Noncommutativity As Well As Observer Dependence Of Qm Are Morally Equivalent

I desire to add together a playful "research" twist to my explanations of the foundations of quantum mechanics.

In many weblog posts, I argued that the bespeak to limited the facts near Nature relatively to an observer – the "subjective" grapheme of noesis – is a defining belongings of quantum mechanics. Quantum mechanics is the exclusively known consistent framework that makes it impossible to assume (like classical physics did) that all observations may live derived every bit consequences of some shared objective information (data independent of an observer). This belongings of quantum mechanics volition live referred to every bit its observer dependence.

However, most late 2 weeks ago, I also argued that all novelties of quantum mechanics may live reduced to the nonzero commutators. After all, Planck's constant \(\hbar\) is the constant that is approximated yesteryear \(\hbar=0\) inward classical physics (or the classical limits of quantum theory) which agency that it quantifies the deviations from classical physics. At the same moment, all commutators of observables that be classically every bit good every bit quantum mechanically are proportional to \(\hbar\) inward the quantum theory. The nonzero commutators are equivalent to what nosotros volition telephone yell upwardly noncommutativity of quantum mechanics.

OK, these seem similar 2 dissimilar "defining" principles of quantum mechanics. Which of them is right? Needless to say, you lot may say that both principles are right together with must live embraced to build the right framework. However, I volition endeavour to defend a prettier statement. These 2 principles aren't genuinely independent. To some extent, they are equivalent!




How it tin move be? Well, according to classical physics, the truth near everything together with anything "objectively exists somewhere". The observer dependence of quantum mechanics agency that according to quantum mechanics, the truth near something is exclusively physically meaningful "relatively to an observer who has observed whether it's truthful or not".




This disceptation basically agency that

the truth must live lived, non simply believed to exist.

Now, this sounds similar a judgement from a sermon, a verbalise yesteryear a motivational speaker, or at to the lowest degree a lecture yesteryear a philosopher. And I formulated it inward that way deliberately. But what I hateful yesteryear this laymen-style assertion is something rather quantitative.

The truth inward quantum mechanics isn't something that simply exists; it's something that affects an observer together with his knowledge. It transforms his life or at to the lowest degree his brain. In other words, the learning of the truth is a transformation.

And it's the grapheme of the truth every bit a transformation that is responsible for the non-commuting grapheme of observables inward quantum mechanics. I think that these insights are the ground why Dirac has framed the basic insights of quantum mechanics every bit "Dirac's transformation theory" upwardly to some moment. The signal is that for certain objects that had been believed to live \(c\)-numbers of absolute value equal to ane – for illustration \(\exp(i\alpha J_z)\) – were genuinely \(q\)-numbers i.e. \(U(N)\) transformations of the Hilbert space.

So you lot know, when an observer learns 3 truthful facts, his moving ridge component undergoes 3 projection yesteryear projection operators:\[

\ket\psi \to P_C P_B P_A \ket\psi.

\] All the operators \(P_i\) are Hermitian projection operators,\[

P_i = P_i^\dagger,\quad (P_i)^2 = P_i.

\] The observer starting fourth dimension perceives that the suggestion associated amongst the projection operator \(P_A\) was true, so he learns the truth of \(P_B\), together with finally of \(P_C\). Just to live sure, the density matrix would live transformed as\[

\rho \to P_C P_B P_A \,\,\rho\,\, P_A P_B P_C.

\] I am allowing an arbitrary normalization of the nation vectors together with density matrices here. Now, noncommutativity of quantum mechanics agency that, for example,\[

P_A P_B \neq P_B P_A

\] Because the projection operators \(P_i\) generalize the truth values of classical bits that are equal either to \(0\) or \(1\), how is it possible that such truth values tin move turn down to commute inward quantum mechanics?

Well, it's because whenever they don't commute, exclusively ane of the orderings genuinely corresponds to the observations yesteryear the observers. When some regionally localized operators \(P_A,P_B\) don't commute, it agency that they're timelike- or lightlike-separated. And nosotros tin move say which of them occurred first. In our example, the observer starting fourth dimension detected \(P_A\) together with "this truth changed his life" inward some way. And so his noesis together with life was changed ane time once to a greater extent than yesteryear \(P_B\). And finally yesteryear \(P_C\), nosotros said, but that's non relevant for the commutator \([P_A,P_B]\).

Note that I encourage you lot to imagine that \(\ket\psi\) or \(\rho\) depict the "state of one's mind" or "soul" or whatever discussion you lot volition unwrap acceptable or exciting enough. It's the same thing.

H5N1 prissy fact preserving the consistency of this structure is that when the chronology of the events \(P_A,P_B\) isn't well-defined because they are spacelike-separated, nosotros could live worried that both orderings of the observations powerfulness live right according to diverse inertial systems. But it's OK because inward that case, the projection operators have got to commute! Needless to say, you lot know this "happy coincidence" every bit a fact that allows the time-ordering operator \(T\) to stay well-defined inward a relativistic theory. But you lot may frame the same happy tidings a fleck differently, too.

Now, nosotros may purpose some vague terminology of philosophers of consciousness. Qualia are said to live the units of perceptions. The singular shape isn't a qualium but a quale. Well, let's honor this ugly word. I won't reproduce the supposition that each quale has to live "elementary" – because I don't believe that in that location is e'er a unique way to split upwardly perceptions to unproblematic edifice blocks.

With this generalization, nosotros may define a quale every bit follows:

A quale is a Hermitian projection operator.

Our operators \(P_A,P_B,P_C\) were qualia. They decided how the noesis or the soul – given yesteryear the nation \(\ket\psi\) or \(\rho\) – was changed. We are supposed to purpose the qualia so that nosotros exclusively multiply the nation yesteryear the projection operators that the observer saw to have got the eigenvalue \(P_i=1\), i.e. whose corresponding propositions were observed to live true.

When the observer simply learns that \(P_A=1\), his density matrix gets transformed as\[

\rho \to P_A \rho P_A.

\] We desire to adopt the terminology that this alter of \(\rho\) is equivalent to the observer's having a item quale (an amount of perception or witting experience) \(P_A\). Later, the observer may perceive the quale \(P_B\) together with so on. Whether he remembers the feeling of \(P_A\) at a afterwards fourth dimension (and whether he tin move translate it to reconstruct the past) depends on his retentivity together with other "technical" things. But the qualia may occur independently of the memory.

Now, I think it's natural to imagine that each quale is a transformation of your brain, together with thence ane of the objects that shape an associative\[

(P_C P_B) P_A = P_C (P_B P_A) = P_C P_B P_A

\] but non commutative algebra.

Imagine that your encephalon consists of really many marbles together with every fourth dimension you lot observe \(P_C\) which corresponds to \(N\neq 0\) where \(N\) is the release of nuclear blasts inward the Visegrád Group during the in conclusion hr (I am for certain that the PC people grip that such an lawsuit is rather \(P_C\) these days), a item permutation of the marbles inward your encephalon takes place. Whenever you lot observe the happier quale \(1-P_C\), i.e. the absence of a nuclear blast, some other permutation takes place. And you lot simply orbit to experience the permutation.

However, the 2 permutations of your encephalon don't commute inward general. That's why the qualia don't commute, either.

You may endeavour to extend this classical model for the non-commutative perception every bit much every bit you lot can. I don't desire to exercise that because it's clear that at the end, whatsoever classical model fails. The work is that every bit long every bit you lot embed your encephalon to a classical world, you lot may imagine that some objective classical facts create upwardly one's take away heed near the permutations of marbles inward your brains. And the locations of all the marbles inward your encephalon are given yesteryear objective bits of information, too.

But quantum mechanics simply doesn't allow you lot to think that the encephalon is embedded inside a classical world. The private qualia are what you lot genuinely perceive together with the appearance of these perceptions can't live reduced to whatsoever objective information every bit inward classical physics. You may imagine that your soul or consciousness given yesteryear \(\ket\psi\) or \(\rho\) is connected to some external "black box" through some USB cables (I wanted to live specific – together with avoid Apple's novel wireless formats of communication). Using these cables, your soul may shipping some "question" to the external device – you lot specify which \(P_i\) you lot attention near – together with you lot have a fleck dorsum through the cable.

Imagine that you lot are a figurer who tries to observe the earth about it using the USB devices (this is the sort of a Sheldon-Cooper-style empathy that most of the superficial, fossilized, uncreative people inward the "quantum foundations" community are incapable of but I promise that you lot are far better). You shipping a ascendence to a device that you lot previously decided to live called a spider web photographic tv camera (because it seemed to acquit similar that) – you lot desire to know whether a pixel is to a greater extent than brilliant than some threshold – together with acquire an answer, ane bit. In that way, you, a computer, tin move larn something near the earth about you. Most people assume that spider web cameras roughly mensurate some "objective reality" every bit understood inward classical physics so that's how you, the computer, create your starting fourth dimension theories explaining what you lot observe.

However, at some moment, you, the computer, finds out that the theory doesn't quite work. So what you're connected to yesteryear these USB cables isn't genuinely a classical earth amongst objective data. It's something optimized for you. In particular, you lot can't ever live a completely passive observer. Whenever you lot shipping the query "I desire \(P_B\)" through the USB cable, you lot also alter the nation of the device. Also, the device won't live deterministic. The best predictions you lot tin move brand are probabilistic.

I am non for certain whether this painting demo of "qualia" together with "you every bit a figurer connected yesteryear USB cables to the external reality" helps anyone. If person is dogmatically convinced that in that location has to live some "classical physics" higher upwardly or beneath everything, he volition simply dismiss whatsoever analogy that would refute the point. But the "computer amongst USB cables" is plenty to select e.g. the signal that the observations might have got some unavoidable effect on the nation of the external physical objects.

Even if you lot embed the figurer inside a completely classical world, it's slow to empathise that every "query" that the figurer sends through the cables powerfulness bear upon the external world. If person volition live able to derive something to a greater extent than interesting inside these analogies together with formalisms, maybe fifty-fifty near quantum gravity, I would live really interested.