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I'm a maths student, and I've studied quite a lot of mathematical physics. All my courses have a similar style - we state the laws of the system, and then deduce the physical consequences as theorems. It has become more and more apparent to me (especially studying electromagnetism and QM) that I have no idea why you would expect these equations to be true.

What I'm looking for is either a kind of physical justification "this is how object X behaves, here are mathematical statements that say this precisely" (similar to momentum-based arguments in fluid dynamics), or some kind of experimental justification (the more direct the better), for physical laws. I'm particularly interested in this with respect to Maxwell's equations, but I'm interested in any others as well.

Edit for clarification:
Perhaps I worded myself poorly. The axioms of a physical theory (which is what I meant by 'laws'), like any set of axioms, are invented by human beings with the aim of capturing certain intuitive properties of an object. For example, the axioms defining a vector space in mathematics are not completely arbitrary - they aim to capture the properties of various objects ($\mathbb{R}^{n}$, or function spaces for example) that we have an intuitive idea of.

I am asking the equivalent question with respect to physical theories, and especially in terms of Maxwell's equations, e.g.: what are the intuitive properties of charges and currents that lead one to write down Maxwell's equations? Note that the answer 'they match up with experiment' doesn't fit, because many theories could match the same observations and be mutually contradictory (e.g. Newton vs relativity at small velocities). There also is no condition on the correctness of the interpretation - any explanation of, say, Newtonian mechanics, would necessarily not be an absolutely correct picture of the world (or it would be the correct physical theory, too!).

Obviously such a justification would have to be empirical to have any grounding in reality. For instance if we want to know if Newton's second law is true, we can set up an experiment to check it. Less experimentally, most people take the existence of absolute time, or inertial frames of reference, to be intuitively obvious. Similarly, properties like acceleration and momentum have physical interpretations, and a good explanation could be given in terms of such quantities.

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closed as off-topic by ACuriousMind, Emilio Pisanty, tfb, CuriousOne, David Z Jul 23 '16 at 18:38

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    $\begingroup$ Um: the laws make predictions about behaviour, we test those predictions with experiments, the laws pass or fail the tests. We keep devising new experiments until they fail the tests, then think up some new laws. So the reason you would expect the equations to be true is their predictions agree with experiment. $\endgroup$ – tfb Jul 23 '16 at 12:08
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    $\begingroup$ I'm sorry, but you have lost me a bit in your post. I have no idea why you would expect these equations to be true. Personally, I don't expect equations to be true, but if it follows logical math principles and its predictions are confirmed out by experiments, then so be it. $\endgroup$ – user108787 Jul 23 '16 at 12:11
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    $\begingroup$ @DanielLittlewood I think borrowing examples from mathematics is almost the worst thing you can do: Physics is an experimental science and it is agreeing with experiment that matters. $\endgroup$ – tfb Jul 23 '16 at 13:40
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    $\begingroup$ I'm voting to close this question as off-topic because in its current form it is poorly researched and can be answered by looking at introductory textbooks. $\endgroup$ – Emilio Pisanty Jul 23 '16 at 13:44
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    $\begingroup$ @DanielLittlewood You guess, test and iterate. If there was some effective procedure for generating laws of physics based on experimental data then physics would not be a subject. $\endgroup$ – tfb Jul 23 '16 at 14:30
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Rather than saying the physical laws are based just on experiments I would say they are based on the Scientific Method.

There are steps that must be checked:

  1. Observation - We observe some unexplained phenomena. At this stage one does a lot of experiment and if all of them gives the same result we can state an experimental fact. For instance: Water cannot be sucked to a height greater than $10.3\, \mathrm{m}$.
  2. Hypothesis - We conjecture some physical idea or concept that explain the observations already done. For example: Air has weight. The atmospheric pressure matches the pressure of a $10.3\, \mathrm{m}$ tall column of water.
  3. Prediction - Based on the hypothesis that explained some particular results we shall predict the behavior of more general results, e.g. The air pressure matches a $0.76\, \mathrm{m}$ tall column of Mercury.
  4. Testing - We do the actual experiments and verify the predictions.
  5. Analysis - The last step concludes whether our hypothesis succeeded or not. If it succeeded we can formulate a law based on it. For instance, after a bit of math we arrive at the law for static fluids: $$\frac{\partial p}{\partial z}=-\rho g,$$ where $p$, $z$, $\rho$ and $g$ are pressure, height, mass density and gravity acceleration, respectively.

If you take electromagnetism you see a similar process although it lasted longer. There were many observations carried by decades and made by different people that support the hypothesis. For instance Faraday observed that under some circumstances electric currents were induced even though there was not a chemical battery in the circuit. He made the hypothesis that a variable magnetic flux leads to an induced electric field. This is known as the Faraday-Lenz law and explain the phenomena of induced currents. Combined with the Ampere-Maxwell law it is able to predict the very new phenomena of electromagnetic waves which were later verified. The conclusion is that we now accept the Maxwell laws as physical laws.

Notice that the formulation of the physical laws in terms of mathematical language evolves in time. The Maxwell laws were not always written in the beautiful differential form. Moreover the physical laws are to be valid in some regime. We know that Newton's laws fail for high energies so we just accept them at low energies and seek for other laws valid at higher energies.

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There is no absolute justification for physical laws. In fact it is best to drop the term law. With Isaac Newton's De motu corporum, book 1 of the Philosophiæ Naturalis Principia Mathematica, the concept of physical laws emerged. Newton's laws were seen in the 18th and 19th century as close to what Moses brought down from Mount Sinai. This idea carried to thermodynamics, where again there are three laws, and somewhat into electromagnetism. With the 20th century the idea of laws was replaced with theory. A theory is a consistent hypothesis on some order of nature that is supported by measurement and observation.

The advances in relativity and quantum mechanics made it clear that Newton's laws are approximations that hold in some domain of experience that is limited. This domain of experience require, velocities much slower than light and comparatively small masses or masses much larger in radius than the Schwarzschild radius for relativity and systems with action $S~>>~n\hbar$ and much larger than wavelength for quantum physics. Outside of that domain of experience Newtonian mechanics fails, and as such they can't really be regarded as laws. They are called laws more as a matter of tradition than anything else.

The state of affairs is not likely to change in any philosophical sense. It is possible that there is final physical theory, which in some operational sense could be called a set of physical laws. This is quantum mechanics, which is not formulated according as laws, but as physical axioms we call postulates. I say this in light of the fact this could be what builds up spacetime. I wrote a post here on SE on how this can be in some fundamental way with an isomorphism between the Tsirelson bound with a line element . Other approaches involve van Raamdsdonk's idea and Sean Carroll has advanced the idea of a combinatorial network of Hilbert spaces in general entanglements. These hypotheses could solidify into more solid theory in the coming decades, and they might be as close to being a law as physics will ever come.

Maxwell's equations are instances of Yang-Mills equations. These are really best understood quantum mechanically. There is a rich underlying theory and mathematics on this. In particular there are the theorems of Uhlenbeck, Freed, Atiyah and Donaldson on the structure of four dimensional manifolds. In addition the gluon in QCD, which is a color charged gauge boson (two charges) can form a colorless di-gluon, which opposite color charges, so that the triplet entangled state is identical to the graviton. It is the possible that the S-dual of the gluon forms in this way the actual graviton. By these means the Maxwell equations, an abelian $U(1)$ gauge principle, and the other gauge fields are a feature of quantum foundations.

We might then say that quantum mechanics is a sort of founding logic to the physical world. The questions are not over, for we have difficulties understanding what is meant by physical reality in quantum physics. Is quantum physics $\psi$-epistemic, as with Bohr and the Copenhagen interpretation, or is it $\psi$-ontological as with the Everett Many Worlds? It seems the human brain that constructs our intuitive understanding of the world in a macroscopic and classical way has troubles understanding some quantum foundations. Physicists have then had a history of imposing quantum interpretations, all of which involve some extra ingredient.

Finally, physics differs from mathematics in that mathematics is a quest for any possible set of relationships between abstract objects. Mathematics is then in a way almost a logical form of art. Physics is more concerned with addressing very similar questions over and over and finding what set of principles can be appealed to that are supported by new observations. Physics is an empirical science, while mathematics is not so much. However with computers there does seems to be a sort of synthetic empiricism developing in the math world.

As for the relationship between physics and mathematics, that is something we are not likely to every understand. The earliest idea of Plato's idea of ideal and real forms, with language (logos) spanning the two. That gets a bit mystical for most physicists, though some mathematicians hold to something like that to justify objectivity to mathematics. In other world mathematics is not only just a sort of word-symbol game, but does involve things that are "real." Max Tegmark has a sort of final multiverse concept of world based on all possible mathematics, but I have a hard time seeing how that can be supported very well by observations. We will have to see how that develops, and there are people who take Max seriously. We may though find that we can never understand how physics and mathematics are unified in some whole way.

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You've asked a complex question and I really don't know if I'm getting it. One thing you asked gives me an idea of how to answer your overall question. You asked:

What are the intuitive properties of charges and currents that lead one to write down Maxwell's equations?

Strictly speaking about Maxwell's Equations, I don't think there is anything at all intuitive about the properties of charges and currents.

Keep in mind, we are or have been students who have learned about Maxwell's Equations by solving college level homework problems on the subject. But these theories were developed by extremely smart people who "bashed their brains in" to come up with these mathematical descriptions. They were true geniuses who had to put in many, many hours of grueling thought to come up with this stuff.

We look at the textbook descriptions today and we say things like, "Oh yeah! That's understandable or intuitive." Back in Maxwell's time, this was on the cutting edge. There was nothing intuitive about it. The unknown is never intuitive. When Einstein came up with his Special Theory of Relativity, it confused almost every genius out there and wasn't fully accepted until multiple decades later.

Another thing that's important to remember is that science is always wrong. It simply becomes less wrong over time. Just because we have described something precisely with mathematics and the description works, it rarely works all of the time. The progression of physics from Newton to Einstein is an example of this. Newton was "correct", but Einstein was "more correct". We now know that Einstein will have to yield to something "even more correct".

The main problem is that humans are unable to physically see the phenomena we want to study. We can't see electrons. We can't see quarks in action. We can't see the quantum foam. We have to be able to "see" in order come up with the mathematical descriptions.

With each passing year, it gets harder and harder for us to make major progress in the fundamentals. How many years beyond the Higgs Boson discovery do we have to wait for the next huge breakthrough? And it's taking larger and larger teams of mega-minds to achieve those breakthroughs.

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    $\begingroup$ I think you've come closer than anyone else to what I was getting at! I think I misused the word 'intuitive'. What I mean is "How can we modify our intuition about charges and currents so that the Maxwell equations become intuitive?". For instance, there's no a priori reason to expect energy or momentum to be conserved, but ince you incorporate that into your intuition, the strange behaviour of mechanics becomes "obvious'. As I mentioned in my edit, this intuition will one day turn out to be incorrect, but that doesn't stop it from being helpful. $\endgroup$ – preferred_anon Jul 23 '16 at 15:38
  • $\begingroup$ @DanielLittlewood There is an extremely strong reason for conservation laws: Noether's theorem. $\endgroup$ – tfb Jul 23 '16 at 15:39
  • $\begingroup$ @tfb Noether's theorem is precisely what I mean by a modification of intuition (unless you find Noether's theorem intuitively obvious?) $\endgroup$ – preferred_anon Jul 23 '16 at 15:44
  • $\begingroup$ @Daniel Littlewood I love the way you think. You think far more deeply than most. Unfortunately, we live and breathe in the non-electromagnetic world. We, instead, live and breathe in the non-relativistic Newtonian world. We are biologically biased and I don't think we'll ever intuitively come to grips with why magnetic fields are perpendicular to the direction of the current (if I'm remembering that correctly). :>) $\endgroup$ – Inquisitive Jul 23 '16 at 15:54
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    $\begingroup$ I think that might be one of the things we have to take as "that's the way the world is". For example, I don't expect to be able to explain why the energy of a particle has a particular mathematical expression, but i hope to be able to explain the dynamics of particles in terms of it. Similarly, I don't know why the magnetic field is there, but I know it pushes moving charges around. $\endgroup$ – preferred_anon Jul 23 '16 at 16:13
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I think the way you posed your question implies a weird concept of physics. As I understand it, your question is "why do we expect the axioms we write down to yield physically reasonable results?" The point is, similar as in math, those axioms are usually not just written down randomly, following by their implications being explored. To the contrary, we usually have a lot of experimental data and single descriptions of those experiments in the form of empirical laws and with time, theories that can predict more and more of those experiments are developed and for those theories an axiomatic approach/formulation is then developed. We expect the axioms to yield reasonable results because they come out of a process of analysing theories that have shown to be reasonable or valid.
Of course, theoretical physics is most fun based on axioms formulated in a good mathematical framework. But I think it is important to recognise that this is not the way theories are developed.

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