What Happens inwards a Measurement?The newspaper implicitly suggests that some conceptually of import work may endure solved past times comments involving the Lindblad equation. I think that inwards this case, he never says things that are sharply wrong (at most distortions of the history). But inwards betwixt the lines, he seems to write that sure steps are useful to clarify the province of affairs fifty-fifty though they aren't.
I think that fifty-fifty though Weinberg avoids clearly wrong statements, relatively to the clear in addition to groundbreaking insights in addition to formulations of the Copenhagen school, his newspaper is a stair backwards away from the proper in addition to problem-free agreement of quantum mechanics. The principal work is that he tries to "reduce" the measurement. But it solely makes feel to "reduce" things that are composite.
The measuring is ultimately an irreducible, cardinal procedure required to apply whatever QM theory, so it just makes no feel to attempt to "decompose" it, to attempt to write whole papers nearly "what is happening inside".
The source problems arise when Weinberg offers his ideas what the Copenhagen Interpretation means. He claims that the Copenhagen Interpretation says that the initial density matrix undergoes the collapse\[
\rho_{\rm initial} \mapsto \rho_{\rm final} = \sum_\alpha p_\alpha \Lambda_\alpha
\] into a combination of Hermitian projection operators where probabilities are the weights. Now, y'all tin say that the claim nearly physics is truthful to some extent or inwards some feel or context – except that i should ameliorate say that it's the "von Neumann pre-measurement", non the collapse itself (the collapse occurs solely when the affair is observed and, equivalently, all the numbers \(p_\alpha\) move either null or one).
But what is misleading is the proposition that this claim is a part, allow lonely a cardinal part, of the Copenhagen Interpretation.
Weinberg's reference #1 is a 1928 Nature paper past times Bohr which Weinberg agrees to consider the "original Copenhagen Interpretation". Note that the newspaper based on a 1927 lecture past times Bohr has its ain spider web page at Nature. You tin regain the total text at Google Books.
Just present me some non-unitary transformation of the density matrix inwards Bohr's newspaper that would endure claimed to "be" the measuring or the collapse. I think that y'all won't fifty-fifty regain whatever reference to the density matrix at all. After all, the density matrix was introduced past times von Neumann inwards 1927 in addition to it could receive got been later Bohr gave his talk. It was a novel materials inwards 1927 – in addition to non essential stuff.
The difficult truth is that none of these things is truly needed or cardinal to formulate the rules of QM. The density matrix may endure viewed every bit a supplementary mathematical tool to bargain alongside the extra incertitude or ignorance that has existed inwards classical physics every bit well. You don't truly demand density matrices to formulate the cardinal QM laws at all – y'all may assume that QM is fundamentally nearly pure states. This supposition is completely analogous to the supposition that inwards classical physics, some values of \(x(t),p(t)\) objectively be fifty-fifty if people are ignorant nearly these values.
The cognition of \(\ket\psi\) inwards QM or \(x(t),p(t)\) inwards classical physics stand upwards for the maximally known i.e. least uncertain states that the apparatuses of the theories allow. We may also bargain alongside the less sure descriptions of the physical objects inwards damage of the stage infinite distribution role or density matrix \(\rho\) but nosotros don't receive got to. Both inwards classical physics in addition to QM, the probabilistic laws for \(\rho\) tin endure straightforwardly derived past times applying the universal probability calculus to the dynamical laws governing the maximum-knowledge states, either \(x(t),p(t)\) classically or \(\ket\psi\) inwards QM. The incertitude implied past times the incertitude regulation is already nowadays inwards predictions based on pure states \(\ket\psi\) in addition to is unavoidable inwards QM; on the other hand, the extra incertitude or ignorance that the density matrix formalism adds is avoidable in addition to hence non-fundamental.
Influenza A virus subtype H5N1 similar comment holds fifty-fifty to a greater extent than strongly for the Lindblad equation. (Incidentally, Weinberg has been playing alongside this equation for some fourth dimension – but he had to withdraw the previous paper on the equation 4 months agone because of a mistake.) This equation is fifty-fifty less cardinal than the concept of the density matrix. We just don't demand it at all to do physics. The Lindblad equation is the most full general linear equation that governs the development of the density matrix \(\rho\) inwards a trend that preserves the line in addition to the positivity of the eigenvalues. For almost all choices of the parameters, the development of \(\rho\) is nonunitary i.e. can't endure written every bit \(\rho\to U \rho U^\dagger\) for some \(U\). The unitary development of \(\rho\) may endure derived from an activity on the "simpler" Hilbert infinite of column (ket) vectors which may endure imagined to endure \(N\)-dimensional; the full general non-unitary development resulting from the Lindblad equation tin transform the \(N^2\) components of the density matrix to a greater extent than to a greater extent frequently than non i.e. access the matrix entries "individually", largely ignoring their organization inwards a matrix.
But at the cardinal level, when dealing alongside the exact description of the physical systems, nosotros solely demand to consider shut systems; in addition to nosotros solely demand to consider their unitary evolution. Any non-unitary development is non solely ugly but it causes additional problems. For example, inwards a 1984 paper published inwards NPB (300 cits), Banks, Peskin, in addition to Susskind showed that whatever non-unitary equation for \(\rho\) violates either locality or the energy-momentum conservation. It's but bad every bit a cardinal equation for Nature. That's also why naught such every bit the Lindblad equation has ever been used past times the QM founding fathers to define the basic rules of the game. It would endure dizzy to do so.
When it comes to its important inwards physics, the Lindblad equation is at most some effective jurist equation that may endure deduced from the cardinal equations; in addition to that may endure useful to bargain alongside opened upwards systems. But it just can't endure cardinal in addition to it can't endure needed to formulate the cardinal postulates of QM.
Imagine that y'all mensurate the spin \(s_z\) of an electron. The electron may endure inwards a pure state\[
\ket\psi = 0.6 \ket{\uparrow} + 0.8 i \ket{\downarrow}.
\] The density matrix is pure, \(\rho_{\rm initial} = \ket\psi \bra\psi\). The "collapse" postulated past times Weinberg replaces this pure density matrix past times the mixed one\[
\rho_{\rm final} = 0.36 \ket{\uparrow}\bra{\uparrow} + 0.64 \ket{\downarrow}\bra{\downarrow}.
\] This density matrix has eigenvalues \(0.36,0.64\) instead of the \(0,1\) eigenvalues of the master copy pure density matrix. It's because the mixed damage \(\ket{\uparrow}\bra{\downarrow}\) in addition to its complex conjugates receive got been erased. They were erased past times the interaction alongside the apparatus that is solely sensitive to \(s_z\). Because of this erasure, the probabilities for all quantities refusing to commute alongside \(s_z\), e.g. \(s_x\), receive got been modified past times the interaction alongside the apparatus. That's how nosotros tin encounter that the measuring has changed the nation of the electron – which is unavoidable.
On the other hand, this is solely truthful if nosotros convey "Weinberg's" mixed density matrix to endure the existent deal. In principle, nosotros may pull the electron in addition to the apparatus exactly. If nosotros do so, the density matrix of the whole scheme remains pure at all times. Fine correlations within the apparatus retrieve the relative stage of "up" in addition to "down". We could inwards regulation contrary all the interactions alongside the apparatus in addition to mensurate \(s_z\) just similar if no interaction betwixt the electron in addition to the apparatus has ever taken place.
This contradicts the outcome of the Lindblad equation which says that the "pure to mixed" transition is irreversible. But the right appraisal of the contradiction is that inwards principle, the Lindblad equation just isn't exact. The "pure to mixed" transformation in addition to the accompanied loss of data nearly the relative phases is an artifact of approximations. That's why no newspaper assuming the Lindblad equation is studying the cardinal laws. The Lindblad equation may endure useful inwards practise – particularly if the "ignored" scheme is a hopelessly messy thermal bath. But sometimes some correlations within this scheme may endure traced in addition to reversed inwards which instance the Lindblad equation volition hit wrong predictions for the scheme of interest.
But my principal electrical load isn't that the founding fathers had naught to do alongside the Lindblad equation which is but non cardinal in addition to it is non needed. Much to a greater extent than generally, the efforts of "decompose" the measuring are pointless. It seems that Weinberg truly realizes that when he writes
Of course, this persuasion of the Copenhagen interpretation just pushes the difficult problems of interpreting quantum mechanics to a larger scale.OK, if he knows that he's just pushing (what he considers) the problems around, why is he doing these things? The of import fact he can't circumvent is that inwards QM, an observation past times an observer is ultimately needed, anyway. Instead of measuring the voltage straight past times his skin, the observer may await at the pointer of the voltmeter. The technical protocol for the measuring may alter if nosotros insert the apparatus inwards betwixt the observer's senses in addition to the measured object but naught changes qualitatively nearly the fact that an observer is needed at the goal in addition to those who (irrationally) consider the observation "mysterious" volition ever do so. The extra inserted layers tin inwards no trend trim down the "amount of mystery" hiding inwards the procedure of the observation.
Weinberg's jurist Lindblad-equation-based description of the interactions betwixt the measured scheme in addition to the measured apparatus may explicate why the data nearly the measured scheme is transferred to the apparatus every bit well. But this transfer is just pushing things around. The possible terminal states of the apparatus volition withal endure predicted just probabilistically – just similar the nation of the measured electron – so at that topographic point withal has to endure an observer who just looks.
Now, if Weinberg or someone else analyzes why in addition to how a particular measuring apparatus works, it would endure useful. We demand to know what's happening within a telescope or LIGO or a voltmeter to empathise these machines in addition to to endure able to improve them, create them, or build ameliorate ones. But if i solely studies all the measuring apparatuses inwards general, the physical content of this inquiry never goes beyond the nearly tautological fact that "the relevant data nearly the measured object is imprinted to the apparatus inwards some way". Great. It does. It's truthful pretty much past times Definition of the measuring apparatus. The voltmeter is defined every bit a physical scheme that tin interpret the voltage betwixt 2 points of a wire to the location of a pointer. As I said, it may brand feel to present that a voltmeter exists, in addition to how it works. This special exercise is also a proof that at to the lowest degree i measuring apparatus exists. But it's vacuous to attempt to utter over "the being of measuring apparatuses" inwards general.
At the end, the basic argue why futile exercises such every bit Weinberg's are repeated all the fourth dimension is that many folks, apparently including Weinberg himself, are just non willing to reconcile themselves alongside the fact that QM is fundamentally unlike than classical physics in addition to it predicts observations for observers. QM just cannot piece of work without observers. When nosotros utter nearly observers, it agency that they realize that they're observers in addition to they've observed something, in addition to they realize what they just observed, in addition to this realization must endure assumed to be – every bit an uncomplicated edifice block for physics to endure possible. It's this perception that QM (probabilistically) predicts. The perceptions are the ultimate client who uses the laws of QM. If this basic setup looks every bit good idealist to you, it's your problem. It's certainly how Nature works.
Even if in addition to when Weinberg avoids downright wrong claims nearly physics (he doesn't avoid wrong claims nearly the history), his newspaper is withal a constituent of the misguided efforts to regain some "classical wheels in addition to gears" within QM. But at that topographic point are no classical wheels in addition to gears within QM. QM describes how Nature industrial plant at the fundamental level. The claim that "the squared absolute value of a complex probability aAmplitude calculates the probability of i outcome of an observation or another", it's a clear, unambiguous, true, in addition to consummate police force of physics. There is naught "deeper" to endure flora here. Influenza A virus subtype H5N1 instructor may attempt to explicate this controversy rattling tardily to a student. But it's dizzy to write whole inquiry papers attempting to claim that at that topographic point are tens of pages of explanations that receive got to endure "added" for the judgement to move bully science. It's but non true. Nothing needs to endure added here.
The distortions of the views of the QM founding fathers seems to endure omnipresent inwards Weinberg's newspaper – in addition to similar papers. For example, Weinberg writes:
In the master copy formulation [1] of the Copenhagen interpretation it was but accepted that the alter inwards a scheme during measuring inwards regulation departs from quantum mechanics.Oh, really? Just opened upwards the full 1928 newspaper past times Bohr in addition to present me where he says that at that topographic point be changes during the measuring that depart from quantum mechanics.
If y'all read the actual paper, Bohr, on the contrary, explains rattling clearly that Planck's quantum of action, \(\hbar\), quantifies the unavoidable deviation of the laws of Nature from the framework of classical physics, namely from the supposition that properties of physical objects be independently of observers in addition to observations. They just don't be independently of those. To examine that a quantity has a important inwards science, i has to formulate an operational method to utter nearly it, at to the lowest degree inwards principle. But because inwards QM, it's guaranteed that whatever observations alter the physical system, it's clear that no values of whatever dynamical variables tin objectively be without observers inwards the same feel every bit they did inwards classical physics.
Bohr says that QM requires the "quantum postulate" which equips whatever private scheme alongside the discontinuity, or rather individuality, that is completely unknown to classical physics. Bohr is arguably non rattling clear nearly what the "quantum postulate" just says but he is extremely clear nearly what it does not say.
QM in addition to Bohr's text (which is non such a bad or incomprehensible summary of the graphic symbol of QM, I think) is all nearly the inadequacy of classical physics, non the breakdown of quantum mechanics. I can't believe that Weinberg can't read. Someone who truly reads Bohr's newspaper must see that Bohr wrote pretty much the opposite than what Weinberg tries to ascribe to him.
The solely "similar" affair that Bohr says is that the procedure of measurement/observation yields the data of the same variety that nosotros knew inwards classical physics. But that's non a violation of QM. It's just a controversy that classical physics in addition to QM hateful just the same affair past times the result of a measurement, e.g. the measuring of \(s_z\) I mentioned above. Classical physics has no monopoly over the "outcomes of observations" in addition to their definition. On the contrary, QM happily uses the same concept – in addition to the concept is to a greater extent than of import inwards QM than it was inwards classical physics.
But classical physics in addition to QM receive got completely unlike methods to predict the outcomes of the measuring – and, consequently, unlike values of the probabilities that they truly predict (sometimes rattling different, sometimes slightly different). Classical physics assumes that independently of whatever observer, at that topographic point exists an objective nation of Nature – a betoken on the stage infinite – piece QM postulates that such a affair doesn't exist, all data must endure carefully reduced to observations, in addition to those depend on what nosotros hateful past times the observer which cannot endure defined inwards whatever objective trend because the observation is ultimately a subjective procedure (an observer knows that he has seen something or learned something, but others can't strictly speaking examine that the observer felt anything at all). QM calculates the probabilities of outcomes of observations "directly" using a formula involving the complex matrix elements of linear operators on the Hilbert infinite – those are associated alongside the measurable information. That's how Nature works. If classical physics were correct, i could calculate the probabilities from probabilistic laws that may endure classically derived from a laid of classical deterministic (or deterministic plus random noise) equations. But QM states that it isn't possible. There don't be whatever "classical laws" beneath the quantum ones.
When people similar Weinberg – in addition to particularly people who are much to a greater extent than confused than he is – incorrectly say that the measurement violates unitarity or the laws of QM (it doesn't do either), they hateful that
if the dynamical equation governing the development of the moving ridge role or the density matrix were a classical development equation, i would receive got to add together modifications of this equation during the procedure of the observation/measurement/collapse.But what they actually say isn't truthful because the status "if" inwards the quote just isn't satisfied inwards Nature. Neither the moving ridge role nor the density matrix is a collection of classical degrees of freedom. Instead, they are packages of probability amplitudes whose solely physical important is that probabilities of outcomes may endure calculated from them (as bilinear functions of the moving ridge function; or linear functions of the matrix elements of the density matrix). Consequently, Schrödinger's equation for the moving ridge role (or the related Liouville-von Neumann equation for the density matrix) isn't a classical dynamical equation for observables, either.
When nosotros regain a detail outcome of a measurement, e.g. that "the spin is up", nosotros are just confirming a demand of QM that observations be (and for every linear Hermitian operator, at that topographic point exists a possible "type of a measurement"). We are non "debunking" QM. Instead, nosotros are confirming its postulate. What Schrödinger's or equivalent equations inwards QM say is how the probabilities may endure calculated. And these predictions for the probabilities may endure verified e.g. past times repetitions of the same experiment. But the rattling fact that private experiments receive got sharp outcomes isn't a violation of QM. It's a confirmation of QM in addition to the outcomes are sudden according to QM because solely sudden outcomes are said to occur alongside nonzero probabilities past times the rules of QM.
I think that Weinberg is a smart human being but it seems obvious to me that when it comes to "conceptually truly new" insights that had to endure made when QM was discovered, relatively to Bohr, Heisenberg, Born, in addition to others, Weinberg is an intellectual dwarf. He couldn't receive got perhaps made the transition from classical physics to quantum physics because he has some problems alongside it fifty-fifty now, xc years later the discovery.
Weinberg's newspaper also discusses i evergreen from similar newspaper – the proposition that he may derive the Born rule. But all these games inwards all similar papers are self-evidently examples of soiled circular reasoning. The betoken is that a basic demand of QM is that the probabilities – something that may endure "measured" past times the frequentist formula if nosotros repeat experiments many times – is a linear role of the density matrix,\[
{\rm Prob}_A = {\rm Tr} (\rho P_A)
\] where \(P_A\) is the projection operator corresponding to the "Yes" statement. This controversy of QM is a necessary link to connect the mathematical symbols alongside the actual observations past times the experimenters. QM says that solely probabilities of outcomes may endure predicted; in addition to it just gives y'all the right formulae that quantify (predict) these probabilities.
If y'all omit a police force that connects the probability (something that experimenters know from their rattling practical life) alongside the moving ridge role or density matrix (something that the workers alongside the mathematical apparatus of QM know, particularly from their piece of work on paper), the mathematical apparatus of QM ceases to receive got any relevance for natural science.
So to brand the equations of QM physical, y'all but need to say that probabilities are connected alongside the mathematical formulae inwards the known way. You demand to add together the Born rule. Once y'all add together it, y'all tin rederive it inwards many other ways, but every bit a "derivation" of the Born rule, whatever such exercise is clearly just a instance of circular reasoning. There can't endure anything to a greater extent than cardinal nearly the "derivation" of the probabilistic graphic symbol of QM than the controversy that the probabilities (known to experimenters) may endure calculated every bit bilinear functions of the moving ridge role or linear functions of the density matrix.
You demand this supposition – the Born dominion or telephone telephone it whatever y'all desire – to do physics in addition to this supposition can't endure "simplified" or "reduced" further. And i time y'all assume that probabilities (that experimenters know) are linear functions of the density matrix, so it follows that it's truthful at all times because all the transformations that the density matrix undergoes are linear, too. There's naught nontrivial in addition to novel to endure flora here. It's a pure waste product of time.
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