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In the mid 1920s, thank you lot to the conservative yet revolutionary piece of work past times giants such as Heisenberg, the foundations of physics receive got switched from the old framework of classical physics to the postulates of quantum mechanics.

The novel full general rules receive got been completely understood. Since that time, solely the search for the "right Hamiltonians" in addition to "their implications" was open. The novel philosophical underpinnings were shown to live on consistent, complete, goose egg has changed nearly them since 1925 (or nosotros powerfulness pose the threshold to 1927 so that the papers clarifying the incertitude regulation etc. are included), in addition to all the evidence suggests that there's no argue to await whatsoever alter to these basic philosophical foundations of physics inwards the future.

Florin Moldoveanu doesn't similar these facts because much of his piece of work (and mayhap most of it) is based on the denial of the fact that quantum mechanics plant in addition to it plant remarkably well. So he wrote, amidst other things:
Luboš sees no value inwards the quantum foundations community because the proper interpretation was settled inwards his persuasion long fourth dimension agone in addition to all quantum foundations practitioners must live on crackpots (obviously at that spot is no dear lost betwixt the quantum foundation community in addition to Luboš).
Apparently to demo that something hasn't been clear nearly the basic rules of the game since the 1920s, he wrote a weblog shipping service dominated past times the basic introduction to the Leibniz identity, the Jacobi identity, in addition to tensor products. Are you lot joking, Florin? While the universal postulates of quantum mechanics receive got been known since the 1920s, the "fancy novel hi-tech topics" that you lot discussed at nowadays receive got been known at to the lowest degree since the 19th century!




Moldoveanu wants to print you lot past times the (freshman undergraduate algebra) insight that the Hermiticity is related to unitarity in addition to the Leibniz identity, in addition to so on. The precise "equivalences" he describes are confused – it is the Hermiticity (of the Lie algebra generators), in addition to non the Leibniz identity, that is equivalent to the unitarity (of Lie grouping elements). The Leibniz identity plant fifty-fifty for non-unitary operations in addition to it is how the differential generators ever human activity on production representations (e.g. composite systems inwards quantum physics), according to the lexicon betwixt the Lie groups in addition to Lie algebras.

But I don't desire to analyze his technical confusion which is intense. It would live on easier to brand him forget nearly everything he knows in addition to learn him from scratch than to seek to create all his mistakes.




I desire to focus on the large flick – in addition to the historical picture. To fence that something hasn't been settled since the 1920s, he talks nearly the Leibniz identity, the Jacobi identity, the rules of the Lie groups in addition to the Lie algebras. Are you lot listening to yourself, Florin? Leibniz lived betwixt 1646 in addition to 1716 so i should live on able to figure out that the Leibniz identity in all probability wasn't born afterwards 1925.

Even to a greater extent than relevant is the history of Lie groups in addition to Lie algebras. Sophus Lie did most of this piece of work betwixt 1869 in addition to 1874. The Lie algebra commutators were understood to obey the Jacobi identity which has been known earlier Lie did his key contributions. Most of Jacobi's piece of work was published inwards the 1860s but the publication was posthumous: Jacobi lived from 1804 to 1851. Killing in addition to Cartan added their noesis nearly the Cartan subalgebra in addition to maximal tori etc. inwards the 1880s. All this mathematical apparatus was laid upwardly decades earlier physicists made their novel insights nearly the foundations of physics inwards the 1920s.

In the same way, mathematicians understood the representation theory. For example, if at that spot are ii independent objects, their properties are described past times ii sets of operators. The ii sets commute alongside i some other – this is clearly needed for the ii objects to be independently. The Hilbert infinite is a representation which way that the minimum Hilbert infinite transforming nether both sets of observables has to live on a tensor product. The relevance of the tensor production \({\mathcal H}_A\otimes {\mathcal H}_B\) for the quantum description of a composite arrangement has been straight off obvious when quantum mechanics was presented. The mathematical underpinnings had been known for decades – and, inwards fact, Heisenberg had no problem to rediscover the matrix calculus when he needed it. The tensor production Hilbert infinite appears because it's a representation of the grouping \(G_A\times G_B\), a direct production that is needed to depict the observables of ii parts of the composite system.

Florin, are you lot genuinely serious when you lot nowadays these basic things as justifications of your claim that something substitution nearly the full general rules of quantum mechanics hasn't been clear from the 1920s?

Even though most of his weblog shipping service is dedicated to these basic, generally 19th century, mathematical insights, the championship reads
Why are unitarity violations fatal for quantum mechanics?
Indeed, unitarity violations would live on fatal for a quantum mechanical theory – they would preclude the gist of all probabilities of mutually exclusive outcomes from existence equal to 100 percent. However,
there is only no violation of unitarity inwards the theories nosotros genuinely usage to depict Nature.
Unitarity is indeed a universal dominion – it is the quantum counterpart of some axioms inwards the commons probability calculus (where the gist of probabilities of dissimilar options is ever 100 percent). Why does Moldoveanu call upwardly otherwise?

He thinks otherwise because he believes that the standard introduces non-unitarity to quantum mechanics. The give-and-take "non-unitarity" solely appears inwards the next judgement of his text:
Because the standard employment must explicate the non-unitary collapse, in addition to since non-unitarity makes the mathematical framework of quantum mechanics inconsistent, the mathematical solution ultimately points out the right interpretation.
Sadly, this critical judgement is completely wrong. This has some implications. For example, this wrongness invalidates almost all papers past times Moldoveanu that usage the give-and-take "unitarity" because he only doesn't know what this status is, when it holds, in addition to whether it holds.

The unitarity is a status inwards quantum mechanics that imposes the dominion that "probabilities add together upwardly to 100 percent" inside the quantum formalism. But what is unitarity to a greater extent than accurately? By unitarity, quantum physicists hateful exactly the same thing as the 19th century mathematicians. So the matrix \(U\) alongside the matrix entries \(U_{ij}\) – in addition to similarly for an operator that may live on defined without a pick of a reason i.e. without indices – is the condition\[

U U^\dagger={\bf 1}\quad {i.e.} \quad \sum_j U_{ij}U^\dagger_{jk} = \delta_{ik}.

\] This is the unitarity. In quantum mechanics, it holds whenever \(U\) is an development operator (by a finite or infinite fourth dimension i.e. the S-matrix is included) – or the operator of whatsoever finite transformation, for that thing (e.g. the rotation of a arrangement past times an angle).

The development operators inwards non-relativistic quantum mechanics, Quantum Electrodynamics, the Standard Model, in addition to string theory (among many other theories) perfectly obey this condition. That's why nosotros say that all these quantum mechanical theories are unitary – they travel past times this especial wellness test.

Moldoveanu in addition to tons of other anti-quantum zealots desire to contradict this declaration past times pretending that the standard of a quantum arrangement is a modification of Schrödinger's equation that deviates from the activity of the unitary development operators above, in addition to is hence non-unitary.

But that's a resultant of a completely incorrect in addition to sloppy thinking nearly all the concepts.

The collapse doesn't mean that at that spot is a violation of unitarity. To sympathise this uncomplicated sentence, nosotros must live on careful in addition to facial expression "what are the objects that are unitary". The response is that the unitary matrices such as \(U\) to a higher house are the
matrices whose entries are the probability amplitudes.
The full general take away inwards quantum mechanics that nosotros receive got referred to is that the matrices of development operators' probability amplitudes – betwixt the reason of possible initial states in addition to the reason of the possible concluding states – are unitary. And live on sure that they are in addition to the measurements don't alter anything nearly it.

Why don't they alter anything nearly it? Because the "sudden collapse of the moving ridge function" that the standard induces isn't a modification of the development operator or a deformation of Schrödinger's equation. Instead, the "sudden collapse" is an interpretation of the moving ridge function.

Quantum mechanics says that afterwards the measurement, i of the possible outcomes becomes true. It "even" allows us to calculate the probabilities of the private outcomes. But the real fact that quantum mechanics says something nearly the probabilities of the outcomes implicitly means that i of the outcomes volition travel the truth afterwards the measurement. This uncomplicated claim is implicitly included inwards all the rules of quantum mechanics. We may patently add together it explicitly, too.

When nosotros mensurate whether a truthful cat is dead or alive, in addition to quantum mechanics predicts the probabilities to live on 36% in addition to 64%, at that spot can't live on whatsoever "vague mixed semi-dead semi-alive" dry soil of the truthful cat afterwards the measurement. This claim logically follows from the declaration that the "probabilities of dead in addition to live on are 36% in addition to 64%" in addition to it doesn't demand whatsoever additional explanation.

If it were possible for the standard of the truthful cat to yield some vague "semi-dead semi-alive" outcome, the probabilistic declaration would receive got to allow this option. To do so, quantum mechanics would receive got to predict that the "probability is 30% for dead, 60% for alive, in addition to 10% for some semi-dead semi-alive fuzzy mixture". But when the laws of quantum mechanics omit the 3rd option, it way that this option's probability is 0% which way that it is impossible for the post-measurement dry soil to live on semi-dead, semi-alive. If you lot demand some extra explanations or repetitions of this fact, that the ill-defined post-measurement outcomes are banned past times quantum mechanics, in addition to then it is because you lot are retarded, Florin, non because the foundations of quantum mechanics demand some extra work.

The ultimate argue why Moldoveanu in addition to others spend upwardly to sympathise uncomplicated points similar that – e.g. the signal that at that spot is no non-unitarity added past times the standard – is that they are refusing to call upwardly quantum mechanically. When nosotros say that the matrix entries of an development operator are probability amplitudes, nosotros sympathise it but the likes of Moldoveanu don't. They may hear the words but they ignore their content.

They totally overlook the fact that the matrix entries that create upwardly one's heed nearly the unitarity are probability amplitudes. They only call upwardly that they are some classical degrees of liberty (that objectively be in addition to don't require observers), Schrödinger's equation is a classical development equation, in addition to the standard must live on "modelled" as an exception for the Schrödinger's equation or its deformation of a sort.

But all these assumptions are completely wrong. The moving ridge component is not a classical wave. It is not a laid of classical degrees of freedom. Schrödinger's equation isn't an instance of a classical development equation. And the standard isn't described past times anything that looks similar an equation for the development at all. The standard yields precipitous outcomes because quantum mechanics postulates that at that spot are precipitous outcomes – the spectrum of an operator lists all the a priori possible outcomes – in addition to it tells you lot how to calculate their probabilities from the complex probability amplitudes.

It's solely the probability amplitudes that may live on meaningfully organized into linear operators in addition to hence matrices. If you lot desire to engineer some "action on a moving ridge component that also visualizes the collapse", in addition to then you lot are trying to create a classical model describing the reality. You are non doing a proper quantum analysis of the problem. And if you lot created such a model where the moving ridge component is a classical champaign that "collapses" according to some equations, the "operation" wouldn't fifty-fifty live on linear, so it wouldn't brand feel to inquire whether it's unitary.

In fact, the performance on the initial moving ridge component wouldn't fifty-fifty live on a map because fifty-fifty if the initial dry soil is just the same twice, the concluding outcomes may live on dissimilar – because of the quantum indeterminacy or randomness. Because this performance assigning the concluding dry soil isn't fifty-fifty a map (because the concluding outcomes of the measurements aren't uniquely determined past times the initial state), it makes absolutely no feel to verbalize nearly its existence unitary. Of class it can't live on correctly shown to live on unitary. It can't live on unitary if it is non fifty-fifty a map! Only for maps, in addition to yes, you lot demand linear maps, you lot tin meaningfully verbalize nearly their existence unitary. For other "processes", the describing word is ill-defined (like "whether the number v is green"). The "operation of the collapse" on the moving ridge component isn't unitary but it isn't non-unitary, either. It isn't a map so it's meaningless to verbalize nearly its existence unitary.

And if you lot managed to "redefine" the transformations inwards some way so that the human activity of the standard would count as "non-unitary evolution", despite its randomness (failure to live on a map) in addition to nonlinearity, in addition to then it wouldn't live on a problem, anyway. What's needed for consistency of the theory is the unitarity of the pre-measurement probability amplitudes (because the unitarity plays the same role as the weather condition for probabilities that should add together to 100 per centum etc.), non some probability amplitudes modified past times random-generator-dependent "collapses". So fifty-fifty if the collapse were redefined as a "non-unitary development of a sort", it only wouldn't hateful that at that spot is a employment to worry nearly or to solve.

Again, inwards the normal approach, the object whose unitarity is a meaningful enquiry is the matrix/operator of the probability amplitudes (defining an development or a transformation). Those don't comprise whatsoever "collapses" because the real pregnant of the give-and-take "probability" is that nosotros substitute the "widespread" distributions "before" nosotros know a especial outcome i.e. without whatsoever collapses. And the matrices of probability amplitudes for development operators must live on unitary inwards all logically consistent quantum mechanical theories.

Even if you lot are a flake confused nearly the logic, you lot should live on able to sympathise that at that spot is almost for sure "nothing intelligent in addition to deep" waiting to live on establish here. Moldoveanu's in addition to similar people's "work on the foundations" is only an artifact of their inability to sympathise some real uncomplicated logical arguments fully described to a higher house – in addition to at many other places. They're crackpots but similar most crackpots, they piece of work alongside the supposition that they tin never live on wrong. That's non a practiced starting signal to sympathise modern physics.

Pedagogic bonus: from classical physics to quantum mechanics

I am afraid that I receive got written real similar things to this appendix inwards the past. But fifty-fifty if it is the instance in addition to the text below fails to live on original, repetition may sometimes live on helpful. Here's a way to meet inwards what way quantum mechanics generalizes classical physics – in addition to why it's foolish to seek to facial expression for some "problems" or "cure to problems" inwards the procedure of the measurement.

Influenza A virus subtype H5N1 theory inwards classical mechanics may live on written inwards price of the equations for the variables \(x(t),p(t)\)\[

\frac{dx}{dt} = \frac{\partial H}{\partial p}, \quad
\frac{dp}{dt} = -\frac{\partial H}{\partial x}

\] for some Hamiltonian component \(H(x,p)\), OK? Now, classical physics allows the objective dry soil at every minute i.e. the functions \(x(t),p(t)\) to live on fully determined. But you lot may ever switch to the probabilistic description which is useful in addition to relevant if you lot don't know the exact values of \(x(t),p(t)\) – everything that may live on known. Introduce the probability distribution \(\rho(x,p)\) on the stage infinite that is existent in addition to normalized,\[

\int dx\,dp\, \rho(x,p)=1.

\] It's piddling to receive got many copies of \(x,p\), only add together an index, in addition to rename some of the variables etc. Fine. What is the equation obeyed past times the probability distribution \(\rho(x,p;t)\)? We are only uncertain nearly the initial dry soil but nosotros know the exact deterministic equations of motion. So nosotros may unambiguously derive the equation obeyed past times the probability distribution \(\rho\). The resultant is the Liouville equation of statistical mechanics.

How do nosotros derive in addition to what it is? The derivation volition live on addressed to adult readers who know the Dirac delta-function. If the initial microstate is perfectly known to live on \((x,p)=(x_0,p_0)\), in addition to then the distribution at that initial minute is\[

\rho(x,p) = \delta (x-x_0) \delta(p-p_0).

\] With this initial state, how does the arrangement evolve? Well, the \(x,p\) variables are known at the commencement in addition to the development is deterministic, so they volition live on known at all times. In other words, the distribution volition ever live on a delta-function located at the right location,\[

\rho(x,p;t) = \delta [x-x(t)] \delta[p-p(t)]

\] What is the differential equation obeyed past times \(\rho\)? Calculate the partial derivative alongside abide by to time. You volition get, past times the Leibniz dominion in addition to the dominion for the derivative of a composite function,\[

\eq{
\frac{\partial \rho (x,p;t)}{\partial t} &= \delta'[x-x(t)] \dot x(t) \delta[p-p(t)]+\\
&+ \delta[x-x(t)] \delta'[p-p(t)] \dot p(t)
}

\] or, equivalently (if nosotros realize that \(\rho\) is the delta-function in addition to substitute it back),\[

\frac{\partial\rho}{\partial t} = \frac{\partial \rho}{\partial x}\dot x(t)+\frac{\partial \rho}{\partial p}\dot p(t).

\] This is the Liouville equation for the probabilistic distribution on the stage space, \(\rho\). The funny thing is that this equation is linear inwards \(\rho\). And because every initial distribution may live on written as a continuous combination of such delta-functions in addition to because the concluding probability should live on a linear component of the initial probabilities, nosotros may only combine all the delta-function-based reason vectors \(\rho(x,p;t)\) corresponding to the classical trajectories \(x(t),p(t)\), in addition to nosotros volition acquire a full general probability distribution that behaves properly.

In other words, because of the linearity inwards \(\rho\) in addition to because of the validity of the equation for a reason of functions \(\rho(x,p;t)\), the finally displayed equation, the Liouville equation, holds for all distributions \(\rho(x,p;t)\).

Excellent. I emphasize that this Liouville equation is completely determined past times the deterministic equations for \(x(t),p(t)\). Aside from the totally universal, mathematical rules of the probability calculus, nosotros didn't demand anything to derive the Liouville equation. Nothing is missing inwards it. But when nosotros mensurate an atom's location to live on \(x_1\), in addition to then the distribution \(\rho(x,p;t)\) "collapses" because of Bayesian inference. We receive got learned some detailed data so our incertitude has decreased. But this collapse doesn't demand whatsoever "modifications" of the Liouville equation or farther explanations because you lot may nonetheless assume that the underlying physics is a deterministic equation for \(x(t),p(t)\) in addition to all the \(\rho\) materials was solely added to bargain alongside our incertitude in addition to ignorance. The shape of the Liouville equation is exact because it was the probabilistic counterpart straight derived from the deterministic equations for \(x(t),p(t)\) which were exact, too.

What changes inwards quantum mechanics? The solely thing that changes is that \(xp-px=i\hbar\) rather than zero. This has the of import number that the deterministic flick beneath everything inwards which \(x(t),p(t)\) are well-defined \(c\)-number functions of fourth dimension is no longer allowed. But the equation for \(\rho\) is nonetheless OK.

Before nosotros switch to quantum mechanics, nosotros may substitute the Hamilton equations to get\[

\frac{\partial\rho}{\partial t} = \frac{\partial \rho}{\partial x}\frac{\partial H}{\partial p}-\frac{\partial \rho}{\partial p}\frac{\partial H}{\partial x}

\] in addition to realize that this shape of the Liouville equation may live on written inwards price of the Poisson bracket\[

\frac{\partial \rho(x,p;t)}{\partial t} = \{\rho(x,p;t),H(t)\}_{\rm Poisson}.

\] That's peachy (up to a conventional sign that may differ). This equation may live on trusted fifty-fifty inwards quantum mechanics where you lot may imagine that \(\rho\) is written as a component (imagine some Taylor expansion, if you lot receive got a psychological employment that this is likewise formal) of \(x,p\). However, \(x,p\) no longer commute, a technical novelty. But the density matrix \(\rho\) inwards quantum mechanics plays the same role as the probability distribution on the classical stage infinite inwards classical physics. You may imagine that the latter is obtained from the one-time as the Wigner quasiprobability distribution.

Because of the usual, purely mathematically provable human relationship betwixt the Poisson brackets in addition to the commutator, nosotros may rewrite the finally shape of the Liouville equation as the von Neumann equation of quantum mechanics\[

\frac{d\rho(t)}{dt} = i\hbar [H,\rho(t)]

\] that dictates the development of the density matrix or operator \(\rho\). (Thankfully, people grip nearly the sign conventions of the commutator.) It tin no longer live on derived from a deterministic starting signal where \(x(t),p(t)\) are well-defined \(c\)-numbers – they cannot live on sharply well-defined because of the incertitude regulation (i.e. nonzero commutator) – but the probabilities nonetheless be in addition to no modifications (let solitary "non-unitary terms" etc.) are needed for the measurement. The standard is only a version of the Bayesian inference. It's nonetheless basically the same thing but this inference must live on carefully described inwards the novel quantum formalism.

If you lot similar Schrödinger's equation, it is non hard to derive it from the von Neumann equation above. Any Hermitian matrix \(\rho\) may live on diagonalized in addition to hence written as a superposition\[

\rho = \sum_j p_j \ket{\psi_j}\bra{\psi_j}

\] Because the von Neumann equation was linear inwards \(\rho\), each term inwards the gist to a higher house volition evolve "separately from others". So it is plenty to know how \(\rho=\ket\psi \bra\psi\) evolves. For this special shape of the density matrix, the commutator\[

[H,\rho] = H\rho - \rho H = H\ket\psi \bra \psi - \ket\psi \bra \psi H

\] in addition to these ii price may live on nicely interpreted as ii price inwards the Leibniz dominion assuming Schrödinger's equation\[

i\hbar \frac{d\ket\psi}{dt} = H\ket\psi

\] in addition to its Hermitian conjugate\[

-i\hbar \frac{d\bra\psi}{dt} = \bra\psi H.

\] So if the moving ridge component \(\ket\psi\) obeys this equation (and its conjugate), in addition to then the von Neumann equation for \(\rho=\ket\psi\bra\psi\) volition follow from that. The implication plant inwards the contrary way as good (Schrödinger's equation follows from the von Neumann equation if nosotros assume the density matrix to depict a "pure state") – except that the overall stage of \(\ket\psi\) may live on changed inwards a full general time-dependent way.

The pure dry soil \(\ket\psi\) corresponds to the "maximum knowledge" inwards the density matrix \(\rho=\ket\psi\bra\psi\). In quantum mechanics, it nonetheless leads to probabilistic predictions for most questions, because of the incertitude principle. Mixed states are superpositions of price of the shape \(\ket{\psi_i}\bra{\psi_i}\). The coefficients or weights are probabilities in addition to this way of taking mixtures is completely analogous (and, inwards the \(\hbar\to 0\) limit, reduces) to classical probability distributions that are also "weighted mixtures".

Because nosotros receive got deduced the quantum equations from the classical ones, it's as lightheaded as it was inwards classical physics to demand some "further explanations" of the measurement, some "extra mechanisms" that allow the unambiguous resultant to live on produced. In classical physics, it's manifestly lightheaded to do so because nosotros may ever imagine that the exact positions \(x(t),p(t)\) receive got ever existed – nosotros only didn't know what they were in addition to that's why nosotros receive got used \(\rho\). When nosotros learn, the probability distribution encoding our noesis all of a precipitous shrinks. End of the story.

In quantum mechanics, nosotros don't know the exact values \(x(t),p(t)\) at a given time. In fact, nosotros know that no i tin know them because they can't simultaneously exist, thank you lot to the incertitude principle. But the probabilistic statements nearly \(x,p\) do be in addition to do work, only similar they did inwards classical statistical physics. But the Schrödinger or von Neumann equation is "as complete" in addition to "as perfectly beautiful" as their counterpart inwards classical physics, the Liouville equation of statistical physics. The latter was ultimately derived (and no adjustments or approximations were needed at all) from the deterministic equations for \(x(t),p(t)\) that the critics of quantum mechanics approve. We only allowed some ignorance on superlative of the equations for \(x(t),p(t)\) in addition to the Liouville equation followed via the rules of the probability calculus.

So the Liouville equation only can't live on "less satisfactory" than the classical deterministic laws for \(x(t),p(t)\). Nothing is missing. And the von Neumann in addition to Schrödinger equations are exactly analogous equations to the Liouville equation – but inwards systems where \(xp-px=i\hbar\) is no longer zero. So the von Neumann or Schrödinger equations must unavoidably live on consummate in addition to perfectly satisfactory, too. They nonetheless depict the development of some probabilities – and, nosotros must acknowledge because of the imaginary nonzero commutator, complex probability amplitudes. Because of the incertitude principle, some ignorance in addition to incertitude – in addition to probabilities strictly betwixt 0 in addition to 100 per centum – are unavoidable inwards quantum mechanics. But the arrangement of laws is just as consummate as it was inwards classical statistical physics. No special explanation or machinery is needed for the standard because the standard is nonetheless goose egg else than a procedure of the reduction of our ignorance. In this process, \(\rho\) all of a precipitous "shrinks" because it's i stair inwards Bayesian inference. It has ever been.

In classical physics, this Bayesian inference may live on idea of as our attempt of learning nearly some "objectively existing truth". In quantum mechanics, no objective truth nearly the observables may be because of the incertitude principle. But the standard is nonetheless a procedure analogous to the Bayesian inference. It improves our subjective noesis – shrinks the probability distribution – as a component of the measured quantity. But because of the nonzero commutator, the standard increases the incertitude of the observables that "maximally" neglect to commute alongside the measured one. So the standard reduces (well, eliminates) our incertitude nearly the thing nosotros measure, but it affects other quantities in addition to increases our incertitude nearly other quantities.

In quantum mechanics, our measurements are non informing us nearly some "God's in addition to everyone's objective truth" (as inwards classical physics) because none exists. But they're steps inwards learning nearly "our subjective truth" that is damn existent for us because all of our lives volition depend on the events nosotros perceive. In most practical situations, the truth is "approximately objective" (or "some gauge truth is objective"). Fundamentally, the truth is subjective but as of import for each observer as the objective truth was inwards classical physics.

But only seek to call upwardly nearly individual who says that a "special modification of the Liouville equations of motion" is needed for the lawsuit when nosotros facial expression at a leave of absence that was tossed in addition to meet a number. The probability distribution \(\rho\) collapses. Well, at that spot is goose egg magic nearly this collapse. We are only learning nearly a belongings of the leave of absence nosotros didn't know nearly – but nosotros do know it afterwards the measurement. The precipitous collapse represents our learning, the Bayesian inference. In classical physics, nosotros may imagine that what we're learning is some "objective truth nearly the observables" that existed independently of all observers in addition to was the "ultimate beacon" for all observers who desire to larn nearly the world. In quantum mechanics, no such "shared objective truth" is possible but it's nonetheless truthful that the standard is an lawsuit when we're learning nearly something in addition to the collapse of the moving ridge component (or density matrix) is no to a greater extent than mysterious than the alter of the probabilities afterwards the Bayesian inference that existed fifty-fifty inwards classical physics.

I am confident – in addition to I saw evidence – that many of you lot receive got understood these rather crystal clear facts nearly the human relationship betwixt classical physics, quantum mechanics, measurements, in addition to probabilities. But mayhap people similar Florin Moldoveanu don't desire to understand. Maybe it's natural to await them non to sympathise these uncomplicated things because their jobs ofttimes depend on their continued ignorance, confusion, in addition to stupidity.

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