## What is Horizontal?

John Platts’s answer to Why does a mirror reverse things horizontally but not vertically? includes some nice illustrations and discussion but declares that the front-to-back reversal is not horizontal.

Interesting point. But I think we use the word “horizontal” to refer to any part of a line or plane that is perpendicular the line joining it to the Earth’s centre, regardless of whether or not our view of it actually appears parallel to the horizon. We often test for this property by looking at it from a position where it lines up (left-to-right) with the horizon but two people building a house will generally agree that a beam is horizontal even when they are not looking at it from such a position. (If that were not the case the word would be a lot less useful because the property of a beam being horizontal would depend on the observer and would be lost whenever she changes her orientation.)

So, in my understanding, horizontal lines aren’t necessarily parallel to the horizon in a perspective view, but (if the world was flat) their infinite extensions (would) all terminate on it (though due to the Earth’s curvature that actually happens a tiny bit above what we see as the visible horizon).

## Violating the Exclusion Principle?

Naive students sometimes ask what would happen if electrons or other fermions were “forced” to violate the Pauli Exclusion Principle, but they never suggest any idea of what this would actually mean.

Whatever they are imagining probably involves thinking of those fermions as independent particles such that you can somehow keep track of which is which; but in quantum theory there is no way of doing that, and no-one has ever suggested an alternative theory that fits the facts and in which such separate identities make sense.

## Blue Shift and Time Dilation

A Quora question asks: Can you deduce time is dilated based on blue shifted light? Or can light be blue shifted without the phenomenon of time dilation?

Yes, and yes.

Taking the questions in reverse order, the non-relativistic Doppler effect (ie neglecting time dilation) is a blue shift for an approaching source and a red shift for one that is receding; and the non-relativistic calculation is reasonably accurate for speeds that are not too great. So, yes, light would be blue shifted without the phenomenon of time dilation. (But this is just a qualified “yes” because in fact no relative motion can occur without the phenomenon of time dilation, so I had to say “would be” rather than “can be”.)

With regard to the first question, the relativistic correction (attributable to time dilation) is basically to make a small red shift adjustment to the non-relativistic calculation (ie to make red shifts slightly stronger and blue shifts slightly weaker). So, by noting the difference between an observed blue shift and what would be predicted classically from the observed rate of approach, we can see evidence of the time dilation effect.

## Orbitals not Orbits

If the electron was orbiting they would be called “orbits”.

The word “orbital” therefore correctly indicates that they are not orbits (but suggests that they are related to what would be the allowed orbits if the electron was a particle with the only quantum effect being to confine it to discrete energy levels).

## Colour

The science of colour (or for Americans color) shares with that of music the property of being an attempt to describe aspects of our physiological and psychological reactions to physical stimuli in situations where the physical stimuli have fairly simple mathematical relationships but the limitations and complexities of our sensory (and signal processing) apparatus cause us to try to fit them into a simpler framework in a way that paradoxically actually makes the analysis even more complicated. (The complication seems to me exacerbated by the fact that people have come up with somewhat conflicting models for simplifying different aspects of the situation but some people are led by popular accounts to treat these simplified models as absolute dogma – and so lots of effort gets expended in going from one to the other rather than seeing how they all arise from different ways of approximating the true story.)

In music, this is exemplified in the issues of harmony, chord, and discord (well-tempered vs equal-tempered scales for example), but here I want to concentrate on colour (where arguments about “primary” colours and which is the true “colour wheel” can waste a lot of time).

Observed Facts and The Physical Basis of Colour

White(ish) light from any source, when passed through a prism (or drops of water in the air), spreads out into a range or spectrum of coloured components.

People sometimes interpret the colours of this rainbow as distinct named bands (ROYGBIV, for example), but different people and cultures have come up with different numbers of bands and it appears that the range is truly continuous. Once light has passed through a prism, the output from a narrow part of the spectrum does not spread further, and this suggests that each point in the spectrum corresponds to a different basic kind of light – each with its own refraction angles.

The fact that similar spectra can be produced by passing the light through (or reflecting it off) a grid of parallel lines suggests that light is transmitted like a system of waves with the various kinds of light in the spectrum just corresponding to different wavelengths, and that the set of all physically possible kinds of light just corresponds to the set of all possible frequency distributions in the “fourier transform” of the signal.

Basic Psychology of Colour Perception

The pure frequencies of the spectrum are all seen by (most of) us as having different colours, but similar-looking coloured light from other sources sometimes can be split. For example, yellow light from the spectrum can be matched by a mixture of red and green, and violet light from the spectrum can be matched by a mixture of blue and a small amount of red. But there are also deeper purples and magentas that can be achieved by mixing red and blue which do not seem to match anything in the spectrum. By reviewing our own experience and comparing with that of experimental subjects we find that every kind of light gives a colour experience that can be at least approximately matched by some combination of three given sources so long as they include one that is predominantly at each end of the spectrum and one in the middle. The question of how to compare quality of matching (eg the number and choice of samples to compare and the brightness at which they are shown to the experimental subjects) is not very well defined and so it is not surprising that results differ, but there are a few widely used choices of Red, Green, and Blue phosphors that have over time come to be accepted as adequate.

Biological Facts and the Neurological Basis of Colour Perception

Examination of the human retina under a microscope shows the presence of various kinds of cells which further study shows to emit electrical signals in response to light. Some of these (the “rods”) are sensitive to all kinds of light and give about the same strength of signal to all of the frequencies in the spectrum of sunlight. Others (the “cones”) are more sensitive to some frequencies than others (and generally less sensitive overall).

Most people have three kinds of cones. One kind have their peak response towards the long wavelength end of the spectrum; some people used to call these the “Red-sensitive” or R cones but nowadays the fashion is to call them the L cones because even though they peak at longer wavelengths than the others they still fall off in the region of the spectrum that we see as bright red (so the strength of the L cone signal cannot be all that contributes to our sense of redness). The second group, which peak closer to the middle, are sometimes called “Green-sensitive” or G cones, but I will join those who refer to them as M cones instead. And for the third group, which peak nearer the short wavelength end and are sometimes called “Blue-sensitive” or B cones, I will follow the convention of calling them S cones and denote the strength of their signal by S.

It is not easy to measure the actual signals output from cone cells, but it is possible to extract the light sensitive chemicals and measure how their light absorbing properties depend on frequency and intensity of the incident light, and we may perhaps take this as a plausible proxy for the actual output signals.  Here’s one plot of how the energy absorption of each type of cone depends on the wavelength when the intensity is relatively low so as not to “saturate” the effect.

(Other sources may give somewhat different shapes because of using different methods for extracting the relevant parts of the cones and/or different illumination levels but the overall pattern seems to be similar in all cases.)

Note that the S signal (or at least the absorbance that we are taking as its proxy) is relatively weak, but this doesn’t necessarily mean that we see blue colours less vividly (though they do appear to fade more quickly in dim light) because our brain may respond more strongly to signals from the S cones  . For this reason the above graph also shows the normalized values L^,M^, and S^, where for each of X=L,M,S,  X^=X/X_max

Also, perhaps more significantly, we can see that the L signal is peaked quite close to the M and falls off a lot before the red end of the spectrum. So why do we nonetheless see true reds as quite vividly coloured?

One possible explanation (partly but not yet completely confirmed by actual analysis of optic nerve and brain neural connections – and also supported by psychological experiments in which people report how vividly coloured different light sources appear) is that the signals we identify as colours are calculated in terms of the percentage differences between L, M, and S signals rather than as those numbers themselves.

For example if we define R=2(L^-M^)/(L^+M^) , R^=R/R_max, B=2(S^-R^)/(S^+R^), and B^=B/B_max, (which can all be produced by running the LMS signals through a fairly simple sequence of neurons)

then the positive values of the R^ signal peak much further to the red end of the spectrum (and the negatives G^=-R^ and Y^=-B^

[insert graph of R^ and B^ signals, along with G^=-R^ and Y^=-B^]

This means that for any light source our sense of the colour and brightness of that source can be represented either by three positive numbers L,M,&S (actually four if we count the rods response) or by two signed numbers, RG and BY, for the colour and one positive number for the brightness.

Working in terms of the cone signals directly, we could represent all possible cone signals from each patch of the retina by points in (the first octant of) a 3d space (with one axis corresponding to the signal strength from each type of cone). Since scaling up all the strengths just corresponds to an increase of overall brightness we can identify the colour hues with the directions or with points on the first octant of a sphere of some fixed radius – which viewed from the middle direction looks like a triangle.

[insert rotatable 3d image of spherical surface in 1st octant of LMS space

and its triangular projection]

The corners of this triangle represent excitation of only one kind of cone – which might in principle be achievable by some chemical stimulus, but not by any kind of actual light since light of every pure frequency causes at least some stimulation of all three cone types (and in particular the L and M responses are generally quite close to one another)

This means that the set of response triples that can actually occur as a result of  stimulation by light is just a small part of the triangle. It includes, of course all of the response triples that can be stimulated by pure frequencies (which correspond to a curved line in the response space – curved because most spectral frequencies that stimulate both L and S also stimulate M so the corresponding points are not on the straight line joining the two endpoints) as well as any kind of weighted combination of those points. For any light source with spectrum of energy per unit of wavelength given by s(

which gives us what is called the “convex hull” of that curve of pure frequencies.

Except for the pure spectral colours, every physical colour spectrum has many others that give the same LMS responses and so look the same to us.

On the other hand, if we represent each LMS triple by its corresponding RG and BY values then we get a “map” of the spherical surface which  spreads things out much more clearly

[insert image of the RG vs BY horseshoe]

Every point in the original LMS triangle now corresponds to a point in the R^G^xB^Y^ coordinate plane but the coordinates are now much more closely related to our actual sense of “primary” colours and the part that corresponds to actual colour responses is more symmetrical (closer to being a circle)

Complementary and Contrasting Colours

Two coloured light sources that combine to make something that looks white are said to be complementary (where the word “complementary” with an ‘e’ refers to parts making up a whole, as opposed to “complimentary” with an ‘i’ which refers to making the other look good – or at least saying that it does).

While many people think of complementary colours as the most natural way of defining contrast, there may also be some reason for thinking that our sense of contrast is driven more by the results of the RG YB transformation and so to define contrast in terms of an RYGB colour wheel

Colours of Objects and Pigments

So far I have only been discussing the colours of light. But the colours of objects and pigments are a lot more complicated. This is because, although at the physical level the absorption/reflectance spectrum of an object or pigment is no more complicated than the spectrum of a light source, the colour we actually see from it depends on both its absorption/reflectance spectrum and the spectrum of the light with which it is illuminated. So, by seeing it under different lighting conditions we can with just our three kinds of colour cones learn far more about the detailed absorption/reflectance spectrum than just three weighted integrals.

Also, the mixing of pigments does not always just correspond to adding their absorption spectra. Depending on how the coloured particles are distributed and on whether the medium in which they are embedded is light dark or clear it may be either the absorptions or the reflectances (or some combination) that get added.

Most printer inks are designed to add their absorbancies in proportion when mixed and to be seen under a standard daylight spectrum and for that purpose is serves well to have fundamental pigments that are complementary to the RGB components used for analysis of the reflected light – namely Cyan, Magenta, and Yellow. But going beyond that basic statement takes us into the domain of the club of serious colour specialists – of which I am not a member.

## Purely Probabilistic Positions?

What we interpret as the locations of elementary particles can certainly be compared with the predictions of regular mechanics. And they will often be quite close, so the classical predictions are actually useful. But the pattern of (usually small) variations from those predictions, while not “purely” probabilistic, does seem to have a component which cannot be explained in terms of some more precise classical properties that we just have not been able to properly measure. So our idea of a regular particle may just be something that does not really exist and what we interpret as its position may indeed by something that has an essentially probabilistic component.

## Maxwell’s Equations for Photons

In the quantum theory of electromagnetic fields Maxwell’s equations play two roles.

One is to describe the behaviour of the actual field observables which measure the combined effects of all possible numbers of photons, and the other is that they are satisfied by something that is as close as possible to being the “wave function” for a single photon.

I say “as close as possible” and put “wave function” in scare quotes because it does not satisfy all the properties of a non-relativistic wave function. In relativistic theories, the concept of strict localization does not exist. It can only be approximated for massive particles in frames where they have low momentum, and cannot be done at all for photons. But nonetheless, (as discussed in this survey article (.pdf) by Iwo Bialynicki-Birula) with appropriate normalization, a function satisfying the complex form of Maxwell’s equations can be used to generate probabilities for detection of a single photon in various experimental contexts.

## Photons in a Refractive Medium

A Quora question asks:Given that light is massless, and that all massless particles travel at the speed of light, it should follow that in a medium with a refractive index >1 (where light slows down), it acquires mass and experiences time. Why is this not the case?

It is not always true that “light is massless”. For example light trapped in a reflective container contributes to the rest mass of the system consisting of the container and its contents.

It is not obvious that massless particles always travel at the speed of light (but unless they are doing so they have zero momentum and so don’t change the momentum of things they collide with).

The speed of a photon is always equal to the vacuum speed of light in between its interactions with matter, but the probability of detecting a photon travelling through a medium is calculated from a sum of probability amplitudes associated with all possible paths including those which involve interacting with atoms in the medium. Since many of these paths are indirect, their lengths are greater than the straight line distance and so the average time taken corresponds to a speed less than that of light in a vacuum.

[Some answers have suggested also delays due to absorption and re-emission but if these really happened with random delays they would destroy the coherence and so in a perfectly clear medium the interactions are all effectively just instantaneous reflections off bound electrons (with minimal energy transfer due to the masses of the nuclei).]

One might be tempted to look for a way of describing the result in terms of effective photons with mass; but we can’t expect any proper Lorentz covariant theory of such particles since the medium is only stationary in a particular inertial frame, and in relatively moving frames it appears contracted which changes the density and so the index of refraction (in a direction dependent way).

## Wave Momentum

How do waves have momentum?” is a very good question, but like many good questions it seems to attract a lot of over-confident incomplete answers.

It is in fact true that many kinds of travelling waves do transfer momentum to anything that actually absorbs or reflects them, and the momentum transfer is often proportional to the energy density and speed of the wave; but just stating that something is true is not an explanation of why it is true, and if the mere fact of carrying energy explained why waves have momentum then a moving charged battery would have more momentum than an uncharged one.

Indeed, it is perfectly reasonable to not be immediately convinced that waves have any momentum at all in the direction of propagation. For transverse waves the primary motions are perpendicular to the direction of motion and for compression waves the motions forwards and backwards mostly cancel out. And the fact that we can get pushed inwards by a water wave doesn’t tell us anything about net momentum transfer, since anyone who has experienced that inward push has probably also experienced the outward suction of the receding wave; and although waves seem to bring flotsam in to the shore it is not obvious that this is due to the waves themselves rather than the wind that gives rise to them.

When it comes to the often mentioned pressure and momentum of electromagnetic radiation, while we can see the effect of light pressure on the tails of comets, the derivation from Maxwell’s equations is rarely given completely. Many sources (such as this one) explain how the perpendicular electric and magnetic fields lead to a force on any charged particle that is perpendicular to both of them, but don’t give any proof that this is in the forward direction of wave propagation rather than backwards; and even when such a proof is given it is usually shown just as a formal calculation without any physical motivation as to why it is working.

A google search for “wave momentum” is unfortunately overwhelmed by ads and reviews for a popular brand of volleyball shoes, but if we change the order and/or add words like “electromagnetic” or “water” we do get a lot of useful hits. The best I’ve seen so far is

https://as.nyu.edu/content/dam/nyu-as/as/documents/silverdialogues/SilverDialogues_Peskin.pdf

This gives pretty complete arguments for the momentum content of various kinds of waves, (and also includes examples of waves that carry energy but do not have momentum – which shows that your skepticism is not at all unreasonable). But it is at a fairly high mathematical level and so takes a pretty advanced reader to see the physical motivation for why its results are true.

So what I want to do in the rest of this answer is provide a bit of a handwavy argument to give some physical motivation for the momentum content of one particular kind of wave. It is not to be taken too seriously, but just as a hint of what might actually be shown by a proper detailed analysis.

Consider a rope tied to a wall or post at one end, with you holding the other end and moving it up and down to project a wave along the string. If the wave carries momentum then during at least part of the cycle your hand must be applying a forwards push (or at least a reduced amount of tension compared to the starting situation) – and I suspect that, even before thinking about this, it has indeed felt that way when you tried it. That may of course just be a psychological effect rather than anything real, but perhaps we can think of an actual physical reason for it. When your hand is at the extreme top of its motion the rope near the end you are holding is bent up a bit, and as you move it back down the tension in the rope tends to straighten it (even if you just let it go free rather than pulling it down). This pulls up the lower part of the bend, and to counter that pulls down the part nearest your hand. But this swinging down of the end would, if tension were maintained, cause it to project outwards a bit – and so maintaining the original distance from the far end would require a bit less tension (or equivalently a slight push forward relative to the starting level of tension). As I said, this is not a real argument, but it’s the best I can do short of a proper mathematical proof as given in the paper linked to above.

## Sound from the Big Bang

A Quora question asks about the possibility of sound from the “Big Bang”:

I’ve been thinking… if we can look back and see the Big Bang and calculate when the universe was created and can see this because of the light traveling through space at the maximum speed that exists. The moment the Big Bang happened the sound had to be extremely loud and the first and only sound ever created which would still be traveling through space like everything else I would think right? So eventually it would make it to our universe and possibly be the catastrophic event that would destroy earth don’t you think?

It may be even more tempting to ridicule the idea of sound in the vacuum of space than the conception of the Big Bang as a localized event that happened someplace far away. But while the latter is a naive misconception, the former may not be so far fetched – at least if we “stretch” our definition of “sound”.

The theory of the “Big Bang” is not that it happened at some particular place, but rather that it happened *everywhere* 14 billion or so years ago; and just as we are now receiving “light” from when it happened 14billion or so light years away, so also whatever stars and beings evolved from what we see at that distance are just now receiving light that started with the part of the bang that happened right here. But although the “bang” was very hot and bright with lots of high energy short wavelengths, what we “see” of it now is very dim and of long wavelength (actually invisible) microwave radiation because the universe has expanded so much between then and now.

The same attenuation and extension of wavelength would apply also to sound waves if they could actually propagate through the vacuum of space, except that the source of what we would “hear” now would be much closer. In fact, back when the universe was denser there may have been variations of density which behaved like sound waves, but the expansion of the universe would have stretched them out by now so that their remnants would not be like audible sound waves (which in any case cannot exist in the vacuum that now fills most of space). Instead they would correspond to density variations on a much bigger astronomical scale. This may indeed be the source of some of the large scale non-uniformity we see in the density and distribution of gas and galaxies on a cosmic scale, but checking to what extent that is the case is a more serious exercise than just speculating that it may be so.