Quantum mechanics is more beautiful than classical physics

Luboš Motl, November 07, 2016

One of the numerous unjustifiable and negative adjectives that Steven Weinberg attached to quantum mechanics in an interview was «ugly». It’s ugly when there isn’t an objective state of affairs at every moment, we were told. Equivalently, it’s ugly when the world doesn’t obey the laws of classical physics.

The aesthetic discussions are unavoidably subjective to some extent and people’s differences cannot be completely settled. Nevertheless, I am confident that his view – and the view of many others who use murky excuses to sling mud on quantum mechanics and its brilliant fathers – is completely misguided. Here, I want to sketch a few dozens of reasons why quantum mechanics is prettier than the framework it has superseded, that of classical physics.

Commutators are prettier than Poisson brackets

I decided to start with this somewhat «obscure and technical» example because the beauty may be immediately translated to the compactness of the formulae. Long time before quantum mechanics was discovered, people were describing the foundations of classical physics in terms of the Poisson bracket:

\[ \{F,G\} = \sum_i \frac{\partial F}{\partial x_i} \frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i} \frac{\partial G}{\partial x_i} \] Hamilton’s equations, canonical transformations etc. were naturally written in terms of it. Well, this structure was replaced by the commutator of two operators in quantum mechanics:

\[ [F,G] = FG — GF. \] Which of the right hand sides is prettier? There’s no contest. You may say that the classical expression is longer because it had to be «reverse-engineered» to have certain helpful properties. In quantum mechanics, simple well-defined questions often have straightforward answers. Note that the Poisson bracket may be obtained from the commutator as

\[ \{F,G\} = \lim_{\hbar \to 0} \frac{1}{i\hbar} [F,G] \] Fine. You may say that the complicated formula for the Poisson bracket was a men’s effort to «fake» the beauty of Nature while the commutator shows the beauty of Nature in its naked form.

The set of possible evolution transformations is prettier in QM than in classical physics

In classical physics, the evolution \(t\to t+\Delta t\) moves points along some lines in the phase space. When you look at two moments and what the evolution does to the points, you may see that the evolution map is a permutation of the points of the phase space, hopefully a permutation that is differentiable at the same moment. It is a one-to-one (invertible) map between the points of the phase space. Moreover, it must be one that preserves the symplectic structure on the phase space (effectively the Poisson bracket above).

Quantum mechanics, the evolution transformation and all analogous operations is given by a unitary linear operator \(U\) i.e. one that obeys \(U U^\dagger = 1\). The unitary group where \(U\) a priori belongs is a much prettier and less singular object than the «set of permutations on the phase space». In particular, as I will discuss below, the effective dimension of the Hilbert space may be chosen to be finite in any realistic setup. So the evolution operators come from a \(U(N)\) group.

That’s much prettier than the permutation group \(S_\infty\) permuting an infinite number of points, especially when this infinity is uncountable (a continuum or worse).

Quantum mechanics avoids the threat of UV singularities on the phase space

In classical physics, one may in principle distinguish two arbitrarily close points \((x_i,p_i)\) and \((x’_i,p’_i)\) on the phase space. This is a pathology of a sort because some new processes could always hide inside these very short areas of the phase space. In quantum mechanics, the uncertainty principle guarantees \(\Delta x \cdot \Delta p \geq \hbar/2\). So if the two points are too close in the \(x\) direction(s), they can’t be close in the \(p\) direction(s), and vice versa.

Everyone who thinks that the «smearing» of the black hole and other singularities is a «good thing» must obviously like what quantum mechanics does to the short distances in the phase space – it makes all of them harmless.

The non-existence of an objective state before a measurement in QM is a sign that QM respects, economically deals with the resources

This is the actual difference that made Weinberg call quantum mechanics «ugly». He thinks that classical physics is «pretty» because a state of the physical system is in principle knowable at all times while quantum mechanics makes it impossible to make any sharp statements about the physical system before or in between observations.

Which one is prettier?

Classical physics assumes that there exists a preferred history or trajectory \((x_i(t),p_i(t))\) that picks a preferred point \((x_i,p_i)\) on the phase space at every moment \(t\). And this information doesn’t care about who the observer is. The observer plays at most a passive role. What is analogous to?

Well, it’s analogous to a single central government authority, like the IRS, that forces everyone (and every particle) to send tax returns at every moment. Those must contain the information such as \(x_i(t)\) and \(p_i(t)\). Sending this complicated pile of paper junk once a year isn’t enough. You must send it every picosecond – infinitely many times each picosecond. And the IRS declares itself the authority that knows about the objective status of all financial subjects at every moment.

On the contrary, quantum mechanics acknowledges that the physical systems and people must have the freedom to breathe and live. They live while no one watches them most of the time. (I couldn’t avoid bringing this metaphor because the totalitarian real-time monitoring of all cash operations in Czechia may already start on December 1st.) A world that wouldn’t be watched by anyone would be effectively non-existent. So quantum mechanics allows observations. But the knowledge only arises when something is actually observed by an observer. The results can depend and do depend on the identity of the observer. Also, the observation has additional consequences – it changes the state of the physical system (wave function «collapse»).

The fact that the results and optimal description depends on the observer may be viewed as a physics analogy of the individual rights. The world exists for the people, not for the central IRS. And the fact that the state vector or density matrix «collapses» is just quantum mechanics’ appreciation that it requires some hard work to file a tax return (or a report about a beer that someone bought in a rural pub). One can’t assume such things for granted. So whenever someone observes a physical system, he changes it. It’s prettier for a physical theory to acknowledge that «observations aren’t for free». The price – the collapse – one «pays» for the observation is a natural protection against too obtrusive supervision. You may observe but it has consequences.

A fan of fascism, Nazism, socialism, or the Big Government or any similar political pathology like that could argue that the individual rights – and the complain that the duties towards the government do affect our lives – are «ugly». But thankfully, I am not one of them. It’s the central authorities, permanent monitoring, and disrespect towards the individual rights and the work needed to file a tax return that are «ugly». Correspondingly, classical physics is ugly for analogous reasons.

You just don’t need to monitor someone at all times – and you really can’t, without greatly affecting the system – and quantum mechanics appreciates that the only real data are the results of observations. And it works with what is available, not attempting to get much more than it needs. Quantum mechanics is humble and thrifty in this sense. On the other hand, classical physics is like a totalitarian state that monitors everyone at all times, allows the only truth, and forces everyone to parrot the only truth even if and when he hasn’t observed any evidence for it.

Quantum mechanics allows much more diverse ways to study physical systems, interpolates between particles and fields etc.

In classical physics, particles and fields were inevitably inequivalent systems. After all, the function \(\Phi(x,y,z,t)\) describing a classical field seems to carry much more information – an infinite collection of real numbers – than the coordinates \(x_i\) of \(M\) particles. But quantum mechanics allows you to have a single physical system – a single Hilbert space – and define the observables such as \(x_i\) and/or \(\Phi(x,y,z,t)\) that act on this Hilbert space (think of a Hilbert space for a single particle species; the particles’ positions are defined in a sector with a fixed number of particle excitations).

It’s much prettier when quantum mechanics allows you all these different operators probing the «same object». It’s like a painting from a top artist that may be looked at from many directions, that can always make you think about hidden meanings in a new way. Quantum mechanics is extremely good exactly at this kind of art. It’s another reason why it’s so beautiful.

Not just wave-particle dualities but stringy and related dualities are enabled by and require quantum mechanics

In string theory, we have many dualities – equivalences between two or many string-theoretical descriptions that are seemingly inequivalent but when you list all the possible objects and processes and probabilities of transitions in between them, the two descriptions may be mapped to one another. The two physical theories are precisely equivalent even though the starting points to define them look significantly different.

Examples include S-dualities (also exist in quantum field theories), T-dualities (plus mirror symmetry), U-dualities, bosonization and fermionization on the world sheet, AdS/CFT holographic correspondence, perhaps the ER-EPR correspondence, and others. It would take us too far to explain what these mean. You should study it elsewhere because they’re cool relationships.

But here I want to emphasize that every single duality in this list absolutely requires the world to fundamentally follow the laws of quantum mechanics, not classical physics. In the previous section, I argued that quantum mechanics allows you to define many sets of operators on the same Hilbert space. Dualities are analogous except that the «two completely different sets of operators» that you may define overlap and have e.g. the same Hamiltonian (or other operators defining the evolution).

So you may imagine the physical system in two or more completely different ways – it’s a system of closed heterotic strings only, or a system of open and closed type I strings, and so on – but the evolution is exactly the same when you map the states on their partners in the other description correctly.

I believe that the dualities – for which quantum mechanics is essential – are beautiful, and for numerous reasons. One of them is that they reduce the number of independent theories (examples of a quantum mechanical theory) and therefore achieve some deeper kind of a unification.

Feynman’s democracy between all trajectories is prettier than a single preferred trajectory

In classical physics, there is one «only correct» history and all the others, including those infinitesimally separated from the correct one, are wrong. The functional \(K[x_i(t)]\) encoding the correctness of the trajectory \(x_i(t)\) is therefore mostly zero and it is equal to one (or infinity) for a single very special trajectory only. It’s a very ugly, singular functional.

On the other hand, the right way to talk about the «space of trajectories» in quantum mechanics is Feynman’s path integral approach to quantum mechanics. All trajectories and histories are really equally kosher. They are just weighted by \(\exp(iS[x_i(t)]/\hbar)\) in the path integral, a number that is a pure phase (i.e. it has the same absolute value for all trajectories). All the results of the sort that «some histories look more real than others» is coming from the constructive (and destructive) interference.

The democratic functional \(\exp(iS[x_i(t)]/\hbar)\) in quantum mechanics is much less singular – and therefore much prettier – than the delta-function-like «correctness» functional \(K[x_i(t)]\) that we saw in classical physics.

The discrete spectra require QM, are beautiful, and probably needed for life and similar structures

In nice enough laws of classical physics with a continuous time \(t\), the physical quantities such as the positions \(x_i\) must be a priori continuous, too. There’s really no truly qualitative difference between two states of an object in classical physics. There are always states in between. You may always interpolate.

On the contrary, very natural and important observables in quantum mechanics often have a discrete (or mixed) spectrum. This is a pretty characteristically quantum mechanical feature that allows Nature to separate the states – and carry well-defined information – really nicely. The list of known nuclei, atoms, and molecules is basically discrete – thanks to quantum mechanics. Genetic information is basically discrete and some information carried by the brains etc. is largely discrete, too. All these possibilities are much more likely and elegant in quantum mechanics. In classical physics, there would never be any strictly separated nuclei, atoms, molecules, DNA codes – one could always mix them to mushed potatoes in between.

A particular consequence of the discreteness of the spectra is the stability of atoms and other objects. A classical atom would allow the electron fall into the nucleus in a picosecond. In quantum mechanics, the electron in the ground state can’t fall deeper because it has the lowest possible energy eigenvalue. Stable objects are of course more beautiful than those who collapse immediately.

The perfect uniqueness and well-definedness of the low-lying states is pretty

As I discussed in some blog posts about the low heat capacities, quantum mechanics at low temperature carries an extremely low amount of information or entropy. Because all the states that are mutually exclusive with the ground state must be orthogonal and there’s an energy gap in between the ground state and the first excited state, the state of a physical system (such as a molecule) at a very low temperature is absolutely unique and precisely given. It’s the ground state.

Classical physics wouldn’t allow such a clean outcome. There would always be some continuous degrees of freedom that could oscillate around the energy minimum – even at arbitrarily low temperatures. That’s why classical theories with «enough stuff to imitate quantum mechanics» would always imply heat capacities that are vastly higher than \(O(k_B)\) per atom i.e. vastly higher than what is observed (usually infinitely higher).

When we focus on the beauty, it would also mean that atoms, molecules, and physical systems in general can never become «clean». There would always be some dirty noise on top of the objects. The possibility to make the objects absolutely clean is another feature that makes quantum mechanics prettier than its classical predecessor.

Symmetry often decides about forbidden processes and/or big decreases of energy in quantum mechanics, and that’s pretty

Everything that can be measured in quantum mechanics is given by the linear operators and there are operators associated with the symmetry transformations, too. The initial state may be an eigenstate – and if the symmetry transformation commutes with the Hamiltonian, it will be an eigenstate forever.

So in quantum mechanics, we have numerous operators such as the parity, CP etc. that imply that some (many/most) processes are strictly forbidden. Symmetry may be used to deduce things in classical physics as well. But in quantum mechanics, the power of symmetries is much stronger. For example, quantum mechanics allows states with \(P=+1\) or \(P=-1\) even when the number of particles is odd. In classical physics, you would need the «odd» particle sit at \(\vec x=0\) if you wanted the configuration to be parity-symmetric.

Quantum mechanics makes proofs of Noether’s and other theorems crisp and pretty

Emmy Noether was a top female mathematician and theoretical physics and she’s primarily known for her theorem relating symmetries and conservation laws. Her paper is somewhat unreadable. In quantum mechanics, it’s really trivial to see why conservation laws and symmetries are linked to each other.

It all boils down to the claim

\[ [H,L]=0. \] There are two ways to interpret this vanishing commutator. The commutator may be seen to be the right hand side of the Heisenberg equation of motion for \(L\). When it’s zero, it means that \(L(t)\) doesn’t depend on time: it’s conserved. Alternatively, you may describe the commutator as the infinitesimal change of the Hamiltonian \(H\) under the transformations generated by the generator \(L\). When the commutator is zero, we may say that \(L\) really generates a symmetry.

So whenever there is a symmetry, generated by \(L\), the quantity \(L\) is conserved in time, and vice versa. Noether’s theorem becomes trivial because the conserved quantity is nothing else than the generator of the symmetry. And the existence of two ways to read the commutator follows simply from its antisymmetry under the permutation of \(H\) and \(L\).

The very linearity of the Hilbert space is beautiful

The possible states of «maximum knowledge» which implies «complete knowledge» in classical physics are represented as points on the phase space. A phase space is really the «set of all possible configurations». It may be a distorted, arbitrary, singular space.

In quantum mechanics, the set of possible states of «maximum knowledge» is given by a Hilbert space which is a linear vector space. That is much more beautiful than the arbitrary, unconstrained phase space – for the same reason why diamonds are more beautiful than potatoes. The exactly linear shapes – exactly because they seem so constrained and therefore a priori precious or unexpected – are much prettier than the unconstrained, arbitrarily chosen and curved, shapes of potatoes or the classical phase spaces.

Quantum mechanics prettily unifies the «maximum knowledge» description with the probabilistic one, uses «all spaces of ideas in between»

Again, in classical physics, you may always assume that there’s some particular trajectory \(x_i(t)\) on the phase space. If you’re not exactly sure about the initial conditions, you may describe the system in terms of some probability distributions \(\rho(x_i,p_i;t)\) on the phase space.

But the description in terms of \(\rho(t)\) is just directly derived from the description in terms of \(x_i(t)\) and the general probabilistic calculus. So the ignorance is always separated from the fundamental laws of Nature.

In quantum mechanics, things are different. The uncertainty principle guarantees that the observer is ignorant about the values of most observables at each moment, even in the pure state i.e. a state of «maximum knowledge». So he needs to describe results in terms of predicted probabilities.

The beautiful thing is that when you construct the quantum counterpart of the classical \(\rho(x_i,p_i;t)\) description, you obtain the density matrix and this density matrix pretty much obeys the same Schrödinger-like equation as the state vector (except that there are two terms or a commutator). In classical physics, the equations for the trajectory \(x_i(t)\) were very different from the equations for the probability distributions \(\rho(x_i,p_i;t)\). In quantum mechanics, these two systems of equations basically become the same.

I think that this very fact is beautiful because the strict separation between the fundamental laws and ignorance about the initial conditions that existed in classical physics meant that the intermediate area in between the strictly known and incompletely known wasn’t used. But in practice, you always have some uncertainties or incomplete knowledge. So the most fundamental «generating» laws of classical physics were unavoidably separated from the real-world applications by a finite gap.

That’s no longer the case in quantum mechanics. Quantum mechanics forces you to include the probabilistic character of predictions and when you do so, you will find out that exactly the same kind of probabilistic descriptions is also the right tool that also deals with the ignorance that was «strictly optional» (yet unavoidable in practice) in the era of classical physics.

Virtual particles (propagators) are a beautiful way for quantum mechanics to derive forces from a deeper starting point

In quantum field theory (or string theory), you may draw Feynman diagrams. An internal line – a propagator – allows the particles depicted by the external lines interact. For example, the electrostatic repulsion may be reduced to an exchange of a virtual photon in between real electrons.

This would be impossible in classical physics because this whole concept of a «virtual particle» is an example of the breathing or life that parts of Nature are allowed to do in between the observations (in between sending the tax returns). Again, in classical physics, you have to be sending tax returns 24 hours a day so you just can’t find any time for life and breathing. Consequently, there are also no virtual particles in classical physics.

For this reason, electrostatic and other interactions can be derived from some other physics – and tightly connected with some other phenomena, e.g. observations of photons – in quantum mechanics but not in classical physics.

I wanted to write a separate but too similar section about instantons but it would be too similar to this one. Quantum mechanics allows additional interactions induced by instanton configurations – local minima of the action in the path integral that aren’t the global minimum. To summarize, virtual particles and instantons show that the «freedom from the tax returns» in quantum mechanics allows the physical objects to do useful things and it allows many interactions to be explained without introducing new arbitrary building blocks to the theory.

The mathematics of low-dimensional Hilbert spaces is beautiful, powerful, and omnipresent in Nature

Quantum mechanical problems in extreme conditions often simplify to two-dimensional or other low-dimensional Hilbert spaces. A big part of the quantum volume of Feynman’s Lectures on Physics was dedicated to two-level systems. All of them are mathematically isomorphic to each other – which is a pretty form of kinship.

All transformations in two-level systems are given by matrices in \(U(2)\). One can always discuss the same Lie group, Lie algebra, spheres that parameterize the normalized vectors or operators etc. It’s pretty, has some sharp consequences, and this powerful mathematics is recycled at many places of Nature.

Quantum field theories are more constrained which is beautiful

In classical field theory, you could imagine all sorts of complicated and perhaps singular terms – with an arbitrary functional dependence on fields – that are added to the Maxwell-like differential equations governing the evolution of the fields in time.

In quantum mechanics, you will find out that all such attempted additions of terms have consequences. It turns out that most of such interactions would be non-renormalizable and make the theory inconsistent or unpredictive at high enough energies. Also, you must cancel gauge anomalies in every quantum field theory, otherwise it’s inconsistent.

These extra constraints wouldn’t arise in classical field theory. For that reason, the number of mathematically possible modifications and distortions of a classical field theory would be infinite. It would be a huge infinity – surely one that allows you to adjust infinitely many new parameters.

The new quantum mechanical effects and the survival skills in their presence greatly reduces the allowed interactions. Quantum field theories such as the Standard Model effectively only allow you a few dozens of parameters. All other interactions are either inconsequential at low, accessible energies; or they would make the theory inconsistent if you increased the energy just a little bit.

So quantum field theories are always more predictive and constrained which is beautiful. When a theory is constrained, it tells you that it was painted by an artist who realized that almost every pattern on the canvas matters and has to be optimized. The ability of quantum mechanics to single out the «only correct painting», or the only consistent theory, becomes maximal in the case of string theory which allows no adjustable continuous dimensionless parameters whatsoever. This near-uniqueness of field theories and complete uniqueness of string theory would be impossible in the world of classical physics.

Summary

I will add additional sections if I realize that I have always wanted to include some point but I forgot about it today.

But the summary is clear. The transition from classical physics to quantum mechanics has made physics more beautiful – much like the transition from non-relativistic physics to the relativistic one and perhaps a small number of other big advances in physics. The very defining principles and postulates of quantum mechanics are pretty.

The people who claim otherwise either don’t understand quantum mechanics or have a very poor aesthetic taste.