Wednesday, April 04, 2018

We're all quantum mechanics, only we didn't know it

Off-topic, Hawking's resurrection: Stephen Hawking's death and resurrection encouraged him to write lots of papers. In a new paper with Gordy Kane, they argue in favor of a Chinese collider.

Luke and Edwin wrote this nice promotional summary of quantum mechanics:
Like I said, we're all quantum mechanics only we didn't know it!

All the vague but tremendously useful "intuitive physics" we do in our heads, all the knowledge we have about the world, all the fuzzy understanding of what things are: it has been quantum mechanics all along. "We" just didn't notice until 1926 and we just didn't notice until we were told about the great papers from 1926.

If you want to know how a real table feels in the quantum world, just stretch out your hand and press down on the table in front of you. That's how it feels to live in a quantum world!
This is the perfect reaction to the anti-quantum zealots' claims such as "How do you fix the ill quantum mechanics to turn it into classical physics which is what we ultimately perceive, which governs the world around us?" Well, we don't perceive classical physics and classical physics doesn't govern any world we know. The world is governed by quantum mechanics and our perceptions and justifiable logical reasoning about the world are only justified by quantum mechanics. Only in some very special limits, both quantum mechanics and classical physics are usable. But there's no situation in which only classical physics would be right.

OK, start with the following meme by the zealots: "Our observations are classical." This statement may be understood as one of the exercises sparked by these people's unstoppable urge to obediently fight against quantum mechanics for the resurrection of classical physics.

More specifically, this statement is meant to say that "observations" – with some clear outcomes – are something that "doesn't belong" to quantum mechanics and that shows that you need to return all physics to the classical framework at the end.

Well, this assertion is completely untrue. Observations are needed to study the world – and to verify the laws of Nature – both in classical physics and in quantum mechanics. A difference is that the observations are more fundamental in quantum mechanics. What is the observation? The observation is the process of learning a value of an observable \(L(t)\):\[

L(t) = ???

\] We do something so that we replace the question marks by a number. In both cases, i.e. in classical and quantum physics, we have some apparatuses that convert \(L(t)\) to some predictable "location of a needle of a voltmeter" or "the state of a display" (at the very end, they're translated to different states of our brain cells that we're able to distinguish through out consciousness). But quantum mechanics differs from classical physics in the character of the object \(L(t)\), the possible outcomes, and the ways to predict the outcomes.

In classical physics, the observable \(L(t)\) is a function on the phase space:\[


\] So it may be one of the canonical coordinates or momenta \(x_i\) or \(p_j\) themselves. The phase space – spanned by the coordinates \(x_i,p_j\) – determines the full information about the physical system at a given moment (e.g. the initial state). So everything that is measurable must be a function of this information which I describe as continuous canonical coordinates and momenta. In principle, they could be discrete, too.

In quantum mechanics, the observable is a linear Hermitian operator\[

\hat L(t).

\] If the Hilbert space is \(N\)-dimensional, there is a real \(N^2\)-dimensional space of possible Hermitian operators. Each of them corresponds to an observable that may be measured. If you consider a spinless particle on a line, \(x,p\) may be considered "basic observables" and there is indeed a way to write every \(\hat L\) as a "function" of \(x,p\). I wrote "function" in quotation marks because it's not a real function in the sense of a map (assigning a result to every arrangement of the values of the parameters). Instead, the parameters are non-commuting objects and a function is some (possibly infinitely long) polynomial where the ordering of the factors in every term matters.

In classical physics, the possible outcomes are typically\[

L(t) \in \RR

\] or in \(\RR^+\). In quantum mechanics,\[

\hat L(t) = \lambda_k

\] the outcome must be one of the eigenvalues of the operator. In fact, we need to write the equation more carefully, with the state vector:\[

\hat L(t) \ket\psi = \lambda_k \ket\psi

\] The measurement means that we learn that the state vector is an eigenstate of the measured operator associated with the eigenvalue \(\lambda_k\) that we just obtained. In fact, this post-measurement situation is always a consequence of the measurement because the pre-measurement state generically wasn't an eigenstate and the measurement always influences the measured system. In the latest measurement, we learn some information that may partially overwrite some previous information. The latest measured observable has a certain value while some previously measured observables may become uncertain.

The spectrum of possible values \(\{\lambda_k\}\) may be a continuous set – interval, collection of intervals – or a finite or countable discrete set, or a mixed set with intervals plus discrete exceptions. By the way, if you deal with a density matrix, the measurement says\[

(\hat L(t)-\lambda_k)\hat\rho = 0

\] from one side. Because \(\hat L(t),\lambda_k,\rho\) are Hermitian, you may also Hermitian-conjugate the identity above to get\[

\hat\rho (\hat L(t)-\lambda_k) = 0.

\] Now, the laws of physics typically dictate the evolution from one moment of time to another. In classical physics, for every point of the phase space in the initial state at moment \(t\),\[

(x_i,p_j)_t \to (x_i,p_j)_T

\] we may calculate the point of the phase space in the final moment \(T\). The phase space gets transformed or deformed in some way. One may introduce some uncertainty to classical physics. In that case, the state isn't described by \(x_i(t),p_j(t)\) but by \(\rho(x_i,p_j;t)\), a probabilistic distribution on the phase space. Note that \(\int dx_i dp_j \rho = 1\). The distribution at the initial moment may also be evolved to one at the final moment:\[

\rho(x_i,p_j;t) \to \rho(x_i,p_j;T)

\] The evolution of this probabilistic distribution on the phase space is directly calculable from the evolution of \((x_i,p_j)\) themselves. You just assume that the "delta function on the phase space" that is located somewhere evolves to the "delta function on the final point of the phase space". And the evolution of all other probabilistic distributions is determined by linearity: Every function is a "linear combination" of delta-functions and you already know how they evolve.

It's only the evolution for \(\rho(x_i,p_j;t)\) that quantum mechanics replaces. So quantum mechanics allows you to start with the \(\rho\)-like knowledge at the initial moment, the density matrix \(\hat \rho(t)\), and calculate \(\hat \rho(T)\) at the final moment:\[

\hat \rho(t) \to \hat \rho (T)

\] The probabilistic distribution on the phase space is directly the classical limit of the density matrix. Everyone who learns quantum mechanics should understand how it works, e.g. on the Hilbert space for a spinless particle on a line.

The density matrix \(\hat \rho(t)\) in quantum mechanics is a matrix so it's not coming in a fundamental representation of \(U(\infty)\), the group rotating the Hilbert space while preserving the norm. Instead, one may work with the vectors in the fundamental representation of \(U(\infty)\), namely with the pure states \(\ket\psi\), i.e. with the so-called Hilbert space. A pure state is associated with every density matrix whose eigenvalues are \((0,0,1,0,0,0,\dots)\) in some order. And all density matrices may be written as linear combinations of such elementary ones.

So quantum mechanics allows you to calculate the evolution of the pure states as well:\[

\ket{\psi(t)} \to \ket{\psi(T)}

\] This evolution is given by a linear unitary operator. Everything is linear in quantum mechanics because all these operators are building blocks for the evolution of the density matrix which generalizes the classical probabilistic distribution on the phase space. And that evolution was linear. And the evolution of a pure state has to be unitary in order for the total probability to be conserved – it has to be 100% at all times.

Great. So what's the difference between classical and quantum physics? In classical physics, you obtain \(L(t)=\ell\) through a measurement which tells you that your state is one of the "slices" on the phase space where the condition is obeyed. By several measurements like that, you may pinpoint the point on the phase space exactly. Or partially – as some distribution that is nonzero in some region etc. And you may evolve this point on the phase space or the distribution using the classical laws.

In quantum mechanics, you get \(\hat L(t) = \lambda_k\), one of the eigenvalues, which allows you determine the pure state \(\ket{\psi(t)}\) or something about it or some density matrix \(\hat \rho(t)\) etc. This knowledge is transformed in time and you may calculate the probabilities of any outcome of a measurement at the final moment using Born's rule.

The differences have been enumerated – there's a difference in the possible outcomes (eigenvalues), non-commuting character of observables in quantum mechanics, and a more general difference in the calculations needed to compute the evolution in time. They're relatively "technical" differences that change nothing about the logical reasoning. If you decide that some basic aspects of reasoning "have to be" classical and non-quantum, then you have misunderstood quantum mechanics completely.

All these basic things that are needed in science – measurement, logical reasoning, calculation of quantitative predictions etc. – may be done in quantum mechanics just like it may be done in classical physics. And, after all, it's only the quantum mechanical reasoning that is actually correct. If your thinking about various phenomena in the real world has been more or less correct, it had to be effectively quantum mechanical reasoning, not a classical one.

For example, look into a telescope and determine the relative position of three planets. To say something about the planets, one uses the insight that the position of the dots on the retina is some simple transformation of the locations of the celestial bodies. Now, this "dictionary" is true in classical physics – because the "rays" propagate in a certain way. But it's also true quantum mechanically. In fact, it holds as an "operator equation". The operators for some coordinates of the three planets are literally equal to (after a trivial, almost linear transformation) to the operators describing the locations of the excited retina cells.

So every valid statement about the "inner workings of the telescope etc." that may be done classically may obviously be done quantum mechanically, too. In fact, lots of the details only work in quantum mechanics.

An anti-quantum zealot may say that according to classical physics, a photon moves along a path through a telescope, and that shows how great classical physics is for similar reasoning. Except that this statement is upside down. The motion of a photon and even the very existence of a photon is only implied by quantum mechanics, not by classical physics.

In classical physics, you may replace photons by classical waves – in which case they won't produce isolated points in the double slit experiment. Or you may replace them by individual particles – in which case they won't correctly interfere with itself, and they won't respect the right "index of refraction" in various materials. So the lenses won't work. Only quantum mechanics predicts the objects that are ready to behave both as particles and waves.

There are lots of other key phenomena that take place when you look into the telescope. An atom in your retina gets excited. An atom is stable. Well, in classical physics, atoms crash within a picosecond. Even if you fixed the crash by some extra classical force, the classical atom won't have discrete levels. Quantum mechanics implies the discrete levels. Also, the entropy of one atom may be measured from the heat capacity in the real world to be comparable to one bit (times Boltzmann's constant). That's what quantum mechanics predicts, thanks to the finite number of accessible low-energy microstates. The entropy and heat capacity of an atom would be infinite and/or huge in classical physics (which would include any realistic theory, from Bohmian to Everettian ones) because there would be infinitely many or hugely many microstates of the atom that would be indistinguishable in practice but distinguishable in principle (and the huge number would grow with the temperature).

So when someone wants to show how classical reasoning works nicely and he says that an atom or a retina cell gets excited from one state to another, that's quite chutzpah. It's only quantum mechanics that allows truly discrete levels of atoms, molecules, and retina cells – and discrete jumps in between them. So when we talk about any discrete levels or even discrete jumps between them, we just absolutely need quantum mechanics. Whenever we assume that statistical physics only depends on some finite amount of bits about a physical object, we implicitly require quantum mechanics, too.

The anti-quantum zealots try to preserve the brutal misconception that every logical reasoning about Nature is unavoidably classical in character. But this brutal misconception has a simple reason: they haven't understood quantum mechanics at all. They haven't ever succeeded in using quantum mechanics to find a single correct fact about their observations! Quantum mechanics allows all the logical reasoning that actually works in physics, and it works much more correctly in quantum mechanics than it does in classical physics.

We need logical reasoning to be applied to elementary propositions such as "if something holds at moment \(t\), then something else will hold at the moment \(T\), or it will hold with some high probability \(p\)". And some people assume that every statement like that may only be derived from "classical physics" where the state of the system is assumed to be objective.

But that's completely wrong. There is absolutely no need for the state of the system to be objective. The propositions of the form "if... then..." as in the previous paragraphs are typical examples of the statements that one may extract from quantum mechanics. The "if..." part of the proposition is mathematically translated to the eigenstate equation\[

\hat L(t) \ket\psi = \lambda_k \ket\psi

\] while the "then..." part is translated to a similar proposition at time \(T\). Or if the probability is less than 100%, some particular number, the "then..." statement may be translated to a statement about the norm of the wave function at moment \(T\) with a projection operator inserted in between. Many such "if... then..." statements (well, in principle, if you're good enough with the formalism, all of them) may be derived directly in the Heisenberg picture, without introducing any ket vectors or density matrices at all.

So these "if... then..." statements about objects in Nature, apparatuses, human organs, nerves, and human brains are exactly what is supposed to be derived from the quantum mechanical evolution combined with Born's rule. The claim that quantum mechanics isn't a sufficient framework to talk about the inner workings of photons, telescopes where the photons propagate, glass with a refraction index, retina, cells in retinas, nerves, brains... which is why some classical physics needs to be "restored" is just totally and absolutely wrong.

Classical physics means that the observables \(L(t)\) commute with each other and that's why it's always possible to assume that all these observables have some actual values according to all observers, even if the observers don't (quite) know these values. Quantum mechanics invalidates this assumption by saying that the generic operators don't commute with each other which is why all of them can't simultaneously have well-defined values. That's why it's important to specify what observable we want to be predicted – and different observers will simply define their questions differently in general which is why their knowledge generically can't be reconciled to any "objective" picture.

But we don't need the vanishing commutators – i.e. the objective character of all the observables – to produce statements of the "if... then..." type, e.g. "if a planet is there, then you will see a dot there". These statements are perfectly derivable from quantum mechanics. And in all sufficiently modern examples, it's only quantum mechanics where the derivation actually produces the right result.

As I mentioned, quantum mechanics is needed to have stable atoms, discrete states of atoms, molecules, and cells, discrete jumps between them, objects that exhibit both wave-like and particle-like properties. Quantum mechanics is necessary in the derivation of the behavior of all basic types of matter – some details of gases but especially conductivity of some solids, behavior of crystals, diamagnetism, paramagnetism, superconductivity, and so on.

Quantum mechanics is also essential for making statistical physics fully consistent. In classical physics, as Boltzmann's tomb says,\[

S = k\cdot \log W.

\] The entropy is (Boltzmann's constant times) the logarithm of the number of states \(W\). Except that in classical physics, the number of states is continuously infinite – it's some volume in the phase space. So the argument of the logarithm is dimensionful in classical physics. You really need to introduce a unit of the volume of the phase space which I will denote \(h^N\), and the tomb should have said\[

S = k\cdot \log (W/h^N).

\] In classical physics, the choice of \(h\) is arbitrary which means an unknown additive shift in the entropy. In quantum mechanics, we have \(h=2\pi \hbar\) – this unit volume really has to be linked to Planck's constant. That's why the unit volume of the phase space naturally produces \(S=0\). That's very helpful especially in the third law of thermodynamics. In classical physics, the third law says that at \(T\to 0\), the entropy goes to a universal constant but that constant depends on \(h\). Only in quantum mechanics, you may say that \(S\to 0\) for \(T\to 0\) – and you really need the entropy of the totally frozen solids at absolute zero to be zero.

In much of the logical thinking about Nature, you really "intuitively assume" the number of states to be finite or countable – you assume the states to be discrete. This assumption is inconsistent with classical physics but it is consistent with quantum mechanics. So any reasoning about Nature that has any "discrete information" in it is therefore reasoning that only works in quantum mechanics!

As Edwin said, if you want to know how a table feels in quantum mechanics, touch the real table in front of you. I think that a famous physicist said it before him ;-) but I forgot who it was. Quantum mechanics is the state-of-the-art description of everything in the real world – quantum mechanics is how the world around us actually works.

The people who say that we need the classical framework to reconcile the laws of physics with our everyday lives or with our mundane reasoning about mundane things are completely wrong – these statements only prove that they have understood 0% of quantum mechanics.

Almost all these people have learned completely wrong things under the title "quantum mechanics". They learned that quantum mechanics is a new type of a classical theory which contains some new classical waves, wave functions, and these waves spread to superpositions that clearly disagree with the well-defined character of our observations. And that's why quantum mechanics doesn't work and we need to return to some classical physics to describe mundane things such as the sharpness of the observations.

But that's completely wrong. You have completely misunderstood what quantum mechanics is and what the wave function is. And frankly speaking, it was totally wrong for you to learn that quantum mechanics is all about the wave functions – because this focus leads the people into thinking that the wave function is a classical wave which is totally wrong. Instead, quantum mechanics is all about observables which are associated with linear Hermitian operators.

The wave function is a complexified generalization of the probabilistic distributions on the phase space – it's generalized to deal with probabilities of statements about observables that don't commute with each other. But otherwise the wave functions and density matrices allow us to perform every "kind" of reasoning that worked classically – quantum mechanics just does so correctly while classical physics did it incorrectly. The wave functions or density matrices "spread into diluted superpositions" because they're just probability distribution. But there's no contradiction between this fact and sharp results of the measurements. Diluted probability distributions don't imply dissolved, melted objects. Objects remain sharp but we may be ignorant about their locations. That's what the spreading wave functions in the position representation represent. There's nothing mysterious about it – in particular, there's no contradiction between this "spreading" and basic empirical facts that we know from everyday situations such as the sharpness of our measurements. Measurements produce sharp outcomes because, according to quantum mechanics, the outcome of a measurements must be an eigenvalue of an operator. Eigenvalues of an operator are sharp, separated from each other, and mutually exclusive – despite the nonzero commutators between operators – which is why quantum mechanics predicts that outcomes of measurements are sharp!

Everyone claiming that quantum mechanics predicts some fuzzy observations and that's why there's something wrong about quantum mechanics is an absolute idiot. An absolute idiot who still thinks classically and who believes that the only modification that a piece of science named "quantum mechanics" is allowed to make is to add a new classical wave with a new classical equation. But that's not what quantum mechanics means at all. Quantum mechanics changes something deeper about the meaning of the mathematical objects and their relationships with our observations and with our thinking about Nature. The change is profound but is so clever that it works whenever classical physics worked well enough – and it also works in cases where classical physics was failing.

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