All Novelties Of Quantum Mechanics Are Consequences Of Nonzero Commutators

In the give-and-take nether his confused musings most the CHSH inequality, a volatile anti-quantum zealot had serious problems alongside the next of import wisdom:

All differences betwixt classical physics too quantum physics are consequences of the dubiety regulation i.e. of the nonzero commutators betwixt observables.

Well, the controversy higher upwards is truthful too important. The misunderstanding of this controversy – oftentimes arrogantly masked equally a unlike opinion, ane that may live on presented assertively – may live on considered ane of the defining characteristics of the anti-quantum zealots.

The anti-quantum zealots are inventing lots of "additional" differences betwixt classical too quantum physics – such equally the purported "extra nonlocality" inherent inwards quantum mechanics, or some completely novel "entanglement" that is something totally unlike than anything nosotros know inwards classical physics, or other things. Except that all these purported additional differences are non-existent.

The argue why they're inventing these non-existent differences is that they would similar to rephrase quantum mechanics equally "another classical theory alongside lots of unlike details" (by details, nosotros hateful the laid of observables too the dynamical equations of motion etc.). But that's non what quantum mechanics is. Quantum mechanics is a fundamentally unlike theory which may accept the same details equally the classical counterpart, however. ;-)




The discrimination of classical physics too quantum physics may actually live on described through a simple parameter, Planck's constant:\[

\begin{array}{|c|c|c|}
\hline {\rm physics}& {\rm classical} & {\rm quantum} \\
\hline {\rm value}& \hbar = 0 & \hbar\neq 0 \\
\hline
\end{array}

\] Assuming that you lot know how to piece of job both alongside classical too quantum physics, the value of this unmarried parameter is plenty to discriminate betwixt them. It's that simple.




Now, quantum mechanics is a to a greater extent than full general framework piece classical mechanics is a special case. Influenza A virus subtype H5N1 classical theory may live on typically obtained equally the \(\hbar\to 0\) bound of a quantum mechanical theory. When a minor \(\hbar\) becomes min too is "really" sent to zero, the resulting limiting theory may live on said to accept \(\hbar=0\) too it's thence a classical theory if the bound exists at all.

In classical physics, there's just no way to accept \(\hbar\neq 0\). This is truthful past times definition because \(\hbar\) is the constant that measures the difference of a physical theory from the degree of theories of classical physics. Because classical theories aren't sufficient to hash out the quantum mechanical ones, nosotros must usage the to a greater extent than full general framework – the quantum mechanical framework – if nosotros desire to compare classical too quantum mechanics.

This demand is completely analogous to the demand to usage relativistic theories when nosotros desire to compare relativistic too non-relativistic theories. In that case, the difference from a non-relativistic theory is measured past times the parameter known equally \(1/c\), the inverse maximum speed inwards Nature. The bound \(1/c\to 0\) i.e. \(c\to \infty\) is the non-relativistic bound too it is analogous to the \(\hbar\to 0\) classical bound of a quantum mechanical theory.

In both cases, the limiting theory obeys some extra axioms that are non valid inwards the to a greater extent than full general theory. In particular, the non-relativistic theories (\(c\to \infty\) limits of relativistic ones) obeys the "absolute graphic symbol of simultaneity" piece the \(\hbar\to 0\) classical limits of quantum theories obey the extra axiom most the objectively well-defined values of observables (independently of observers or earlier the observations).

The to a greater extent than full general theories or frameworks spend upwards these extra axioms. Relativity says that the simultaneity is relative i.e. subject on the inertial system; quantum mechanics says that the values of observables are only well-defined relatively to an observer too what the observer considers to live on observations (events inwards which the observer acquired the information).

OK, what is Planck's constant operationally?

Planck's constant \(\hbar\) is omnipresent inwards quantum mechanical theories. Well, it's omnipresent if nosotros usage the scheme of units inwards which \(\hbar\) has to live on inserted at all. Particle physicists too some other adult physicists oftentimes usage units inwards which \(\hbar=1\) which simplifies most of the formulae inwards whatever quantum mechanical research.

However, if nosotros desire to compare quantum mechanical theories alongside their limits, nosotros need to usage units inwards which \(\hbar\) is variable – i.e. it is non laid to ane (or some other constant) – because a especial variation, the \(\hbar\to 0\) limit, is how nosotros instruct from the full general quantum mechanical theories to the classical ones (the limits).

Because \(\hbar\) appears at so many places, nosotros may define its value inwards many ways. But ultimately all of them may live on shown to live on equivalent to the commutator of some observables – the refusal of observables to commute alongside each other. For example, nosotros may define \(\hbar\) as\[

\hbar = i(px-xp)

\] inwards whatever theory that contains positions \(x\) too momenta \(p\). In classical physics, the commutator is zero, inwards a quantum mechanical theory, it's not. The precise value dictates the shape of the moving ridge associated alongside a momentum \(p\) particle. The wavelength (period) of this moving ridge is just \(2\pi\hbar / p\).

Equivalently, nosotros may define \(\hbar\) as\[

\hbar = -i \cdot J_z^{-1} \cdot (J_x J_y - J_y J_x)

\] The commutator betwixt diverse components of the angular momentum are null inwards classical physics, nonzero inwards quantum mechanics, too these commutators may live on easily derived from the formulae for \(J\) equally a business office of \(x,p\) too from the commutator \([x,p]\) that nosotros discussed a min ago.

Now, the commutators such equally \([J_x,J_y]\) are plenty to determine that the eigenvalues of each factor are quantized. \(\hbar/2\) is the smallest allowed nonzero value of \(J_z\) or whatever factor of the angular momentum, for example. Obviously, nosotros could hash out equally many examples of commutators of observables – "elementary" or "composite" ones – inwards many mechanical or other theories.

The constant \(\hbar\) also appears inwards the Schrödinger equation. Does this appearance accept anything to practise alongside nonzero commutators? You bet. Influenza A virus subtype H5N1 to a greater extent than conceptual sensible way to rewrite the Schrödinger equation is to apply the time-dependent unitary transformation too derive the Heisenberg equation alongside the Heisenberg equations of motion. In that picture, the observables obey \[

i\hbar \frac{\dd L(t)}{\dd t} = [L(t),H]

\] This equation explicitly says that the commutators alongside the Hamiltonian are the fourth dimension derivatives of the operators multiplied past times the tiny constant \(i\hbar\) ane time again. The dynamical equations of the Heisenberg motion-picture demonstrate are examples of the to a greater extent than full general fact that the commutators are proportional to \(\hbar\).

Classical physics emerges inwards the bound \(\hbar\to 0\). It recognizes that the commutators are minor because they're proportional to \(\hbar\) which is tiny inwards the units used to written report problems where classical physics becomes a adept approximation. So classical physics "amplifies" these commutators, multiplies them past times \(1/i\hbar\), too this production is called the "Poisson bracket". Effectively, only the leading damage inwards a mightiness expansion inwards \(\hbar\) are kept inwards the classical limit. That's how nosotros create a classical theory.

The volatile anti-quantum zealot I accept referred to has written things like

What distinguishes quantum from classical mechanics is how the observables of a composite scheme are related to the observables of the private systems.

But this sentiment is completely wrong – the bold confront fonts used for that judgement only highlight how much wrong he is. When nosotros accept a conventional composite system, the observables \(A_j,B_k\) describing the subsystems \(A,B\) of this composite scheme commute alongside all operators inwards the other group:\[

[A_j,B_k] = 0.

\] The commutator is zero, just similar inwards classical physics, so the mere additional of "subsystems" to the whole scheme just cannot brand the physical scheme to a greater extent than quantum. The composite scheme may display lots of characteristically quantum demeanor but that demeanor only arises due to the nonzero commutators, i.e. \[

[A_j,A_{j'}] \neq 0, \quad [B_{k},B_{k'}]\neq 0.

\] All the novel quantum demeanor appears "inside" \(A\) or within \(B\), within the private subsystems of the composite system!

Now, the "quantum mechanics is non-local" crackpots dearest to tell non only crazy things most the extra non-locality of quantum mechanics – which I accept thoroughly debunked inwards many previous specialized weblog posts – but they also dearest to acquaint the quantum entanglement equally some absolutely novel voodoo, a supernatural phenomenon that has absolutely no counterpart inwards classical physics too whose divine content has zilch to practise alongside the dubiety regulation or the nonzero commutators.

Except that all this voodoo is just fog.

Quantum entanglement is zilch else than the most full general pure-state description of a correlation betwixt ii subsystems inwards quantum mechanics. Assuming that nosotros usage the right laws too formalism of quantum mechanics to depict systems that obey quantum mechanics,

an entangled nation too a nation alongside a correlation betwixt ii subsystems are absolutely synonymous.

If you lot written report a singlet nation of ii spin-1/2 particles, you lot may stair out \(J_z\) of both entangled particles. As I accept discussed inwards numerous recent weblog posts, this anticorrelation betwixt the ii spins is absolutely equivalent to the anticorrelation betwixt the colors of ii socks of Dr Bertlmann – a scheme nosotros may depict past times classical physics.

The thought that the correlation betwixt the ii electrons is "something fundamentally different" from the correlation betwixt the ii socks' colors – an thought pioneered past times John Bell – is totally too absolutely wrong. After all, inwards the existent globe closed to us, fifty-fifty socks of Dr Bertlmann are accurately described past times quantum mechanics. Classical physics isn't quite right, fifty-fifty for socks. So if nosotros accept measured a consummate laid of observables to convey the ii socks inwards a pure nation – but an entangled ane (and live on sure enough it's possible) – so the socks volition inevitably live on inwards an entangled state. If you're accurate most socks inwards the existent world, you lot demand to depict them equally an entangled state, just similar the ii spins, too non just equally a classical correlation. It's truthful just because the classical theory is never quite right inwards the globe closed to us!

So the sentiment that the correlation of (real world) Bertlmann's socks is something "totally too fundamentally different" from the entanglement of the ii spins contains the self-evidently wrong supposition that socks inwards Republic of Austria don't obey the laws of quantum mechanics. But live on sure enough that they do. Everything inwards this damn Universe does.

What's novel most the quantum entanglement – relatively to the "ordinary" classical correlation – is that the ii subsystems such equally the ii spins may accept correlated many other properties at the same moment. The singlet nation has \[

\vec J_1 = -\vec J_2

\] which just way that \((\vec J_1+\vec J_2)\ket\psi = 0\). So you lot may stair out \(J_{1z},J_{2z}\) too obtain a perfect anticorrelation. But if the ii experimenters come about to stair out \(J_{1x},J_{2x}\) instead, they instruct a perfect anticorrelation, too. And similarly for \(J_{1y},J_{2y}\). You couldn't invent a classical model alongside ii classical bits that would emulate this behavior. And the components of the spin \(J_x,J_y\) are exclusively unlike than or independent from \(J_z\).

The real-world socks are only simpler because of ane fact: due to the complexity that produces decoherence etc., it is extremely difficult too practically impossible to stair out anything such equally \(J_x,J_y\) for the socks – observables that don't commute alongside the sock color i.e. are non-diagonal inwards a footing of the color-of-sock eigenstates. These non-diagonal observables operationally don't be for socks.

But all this novel demeanor depends on the nonzero commutators \([J_{1z},J_{1x}]\) too similar ones! What prevents you lot from this actually "intimate" correlation betwixt the ii bits inwards classical physics is that inwards classical physics, all observables are simple functions on the stage infinite \(F=F(x_j,p_k)\) – or the laid of possible values of all the bits if the data is described inwards damage of bits. For this reason, inwards classical physics, it's ever plenty to stair out the values of \(x_j,p_k\) etc. too you lot know everything.

However, this is non the instance inwards quantum mechanics. You can't stair out all "elementary" observables \(x_j,p_k\) at the same 2d because of the dubiety regulation – because of the nonzero commutators. Instead, you lot must determine what you lot stair out too the post-measurement nation volition live on an eigenstate of this measured observable \(L\). Moreover, you lot actually demand to listing all conceivable observables which accept all conceivable eigenstates if you lot desire to exhaust all the options. And the laid of observables – Hermitian matrices – is really large.

In a classical description of ii bits, the stage infinite would accept iv points \(00,01,10,11\) which gibe to the iv scheme of the values \(J_{1z},J_{2z}\), to pick the criterion convention. Once you lot would know that \(J_{1z},J_{2z}\) may only accept \(2\times 2 =4\) values, i.e. they are ii bits, too the cognition of these ii bits is the "maximum" you lot may know most the system, inwards classical physics, it would follow that no other interesting correlation may live on acquaint inwards the nation of the ii bits. All observables are functions of \(J_{1z},J_{2z}\). It way that you lot may at most stand upwards for the possibilities \(00,01,10,11\) past times iv arbitrary numbers \(e,f,g,h\). There are just iv fixed options too a perfect correlation may at most hateful that some of these iv options are ruled out.

However, inwards quantum mechanics, in that location be operators such equally \(J_{1x}\) which don't commute alongside ane of the observables \(J_{1z},J_{2z}\) – inwards this instance alongside \(J_{1z}\). It's just this nonzero commutator that makes \(J_{1x}\) sensitive on the relative stage betwixt the complex probability amplitudes that know most the options \(J_{1z}=+\hbar/2\) too \(J_{1z}=-\hbar/2\). Also, it's just the nonzero commutator \([J_{1x},J_{1z}]\) that says that \(J_{1x}\) isn't a diagonal matrix inwards the footing of the \(J_{1z}\) eigenstates. And it's the non-diagonal matrices for observables that comprise all the novelties of quantum mechanics, including the "tighter" correlation that the quantum entanglement may guarantee inwards comparing alongside the correlations inwards classical physics.

Aside from the consequences of the nonzero commutators – i.e. of the demand to usage off-diagonal matrices for the most full general observables that may actually live on measured – in that location is just zilch "qualitatively new" inwards quantum mechanics. Whoever fails to sympathise these points misunderstands the graphic symbol of quantum mechanics too the human relationship betwixt quantum mechanics too classical physics, too.