Even though a whole "community" of wouldbe scientists nurture a religion based on the (scientifically debunked) dogma that there is something incomplete or unsatisfying about quantum mechanics, quantum mechanics has worked extremely well for 90 years and it really takes a few lines to fully and rigorously describe its general rules and check that they don't suffer from any flaw.
Quantum mechanics is a framework to produce predictions – or construct valid statements – about Nature our of some known facts about Nature. According to this framework, every physical system is described by a Hilbert space \(\HH\) on which linear operators act. Everything we know or want to know – an observable – is associated with such an operator (or operators).
The prediction works as follows. We want to know something about the next measurement – new information we are going to learn if we "live". Without a loss of generality, this information may always be decomposed to answers to Yes/No questions which are associated with Hermitian projection operators \(P\). Because \(P^2=P\), the eigenvalues are either zero (No) or one (Yes).
Quantum mechanics answers the \((N+1)\)st question linked to the operator \(P_{N+1}\) probabilistically, by calculating the probability that the answer is going to be Yes. The probability is \[
{\rm Prob} = \abs{ P_{N+1} \ket \psi }^2
\] or, in the case of mixed states,\[
{\rm Prob} = {\rm Tr} ( P_{N+1} \rho P_{N+1} )
\] where I could have omitted one of the copies of \(P_{N+1}\) because it's a projection operator, but I chose to keep it for symmetry reason.
The pure state \(\ket\psi\) that we need to substitute to the formula for the probability above is\[
\ket\psi = {\rm Norm} \zzav{ T\zav{ \prod_{i=1}^N P_i } \ket{\phi_{\rm in}} }
\] where \(P_i\) are the projection operators corresponding to all the questions whose answers were measured and found to be Yes and \(\phi_{\rm in}\) is the "truly" initial state before all these measurements – which becomes irrelevant if the sequence of the operators \(P_i\) is sufficiently long.
The symbol "Norm" means "normalization" (the division of the ket vector inside by its length) – for the sake of simplicity, we keep \(\ket\psi\) normalized to unity. The metaoperator \(T\) is the time ordering: the operators \(P_i\) are sorted so that the measurements that took place first appear on the right, closer to \(\ket{\phi_{\rm in}}\). Those that took place later are on the left side from them.
Note that if the operators \(P_i\) are constructed from field operators in a relativistic quantum field theory, they (super)commute at spacelike separation, so a pair of operators \(P_i\) and \(P_{i+1}\) that are spacelikeseparated may be ordered in both ways. The ordering becomes irrelevant for them.
On the other hand, if \(P_i\) and \(P_{i+1}\) don't (super)commute with one another, their ordering is extremely important. And if they're associated with the same value of time \(t\), like if the two adjacent projection operators depend on \(x(t)\) and \(p(t)\), the ordering becomes illdefined if these operators don't (super)commute. This is why the position and momentum (and any pair of noncommuting operators) can't be measured at the same moment. If you try to do such things on paper, the formalism breaks down and you fail as a theorist. If you try to measure them (noncommuting operators) simultaneously in reality, you will fail as a practician. Noncommuting observables can't be simultaneously measured – which is why their products won't appear in the expressions that predict probabilities of anything meaningful, either.
Again, I emphasize that the formalism of quantum mechanics depends on the chronology. One must know what events or measurements occurred first and which occurred later because the ordering is translated to the ordering of the application of the projection operators – or "collapses" of the wave function, if you like this problematic jargon – and the ordering matters. Also, the arrow of time matters. The past operators are always closer to the kets (or bras, if there is another copy of all these operators in the conjugate part of the formula) – there is simply no \(\ZZ_2\) symmetry between the future and the past in the full formalism that predicts probabilities. The Heisenberg equations for operators may be timereversalsymmetric (or CPTsymmetric, which is always the case in quantum field theories) but the full formalism that actually calculates probabilities out of them is simply not! This is nothing else than the quantum analogy of the fact that the hypotheses and evidence play asymmetric roles in ordinary Bayes' theorem.
The density matrix \(\rho\) to substitute is similarly\[
\rho = {\rm Norm} \zzav{ T\zav{ \prod_{i=1}^N P_i } \rho_{\rm in} \, T\zav{ \prod_{i=1}^N P_i }^\dagger }
\] where the dagger \(\dagger\) simply reverts the ordering of the chronological metaoperator \(T\). Again, the "past" operators on both sides are closer to \(\rho\) than the future operators. The normalization \({\rm Norm}\) of the density matrix is nothing else than the division of the argument by its trace.
(If we weren't sure about the results of measurements of \(P_i\), we could write down a more complicated form of the density matrix \(\rho\) with many terms weighted by probabilities.)
That's it. Quantum mechanics tells us that we must know what we want to know – we must specify the questions (the relevant observables that are measured) – before quantum mechanics may tell us the (probabilistic answers). Without a welldefined question (and a choice of this question depends on an observer who can "perceive" the results or who "cares" about the results), there can't be a welldefined answer. Unlike classical physics, there is no engine that would provide us with the "right questions" at the same moment.
There are many different questions that may be asked about a physical system – but due to the complementarity or the uncertainty principle, they can't be simultaneously meaningful. You either want to know the position of a particle after some process; or its momentum. You just can't ask about both at the same moment. These questions are only meaningful to the extent to which they are measured and if one measures the position, the value(s) of the momentum change, and vice versa.
Every question about an arbitrary physical system in Nature that can't be translated to the template above – or some of its allowed modest generalizations which still respect the Hilbertspace and probabilistic character of quantum mechanics, I don't want to go into it – is physically meaningless. There's nothing wrong with this proposition. Every theory must decide which sentences are meaningful and which are not (and there always exist sentences in both groups). Quantum mechanics forces us to phrase everything we insert as input – the initial state etc. – and everything we want to know in terms of observables i.e. operators and their measured values. This condition is in no way constraining (or a rule that would be making quantum mechanics incomplete) because we can't really know anything about Nature without some kind of measurement. So it's just fine if quantum mechanics assumes and depends on this selfevident fact. And be sure that it does.
That's it. There is no justification for writing dozens of articles full of doubts and dissatisfaction. Who can't understand, in 2015, the rules above and the fact that they represent a totally internally consistent and logically complete framework that may be used to explain everything we know about Nature, is a retarded imbecile, euphemistically speaking.
Boolean logic
Note that the formalism above was still answering Yes/No questions – one could have reformulated the framework in terms of general observables with arbitrary spectra but I chose the Yes/No approach which is enough and equivalent. It means that it was assigning truth values to logical propositions. The observations done in the past already have welldefined truth values. The observations that will be done in the future don't have welldefined truth values yet. Quantum mechanics allows us to calculate the probability that the value is "zero" or "one".
Boolean logic is totally OK because the measurements assign truth values to particular propositions and these truth values obviously behave just like they always did in Boolean logic. It is really misleading to use the term "classical logic" because there is nothing in Boolean logic that depends on classical physics – on the framework of physics which assumes that there is a phase space of objective states with some dynamical laws how the point on this phase space evolves. Quantum mechanical physicists, as I have shown, use the same Boolean logic to study Nature. So Boolean logic is both classical and quantum; its applicability is not constrained in similar ways.
A Romanian blog post "Boolean logic and quantum mechanics" has claimed that quantum mechanics forces us to abandon Boolean logic but it simply ain't so. The measurement is what connects the new "engine" inside quantum mechanics to the usual truth values of propositions we used to use in classical physics as well.
The projection operators \(P_i\) in the formalism at the top generically don't commute with each other. In some sense, the algebra they represent is "nonclassical" in the sense of "noncommuting". But when we normally talk about logic, we talk about the particular truth values of propositions that have already been measured, not the whole operators, and those are commuting. The actual, measured truth values are commuting – they are just not "deterministically" or "unambiguously" calculable from the initial state using a valid physical theory. Quantum mechanics doesn't allow such calculations.
In 1935, Birkhoff and von Neumann proposed their quantum logic, a generalization of the mathematical axioms for logic in which e.g. the distributive law for Boolean values doesn't hold. It was a rather weird paper and it has in no way become a part of the standard physics cannon – because it's not needed. Physicists like those in Copenhagen haven't started to use it because while it can be a mathematically consistent set of axioms and definitions inspired by quantum mechanics, it's not needed in physics. It brings nothing new to physics. Even Matt Leifer has said "No" to the Birkhoffvon_Neumann logic. Also, the Wikipedia article admits that the Birkhoffvon_Neumann work is pretty much just showing some similarity between algebras of operators and logic. Because of the differences, one may "generalize" the concept of logic, but all of it is just a repackaging of known things with no important physical or philosophical consequences.
Boolean logic is just fine in quantum mechanics. Quantum mechanics doesn't differ from classical physics by the logic it uses; it differs by the character of the predictions it allows us to make about the truth value of propositions about future measurements and the general algorithm how to make these predictions.
Sunday, July 12, 2015 ... //
Boolean logic is sufficient to work with quantum mechanics
Vystavil
Luboš Motl
v
10:39 AM



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