## Statistical vs Mathematical Physics

Mathematical Physics is the study of what can be really proved about our theoretical models of physical systems. This differs from other kinds of theoretical physics because physicists often take the lack of experimental refutation of a mathematically invalid calculation as “proof” that the result is correct.

Statistical physics (aka Statistical Mechanics) is the study of physical systems having so many degrees of freedom that it is not feasible to measure a complete set of the individual observables (such as the positions and momenta of all the molecules in a volume of gas or the angular momenta of all the electrons in a crystal), but for which some observables (such as temperatures and pressures), defined as averages of those most naturally considered as forming a complete set, are expected to evolve in a way that does not depend on the specific values of all the variables needed for a complete description.

Many of the expected behaviours of these averages are assumed by physicists without any complete proof; and one important area of mathematical physics is the filling in of these missing proofs. A classic text of this sort is the book ‘Statistical Mechanics: Rigorous Results’ by David Ruelle.

## 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.

## Relativistic Mass

A Quora question asks: What is the equation that states that an object’s observed mass increases with its velocity?

It depends on what you mean by “an object’s observed mass”.

Nowadays the term “mass” is used exclusively for what used to be called the “rest mass” and is a property of the object alone that is independent of the relative velocity of the observer. So no physicist working today would say that “an object’s observed mass increases with its velocity”.

But there was a time in the past when some physicists used the term “mass” (usually, but not always, qualified with the adjective “inertial” or “relativistic”) to identify the multiplier needed to make a relativistically correct equation having the same form as Newton’s third law $#F=ma#$ (albeit only for the special case where the force and acceleration are parallel to the direction of relative motion between object and observer). So the equation you may be thinking of is $#m_{rel}=\frac{m_0}{\sqrt{1-v^2/c^2}}#$, but it is not a statement about what we now mean by “an object’s mass”. (And even adding the adjective “observed” or “apparent” doesn’t change that, as our observation of the “rest” mass is pretty much just as direct as that of the old “inertial” version.)

Despite many strident claims in other answers that it was “incorrect”, the alternative choice of using the word “mass” for $#m_{rel}#$ was in fact perfectly valid if applied correctly. It just wasn’t very useful because the resulting number $#m_{rel}#$ is not a property just of the object itself but depends also on the observer and is different (with a slightly more complicated formula) for accelerations and forces in directions other than that of the relative motion.

## Explaining Relativity Without Equations

Can you explain time dilation and space contraction in relativity without using complex mathematical equations?

Yes. Any decent introductory text on relativity does this – but probably just in one or two sentences before going on to derive the actual formula (for which the apparent level of complexity of the resulting equations may depend on the reader’s experience).
The basic idea is that if two identical side-by-side trains are passing by one another and a light signal is sent when the ends from which it is sent are together, then if the trains are in relative motion the signal will reach the far end of one before the other. So if observers on both trains measure the same speed of light then their units of length and/or time must be different. Working out exactly how the coordinates used by each observer are related to those of the other does involve the use of mathematical formulas and equations, but they are well within the scope of high school algebra so whether or not you call them “complex” is a matter of perspective.

## Time Contraction

Special Relativity tells us that two inertial observers in relative motion each perceive the other to be ageing more slowly – ie each infers that the tick intervals of the moving clock appear to be dilated. But can time contract as well as dilate?

Yes, but with the proviso that the dilation or contraction is just a description of how the progress of one clock appears relative to another and that two observers will not necessarily agree on which events in their lives are simultaneous – and so can only compare average (rather than instantaneous) clock rates using the total time intervals on their clocks between events where they are together.

Two observers who separate and reunite will agree that the total time experienced by the one that felt more forces of acceleration (or of resistance to gravity if spending time near a massive object) will be less than that experienced by the other. This means that from the point of view of the one who was more accelerated (or spent more time at the bottom of a potential well) the clock of the other appears on average to have been speeded up (ie tick intervals appear contracted), while the one who remains unaccelerated interprets this as meaning that that the other’s clock tic intervals were, on average, dilated.

## Quantum “Weirdness”

So far as we can tell everything is probably quantum. But it’s just that some quantum things seem more weird to us than others.

Our minds evolved to cope with situations in which the information most relevant to our survival consists of averages over large numbers of microscopic subsystems. For dealing with such averages the “classical” models of reality that we consider natural are good enough for survival (with the advantage of not requiring too much sophistication of the instincts we follow). It is only in specific (mostly artificial) contexts that the quantum nature of reality becomes relevant; so there has been no need for our instincts to take account of that, and so those instincts tend to be based on the classical approximation – and when that approximation fails it feels to us like “weirdness”.

The weirdness stops, not when things are “not quantum”, but just when (due to the averaging business) their quantum behaviour is well approximated by the classical models that correspond to our evolved instincts.