstates corresponding to different eigenvalues of the energy. You may split the state to pieces and pick the lowest-energy eigenstate contribution in it. And this true ground state will be stationary.

Even though the relativistic equations respect the Lorentz symmetry which is some kind of a "perfect" symmetry between the space and time, it's still true that relativity doesn't question the qualitative difference between objects that are timelike and objects that are spacelike. Indeed, whether e.g. a spacetime interval is timelike or spacelike is a question that all inertial observers will agree upon – the invariant squared length of the interval is either positive or negative and its value is Lorentz-invariant, a reason why Einstein was tempted to use the term Invariantentheorie for the theory that we know as the theory of relativity.

So world lines of true objects in consistent theories have to be timelike (or at most null) and not spacelike. This asymmetric treatment of the two signs is not in any conflict with the Lorentz symmetry because the Lorentz transformations do preserve the qualitative (timelike vs spacelike) character of intervals. For the same reason, one may consistently choose the initial conditions at spacelike Cauchy surfaces, but not surfaces of a mixed signature. As I mentioned, this difference boils down to the fact that fields' commutators vanish at spacelike separation but not timelike separation, so one can only determine the fields independently at spacelike hypersurfaces. Similarly, one can observe or build systems that spontaneously and permanently break the translational symmetry in space but not those that spontaneously and permanently break the translational symmetry in time.

Unless I am wrong, Wilczek's reasoning is probably rooted in "overinterpreting" the Lorentz symmetry in a certain way. Alternatively, we may say that people with common sense know that a perpetuum mobile is impossible – and within quantum mechanics, this fact is just demonstrated using a different formalism. It could of course be conceivable that quantum mechanics changes things so radically that it could allow the perpetual motion machines – after all, it allows the quantum tunneling and many other things that are impossible classically. But in the case of the perpetual motion machines, it just isn't the case.

Spinning nuclei

Under a 2012 blog post about the electron's electric quadrupole moment – which has to be zero (like the tensor properties of all particles with \(j=0\) or \(j=1/2\)) by the Wigner-Eckart theorem – someone asked how it's possible that people often say that uranium-238 is cigar-shaped i.e. has some "quadrupole moment" even though it has \(j=0\). He has also mentioned two \(j=1/2\) nuclei that are sometimes hinted to have a non-spherical shape, too.

It's confusing but if you really had just the state with \(j=0\) and nothing else, the Wigner-Eckart theorem – a general group-theoretical consequence of the addition of the angular momentum in quantum mechanics – would require all (traceless symmetric) tensors to be zero. That includes the ordinary and electric quadrupole moments. The spin \(j=0\) or \(j=1/2\) mean that the particles doesn't even carry enough spin-related information to remember the (sign-independent) axis along which it could be elongated or shrunk.

What's new about the nuclei is that they're composite which means that they have lots of excited states (describing various types of relative motion between the protons and neutrons – or quarks and gluons). In particular, there are states that look like the extra "orbital motion" added to the nucleus' spin. So aside from the \(j=0\) ground state, there is a \(j=2\) and \(j=4\) and \(j=6\) excited state of uranium-238. The dependence of the energy on the angular momentum goes approximately like \(a+bJ^2\) which allows you to extract something like a "moment of inertia" from \(b\). But this \(b\) isn't quite identified with the expectation value of any tensor in the state with \(j=0\) itself – that expectation value simply has to be zero.

Also, the states of uranium-238 with \(j=2,4,6,\dots\) are "excited" which means that they won't survive forever. These excited ("spinning") states of nuclei will emit a photon (gamma-ray) and drop to a lower value of \(j\) very quickly – they will reach the true ground state with \(j=0\) almost instantaneously. The energy and rates of transition of these jumps may also be used to deduce some nonzero values of the "quadrupole moments" (where the quotation marks indicate that you must be careful about the definition of the object because it's a generalization that doesn't necessarily coincide with other meanings of the phrase) – especially if it is true that the dominant emission is some electric quadrupole radiation. But if the transition results from some quadrupole radiation, the transition is determined from the matrix elements such as\[

\bra{\text{U-238}, j=0} Q_{ij} \ket{\text{U-238}, j=2}

\] which may be nonzero but they related two states of the nucleus. The matrix element above isn't an expectation value, it isn't a property of one state only, especially not the ground state.

So Wilczek's perpetuum mobile doesn't work for the nuclei, either. If the nuclei are spinning in a way that has some "classical component" – in the sense that some expectation value of an observable would be time-dependent – then they are superpositions of many energy eigenstates and the higher ones will collapse to the lower ones (e.g. by the emission of gamma quanta). At the end, you are left with the true ground state that simply has to be stationary.

For this reason, the temporal translational symmetry cannot be spontaneously broken, at least not in the sense envisioned by Wilczek. Note that the previous paragraph talks about the collapse to a lower state. This description works because the spectrum of energy is bounded from below. That's how it differs from the spatial counterpart, the momentum which is unbounded – both signs of any component of the momentum are equally good. You could view this "boundedness from below" as another example of the "qualitative differences" between spacelike and timelike entities in relativity. The combination \(k_\mu p^\mu\) of components of the energy-momentum operator \(p^\mu\) has a spectrum that is bounded from below exactly if \(k_\mu\) is timelike i.e. if \(k^\mu k_\mu\leq 0\), assuming the timelike (mostly minus) signature.

This "discrimination against" the timelike operators – and timelike crystals – is totally compatible with the Lorentz symmetry because the Lorentz symmetry only allows to transform spacetime intervals (or components of vectors or tensors, or slices, or other entities) of the same qualitative type to each other.

Another topic: LIGO and axions

Adrian Cho at Science discusses an April 2016 paper by Arvanitaki, Dimopoulos, et al. that just appeared in PRD.

When advanced LIGO sees thousands of black hole mergers in coming years, they say, it could also see signs of axionic (dark matter) waves created with the help of the black hole horizons – assuming that the axion mass is a picoelectronvolt, plus minus (or times over?) two orders of magnitude. Some smart folks say that it's a more exciting new thing that could be observed by LIGO than all the previous "possible future discoveries".

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