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I understand how classical physics leads to the UV catastrophe. But I cannot understand how quantization solves it.
I know this question is similar to many other questions but I need a layman's explanation (without too much mathematical details).
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Concerning point c), on how complex numbers come into quantum theory:
This has a beautiful conceptual explanation, I think, by applying Lie theory to classical mechanics. The following is taken from what I have written on the nLab at quantization -- Motivation from classical mechanics and Lie theory. See there for more pointers and details:
Quantization of course was and is motivated by experiment, hence by observation of the observable universe: it just so happens that quantum mechanics and quantum field theory correctly account for experimental observations where classical mechanics and classical field theory gives no answer or incorrect answers. A historically important example is the phenomenon called the “ultraviolet catastrophe”, a paradox predicted by classical statistical mechanics which is not observed in nature, and which is corrected by quantum mechanics.
But one may also ask, independently of experimental input, if there are good formal mathematical reasons and motivations to pass from classical mechanics to quantum mechanics. Could one have been led to quantum mechanics by just pondering the mathematical formalism of classical mechanics? (Hence more precisely: is there a natural Synthetic Quantum Field Theory?)
The following spells out an argument to this effect. It will work for readers with a background in modern mathematics, notably in Lie theory, and with an understanding of the formalization of classical/prequantum mechanics in terms of symplectic geometry.
So to briefly recall, a system of classical mechanics/prequantum mechanics is a phase space, formalized as a symplectic manifold (X,ω). A symplectic manifold is in particular a Poisson manifold, which means that the algebra of functions on phase space X, hence the algebra of classical observables, is canonically equipped with a compatible Lie bracket: the Poisson bracket. This Lie bracket is what controls dynamics in classical mechanics. For instance if H∈C ∞(X) is the function on phase space which is interpreted as assigning to each configuration of the system its energy – the Hamiltonian function – then the Poisson bracket with H yields the infinitesimal time evolution of the system: the differential equation famous as Hamilton's equations.
Something to take notice of here is the infinitesimal nature of the Poisson bracket. Generally, whenever one has a Lie algebra 𝔤, then it is to be regarded as the infinitesimal approximation to a globally defined object, the corresponding Lie group (or generally smooth group) G. One also says that G is a Lie integration of 𝔤 and that 𝔤 is the Lie differentiation of G.
Therefore a natural question to ask is: Since the observables in classical mechanics form a Lie algebra under Poisson bracket, what then is the corresponding Lie group?
The answer to this is of course “well known” in the literature, in the sense that there are relevant monographs which state the answer. But, maybe surprisingly, the answer to this question is not (at time of this writing) a widely advertized fact that has found its way into the basic educational textbooks. The answer is that this Lie group which integrates the Poisson bracket is the “quantomorphism group”, an object that seamlessly leads to the quantum mechanics of the system.
Before we spell this out in more detail, we need a brief technical aside: of course Lie integration is not quite unique. There may be different global Lie group objects with the same Lie algebra.
The simplest example of this is already one of central importance for the issue of quantization, namely, the Lie integration of the abelian line Lie algebra ℝ. This has essentially two different Lie groups associated with it: the simply connected translation group, which is just ℝ itself again, equipped with its canonical additive abelian group structure, and the discrete quotient of this by the group of integers, which is the circle group U(1)=ℝ/ℤ. Notice that it is the discrete and hence “quantized” nature of the integers that makes the real line become a circle here. This is not entirely a coincidence of terminology, but can be traced back to the heart of what is “quantized” about quantum mechanics.
Namely, one finds that the Poisson bracket Lie algebra 𝔭𝔬𝔦𝔰𝔰(X,ω) of the classical observables on phase space is (for X a connected manifold) a Lie algebra extension of the Lie algebra 𝔥𝔞𝔪(X) of Hamiltonian vector fields on X by the line Lie algebra: ℝ⟶𝔭𝔬𝔦𝔰𝔰(X,ω)⟶𝔥𝔞𝔪(X). This means that under Lie integration the Poisson bracket turns into an central extension of the group of Hamiltonian symplectomorphisms of (X,ω). And either it is the fairly trivial non-compact extension by ℝ, or it is the interesting central extension by the circle group U(1). For this non-trivial Lie integration to exist, (X,ω) needs to satisfy a quantization condition which says that it admits a prequantum line bundle. If so, then this U(1)-central extension of the group Ham(X,ω) of Hamiltonian symplectomorphisms exists and is called… the quantomorphism group QuantMorph(X,ω): U(1)⟶QuantMorph(X,ω)⟶Ham(X,ω). While important, for some reason this group is not very well known, which is striking because it contains a small subgroup which is famous in quantum mechanics: the Heisenberg group.
More precisely, whenever (X,ω) itself has a compatible group structure, notably if (X,ω) is just a symplectic vector space (regarded as a group under addition of vectors), then we may ask for the subgroup of the quantomorphism group which covers the (left) action of phase space (X,ω) on itself. This is the corresponding Heisenberg group Heis(X,ω), which in turn is a U(1)-central extension of the group X itself: U(1)⟶Heis(X,ω)⟶X. At this point it is worth pausing for a second to note how the hallmark of quantum mechanics has appeared as if out of nowhere simply by applying Lie integration to the Lie algebraic structures in classical mechanics:
if we think of Lie integrating ℝ to the interesting circle group U(1) instead of to the uninteresting translation group ℝ, then the name of its canonical basis element 1∈ℝ is canonically ”i”, the imaginary unit. Therefore one often writes the above central extension instead as follows: iℝ⟶𝔭𝔬𝔦𝔰𝔰(X,ω)⟶𝔥𝔞𝔪(X,ω) in order to amplify this. But now consider the simple special case where (X,ω)=(ℝ 2,dp∧dq) is the 2-dimensional symplectic vector space which is for instance the phase space of the particle propagating on the line. Then a canonical set of generators for the corresponding Poisson bracket Lie algebra consists of the linear functions p and q of classical mechanics textbook fame, together with the constant function. Under the above Lie theoretic identification, this constant function is the canonical basis element of iℝ, hence purely Lie theoretically it is to be called ”i”.
With this notation then the Poisson bracket, written in the form that makes its Lie integration manifest, indeed reads [q,p]=i. Since the choice of basis element of iℝ is arbitrary, we may rescale here the i by any non-vanishing real number without changing this statement. If we write ”ℏ” for this element, then the Poisson bracket instead reads [q,p]=iℏ. This is of course the hallmark equation for quantum physics, if we interpret ℏ here indeed as Planck's constant. We see it arises here merely by considering the non-trivial (the interesting, the non-simply connected) Lie integration of the Poisson bracket.
This is only the beginning of the story of quantization, naturally understood and indeed “derived” from applying Lie theory to classical mechanics. From here the story continues. It is called the story of geometric quantization. We close this motivation section here by some brief outlook.
The quantomorphism group which is the non-trivial Lie integration of the Poisson bracket is naturally constructed as follows: given the symplectic form ω, it is natural to ask if it is the curvature 2-form of a U(1)-principal connection ∇ on complex line bundle L over X (this is directly analogous to Dirac charge quantization when instead of a symplectic form on phase space we consider the the field strength 2-form of electromagnetism on spacetime). If so, such a connection (L,∇) is called a prequantum line bundle of the phase space (X,ω). The quantomorphism group is simply the automorphism group of the prequantum line bundle, covering diffeomorphisms of the phase space (the Hamiltonian symplectomorphisms mentioned above).
As such, the quantomorphism group naturally acts on the space of sections of L. Such a section is like a wavefunction, except that it depends on all of phase space, instead of just on the “canonical coordinates”. For purely abstract mathematical reasons (which we won’t discuss here, but see at motivic quantization for more) it is indeed natural to choose a “polarization” of phase space into canonical coordinates and canonical momenta and consider only those sections of the prequantum line bundle which depend only on the former. These are the actual wavefunctions of quantum mechanics, hence the quantum states. And the subgroup of the quantomorphism group which preserves these polarized sections is the group of exponentiated quantum observables. For instance in the simple case mentioned before where (X,ω) is the 2-dimensional symplectic vector space, this is the Heisenberg group with its famous action by multiplication and differentiation operators on the space of complex-valued functions on the real line.
The idea is related to thermal equilibrium of a black body. The classical theory predicts that this type of bodyy will emit light on all frequencies, with energy density increasing with frequency $B_\nu \sim\nu^2$.
And you can actually see this is not case (hence the catastrophe part), a furnace is a good example of a black body, the classical model will conclude that there'll be light with an arbitrarily high energy density coming out of the furnace, which is not observed in reality
Best Answer
Avoiding mathematical formulae to the maximum, and warning for furious hand-waving ahead, I would state it like this:
In the classical picture, there is no quanta concept, so you could have just a little bit of radiation energy at any frequency. However, since quantization appeared, the minimum amount of radiation energy that you could possibly have at a frequency $\nu$ is $h\nu$. Since the temperature regulates the distribution of energy by the Boltzmann equation ($\mathbb{P}(E)\propto e^{-E/kT}$), if you have radiation at a certain frequency such that $h\nu\gg kT$, then having just one photon is "too much", or it is highly unlikely.
Alternatively, we know from statistical mechanics that it is very unlikely to have radiation with energy higher than $kT$, with the precise expression for this statement given by Boltzmann law. But for $h\nu\gg kT$ type of radiation, Planck leaves us no choice: you have to have more than one photon, or none. Therefore, you almost do not have this type of radiation (or these type of photons) when the temperature is low, therefore "depopulating" the UV states.
Now I realize something: Planck could have "solved" the UV catastrophe by postulating not complete quantization, but only that the minimum amount of energy at a certain frequency is $h\nu$. With this "floor", you do not need to go further and say that the energy has to come on integer multiples of this minimum quantity, just that, if energy of radiation at this frequency exists at all, then it has a minimum $h\nu$.