A Quick Visual Intro to QED
Terry Bollinger - 2008-11-23

– Part 1 of 2 –

1. The Question: Why Waves Here, and Particles There?

On November 10, Neil Bates asked a difficult question as part of the Discover Magazine “Quantum Hyperion” physics thread. My paraphrasing of his question is this:

“Why does a photon behave like a wave when it encounters beam splitter, but like a particle when it encounters a particle detector?”

Below is my attempt to answer this question. Since this will be a bit long, I’ll break it up into two parts.

The first part (this one) deals with the mystery of the coupling constants, or what I refer to as the roulette wheels down at the bottom of quantum mechanics.

In the second part I’ll discuss the clockwork photon. This is my adaptation, with a few visualization updates, of Feynman’s explanation of QED. My goal in Part 2 is to show how geometry transforms the simple probabilities of coupling constants into the richness of the physical world that we see all around us.

To make Part 2 more specific, I’ll include a thought experiment in which a single material, silver, both reflects a photon as if it were a wave, and in another part of the apparatus absorbs it as if it were a particle. Using a single material for both components emphasizes the critical role that geometry plays in understanding quantum mechanics.

2. The Roulette Wheel at the Bottom

The best non-mathematical reference to the question of why quantum mechanics sometimes gives wave-like results and sometimes particle-like results is, without qualification, Richard Feynman’s “QED: The Strange Theory of Light and Matter.” I recommend it highly for anyone interested in the more mysterious aspects of how quantum mechanics works.

In his book, Feynman quickly informs the reader without apology that he will not try to explain why reality is ultimately probabilistic. His reason is simple: Although quantum mechanics enables very accurate predictions of how particles such as electrons and photons will interact when in large groups, there is no accepted theoretical explanation for the ultimate source of the probabilities that are intrinsic to such models.

An analogy is that there is a sort of roulette wheel at the very bottom of the physics of electrons and photons. This wheel is spun every time we ask a question about a specific electron and a specific photon, but the details of its construction remain a complete mystery to us to this day.

(To be complete, I should mention that there are actually several such roulette wheels in physics, which are collectively known as “coupling constants.” Only the coupling constant for electrons and photons has much impact on everyday physics, however, so that is the only constant I will discuss here.)

Spinning the roulette wheel for an electron and a photon results in one of two outcomes: “interact” or “ignore.” (A warning: There are some complications in how these values are used. I’ll describe those complications later, in Part 2.)

For photons and electrons, an accident of physics history bequeathed the corresponding roulette wheel with the highly uninformative name of “fine structure constant.” Fortunately, there is another name for it that is much more intuitive: It is the charge of an electron, expressed in certain universal units.

Specifying how much electrical charge an electron has thus is just another way of describing the odds that the electron will interact with a passing photon. A point particle with no charge would ignore such a photon entirely, since its roulette wheel would be rigged only with slots marked “ignore.” Such a particle does exist. It is called the neutrino, and it is rigged in just this way. Because a neutrino cannot see photons, it passes through ordinary matter pretty much as if it wasn’t even there.

The roulette wheel that corresponds to the charge of an electron has a surprisingly small number “interact” slots. The number is about 1 in 137, or less than 1%. This small probability is nonetheless just the right size to give rise to all of the remarkable complexity that we see and interpret as non-nuclear physics and chemistry.

Finally, I cannot emphasize enough that the underlying design of these roulette wheels — these coupling constants of standard physics — is unknown. Attempts to postulate “gears and wheels” to explain these probabilities always seem instead to end up adding complexity without adding any new insights — the sure sign of a bad theory. This is one of those interesting cases in physics where the most mathematically abstract model, in this case a simple probability function, stubbornly remains the best one available. This is true both in terms of overall simplicity, and in terms of its ability to produce verifiable experimental predictions. The probabilistic nature of coupling constants thus remains a true mystery, one into which the physics of Feynman’s time (and I would argue ours also) produced no significant insights.

2. Charge and the Anthropic Principle

I should mention this seemingly arbitrary setting of the photon-electron roulette wheel at 1 in 137 is quite special in some unexpected ways. For example, if you increased it to 1 in 135 or lowered it to 1 in 138, it’s a pretty good bet we would not be having this dialog. The problem is that the ability of carbon to form indefinitely long chains is closely linked to this number, and if you change it even slightly, organic chemistry would likely stop working well enough to support the existence of constructions such as the proteins necessary for life.

As it turns out, pretty much all of the fundamental constants of physics seem to work that way. That is, if you make these seemingly arbitrary numbers just a larger or a little smaller, you still get a universe of some sort, but one that no longer supports organic life as we know it. Or to put it a bit more graphically, nudging fundamental constants is a lot like kicking the foot of a juggler who has ten plates and twelve hoops all spinning at once: Everything comes tumbling down.

This curious link between fundamental physics and life-supporting organic chemistry is called the anthropic principle, and it is one of the most fascinating mysteries of current physics. It is a topic for another time, however. I just did not want to leave an incorrect impression that the value of the electron charge could have been set arbitrarily to almost any value. It is instead fine-tuned in ways that are unexpected and deeply interwoven with the other fundamental constants of physics. Developing a full and convincing explanation of this fine-tuning constitutes one of the great ongoing challenges of fundamental physics.

– End of Part 1 –

=========================================================

My response is below, adapted to the current thread. BTW the discussion in the thread “What’s the Matter with Making Universes?” is directly pertinent to the one here, why not get some word in there too?

I propose Bates’ Corollary to Warnock’s Dilemma: the problem of interpreting a lack of response to a comment in a thread and not just a post. I also propose Bates’ Ancillary Dilemma: why do respondents (repliers? sorry) address some of the key points made in a post or comment, and not others - even if the poster/commenter pleads or insists, and even repeatedly, that the unanswered points are relevant or even more relevant? I have in mind, in the thread http://scienceblogs.com/principles/2008/11/whats_the_matter_with_making_u.php#commentsArea, that [no one here AFAICT] would address my concern about why “collapse” (or whatever) happens so far downchain in the interaction of say a photon, instead of earlier. In particular, why doesn’t the interaction with an initial beamsplitter cause a photon to just collapse and go one way or the other, instead of indeed “splitting” the single photon wave to enable subsequent interference. But then, at the detectors at the far end of the MZ interferometer etc., there is a “hit” at one or the other detector. Er, maybe if [anyone, such as LBC, Terry B?] is reading this comment, you could reply to that question? I thank you in advance for your cooperation .

BTW Lawrence I can be hard on people putting forth what I consider contrived and rationalized attempts to solve problems, which I still think fairly characterizes “decoherence” as a putative explanation of collapse <in general or even “apparent collapse” (whatever that means.) But I do not think you or others are arguing in bad faith or anything like that. I think you just feel too attached to a false hope that is alluring because it seems to resolve a vexing issue, and because the vagaries of meaning in talking about wave amplitudes, probabilities, etc,, lend themselves to misdirection and contrivance. BTW’, the whole idea of “entanglement” is that the mingled photon states literally don’t have a definite polarization in any individual sense, but the polarization is only established as a correlation upon later measurement. Hence I don’t see how entanglement can become a model or metaphor for collapse in general, which usually involves definite wavefunctions (such as 20 degree linear polarization as produced) collapsing into x or y, etc.

It is one reason that ice is harder to heat up in microwaves. Since the molecules are bound in a crystalline lattice the conversion of vibrational energy to translational energy is less efficient. As a result the H_2O atoms saturate quickly with vibrational energy and do not absorb as much microwave energy. For this reason the defrost cycle on microwave ovens is a lower setting, turning on and off the magneton so the fields in the cavity don’t feedback too much. The magneton is feathered to give the ice more time to thermalize its vibrational energy.

The idea for the microwave oven came with radar during WWII. The large antennas tended to collect lots of dead birds. This messy problem was found to be caused because the birds sat on the antenna and got cooked.

I am a bit of a maven for polytopic geometry. I highly recommend Coxeter’s book on convex polytopes. Then if you want to really grab this business by the horns Conway & Sloane’s “Sphere Packing, Lattices and Groups” is recommended. This book gives a decent account of lattice systems, such as E_8 and the Leech lattice, and systems of quaterionions these imply. This then leads up to the Conway & Fischer-Griess group called the “Monster.”

I think these structures are involved with quantum gravity, or in the lattice-tesselation of spacetime and AdS space. The vertices of the lattice system are roots of the gauge group. It is a sort of solid state physics analogue to gauge theory and gravity. I will leave things at this point, lest I am accussed of “theory mongering.”

The occurrence of large crystals is largely a matter of energetics. In an adiabatic situation atoms will align into cyrstals because that is most energetically favorable. It is interesting to note that selenide crystals of truly astounding proportions were found in a mine in Mexico. There are pictures of the cave explorers literally crawliing on them and rappelling off them.

Lawrence B. Crowell

If you are suggesting that liquid-to-solid phase transitions in general could be interpretable as including coherent Bose-Einstein phonon condensation components… that would be an interesting alternative on how to view crystallization, and certainly not one I ever recall bumping into.

Here’s a bit of elaboration on that idea, using brainstorming mode (by which I mean exploring the concept space, but not yet attempting to quantify or disprove the theorem): Crystal faces are macroscopic results of nanoscale assembly, with ratios of emergent sizes to creation component sizes that are truly astronomical. Could these emergent features be coordinated in part by the unrecognized existence of large-scale phonon Bose-Einstein condensates during the crystallization process?

For example, the assembly components that generate natural beryl crystals are in the Angstrom range, consisting of beryllium, aluminum, silicon, oxygen, yet large natural crystals of beryl can have very flat faces in the order of a meter across. That means a highly parallelized atomic-level crystallization process can easily generated well-defined emergent structures with features 10 orders of magnitude larger.

A comparison: This is roughly the same as 1 millimeter ants paving all of Asia, Europe, and Africa with a platform that remains level over that entire area during the construction process. Pretty decent group coordination, that!

(Quick critique: Extremely high relative reaction rates at the layer-addition shelf edges versus the flat surfaces may sufficient to explain the planar face. Thus the quick phonon idea could possibly be whacked away using Occam’s Razor and no some good reaction rate data.)

(Quick counter: The reaction data may inadvertently _include_ hidden coherent phonon that have never been recognized, and thus never adequately analyzed. Familiarity and an unexamined assumption that “this is all well known stuff” could be hiding an interesting phenomenon that has not been adequately examined or quantified.)

…

Brainstorming mode again: Water should be largely transparent to microwaves, since when it is hot it gives off radiation that is mostly in the much higher infrared range (the heat one feels when you put your hand near, but not directly over (that’s steam), a hot cup of coffee. Why do microwaves then heat water so well? To put the issue in terms of an analogy, microwaves heating water to the point where it gives off infrared radiation is a bit like shining an infrared heat lamp on a piece of goal and causing the coal give off blue light. There’s a definite “upping of frequencies” that at a first approximation is a bit hard to explain.

Theorem: The microwaves are actually interacting with a patchy network Bose-Einstein phonon condensates. Many of these phonon condensates include sufficiently large total masses of water molecules that they resonate easily with the comparatively low frequency microwaves. The heating simultaneously causes the condensates to break down, resulting in molecular-level “pieces” (water molecules) whose vibration frequencies are much higher, in the infrared range.

…

Your comments on long-range order: 1D and 2D constrained systems should encourage Bose-Einstein condensation. Some of Peierl’s early work (he actually got a lot of that from a German fellow whose name escapes me at the moment) on 1D effects that lead to alternating single-like and double-like bonds in long polymer chains (they are actually quasiparticle bonds composed of Fermi sea waves, and are _not_ really localized electrons) comes to mind, although that is in the fermion domain mostly.

Cheers,
Terry

That might happen with ice.

The problem is that for this sort of physics to take place it would have to involves the quasi-crystaline structure of polypeptides. Of course biology is not compatible with gamma rays. There are phonon physics associated with how replicase moves on a DNA strand. The ATP to ADP energy exhange with each step causes a quanta of vibration to move along the 5′-3′ strand which by recoil bumps the replicase to the next nucleotide. Okazaki fragments for the 3′-5′ strand replication are put together by more standard chemical processes.

Polypeptides are quasi-crystaline (like) structures. In fact DNA has a 10 nucleotide per 2-pi twist in the A and B conformational forms, and this is also mirrored in dihedral angles in some polypeptides. I think this has some connections with fractal geometry and chaos theory, which if there are quantum aspects to their physics leads to a huge area largely not well known. I could go on about this at considerable length, but work and time (and this is election day) preclude that possibility for now.

Lawrence B. Crowell

Well, yes, I must confess right here to a bad case of off-topic-drifty-thoughtalism!…
> First off the idea that tubulins are quantum signal conduits
> is doubtful.

(?) I would be blunter: Tubulins are complex and structural components that are flatly irrelevant to any serious discussion of whether quantum effects exist in organic systems. Focusing on them has held back for decades any serious analysis of whether or not quantum effects can impact in organic systems.

Tubulins are irrelevant because they are too large and contain too much distinctive state information to participate in quantum effects. I suppose one could propose that quantum-capable quasiparticle waves exist within tubulins, but why in the world would one bother? They are the movable scaffolding of the cell, with well-defined purposes that require no other explanations, especially ones so far afield from their primary purpose. I have remained baffled for decades as to why a mathematical physicist as sharp as Roger Penrose’s has stayed locked in so adamantly to this very poor candidate for room temperature quantum effects.

Let me be more specific about what I did mean:

I am proposing that small molecules, such as ordinary water, have sufficiently small state spaces that even within a quite small volume plausible numbers of them could be assumed to participate in ground-state phonon condensates, in the same sense that nuclei do in ordinary Mossbauer. In the remainder of this entry I refer to this idea as low-energy Mossbauer, since it is not not so much a proposal of new physics — the math does not change in any fundamental way, for example — as it is a translation of existing physics from the energetic domain of nuclei to the lower-energy domain of molecules, with identical use of phonon condensates. While I’ve never seen (and never looked) for the idea in the literature, such extrapolations of scale are sufficiently straightforward that I do think some care is needed to eliminate the possibility.

The second component of low-energy Mossbauer is another translation of scale: Instead of having the phonon-immobilized molecules exchange gamma rays, why not ask whether they might exchange lower-energy photons such as microwave or heat? Or for that matter, higher-order (non-condensate) phonons? This again is not so much new physics as it is a rescaling of the existing Mossbauer model to a lower-energy domain.

The measurable effect of this low-energy Mossbauer Effect would be anomalously high rates of transfer of exact rotational or vibrational molecular energy between molecules at distances (e.g., inches) that are vastly larger than could be explained using a fully classical model. The fully classical model would in contrast predict only noise and locally mediated (molecule-to-molecule) energy transfers in such situations.

In contrast, low-energy Mossbauer predicts the existence of a fairly large class of “impossible” transfers of energy between identical molecules at large distances from each other. The transfers would only occur for the primary rotational and vibrational modes of each class of molecules involved. If the transfers occur at all, the frequencies involved would be very precise, just as in Mossbauer. Finally, the distances involved in the transfers would be far larger than the radii of local thermal (and thus fully classical) effects.

Low-energy Mossbauer Effects would typically be masked thermal noise, since unlike the classical Mossbauer Effect the photons exchanged would be comparable to those of the noisy environment in which they exist. This means that some non-trivial experimental care would be required to detect them. Still, I suspect a clever experimentalist could come up with a good (and probably even cheap) way to look for such effects, since in particular the frequencies involved would be both well-known and would necessarily have sharply defined peaks, like Mossbauer.

Where I wonder about the plausibility of low-energy Mossbauer, though, is that it seems unavoidably to imply some pretty odd constraints on some very well-studied systems. Take water, for example. The existence of low-energy Mossbauer in ordinary water would unavoidably imply that a glass of drinking water contains molecules that are “stitched together” by networks of photon and possibly phonon exchanges that cannot be modeled classically.

At the very least, the existence of such networks in water would have entropic implications, since information would constantly be exchanged in non-local ways that would be better modeled using a collection of simultaneous and intermixed Bose-Einstein condensates. The decay of such structures would necessarily take longer than is possible with a purely classical model, and so would give the water a sort of “memory effect” that should not be there. Also, since different condensates could exist at the same time, a new range of variables would be introduced in which one glass of water is no longer the “same” as another that has a different condensate configuration.

I would think that such effects would have been noticed by experimentalists, at least peripherally. If no such effects have ever been seen, this would argue against the existence of low-energy Mossbauer Effect.

—

Regarding Hyperion: Covered that in my first entry, seriously I did. I just prefer Dr Feynman’s terminology and perspective. Decoherence is fine, but I think it’s fair to say that it is really is just another way of describing how information emerges from a quantum system.

—

Regarding the excellent question of how a photon “decides” whether to be absorbed by one atom (an information-creating event that destroys coherence) or reflected from a huge array of atoms (coherence is maintained):

If you have not already, be sure to pick up a copy of Feynman’s “QED: The Strange Theory of Light and Matter.” Snip out Zee’s intro (just kidding… no, actually, I’m not) and settle in for a good read. Not only will this book _not_ answer your question, it will leave you more frustrated than before. This is what is so great about it! Feynman pulls no punches in describing how difficult and deep your question truly is. Yet bizarrely, by the end of his book he will nonetheless given you the ability to calculate, in principle at least, exactly how many photons will “decide” to do one or the other, for any imaginable experimental setup.

Feynman also points out that even Isaac Newton pondered your question and realized how profound it is –- a remarkable achievement for someone who lived hundreds of years before quantum mechanics came into being.

And if you want to know how Newton managed to contemplate such absorption probabilities for a particle that was not known to exist until Einstein postulated it (his Nobel Prize was for that work, not relativity)… why, then, read QED! (And no, I don’t get a cut, I just like the book a lot.)

Cheers,
Terry Bollinger

Yet with neurons there are a number of problems. First off the idea that tubulins are quantum signal conduits is doubtful. These are the scaffolding of a cell, and where kinesin and dysin polypeptides walk up and down them. These are literally nano-bots of sorts which mechanically walk! They transport various compounds through a eukaryotic cell. Cells conduct their energetics through ion pumps across the membrane. Mitochondria pump protons across their membranes and the ion pump is the cell’s energy source. Similarly with neurons, an action potential is the offset and reset of the 1.6v potential difference across a cell membrane by the opening of Ca and K ion channel gates. These gates are receptors for certain chemicals such as acetylcholine, seritonin, dopamine etc. The action potential is a sort of wave, but it is one which is contantly pumped in a sense. So the action potential propagating down an axon or dendrite is not a conservative wave, but is more like a wave in the bobbing motion of a bucket being passed in a bucket brigade.

Of course quantum mechanics has some role in biology, such as the hydrogen bond between purines and pyramidines in the DNA double helix. The action of a photon on a rhodopsin molecule in a retinal cell has some quantum mechanical interpretations, and so forth. Yet there is not much evidence for any quantization on the large. Of course this blog page was on the quantum properties of Hyperion, a large moon of Saturn, so quantum properties might percolate through quantum systems in certain ways we are not as yet aware of.

Lawrence B. Crowell

vvvvvvvvvv
… There were some ideas about “quantum brains” 10 years ago or more. Penrose sort of got this idea going, and I think the idea is probably flawed … The brain is a warm system which is too messy for coherent wave functions to be running around.
^^^^^^^^^^

I agree heartily with almost all of this. Penrose’s microtubules are very small, to be sure, but they are also quite massive in comparison to the scale of systems in which quantum effects plausibly apply at room temperature.

The “almost all” qualifier is due to this: While matter is a very poor candidate for room temperature quantum, the same statement cannot safely be made for quasiparticles.

(Brief background: Quasiparticles are energetic phenomena that are quantized “on top” of the ordinary matter. They are for the most part composed of energy, and so have very low masses, far less than those of electrons. This means conversely that quasiparticles can participate in quantum phenomena such as Bose-Einstein condensation at temperatures for which even a light-weight electron would behave classically.)

Phonons are a good example. These quasiparticles are the quanta of sound, just as photons are the quanta of light, and they are fully capable of combining into coherent states within ordinary room temperature matter. If this were not the case, the well-known Mossbauer Effect could not exist.

(Brief background: In Mossbauer, room temperature matter supports an extraordinary precise matching up of gamma frequencies between nuclear emitters and receivers. The gamma ray emissions and detections used require exceedingly precise frequency matches, so much so that a relative velocity of just a few centimeters per second is enough to squelch reception. Such precision is impossible in a fully classically room temperature system, since atoms and their nuclei move so quickly that the Doppler effect would blur the relative gamma frequencies of the emitter and receiver far beyond what is detectable.

How then can the Mossbauer Effect even exist? One way to look at the situation is to picture the motions of individual atoms as being controlled by a spectrum of “quanta of vibrations.” These quanta range from no motion at all to very rapid vibration.

Like most pure energy phenomena, phonons are bosons — that is, they obey the “let’s all get together” statistics of Bose-Einstein. Thus not only are the motions of atoms controlled by these phonons, but the phonons themselves can group together to create “super phonons” that all behave in exactly the same way.

Of particular interest in this case is the ground energy set of such phonons for which motion is zero. This condensate in effect “freezes” a certain percentage of atoms in a material, even in one that is otherwise at room temperature. These non-classically motionless atoms are the ones capable of participating in the extremely motion-sensitive emission and receipt of gamma rays.)

While I agree that existing models readily eliminate direct quantum behavior for objects as large as microtubules, this is not the same as proving that no quantum effects of any type are possible. A full proof must also show that there are no configurations of quasiparticles that could transfer of information from a quantum state back into the classical matter component of the system.

The first problem with creating such a proof is the existence of the Mossbauer Effect, which shows that quantum-enabled point-to-point data transfers exist at room temperature. In the case of Mossbauer, such data transfers are enabled by the ability of very lightweight quasiparticles to form Bose-Einstein condensates at room temperature.

At first it would seem easy to eliminate the relevance of Mossbauer. It does after all rely on nuclear isotopes and gamma rays. Caution is needed, however. The problem is that there is nothing in the physics of the Mossbauer Effect that requires use of gamma rays. The gamma rays of the Mossbauer Effect instead provide a convenient way to detect such effects due to the exceptionally sharp detection lines they produce.

The hypothesis to be disproven, then, is whether there exist Mossbauer like non-classical transfers of data that follow the same mathematical model as Mossbauer, but which substitute phonon condensates of larger molecules for nuclei, and lower-frequency mechanical (heat) or electromagnetic vibrations in place of gamma rays to transfer data. The possibility of such effects would need to be disproved explicitly to eliminate the possibility of non-classical point-to-point data transfers in room temperature systems.

Also, a full proof of the irrelevance of quasiparticle-mediated quantum effects in room temperature organic systems would also require a proof that quasiparticles cannot be used to construct qubits, or at least that any qubits constructed in such a fashion cannot then be linked back to the classical components of the system.

If the possibility of molecular-level non-classical data transfers can be eliminated, I suspect that qubits would trivially fall as a direct consequence. On the other hand, if quantum enabled molecule-to-molecule data transfers can be shown experimentally to exist, disproving the relevance of room temperature qubits to organic systems becomes much more difficult. I suspect that if molecular non-classical data transfers and quasiparticle condensates exist, such components could also be configured to build qubits.

In short: To complete the assertion that room temperature systems cannot include quantum behaviors, a rigorous analysis of the quasiparticle issue is required. Since non-classical data transfers via the Mossbauer Effect are part of accepted physics, such a proof would need explicitly to eliminate the possibility of translating the Mossbauer model to larger (molecular) units and lower frequency mechanical or electromagnetic phenomena. If such non-classical transfers of data are in fact possible, the proof would have to show that such transfers are irrelevant to the specific case of room-temperature organic systems such as the brain. Finally, if non-classical molecular data transfers are possible, a further proof would be needed that they cannot also be used to construct qubits, or alternatively that any qubits constructed from quasiparticles will be unable to transfer data back into the classical components of the system.

Cheers,
Terry Bollinger

First, to reiterate: How come a conventional “detector” D is what collapses a wave function (or whatever it is) and not other things that particles come in contact with? For example, the beamsplitter in an interferometer. We know the photon wave splits there and does not collapse because it can interfere later.

2. Lawrence, don’t confuse “real” per existence, about wave functions, with “real” number value versus imaginary. Our being able to assign complex values to the WF is just a procedure, it doesn’t mean nature can’t really hold such a thing. Remember that the complex value is use to show phase difference, which could be represented some other way. Indeed, one can use the analogous complex system to represent relative phase of electrical currents (phasors in that context), that doesn’t keep currents from being “real” per existence. The question still is, what is it that goes through space and how can it condense at a small space even when available detectors are miles apart with no chance of whatever “interference” the decoherence sophistry attempts to imply.

Yes, using formal tools to say things as confidently as we reasonably can be: that’s the general spirit of scientific research! But formal tools apply as well to classical physics as to quantum physics. So if we reject reification for quantum physics, we should also reject it for classical physics. I’m always dubitative when there is an avoidance to think in terms of ordinary objects in quantum physics, while using them intensively in classical physics.