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46

The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system. To understand this answer you have to ...


30

The theory of probability used in QM is intrinsically different from the one commonly used for the following reason: The space of events is non-commutative (more properly non-Boolean) and this fact deeply affects the conditional probability theory. The probability that A happens if B happened is computed differently in classical probability theory and in ...


28

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number ...


23

Why would you ever try to motivate a physical theory without appealing to experimental results??? The motivation of quantum mechanics is that it explains experimental results. It is obvious that you would choose a simpler, more intuitive picture than quantum mechanics if you weren't interested in predicting anything. If you are willing to permit some ...


13

I am late to this party here, but I can maybe advertize something pretty close to a derivation of quantum mechanics from pairing classical mechanics with its natural mathematical context, namely with Lie theory. I haven't had a chance yet to try the following on first-year students, but I am pretty confident that with just a tad more pedagogical guidance ...


10

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples: The finite number of states in a compact region of phase space. See e.g. this Phys.SE post. The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular ...


9

No macroscopic quantum system is described by a pure state. For example, notions like temperature or pressure, which apply to macroscopic systems do not even exist for systems described by a pure state. The description of macroscopic objects (discussed in statistical mechanics) is always in terms of a density matrix (or the essentially equivalent notion of a ...


9

There's two (ultimately related) answers. For the first answer, just forget about $\hbar$ (but say $c=1$), we are doing a classical relativistic field theory. The first is that you can consider the field profile around a static, spherically symmetric source of mass $M$ (you need to add a coupling to the action of the form $g \phi J$, where $J$ is an ...


8

You are right, it is wrong to think that in gauge theory "gauge transformations are just a redundancy". This becomes true only if one abandons locality, ignores all boundary effects, all instanton effects, hence most of what is interesting about gauge theory. Of course forming gauge equivalence classes (say of observables) is something one wants to do every ...


7

I suppose there are two scientific reasons to look into the foundations of QM: As part of checking in finer and finer detail that indeed the world is governed by standard quantum physics. The towering example here is Bell's theorem. From inspection of the foundations this makes some prediction which can be and has been checked by experiment. As part of the ...


7

Well, the problem is still open. Although maybe the axioms were taken as self-evident for mathematics, Hilbert did not really want mathematically self-evident axioms to be the basics for physical axioms. Since Gauß and the hyperbolic space, it is well known that you can get equally valid models from different assumptions that could all be seen as ...


7

I think there are two answers to this, one emprical and one theoretical. First, the theoretical one: What you describe is essentially induction, the belief that we can generalize from a subset of a class events/situations to the whole class of events/situations. This belief is, by necessity, unprovable, only falsifiable, since proving it would require ...


7

An excellent book which does more or less what you ask for is Asher Peres' "Quantum theory:concepts and methods". It starts from the Stern-Gerlach experiments and logical reasoning to develop the basic principles of quantum mechanics. From there, it develops the necessary algebra. Another interesting book for an approach of the conceptual side of quantum ...


7

There are several forms of discreteness in quantum theory. The simplest one is the discreteness of eigenvalues and the associated countable eigenstates. Those arise similarly to the discrete standing waves on a guitar string. The boundary conditions only allow certain standing waves that nicely fit into the enforced region in space. Even though the string is ...


6

I searched for "mixing" and didn't find it in other answers. But this is the key. Ergodicity is largely irrelevant, but mixing is the property that makes equilibrium statistical physics tick for many-particle systems. See, e.g., Sklar's Physics and Chance or Jaynes' papers on statistical physics. The chaotic hypothesis of Gallavotti and Cohen basically ...


6

You should use history of physics to ask them questions where classical physics fail. For example, you can tell them result of Rutherford's experiment and ask: If an electron is orbiting around nucleus, it means a charge is in acceleration. So, electrons should release electromagnetic energy. If that's the case, electrons would loose its energy to collapse ...


6

If I would be designing an introduction to quantum physics course for physics undergrads, I would seriously consider starting from the observed Bell-GHZ violations. Something along the lines of David Mermin's approach. If there is one thing that makes clear that no form of classical physics can provide the deepest law of nature, this is it. (This does make ...


5

Mathematical proof is to physics roughly what syllogism (or some other fundamental inference rule) is to logic. Namely, it begins from assumptions modelling our conception of some physical reality and shows what must be so if the assumptions hold, but it cannot say anything about the underlying assumptions themselves. A simple example was given by dmckee in ...


5

Here is a belated reply. (I come across this question only now, by chance. This was posted right when our daughter was born, which was kind of distracting for me...) The quick answer to the question is the following somewhat remarkable statement Identity types in the new foundations of mathematics in homotopy type theory correspond in physics to spaces of ...


5

To be honest, I think that the route you describe (and which is also used in many textbooks) is not physically well motivated at all. You have begun with a theory of a fermion with a global symmetry which maps physical states to different physical states. This theory has the property that specifying initial conditions on a spacelike surface completely ...


5

Though there are many good answers here, I believe I can still contribute something which answers a small part of your question. There is one reason to look for a theory beyond classical physics which is purely theoretical and this is the UV catastrophe. According to the classical theory of light, an ideal black body at thermal equilibrium will emit ...


5

Let me try to convince you that the density operator is a mathematical convenience and not a fundamental aspect of quantum mechanics by describing a very general setup for states and observables in both classical and quantum mechanics. This may not directly answer your question, but hopefully it will settle whatever motivated this question. Briefly, ...


5

It seems that perhaps you are missing a crucial piece of the puzzle. The stochastic approach you describe is equivalent to performing a unitary which is chosen stochastically (essentially applying a superoperator) and then measuring in some fixed basis. In principle, however, you can go further, by stochastically choosing whether or not to measure at all, ...


4

All the key parts of quantum mechanics may be found in classical physics. 1) In statistical mechanics the system is also described by a distribution function. No definite coordinates, no definite momenta. 2) Hamilton made his formalism for classical mechanics. His ideas were pretty much in line with ideas which were put into modern quantum mechanics long ...


4

Hilbert's Sixth problem is not the same as finding the theory of everything and then making the maths rigorous. This is a very common misconception, and has led to people thinking that making renormalisation in QFT rigorous was the main thing to do. But in fact Hilbert stated explicitly that it would be just as important to axiomatise false physical ...


4

Given a $C^\ast$-algebra $A$, its "Bohr topos" (see there for a survey) is the presheaf topos on its commutative subalgebras. The idea here is that if we think of $A$ as the algebra of quantum operators of a quantum mechanical system (for instance all the bounded operators on the Hilbert space of states of a system), then the commutative subalgebras ...


4

There is no reason why physical laws should be absolute. But observation tells us they are. If you think about it, if the laws of the universe did change from place to place or if they were different at different times, there would be no laws, and there would be no science.


4

I would say that the answer to your question is in the Wigner's theorem http://en.wikipedia.org/wiki/Wigner%27s_theorem. For any quantum system you need to have a "representation" of the Poincarè group on it. By "representation" I mean an homomorphism from the Poincarè group to the group of "Simmetries" (as defined in the above article) given the group ...


4

General relativity is only conformally invariant in two dimensions. This can be proven by making the transformation $g_{ab} \rightarrow \phi g_{ab}$, and seeing what transformation Einstein's equation${}^{1}$ makes. What you will find is that Einstein's equation will MOSTLY transform, but you will get terms proportional to $(d-2)(d-1)$ and derivatives of ...



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