Why the Quantum | A Response to Wheeler’s 1986 Paper
Wheeler’s opening statement in his 1986 paper, “How Come the Quantum?” holds as true today as it did then [1]
The necessity of the quantum in the construction of existence: out of what deeper requirement does it arise? Behind it all is surely an idea so simple, so beautiful, so compelling that when — in a decade, a century, or a millennium — we grasp it, we will all say to each other, how could it have been otherwise? How could we have been so stupid for so long?
In this Insight, I will answer Wheeler’s question per its counterpart in quantum information theory (QIT), “How come the Tsirelson bound?” Let me start by explaining the Tsirelson bound and its relationship to the Bell inequality, then it will be obvious what that has to do with Wheeler’s question, “How Come the Quantum?” The answer (the Tsirelson bound is a consequence of conservation per no preferred reference frame (NPRF)) may surprise you with its apparent simplicity, but that simplicity belies a profound mystery, as we will see.
The Tsirelson bound is the spread in the Clauser-Horne-Shimony-Holt (CHSH) quantity
\begin{equation}\langle a,b \rangle + \langle a,b^\prime \rangle + \langle a^\prime,b \rangle – \langle a^\prime,b^\prime \rangle \label{CHSH1}\end{equation}
created by quantum correlations. Here, we consider a pair of entangled particles (or “quantum systems” or “quantum exchanges of momentum”). Alice makes measurements on one of the two particles with her measuring device set to ##a## or ##a^\prime## while Bob makes measurements on the other of the two particles with his measuring device set to ##b## or ##b^\prime##. There are two possible outcomes for either Bob or Alice in either of their two possible settings given by ##i## and ##j##. For measurements at ##a## and ##b## we have for the average of Alice’s results multiplied by Bob’s results on a trial-by-trial basis
\begin{equation}\langle a,b \rangle = \sum (i \cdot j) \cdot P(i,j \mid a,b) \label{average}\end{equation}
That’s a bit vague, so let me supply some actual physics. The two entangled states I will use are those which uniquely give rise to the Tsirelson bound [2-4] , i.e., the spin singlet state and the ‘Mermin photon state’ [5]. The spin singlet state is ##\frac{1}{\sqrt{2}} \left(\mid ud \rangle – \mid du \rangle \right)## where ##u##/##d## means the outcome is displaced upwards/downwards relative to the north-south pole alignment of the Stern-Gerlach (SG) magnets (Figure 1).
Figure 1. A Stern-Gerlach (SG) spin measurement showing the two possible outcomes, up and down, represented numerically by +1 and -1, respectively. Figure 42-16 on page 1315 of Physics for Scientists and Engineers with Modern Physics, 9th ed, by Raymond A. Serway and John W. Jewett, Jr.
Figure 1. A Stern-Gerlach (SG) spin measurement showing the two possible outcomes, up and down, represented numerically by +1 and -1, respectively. Figure 42-16 on page 1315 of Physics for Scientists and Engineers with Modern Physics, 9th ed, by Raymond A. Serway and John W. Jewett, Jr.This state obtains due to conservation of angular momentum at the source as represented by momentum exchange in the spatial plane P orthogonal to the source collimation (“up or down” transverse). This state might be produced by the dissociation of a spin-zero diatomic molecule [6] or the decay of a neutral pi meson into an electron-positron pair [7], processes which conserve spin angular momentum. For more information about the spin singlet state and the spin triplet states, see this Insight.
The Mermin state for photons is ##\frac{1}{\sqrt{2}} \left(\mid VV \rangle + \mid HH \rangle \right)## where ##V## means the there is an outcome (photon detection) behind one of the coaligned polarizers and ##H## means there is no outcome behind one of the co-aligned polarizers. This state obtains due to conservation of angular momentum at the source as represented by momentum exchange along the source collimation (“yes” or “no” longitudinal). Dehlinger and Mitchell created this state by laser inducing spontaneous parametric downconversion in beta barium borate crystals [8], a process that conserves spin angular momentum as represented by the polarization of the emitted photons. At this point we will focus the discussion on the spin single state for total anti-correlation, since everything said of that state can be easily transferred to the Mermin photon state.
Let us investigate what Alice and Bob discover about these entangled states in the various contexts of their measurements (Figure 2). Alice’s detector responds up and down with equal frequency regardless of the orientation ##\alpha## of her SG magnet. This is in agreement with the relativity principle, aka “no preferred reference frame” (NPRF), where different SG magnet orientations relative to the source constitute different “reference frames” in quantum mechanics just as different velocities relative to the source constitute different “reference frames” in special relativity (see this Insight).
Figure 2. Alice and Bob making spin measurements in the xz plane on a pair of spin-entangled particles with their Stern-Gerlach (SG) magnets and detectors.
Bob observes the same regarding his SG magnet orientation ##\beta##. Thus, the source is rotationally invariant in the spatial plane P orthogonal to the source collimation. When Bob and Alice compare their outcomes, they find that their outcomes are perfectly anti-correlated (##ud## and ##du## with equal frequency) when ##\alpha – \beta = \theta = 0## (Figure 3). This is consistent with conservation of angular momentum per classical mechanics between the pair of detection events (again, this fact defines the state). The degree of that anti-correlation diminishes as ##\theta \rightarrow \frac{\pi}{2}## until it is equal to the degree of correlation (##uu## and ##dd##) when their SG magnets are at right angles to each other. In other words, whenever the SG magnets are orthogonal to each other anti-correlated and correlated outcomes occur with equal frequency, i.e., conservation of angular momentum in one direction is independent of the angular momentum changes in any orthogonal direction. Thus, we wouldn’t expect to see more correlation or more anti-correlation based on conservation of angular momentum for transverse results in the plane P when the SG magnets are orthogonal to each other. As we continue to increase the angle ##\theta## beyond ##\frac{\pi}{2}## the anti-correlations continue to diminish until we have totally correlated outcomes when the SG magnets are anti-aligned. This is also consistent with conservation of angular momentum, since the totally correlated results when the SG magnets are anti-aligned represent momentum exchanges in opposite directions in the plane P just as when the SG magnets are aligned, it is now simply the case that what Alice calls up, Bob calls down and vice-versa.
The counterpart for the Mermin photon state is simply that angular momentum conservation is evidenced by ##VV## or ##HH## outcomes for coaligned polarizers. When the polarizers are at right angles you have only ##VH## and ##HV## outcomes, which is still totally consistent with conservation of angular momentum as ‘not ##H##’ implies ##V## and vice-versa [8]. In other words, a polarizer does not have a ‘north-south’ distinction (longitudinal rather than transverse momentum exchange). In particular, having rotated either or both polarizers by ##\pi## one should obtain precisely ##VV## or ##HH## outcomes again.
Nothing is particularly mysterious about the entangled states for electron spin or photon polarization described here so far because we have been thinking as if conservation of angular momentum holds for each experimental trial, as in classical mechanics. Truth is, since Alice and Bob can only measure +1 or -1 (quantum exchange of momentum per NPRF), we can only get conservation of angular momentum in any particular trial when their SG magnets/polarizers are co-aligned. And, we cannot use classical probability theory to account for the conservation of angular momentum on average.
In particular, the probability that Alice and Bob will measure ##uu## or ##dd## at angles ##\alpha## and ##\beta## for the spin singlet state is
\begin{equation}P_{uu} = P_{dd} = \frac{1}{2} \mbox{sin}^2 \left(\frac{\alpha – \beta}{2}\right) \label{probabilityuu}\end{equation}
And, the probability that Alice and Bob will measure ##ud## or ##du## at angles ##\alpha## and ##\beta## for the spin singlet state is
\begin{equation}P_{ud} = P_{du} = \frac{1}{2} \mbox{cos}^2 \left(\frac{\alpha – \beta}{2}\right) \label{probabilityud}\end{equation}
Using these in Eq. (\ref{average}) where the outcomes are +1 (##u##) and -1 (##d##) gives Eq. (\ref{CHSH1}) of
\begin{equation}-\cos(a – b) -\cos(a – b^\prime) -\cos(a^\prime – b) +\cos(a^\prime – b^\prime) \label{CHSHspin}\end{equation}
Choosing ##a = \pi/4##, ##a^\prime = -\pi/4##, ##b = 0##, and ##b^\prime = \pi/2## minimizes Eq. (\ref{CHSHspin}) at ##-2\sqrt{2}## (the Tsirelson bound).
Likewise, for the Mermin photon state we have
\begin{equation}P_{VV} = P_{HH} = \frac{1}{2} \mbox{cos}^2 \left(\alpha – \beta \right) \label{probabilityVV}\end{equation}
and
\begin{equation}P_{VH} = P_{HV} = \frac{1}{2} \mbox{sin}^2 \left(\alpha – \beta \right) \label{probabilityVH}\end{equation}
Using these in Eq. (\ref{average}) where the outcomes are +1 (##V##) and -1 (##H##) gives Eq. (\ref{CHSH1}) of
\begin{equation}\cos2(a – b) +\cos2(a – b^\prime) +\cos2(a^\prime – b) -\cos2(a^\prime – b^\prime) \label{CHSHmermin}\end{equation}
Using ##a = \pi/8##, ##a^\prime = -\pi/8##, ##b = 0##, and ##b^\prime = \pi/4## maximizes Eq. (\ref{CHSHmermin}) at ##2\sqrt{2}## (the Tsirelson bound). So, we have two mysteries.
First, as explained by Mermin [5], suppose you restrict Alice and Bob’s measurement angles ##\alpha## and ##\beta## to three possibilities, setting 1 is ##0^o##, setting two is ##120^o##, and setting three is ##-120^o##. Eq. (\ref{probabilityud}) says the probability of getting opposite results is 1 when ##\alpha = \beta## (1/2 ##ud## and 1/2 ##du##) and 1/4 otherwise (1/8 ##ud## and 1/8 ##du##). Now, if the source emits particles with definite properties that account for their outcomes in the three possible measurement settings, and we have to get total anti-correlation for like settings, then the particles’ so-called “instruction sets” must be opposite for each of the three settings. For example, suppose we have 1(##u##)2(##u##)3(##d##) for Alice and 1(##d##)2(##d##)3(##u##) for Bob. That guarantees the total anti-correlation for like settings, i.e., 11 gives ##ud##, 22 gives ##ud##, and 33 gives ##du##. And, for unlike settings we get anti-correlation in two combinations, i.e., 12 gives ##ud## and 21 gives ##ud##. In fact, for any instruction set with two ##u## and one ##d## we get anti-correlation for unlike settings in two of the six possible unlike combinations (12,13,21,23,31,32). The only other way to make a pair of instruction sets is to have one with all ##u## and the other with all ##d##. In that case, we get anti-correlation for all six unlike combinations. That means the instruction sets necessary to guarantee anti-correlation for like settings lead to an overall anti-correlation greater than 2/6 for unlike settings, which is greater than the quantum probability for anti-correlation in unlike settings of 1/4. This is Mermin’s version of the Bell inequality [9] (fraction of anti-correlated outcomes for unlike settings must be greater than 2/6) and the manner by which it is violated by quantum correlations (1/4 is less than 2/6). Thus, instruction sets (“counterfactual definiteness”) assumed by classical probability theory cannot account for quantum correlations in this case.
The counterpart to this for the CHSH quantity is that classical correlations give a range of -2 to 2 for the CHSH quantity (“CHSH-Bell inequality”). And, as we saw above, the Tsirelson bound violates the CHSH-Bell inequality. Experiments show that the quantum results can be achieved (violating the Bell inequality), ruling out an explanation of these correlated momentum exchanges via instruction sets per classical probability theory.
The second mystery is that even in cases where we don’t violate the Bell inequality, e.g., ##a = b = 0## and ##a^\prime = b^\prime = \pi/2## which give a CHSH value of 0, we still have conservation of angular momentum. Why is that mysterious? Well, it’s not when the SG magnets are co-aligned, since in those cases we always get a +1 outcome and a -1 outcome for a total of zero. But, in trials where ##\alpha – \beta = \theta## does not equal zero, we need either Alice or Bob, at minimum, to measure something less than 1 to conserve angular momentum. For example, if Alice measures +1, then Bob must measure ##-\cos{\theta}## to conserve angular momentum for that trial. But, again, Alice and Bob only measure +1 or -1 (quantum exchange of momentum per NPRF, which uniquely distinguishes the quantum joint distribution from its classical counterpart [10]), so that can’t happen (Figure 4). What does happen? We conserve angular momentum on average in those trials.
It is easy to see how this follows by starting with total angular momentum of zero for binary (quantum) outcomes +1 and -1 (I am suppressing the factor of ##\hbar/2## and I’m referring to the spin singlet state here [11], Figure 3).
Figure 3. Outcomes (yellow dots) in the same reference frame, i.e., outcomes for the same measurement (blue arrows represent SG magnet orientations), for the spin singlet state explicitly conserve angular momentum.
Alice and Bob both measure +1 and -1 results with equal frequency for any SG magnet angle (NPRF) and when their angles are equal they obtain different outcomes giving total angular momentum of zero. The case (a) result is not difficult to understand via conservation of angular momentum, because Alice and Bob’s measured values of spin angular momentum cancel directly when ##\alpha = \beta##, that defines the spin singlet state. But, when Bob’s SG magnet is rotated by ##\alpha – \beta = \theta## relative to Alice’s, the situation is not as clear (Figure 6).
In classical physics, one would say the projection of the angular momentum vector of Alice’s particle ##\vec{S}_A = +1\hat{a}## along ##\hat{b}## is ##\vec{S}_A\cdot\hat{b} = +\cos{(\theta)}## where again ##\theta## is the angle between the unit vectors ##\hat{a}## and ##\hat{b}## (Figure 2). From Alice’s perspective, had Bob measured at the same angle, i.e., ##\beta = \alpha##, he would have found the angular momentum vector of his particle was ##\vec{S}_B = -1\hat{a}##, so that ##\vec{S}_A + \vec{S}_B = \vec{S}_{Total} = 0##. Since he did not measure the angular momentum of his particle at the same angle, he should have obtained a fraction of the length of ##\vec{S}_B##, i.e., ##\vec{S}_B\cdot\hat{b} = -1\hat{a}\cdot\hat{b} = -\cos{(\theta)}## (Figure 4).
Figure 4. The projection of the angular momentum of Bob’s particle ##\vec{S}_B## along his measurement direction ##\hat{b}##. This does not happen with spin angular momentum due to NPRF.
Of course, Bob only ever obtains +1 or -1 per NPRF, so Bob’s outcomes can only average the required ##-\cos{(\theta)}##. Thus, NPRF dictates
\begin{align*}
P_{uu} + P_{ud} & = \frac {1}{2} \\
P_{ud} + P_{dd} & = \frac {1}{2},
\end{align*}
These equations now allow us to uniquely solve for the joint probabilities
\begin{equation}
P_{uu} = P_{dd} = \frac{1}{2} \mbox{sin}^2 \left(\frac{\theta}{2} \right) \label{QMjointLike}
\end{equation}
and
\begin{equation}
P_{ud} = P_{du} = \frac{1}{2} \mbox{cos}^2 \left(\frac{\theta}{2} \right) \label{QMjointUnlike}
\end{equation}
\begin{equation}
\overline{BA-} = 2P_{du}(+1) + 2P_{dd}(-1) = \cos (\theta) \label{BA-}
\end{equation}
Using Eqs. (\ref{BA+}) and (\ref{BA-}) in Eq. (\ref{consCorrel}) we obtain
\begin{equation}
\langle \alpha,\beta \rangle = \frac{1}{2}(+1)_A(-\mbox{cos} \left(\theta\right)) + \frac{1}{2}(-1)_A(\mbox{cos} \left(\theta\right)) = -\mbox{cos} \left(\theta\right) \label{consCorrel2}
\end{equation}
which is precisely the correlation function for a spin singlet state found using the joint probabilities per quantum mechanics. To see that we simply use Eqs. (\ref{probabilityuu}) and (\ref{probabilityud}) in Eq. (\ref{average}) to get
\begin{equation}
\begin{split}
\langle \alpha,\beta \rangle = &(+1)(-1)\frac{1}{2} \mbox{cos}^2 \left(\frac{\alpha – \beta}{2}\right) + (-1)(+1)\frac{1}{2} \mbox{cos}^2 \left(\frac{\alpha – \beta}{2}\right) +\\ &(+1)(+1)\frac{1}{2} \mbox{sin}^2 \left(\frac{\alpha – \beta}{2}\right) + (-1)(-1)\frac{1}{2} \mbox{sin}^2 \left(\frac{\alpha – \beta}{2}\right) \\ &= -\mbox{cos} \left(\alpha – \beta \right) = -\mbox{cos} \left(\theta \right)
\end{split}
\label{correl}\end{equation}
Thus, “average-only” conservation maps beautifully to our classical expectation (Figures 6 & 7). Since the angle between SG magnets ##\theta## is twice the angle between Hilbert space measurement bases, this result easily generalizes to conservation per NPRF of whatever the measurement outcomes represent when unlike outcomes entail conservation in the symmetry plane [15] (see this Insight on the Bell spin states). However, again, none of the formalism of quantum mechanics is used in obtaining Eq. (\ref{consCorrel2}) or our quantum state Eqs. (\ref{QMjointLike}) & (\ref{QMjointUnlike}). In deriving the quantum correlation function and quantum state in this fashion, we assumed only NPRF.For the Mermin photon state, conservation of angular momentum is established by ##V## (designated by +1) and ##H## (designated by -1) results through a polarizer. When the polarizers are co-aligned Alice and Bob get the same results, half pass and half no pass. Thus, conservation of angular momentum is established by the intensity of the electromagnetic radiation applied to binary outcomes for various polarizer orientations. As with spin angular momentum, this is classical thinking applied to binary outcomes per conservation of angular momentum. Again, grouping Alice’s results into +1 and -1 outcomes we see that she would expect to find ##[\mbox{cos}^2\theta – \mbox{sin}^2\theta]## at ##\theta## for her +1 results and ##[\mbox{sin}^2\theta – \mbox{cos}^2\theta]## for her -1 results. Since Bob measures the same thing as Alice for conservation of angular momentum, those are Bob’s averages when his polarizer deviates from Alice’s by ##\theta##. Therefore, the correlation of results for conservation of angular momentum is given by
\begin{equation}\langle \alpha,\beta \rangle =\frac{(+1_A)(\mbox{cos}^2\theta – \mbox{sin}^2\theta)}{2} + \frac{(-1_A)(\mbox{sin}^2\theta – \mbox{cos}^2\theta)}{2} = \cos{2\theta} \label{merminconserve}\end{equation}
which is precisely the correlation given by quantum mechanics.As before, we need to find ##P_{VV}##, ##P_{HH}##, ##P_{VH}##, and ##P_{HV}## so we need four independent conditions. Normalization and ##P_{VH} = P_{HV}## are the same as for the spin case. The correlation function
\begin{equation}
\begin{split}
\langle \alpha,\beta \rangle = &(+1)_A(+1)_BP_{VV} + (+1)_A(-1)_BP_{VH} + \\&(-1)_A(+1)_BP_{HV} + (-1)_A(-1)_BP_{HH}\label{correlFn2}
\end{split}
\end{equation}
along with our conservation principle represented by Eq. (\ref{merminconserve}) give
\begin{equation}
P_{VV} – P_{VH} = -\frac{1}{2}(\mbox{sin}^2\theta – \mbox{cos}^2\theta)
\end{equation}
and
\begin{equation}
P_{HV} – P_{HH} = \frac{1}{2}(\mbox{sin}^2\theta – \mbox{cos}^2\theta)
\end{equation}
Solving these four equations for ##P_{VV}##, ##P_{HH}##, ##P_{VH}##, and ##P_{HV}## gives precisely Eqs. (\ref{probabilityVV}) & (\ref{probabilityVH}).Notice that since the angle between polarizers ##\alpha – \beta## equals the angle between Hilbert space measurement bases, this result immediately generalizes to conservation per NPRF of whatever the outcomes represent when like outcomes entail conservation in the symmetry plane [15] (again, see this Insight on the Bell spin states).Since the quantum correlations violate the Bell inequality to the Tsirelson bound and satisfy conservation per NPRF while the classical correlations do not violate the Bell inequality, the classical correlations do not satisfy conservation per NPRF. Experiments of course tell us that Nature obeys the quantum correlations and therefore the conservation per NPRF.
Figure 5. A spatiotemporal ensemble of 8 experimental trials for the Bell spin states showing Bob’s outcomes corresponding to Alice‘s ##+1## outcomes when ##\theta = 60^\circ##. Angular momentum is not conserved in any given trial, because there are two different measurements being made, i.e., outcomes are in two different reference frames, but it is conserved on average for all 8 trials (six up outcomes and two down outcomes average to ##\cos{60^\circ}=\frac{1}{2}##). It is impossible for angular momentum to be conserved explicitly in each trial since the measurement outcomes are binary (quantum) with values of ##+1## (up) or ##-1## (down) per no preferred reference frame. The conservation principle at work here assumes Alice and Bob’s measured values of angular momentum are not mere components of some hidden angular momentum with variable magnitude. That is, the measured values of angular momentum are the angular momenta contributing to this conservation.
Figure 6. For the spin singlet state (S = 0). Reading from left to right, as Bob rotates his SG magnets relative to Alice’s SG magnets for her +1 outcome, the average value of his outcome varies from –1 (totally down, arrow bottom) to 0 to +1 (totally up, arrow tip). This obtains per conservation of angular momentum on average in accord with no preferred reference frame. Bob can say exactly the same about Alice’s outcomes as she rotates her SG magnets relative to his SG magnets for his +1 outcome. That is, their outcomes can only satisfy conservation of angular momentum on average, because they only measure +1/-1, never a fractional result. Thus, just as with the light postulate of special relativity, we see that no preferred reference frame leads to counterintuitive results (see this Insight).
Figure 7. The situation is similar for the spin triplet states where outcomes agree for the same measurement in the plane containing the conserved angular momentum vector (S = 1). Reading from left to right, as Bob rotates his SG magnets relative to Alice’s SG magnets for her +1 outcome, the average value of his outcome varies from +1 (totally up, arrow tip) to 0 to –1 (totally down, arrow bottom). This obtains per conservation of angular momentum on average in the plane containing the S = 1 spin angular momentum in accord with no preferred reference frame. See this Insight for details.
So, while conservation per NPRF sounds like a very reasonable constraint on the distribution of quantum exchange of momentum (+1 or -1, no fractions), we still do not have any causal mechanism to explain the outcomes of any particular trial when the SG magnets/polarizers are not co-aligned (Figure 4). And, as I showed above, we cannot use instruction sets per classical probability theory to account for the Tsirelson bound needed to explain the conservation of angular momentum on average. Thus, while we have a very reasonable constraint on the distribution of entangled quantum exchanges (conservation of angular momentum), that constraint has no compelling dynamical counterpart, i.e., no consensus causal mechanism to explain the outcome of any particular trial when the SG magnets/polarizers are not co-aligned and no counterfactual definiteness to explain why conservation of angular momentum is conserved on average. What we have is a “principle” account of entanglement and the Tsirelson bound (see this Insight). I will return to this point after showing how so-called “superquantum correlations” fail to satisfy this constraint as well.
There are QIT correlations that not only violate the Bell inequality, but also violate the Tsirelson bound. Since these correlations violate the Tsirelson bound, they are called “superquantum correlations.” The reason QIT considers these correlations reasonable (no known reason to reject their possibility) is because they do not violate superluminal communication, i.e., the joint probabilities don’t violate the no-signaling condition
\begin{equation}\begin{split}P(A \mid a\phantom{\prime},b\phantom{\prime}) &= P(A \mid a\phantom{\prime}, b^\prime)\\
P(A \mid a^\prime,b\phantom{\prime}) &= P(A \mid a^\prime, b^\prime)\\
P(B \mid a\phantom{\prime},b\phantom{\prime}) &= P(B \mid a^\prime, b\phantom{\prime})\\
P(B \mid a\phantom{\prime},b^\prime) &= P(B \mid a^\prime, b^\prime )\end{split}\label{nosig}\end{equation}
This means Alice and Bob measure the same outcomes regardless of each other’s settings. If this wasn’t true, Alice and Bob would notice changes in the pattern of their outcomes as the other changed their measurement settings. Since the measurements for each trial can be spacelike separated that would entail superluminal communication.
The Popescu-Rohrlich (PR) joint probabilities
\begin{equation}\begin{split}&P(1,1 \mid a,b) = P(-1,-1 \mid a, b)=\frac{1}{2}\\
&P(1,1 \mid a,b^\prime) = P(-1,-1 \mid a, b^\prime)=\frac{1}{2}\\
&P(1,1 \mid a^\prime,b) = P(-1,-1 \mid a^\prime, b)=\frac{1}{2}\\
&P(1,-1 \mid a^\prime,b^\prime) = P(-1,1 \mid a^\prime, b^\prime)=\frac{1}{2} \end{split}\label{PRcorr}\end{equation}
produce a value of 4 for Eq. (\ref{CHSH1}), the largest of any no-signaling possibilities. Thus, the QIT counterpart to Wheeler’s question, “How Come the Quantum?” is “Why the Tsirelson bound?” [12-14]. In other words, is there any compelling principle that rules out superquantum correlations as conservation of angular momentum ruled out classical correlations? Let us look at Eq. (\ref{PRcorr}) in the context of our spin singlet and Mermin photon states. Again, we will focus the discussion on the spin singlet state and allude to the obvious manner by which the analysis carries over to the Mermin photon state.
The last PR joint probability certainly makes sense if ##a^\prime = b^\prime##, i.e., the total anti-correlation implying conservation of angular momentum, so let us start there. The third PR joint probability makes sense for ##b = \pi + b^\prime##, where we have conservation of angular momentum with Bob having flipped his coordinate directions. Likewise, then, the second PR joint probability makes sense for ##a = \pi + a^\prime##, where we have conservation of angular momentum with Alice having flipped her coordinate directions. All of this is perfectly self consistent with conservation of angular momentum as we described above, since ##a^\prime## and ##b^\prime## are arbitrary per rotational invariance in the plane P. But now, the first PR joint probability is totally at odds with conservation of angular momentum. Both Alice and Bob simply flip their coordinate directions, so we should be right back to the fourth PR joint probability with ##a^\prime \rightarrow a## and ##b^\prime \rightarrow b##. Instead, the first PR joint probability says that we have total correlation (maximal violation of conservation of angular momentum) rather than total anti-correlation per conservation of angular momentum, which violates every other observation. In other words, the set of PR observations violates conservation of angular momentum in a maximal sense. To obtain the corresponding argument for angular momentum conservation per the correlated outcomes of the Mermin photon state, simply start with the first PR joint probability and show the last PR joint probability maximally violates angular momentum conservation.
To find the degree to which superquantum correlations violate our constraint, replace the first PR joint probability with
\begin{equation}\begin{split}&p(1,1 \mid a,b) = C \\
&p(-1,-1 \mid a, b) = D \\
&p(1,-1 \mid a,b) = E \\
&p(-1,1 \mid a, b) = F \\ \end{split} \label{PRcorrMod}\end{equation}
The no-signaling condition Eq. (\ref{nosig}) in conjunction with the second and third PR joint probabilities gives ##C = D## and ##E = F##. That in conjunction with normalization ##C + D + E + F =1## and P(anti-correlation) + P(correlation) = 1 means total anti-correlation (##E = F = 1/2##, ##C = D = 0##) is the conservation of angular momentum per the quantum case while total correlation (##E = F = 0##, ##C = D = 1/2##) is the max violation of conservation of angular momentum per the PR case. To get the corresponding result for the Mermin photon state, simply replace the last PR joint probability in analogous fashion, again with ##\theta = \pi##. In that case, the PR joint probabilities violate conservation of angular momentum with total anti-correlation while the Mermin photon state satisfies conservation of angular momentum with total correlation. Thus, we have a spectrum of superquantum correlations all violating conservation of angular momentum.
So, we see explicitly in this result how quantum mechanics conforms statistically to a conservation principle without need of a ‘causal influence’ or hidden variables acting on a trial-by-trial basis to account for that conservation. That is the essence of a “principle theory.” Indeed, the kinematic structure (Minkowski spacetime) of special relativity and the kinematic structure (qubit Hilbert space) of quantum mechanics both follow from NPRF, so we now know that quantum mechanics is on par with special relativity as a principle theory (again, see this Insight).
Therefore, my answer to QIT’s version of Wheeler’s question is
The Tsirelson bound obtains because of conservation per no preferred reference frame.
Whether or not you consider this apparently simple 4-dimensional (4D) constraint (conservation per NPRF [16,17,18]) to dispel the mystery of entanglement and answer Wheeler’s question depends on whether or not you can accept the fundamentality of a principle explanation via patterns in both space and time (see this Insight). While we have a compelling 4D constraint (who would argue with conservation per NPRF?) for our adynamical explanation, we do not have a compelling dynamical counterpart. That is, we do not have a consensus, causal mechanism to explain outcomes on a trial-by-trial basis when the SG magnets/polarizers are not co-aligned, and we cannot use counterfactual definiteness per classical probability theory to account for the fact that we conserve angular momentum on average. So, perhaps we do not need new physics to rise to Wilczek’s challenge [19].
To me, ascending from the ant’s-eye view to the God’s-eye view of physical reality is the most profound challenge for fundamental physics in the next 100 years.
[Note: “God’s-eye view” simply means the blockworld, block universe, “all-at-once”, or 4D view like that of Minkowski spacetime, there is no religious connotation.] Since special relativity already supports that view, perhaps we should accept that adynamical explanation is fundamental to dynamical explanation, so that not all adynamical explanations have dynamical counterparts [20]. In that case, “we will all say to each other, how could it have been otherwise? How could we have been so stupid for so long?” [1]
References
- Wheeler, J.A.: How Come the Quantum?, New Techniques and Ideas in Quantum Measurement Theory 480(1), 304–316 (1986).
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PhD in general relativity (1987), researching foundations of physics since 1994. Coauthor of “Beyond the Dynamical Universe” (Oxford UP, 2018).
I'm sort of in agreement with you that in QM, measurement and preparation seem very similar, but there are some circumstances where it is possible to get particles in a particular state without measuring them. For example, if you send electrons through a Stern-Gerlach device, the ones that are spin-up will go in one direction and the ones that are spin-down will go in another direction. Then if you perform an experiment on just one of the two streams, you can be assured that the electrons are in a specific spin state even though you didn't measure the spin.In fact you have to clearly distinguish preparation (which defines states in an operational sense) and measurements. E.g., the uncertainty principle clearly says that it is not possible to prepare an electron to have determined two spin components in different directions. Nevertheless, no matter in which pure of mixed state the electron might be prepared in you can measure accurately any spin component you like. Often you read wrong statements about these ideas, because people don't precisely distinguish the subtle difference between state preparation and measurement.
Indeed what you describe concerning the SG experiment by filtering out one partial beam is a preparation procedure for the spin component of the particle, i.e., you prepare the particle with a determined spin component in the direction of the magnetic field of the SG apparatus. Then you can measure the spin component in any direction you like.
The circularity of that claim is obvious.
But maybe that "preparation" is yet another kind of physical process I am not aware off, and described in your version of QM that is neither interaction nor measurement.
OK then how do you prepare an entangled pair of electron or photon that have probability 1 to be polarized at such angle along such axes…I've no clue what you want me to prepare. It seems self-contradictory to me what you want me to prepare.
A polarization-entangled pair of photons nowadays is easily prepared by using parametric down conversion using certain kinds of birefringent crystals and a laser:
https://en.wikipedia.org/wiki/Quantum_entanglement
https://en.wikipedia.org/wiki/Spontaneous_parametric_down-conversion
To make a spin-entangled electron-positron pair one way is to use a neutral pion which (however rarely) can decay into a single electron-positron pair with total spin 0. The single electron in the pair is of course not polarized in a certain direction, but for any direction you may measure the spin component you get 50% +1/2 and 50% -1/2. The single-electron spin is in the state ##hat{rho}=hat{1}/2##, i.e., the spin component in any direction is maximally uncertain (i.e., in the state of maximum entropy).
This is indeed one of the errors in Ballentine – he claims that Copenhagen must treat this as a collapse even when no definite outcome is obtained.The book is over 600 pages, can you be more specific with the citation.
Even so, on page 5 Ballentine describes exactly the same recombination setup for neutrons. It is confusing because here he cannot mean that the split is irreversible, surely ?I don't see anything confusing or incorrect on page 5, 6. What do you mean exactly?
Even so, on page 5 Ballentine describes exactly the same recombination setup for neutrons. It is confusing because here he cannot mean that the split is irreversible, surely ?I'll let someone else answer. For me, Ballentine is in such sustained and fundamental error, I ignore his writings on many topics.
This is indeed one of the errors in Ballentine – he claims that Copenhagen must treat this as a collapse even when no definite outcome is obtained.Even so, on page 5 he describes exactly the same recombination setup for neutrons. It is confusing because here he cannot mean that the split is irreversible, surely ?
No, it's not a measurement until someone detects it. The definition of "measurement" is that you have measured some quantity when you have made a persistent record of its value (or something that maps to its value). If that hasn't happened, then a measurement hasn't been made.
In deflecting an electron to the left or to the right, what you've done is set up a correlation between two different properties of the electron: its position (left or right) and its spin (up or down). Every interaction sets up a correlation of that type, but not every interaction is a measurement.
No, not all interactions with a macroscopic/classical apparatus result in a measurement. Only irreversible interactions—interactions that leave the apparatus in a persistent state that records the value being measured.
Not by the definition of "measurement" that I'm using. By what definition is it a measurement? It doesn't collapse the wavefunction.
And yet it is the most prominent example where projected wavefunctions are actually used in practise.
I don't disagree with your terminology but this is something about the measurement problem which I find peculiar. The collapse postulate is only needed for sequential measurements (because only there, we can check the state after a measurement). Textbook examples of sequential measurements mostly involve multiple Stern Gerlach devices or multiple polarizers. After each device, the new state vector is calculated by a projection. But usually, the only actual measurement is provided by a single screen at the end. So in these cases, collapse arguably is just a convenient way to simplify calculations.
I think that a lot of discussions about the measurement problem would gain considerable clarity if people tried to focus on distinguishing these two classes of experiments:
1) Real sequential measurements where outcomes are obtained at each device.
2) Sequential preparations, where state vectors are projected for convenience because nobody cares about what happens inside the devices.
It turns out that there aren't many experiments of type 1 if by "outcome" we mean things which are actually reported by the experimenters. If people agree about the classification of typical experiments, the focus of the discussion can be narrowed. If they don't, the discussion is probably shifted from an issue which is specific to QM to the broader issue of irreversibility first.This is indeed one of the errors in Ballentine – he claims that Copenhagen must treat this as a collapse even when no definite outcome is obtained.
Do you mean "reversible" instead of "Irreversible"?Sorry, I lost control of my fingers. Now corrected, thanks.
Irreversibility is key – for example the splitting in step 2 is irreversible until either beam is decohered by being interrupted for instance.Do you mean "reversible" instead of "Irreversible"?
Thanks for the conversation, it has been very enlightening.I don't know what you are talking about, so it hasn't been very enlightening for me. I ask you a direct question to understand what you're saying, and you don't answer it. I just don't understand.
But you still want to call it a measurement?Thanks for the conversation, it has been very enlightening.
[..]
The collapse postulate is only needed for sequential measurements (because only there, we can check the state after a measurement). Textbook examples of sequential measurements mostly involve multiple Stern Gerlach devices or multiple polarizers. After each device, the new state vector is calculated by a projection. But usually, the only actual measurement is provided by a single screen at the end. So in these cases, collapse arguably is just a convenient way to simplify calculations.
[..]
It turns out that there aren't many experiments of type 1 if by "outcome" we mean things which are actually reported by the experimenters. If people agree about the classification of typical experiments, the focus of the discussion can be narrowed. If they don't, the discussion is probably shifted from an issue which is specific to QM to the broader issue of irreversibility first.In all cases I know we used a macroscopic variable which becomes correlated to the quantum state to make a calculation.
In cavity QED expriments with Rydberg atoms a detector can find the excited state |e> by applying a potential just strong enough to cause ionization and send the state to |g>. This is a projection operator but (again) in order to make a decision we use something that is correlated with the state (ionization) to get a measurement. Is there collapse in this case ?
Irreversibility is key – for example the splitting in step 2 is irreversible until either beam is decohered by being interrupted for instance.
I thought the preparation consist exactly to keep the right beam (by filtering it with a Stern Gerlach in X).
How do you preparation electron in a +X state ?I think I've said the same answer many times now. I don't have any idea why you want more.
If you send the spin-up electrons to the left, and sent the spin-down electrons to the right, then you know that any electrons you find on the left will be spin-up. That doesn't mean that you have detected any electrons at all, so it doesn't mean that you have measured anything at all.
When you detect an electron on the left, at that moment you will (indirectly) be measuring the spin state. But not until then. The measurement does not happen when the electrons are sent one way or the other, but later.
You keep wanting to say that the splitting into two beams is a measurement, even though it has none of the properties of a measurement. It doesn't collapse the wave function. It doesn't result in my knowing the spin. It doesn't produce a probabilistic outcome according to the Born rule. Nothing about measurements apply. But you still want to call it a measurement?
If you arrange for spin-up electrons to be sent to the left and spin-down electrons to be sent to the right, you still don't know whether the electron is spin-up or spin-down. Not until you detect the electron on the right, or on the left. Until you do that, you don't have a measurement.I thought the preparation consist exactly to keep the right beam (by filtering it with a Stern Gerlach in X).
How do you preparation electron in a +X state ?
Is this a joke ? Preparing +X means you know they are +X, if not, what would be the point of "preparation"If you arrange for spin-up electrons to be sent to the left and spin-down electrons to be sent to the right, you still don't know whether the electron is spin-up or spin-down. Not until you detect the electron on the right, or on the left. Until you do that, you don't have a measurement.
I really don't understand what you're saying.
What is the point of such a preparation? It's not an end in itself, it's a PREPARATION for some further experiment. You send the spin-up electrons one direction toward an experimental setup. You send the spin-down electrons another direction toward a different setup. In the analysis of the first experiment, you can assume that any electrons that you find will be spin-up, because only the spin-up electrons are sent there. But until you find the electron, you haven't measured the spin.
Why on Earth would it be a measurement? Isn't it part of the definition of "measurement" that afterward, you know the value of whatever was being measured?Is this a joke ? Preparing +X means you know they are +X, if not, what would be the point of "preparation"
Yes, you have. Sending spin-up electrons to the left and sending spin-down electrons to the right is a preparation, but not a measurement.I see, i see :rolleyes:
But my point is that a measurement did occur, and it can be measured at 4 (but in Z). No willing to do that do not destroy or retroactively nullify the apparatus (it is there, whatever you take note or not).I really don't understand why you want to call it a measurement when spin-up electrons are sent to the left and spin-down electrons are sent to the right. But I can accommodate whatever terminology you want. What point are you wanting to make about measurements?
The significance of measurement in QM (or at least, the usual, informal interpretation) is that:
These three points don't apply to a non-destructive preparation procedure. So lumping all preparation procedures in with measurements seems to be mixing up things that are fundamentally unalike.
I cannot fathom why on earth preparing +X is not a measurement to +X.Why on Earth would it be a measurement? Isn't it part of the definition of "measurement" that afterward, you know the value of whatever was being measured?
Nor have I obtained any example of a preparation that is not a measure.Yes, you have. Sending spin-up electrons to the left and sending spin-down electrons to the right is a preparation, but not a measurement.
If people agree about the classification of typical experiments, the focus of the discussion can be narrowed.That would be great indeed. But I am more inclined to think people will prefer to inject meaning instead. The setup #269 seems pretty clear. There are 3 identical Stern-Gerlach "apparatus". Yet the step1 is call a "preparer" the step2 a "interaction/useless" the step3 a "measurer".
I cannot fathom why on earth preparing +X is not a measurement to +X. Nor have I obtained any example of a preparation that is not a measure. But OK if the terminology requires that identical apparatus working identically (and perfectly exchangeable in the setup) are designated by different word if a start and at end, then OK, I'll do it.
Likewise the step2 is an identical process. But because the angle is different, somewhat some experimenter can decide that "it does not collapse the wave function". My understanding was that it did not bother him to take note and modify its expectation with the projection (because, say, it is a case where it wouldn't change expectation in X anyway).
But my point is that a measurement did occur, and it can be measured at 4 (but in Z). No willing to do that do not destroy or retroactively nullify the apparatus (it is there, whatever you take note or not).
I am not even sure that @stevendaryl is not thinking that the human-mind/or consciousness/or maybe a piece of paper, only constitute a measurement (doing physical projection to eigenvalue).
If they don't, the discussion is probably shifted from an issue which is specific to QM to the broader issue of irreversibility first.Maybe it is what I don't get to get out of this conundrum. Do step 3 actually totally reverse the step2, in the sense that not even data collected after step2 modify some expectation at step4 even in Z?
Or do you mean special measurement that destroy the state (photon absorption, anti-electron anhihilation) making it irreversible ?
Not by the definition of "measurement" that I'm using. By what definition is it a measurement? It doesn't collapse the wavefunction.And yet it is the most prominent example where projected wavefunctions are actually used in practise.
I don't disagree with your terminology but this is something about the measurement problem which I find peculiar. The collapse postulate is only needed for sequential measurements (because only there, we can check the state after a measurement). Textbook examples of sequential measurements mostly involve multiple Stern Gerlach devices or multiple polarizers. After each device, the new state vector is calculated by a projection. But usually, the only actual measurement is provided by a single screen at the end. So in these cases, collapse arguably is just a convenient way to simplify calculations.
I think that a lot of discussions about the measurement problem would gain considerable clarity if people tried to focus on distinguishing these two classes of experiments:
1) Real sequential measurements where outcomes are obtained at each device.
2) Sequential preparations, where state vectors are projected for convenience because nobody cares about what happens inside the devices.
It turns out that there aren't many experiments of type 1 if by "outcome" we mean things which are actually reported by the experimenters. If people agree about the classification of typical experiments, the focus of the discussion can be narrowed. If they don't, the discussion is probably shifted from an issue which is specific to QM to the broader issue of irreversibility first.
But a measurement has been made nonetheless. There is no way for someone not knowing/measuring (that is taking note of which electron when by which path) to assert/prove/measure that a measurement had not been made. Sure he cannot detect it, but it doesn't mean nobody can.No, it's not a measurement until someone detects it. The definition of "measurement" is that you have measured some quantity when you have made a persistent record of its value (or something that maps to its value). If that hasn't happened, then a measurement hasn't been made.
In deflecting an electron to the left or to the right, what you've done is set up a correlation between two different properties of the electron: its position (left or right) and its spin (up or down). Every interaction sets up a correlation of that type, but not every interaction is a measurement.
That thing is a measurement, not an interaction, because the projection is done by a classical apparatus which is the only thing able to set a particle into some eigenvalue. If the apparatus wasn't classical in the first place, you simply could not even set it in some orientation in the first place.
However that process take place, the only formulation of it is the Born rule, which may or may not be deduced in some way (but isn't currently).No, not all interactions with a macroscopic/classical apparatus result in a measurement. Only irreversible interactions—interactions that leave the apparatus in a persistent state that records the value being measured.
Even with your second example in post #269, step 2 is a also a measurement (in another bases, but nonetheless).Not by the definition of "measurement" that I'm using. By what definition is it a measurement? It doesn't collapse the wavefunction.
[..]
But a measurement has been made nonetheless. There is no way for someone not knowing/measuring (that is taking note of which electron when by which path) to assert/prove/measure that a measurement had not been made. Sure he cannot detect it, but it doesn't mean nobody can.
I someone else (aware of the result) come an got a much more accurate result (let's say 100% correct), it does not mean then QM is wrong. It means something did happen to each individual electron, no mater ones ignorance of it.
[..]The experiment that @stevendaryl described has been analysed in terms of projection operators here in post#5
https://www.physicsforums.com/threads/spin-state-recombination.927182/
The separation into streams did not constitute a measurement.OK do you call it an interaction ? but one that nobody "observe" ? The problem is that by you own setup, you are going to work on one of the stream only…
To see that the separation by itself is not a measurementI am trying hard to follow your argumentation. Here I am still wondering how any preparation is different with "knowing/measuring/projecting" some state.
, I could redirect both streams back together into a single stream, and then no measurement of spin would ever be performed.But a measurement has been made nonetheless. There is no way for someone not knowing/measuring (that is taking note of which electron when by which path) to assert/prove/measure that a measurement had not been made. Sure he cannot detect it, but it doesn't mean nobody can.
I someone else (aware of the result) come an got a much more accurate result (let's say 100% correct), it does not mean then QM is wrong. It means something did happen to each individual electron, no mater ones ignorance of it.
That thing is a measurement, not an interaction, because the projection is done by a classical apparatus which is the only thing able to set a particle into some eigenvalue. If the apparatus wasn't classical in the first place, you simply could not even set it in some orientation in the first place.
However that process take place, the only formulation of it is the Born rule, which may or may not be deduced in some way (but isn't currently).
So a preparation does not necessarily count as a measurement (although it can be a preliminary step in a measurement).Even with your second example in post #269, step 2 is a also a measurement (in another bases, but nonetheless). Why should it change step 4 ? But it does change the wavefunction (of this basis, and maybe in other, but then QM would predict it anyway).
Can you try to give another example where no classical apparatus is used to "prepare" a state ? I kind of think it is impossible given the very definition of quanta.
Yes, that's what I meant.Whatever we call steps 2 and 3 we can ignore them and look at steps 1 and 4. The only actual projection ( which some people may call a collapse) happens in step 1. After that there is no further projection so no information is lost or gained. We prepared x+ and we've still got it.
It occurs that due to the idempotency of operators ##hat{S}_xhat{S}_x|phirangle=hat{S}_x|phirangle##
Do you mean step 2 is a preparation but not a measurement ?Yes, that's what I meant.
Yes, I agree that it's not a measurement, but it is a preparation.Do you mean step 2 is a preparation but not a measurement ?
In step 1 there is a measurement. You started with a thermal beam and separated out x+. That was a projection and the previous state is lost. Step 2 is not a measurementYes, I agree that it's not a measurement, but it is a preparation.
I don't know what you mean. I would have guessed that "information about the previous state" would cover "the electrons have spin-up in the x-direction". That information has not been lost.In step 1 there is a measurement. You started with a thermal beam and separated out x+. That was a projection and the previous state is lost. Step 2 is not a measurement, nor is the final step a measurement because there was no projection, so nothing changed.
The whole experiment amounts to prepaing the beam in +x state then measuiring in x and finding +x. The only measurement in this experiment is was the one where you prepared the initial beam. I stand by All projective 'measurements' are preparations. Nothing has been measured and all information about the previous state is lost.I don't know what you mean. I would have guessed that "information about the previous state" would cover "the electrons have spin-up in the x-direction". That information has not been lost.
Perhaps all measurements are preparations, but the issue is whether all preparations are measurements.
I would call it "not a measurement" rather than "a reversible measurement".The whole experiment amounts to prepaing the beam in +x state then measuiring in x and finding +x. The only measurement in this experiment is was the one where you prepared the initial beam. I stand by All projective 'measurements' are preparations. Nothing has been measured and all information about the previous state is lost.
If a coherent state is prepared before the splitting/recombination and coherence is maintained then there will be state reconstruction. In those circumstances the splitting is a 'reversible measurement' because it tells us nothing about the previous state.I would call it "not a measurement" rather than "a reversible measurement".
I'm not sure what you mean. Suppose I do the following:
If step 2 were a measurement, then step 4 would yield spin-up or spin-down, with equal probability. If step 2 is not a measurement, then step 4 will only produce the result spin-up.If a coherent state is prepared before the splitting/recombination and coherence is maintained then there will be state reconstruction. In those circumstances the splitting is a 'reversible measurement' because it tells us nothing about the previous state – i.e. like having no 'which-path' information.
If you recombine the beams you do not get a thermal state, but you may have had one before the projections (depending on your preparation !) .
All projective 'measurements' are preparations. Nothing has been measured and all information about the previous state is lost.I'm not sure what you mean. Suppose I do the following:
If step 2 were a measurement, then step 4 would yield spin-up or spin-down, with equal probability. If step 2 is not a measurement, then step 4 will only produce the result spin-up.
Well, it depends on exactly what is done with the two streams. If I perform a measurement of the electrons that go through one of the streams and get some result, then I'm indirectly measuring which stream the electron went in (since only one of the streams is measured), and so that indirectly counts as a spin measurement. But the measurement occurs at the moment I measure something about the electron. The separation into streams did not constitute a measurement.
To see that the separation by itself is not a measurement, I could redirect both streams back together into a single stream, and then no measurement of spin would ever be performed.
So a preparation does not necessarily count as a measurement (although it can be a preliminary step in a measurement).If you recombine the beams you do not get a thermal state, but you may have had one before the projections (depending on your preparation !) .
All projective 'measurements' are preparations. Nothing has been measured and all information about the previous state is lost.
This is elementary stuff which most people choose to ignore.
Do you mean someone else have chosen which stream (left or right, or apparatus angle) and that you just don't know on which one you are working on ?Well, it depends on exactly what is done with the two streams. If I perform a measurement of the electrons that go through one of the streams and get some result, then I'm indirectly measuring which stream the electron went in (since only one of the streams is measured), and so that indirectly counts as a spin measurement. But the measurement occurs at the moment I measure something about the electron. The separation into streams did not constitute a measurement.
To see that the separation by itself is not a measurement, I could redirect both streams back together into a single stream, and then no measurement of spin would ever be performed.
So a preparation does not necessarily count as a measurement (although it can be a preliminary step in a measurement).
But I don't know which is the case, so I haven't actually measured the spin.Do you mean someone else have chosen which stream (left or right, or apparatus angle) and that you just don't know on which one you are working on ?
But didn't you just describe a measurement ?No. After sending an electron through a Stern-Gerlach device, I know that:
But I don't know which is the case, so I haven't actually measured the spin.
The ensemble doesn't exist only in my head, but it's realized with accelerators. That's why they aim at ever higher luminosities to "collect statistics as quickly as possible".I agree this process is important, and this is where the probabilistic abstractions are attached to physics.
This requires two things to actually make sense:
– The timescale of the processes we observer must be "small" so that we can prepare, decode data, and repeat enough statistis fast on a relative timescale
– The experimental control requires the system of study to be small relative to the lab so that we can control its boundary.
This is certainly true for HEP where we can observe scattering on the boundary, but fails for cosmology (here a new paradigm for inference is needed! which one?)
If we can do this we have good foundation for the probabilistic predictions, as well as extracting timeless patterns that stay constant over trials (symmetries). This how the standard model of particle physics is designed. But if these premises fail, not only do "probability" loose its original meaning, we also loose the ability in inferring symmetries, either because its too much data and limiting processing power or because of insufficient data to with any reasonable accuracy make statistical statements.
/Fredrik
I'm sort of in agreement with you that in QM, measurement and preparation seem very similar, but there are some circumstances where it is possible to get particles in a particular state without measuring them. For example, if you send electrons through a Stern-Gerlach device, the ones that are spin-up will go in one direction and the ones that are spin-down will go in another direction. Then if you perform an experiment on just one of the two streams, you can be assured that the electrons are in a specific spin state even though you didn't measure the spin.But didn't you just describe a measurement ? How can you say you didn't measure their spin ? Or are you saying you are no more interested by spin, but want to measure some other property (maybe loosely coupled with spin) ?
The circularity of that claim is obvious.
But maybe that "preparation" is yet another kind of physical process I am not aware off, and described in your version of QM that is neither interaction nor measurement.
OK then how do you prepare an entangled pair of electron or photon that have probability 1 to be polarized at such angle along such axes…I'm sort of in agreement with you that in QM, measurement and preparation seem very similar, but there are some circumstances where it is possible to get particles in a particular state without measuring them. For example, if you send electrons through a Stern-Gerlach device, the ones that are spin-up will go in one direction and the ones that are spin-down will go in another direction. Then if you perform an experiment on just one of the two streams, you can be assured that the electrons are in a specific spin state even though you didn't measure the spin.
How do you come to these conclusions? We can prepare single electrons, even single photons, very well nowadaysThe circularity of that claim is obvious.
But maybe that "preparation" is yet another kind of physical process I am not aware off, and described in your version of QM that is neither interaction nor measurement.
And an ensemble can (among other ways to prepare them) consist of many repetitions of such single-quanta states. If this was not the case, we couldn't have ever checked that QT is describing things right in terms of the predicted probabilities.OK then how do you prepare an entangled pair of electron or photon that have probability 1 to be polarized at such angle along such axes…
Circular reasoning. You cannot prepare an electron in a pretty well defined state without measuring it first
The ensemble preparation is an laboratory artifact. You cannot, in the real world (or even based on QM phenomenology), propose an experiment to "ask/probe" an electron to find its companions in "an ensemble". This ensemble is not real. And the prediction are only more and more accurate with respect to the ensemble size.
CM does not need an ad-hoc rule to connect the evolution formalism to the lab event. It does not need ensemble either. And CM don't treat measurement and interaction differently.How do you come to these conclusions? We can prepare single electrons, even single photons, very well nowadays. And an ensemble can (among other ways to prepare them) consist of many repetitions of such single-quanta states. If this was not the case, we couldn't have ever checked that QT is describing things right in terms of the predicted probabilities.
Quantum mechanics doesn't treat measurement and interaction differently (I won't again repeat the obvious arguments I've stated several times in this thread again).
Which version of Landau and Lifshitz are you reading? Perhaps the German translation is different from the English one. There is a possibility the English version is biased towarda my views, since John Bell apparently had a role in it.The english translation of that section is faithfull to the original russian text.
Not again this wrong statement. You cannot admit at the same time that the classical behavior is derivable from QT and then claim that there is a cut. That's a contradictio in adjecto!
Which version of Landau and Lifshitz are you reading? Perhaps the German translation is different from the English one. There is a possibility the English version is biased towarda my views, since John Bell apparently had a role in it.
But each individual system produces definite result (or appears to produce definite result). And either you have something to say about that, and then you participate in discussions about "collapse" and alternatives, or you keep agnostic position and do not say anything like "collapse is superfluous"/"collapse is required".That's my feeling. A true minimalist interpretation, in the sense of making minimal assumptions, is not a denial of the collapse interpretation or the Many-Worlds Interpretation or the Bohmian interpretation, but should open to any of those possibilities. It should be silent on the question of what happens during a measurement.
But also #2 doesn't distinguish measurements from other interactions.I think it definitely does.You measure a property and you get an eigenvalue ##lambda## of the operator corresponding to the observable being measured. That means that the measuring device is in a specific state—the state of "having measured ##lambda##". But treating the device as a physical system and treating the measurement as a physical interaction leads to a different state–where the measuring device is not in a specific state, but is entangled with the system being measured. Those are two different situations in QM, and are described by different quantum-mechanical states and those states have theoretically different statistical properties, leading to different predictions for future states. The two possible quantum-mechanical states are different, with different (in theory) observable consequences. They can't both be correct.
Now, I stuck the phrase "in theory" in there, because I think that the difference between an entangled macroscopic system and one that has a specific macroscopic properties may be undetectable in practice, but they are different states in QM. So you get different answers depending on whether you're treating the macroscopic system as a physical system following Schrodinger's equation or as a measuring device obeying the Born rule.
Because "ensemble" can not be defined in terms of QM, minimal QM is not a selfcontained model. It requires CM as a starting platform.
Of course we can handle electrons pretty well in accelerators and thus prepare, e.g., electrons with a pretty well determined energy and momentum to make all kinds of scattering experiments with them for decades. The ensemble doesn't exist only in my head, but it's realized with accelerators. That's why they aim at ever higher luminosities to "collect statistics as quickly as possible".Circular reasoning. You cannot prepare an electron in a pretty well defined state without measuring it first
The ensemble preparation is an laboratory artifact. You cannot, in the real world (or even based on QM phenomenology), propose an experiment to "ask/probe" an electron to find its companions in "an ensemble". This ensemble is not real. And the prediction are only more and more accurate with respect to the ensemble size.
Indeed, I don't see any epistemological difference with CM. There's an ontological difference though.CM does not need an ad-hoc rule to connect the evolution formalism to the lab event. It does not need ensemble either. And CM don't treat measurement and interaction differently.
So far so good…
No you cannot. There is no phenomenon as "state preparation" in nature. You (the observer) only do it in a lab, because you need that to match the ensemble with the esoteric Hilbert space, by using an ad-hoc Born rule. This ensemble exist only in your head. CM does not need any ad-hoc projection for observation, nor does nature (as per QM Schrodinger equation).
You say much more then that. You made make hidden assumptions (often circular) and quite astonishing claim (like QM completeness). You say noticeably that there is no epistemological difference with CM.
But in CM you don't need to make measurement to "create" the value of any observable (out of probability or whatnot).Of course we can handle electrons pretty well in accelerators and thus prepare, e.g., electrons with a pretty well determined energy and momentum to make all kinds of scattering experiments with them for decades. The ensemble doesn't exist only in my head, but it's realized with accelerators. That's why they aim at ever higher luminosities to "collect statistics as quickly as possible".
I don't claim the completeness of any physical theory we have so far. QM is incomplete because there is no satisfactory quantum description of the gravitational field yet. Indeed, I don't see any epistemological difference with CM. There's an ontological difference though.
I don't object to the choice. But after the observer has chosen the device (by whatever rule), there remains the pure quantum problem to show that the device actually produces on each reading the numbers that qualify as a measurement, in the sense that they satisfy Born's rule.
This is the measurement problem! It has nothing to do with the observer but is a purely quantum mechanical problem.Ok, that's true. Of course, it's only possible for very simple cases in a strict way (like the famous analysis of tracks of charged particles in vapour chambers by Mott or the measurement of spin components in the Stern Geralach experiment).
Of course classical mechanics is a limit of quantum mechanics. One can see this in the saddle point approximation to the path integral.
However, what you are not understanding and which Landau and Lifshitz state clearly, is that quantum mechanics cannot be formulated without "classical concepts" also in its assumptions. It is not possible to derive classical physics from "purely quantum" assumptions.
One can use different language to state this assumption, but they are all essentially equivalent – measurement has a different status than the interactions described in the Hamiltonian.There is a difference between this and the statement that QT doesn't make any predictions without a cut.
That's not what I claim. Of course single electrons exist,So far so good…
and we can prepare them in many quantum states quite accurately.No you cannot. There is no phenomenon as "state preparation" in nature. You (the observer) only do it in a lab, because you need that to match the ensemble with the esoteric Hilbert space, by using an ad-hoc Born rule. This ensemble exist only in your head. CM does not need any ad-hoc projection for observation, nor does nature (as per QM Schrodinger equation).
All I say is that within the ensemble interpretation quantum theory only describes the probabilities, and these probabilities can be empirically measured only on ensembles of equally prepared systems.You say much more then that. You made make hidden assumptions (often circular) and quite astonishing claim (like QM completeness). You say noticeably that there is no epistemological difference with CM.
But in CM you don't need to make measurement to "create" the value of any observable (out of probability or whatnot).
Of course single electrons exist, and we can prepare them in many quantum states quite accurately. All I say is that within the ensemble interpretation quantum theory only describes the probabilities, and these probabilities can be empirically measured only on ensembles of equally prepared systems.But each individual system produces definite result (or appears to produce definite result). And either you have something to say about that, and then you participate in discussions about "collapse" and alternatives, or you keep agnostic position and do not say anything like "collapse is superfluous"/"collapse is required".
The observer constructs the measurement device to measure the observable he likes to measure. If you now start to discuss the ability of this free choice of the observerI don't object to the choice. But after the observer has chosen the device (by whatever rule), there remains the pure quantum problem to show that the device actually produces on each reading the numbers that qualify as a measurement, in the sense that they satisfy Born's rule.
This is the measurement problem! It has nothing to do with the observer but is a purely quantum mechanical problem.
Whatever your version of QM is, individual particles have no speed or momentum or whatnot. And yet, nature only "hidden ontology", can only be approached and probed, (and this is even more true in QM), with unique individual event in the lab (and that is a that individual level, that all conservation law work).
Your view that only "ensemble of identically prepared thing" exist, and this is a complete physical phenomenology of nature, is highly incoherent with the fact that quanta do exist, and only them, and their individual interaction, are ever observed in a laboratory, or elsewhere.Quantum objects have all the observables which can be defined on them. For massive particles these are particularly their energy, momentum, angular momentum, and their position. The point in QT is that not all the observables can take determined values at once (some cannot take determined values at all, which is the case for all observables having a continuous spectrum only like energy, momentum, and position).
Your view that only "ensemble of identically prepared thing" exist, and this is a complete physical phenomenology of nature, is highly incoherent with the fact that quanta do exist, and only them, and their individual interaction, are ever observed in a laboratory, or elsewhere.That's not what I claim. Of course single electrons exist, and we can prepare them in many quantum states quite accurately. All I say is that within the ensemble interpretation quantum theory only describes the probabilities, and these probabilities can be empirically measured only on ensembles of equally prepared systems.
There you have made the subjective classical-quantum cut.Not again this wrong statement. You cannot admit at the same time that the classical behavior is derivable from QT and then claim that there is a cut. That's a contradictio in adjecto!
This is not true. Given the measurement device as a quantum object, the observer has no choice which observables to measure – it can only measure the observables that can be read off from (or calculated from reading of) the measurement device.Thus there must be a way to determine the preferred basis directly the quantum device, without choices by a further observer.Well, this is semantics. The observer constructs the measurement device to measure the observable he likes to measure. If you now start to discuss the ability of this free choice of the observer to measure the observable he likes you get into funny discussions about "consciousness" and "free will" and all kinds of esotrical philosophy around it. I was shocked to see that this is even publishable in serious scientific publishing companies like Springer. Well, there's no law forbidding to publish nonsense in serious science publishing companies, which like to make money with anything they can find. A funny anecdote is that when I went into a big bookshop in Munich to look for quantum theory textbooks I couldn't find it in the very small science corner of this bookshop. Asking a nice employee of the bookshop, whether they don't have also quantum theory textbooks, she answered "Oh, you are complete wrong here. That's in the shelf with book on esoterics, which in fact was much larger than the little shelf they sold science textbooks." Well, they had no quantum theory textbooks but all kinds of "quantum nonsense" (as Bricmont calls it in his book "Quantum sense and quantum nonsense", of which I currently read the German translation; it's pretty entertaining to read although sometimes rather imprecise in popularizing QT; I also don't buy the "solution" of the interpretational problems in terms of de Brogli-Bohm theory since so far nobody could make sense of it in context relativistic QFT).
But also #2 doesn't distinguish measurements from other interactions.And yet, you have admitted that you don't use the Born rule inside the Schrodinger equation. That clearly means that there is no "preferred base" picking in QM interaction. None.
The "preferred basis" is just the observer's choice which observable s/he likes to measure.Really ? Just ? So it should be easy for you to provide a derivation (without any cut) of the Born rule. This must be an very interesting proof, because it will DEFINE what a measure is. It would be the first theory to actually create its own version of what an observation is, not related to an event in a lab, but to some VAGUELY defined ensemble of events.
Thus, you'll have to make another theory leap, to explain how measurement that only are coherent for ensemble (and basically, that this is probabilistic or not, is not even relevant here) will still actually physically modify (set in an eigenstate) only individual system, and not ensemble.
That's the only meaning in which a basis has to be chosen to evaluate probabilities: To get the probabilities for the outcome of measurements you need the eigenbasis of the self-adjoint operator representing this observable. It's not more special then the observer's choice in classical physics too. If I measure the position of a particle I need another device than when I measure its momentum.You've just said you cannot measure the position of a particle (only a probability in an ensemble). In fact "speed" it does not exist in the interaction picture (its a complex mixture of imaginary speed)
Whatever your version of QM is, individual particles have no speed or momentum or whatnot. And yet, nature only "hidden ontology", can only be approached and probed, (and this is even more true in QM), with unique individual event in the lab (and that is a that individual level, that all conservation law work).
Your view that only "ensemble of identically prepared thing" exist, and this is a complete physical phenomenology of nature, is highly incoherent with the fact that quanta do exist, and only them, and their individual interaction, are ever observed in a laboratory, or elsewhere.
So indeed I agree with the statement that to understand measurements one needs classical concepts, but that doesn't mean that a measurement is anything different from any other interaction of the measured system with a macroscopic object that's not used as a mesurement device.There you have made the subjective classical-quantum cut.
The "preferred basis" is just the observer's choice which observable s/he likes to measure.This is not true. Given the measurement device as a quantum object, the observer has no choice which observables to measure – it can only measure the observables that can be read off from (or calculated from reading of) the measurement device.Thus there must be a way to determine the preferred basis directly the quantum device, without choices by a further observer.
Of course classical mechanics is a limit of quantum mechanics. One can see this in the saddle point approximation to the path integral.
However, what you are not understanding and which Landau and Lifshitz state clearly, is that quantum mechanics cannot be formulated without "classical concepts" also in its assumptions. It is not possible to derive classical physics from "purely quantum" assumptions.
One can use different language to state this assumption, but they are all essentially equivalent – measurement has a different status than the interactions described in the Hamiltonian.QT is about what's observable in nature, and to observe we need macroscopic objects, which are describable to sufficient accuracy with classical physics. That's all what LL state in their marvelous textbook on QM, and since the validity of the classical limit can be understood from QM there's no contradiction in that, i.e., there are no special laws for macroscopic objects (i.e., no quantum-classical cut) and no specialty of measurement devices in terms of the physical description in theory from any other kind of matter, which is self-evident, because obviously measurement devices must be made of the matter around us. They are only special in the sense that physicists construct them to measure the one or the other observable, but they are still consisting of the matter around us. Of what else shoud they be made?
So indeed I agree with the statement that to understand measurements one needs classical concepts, but that doesn't mean that a measurement is anything different from any other interaction of the measured system with a macroscopic object that's not used as a mesurement device.
As was said in a previous comment, a measurement device plays two different roles in the minimal interpretation:
In role #1, there is no distinction between measurement devices and any other physical system. In role #2, there is a big distinction. Probabilities only appear in quantum mechanics if you have measurements, not for any other interactions.
I don't understand why, when the issue is #2, people keep bringing up that #1 doesn't distinguish measurements from other interactions. That's true, but it's only half the story.But also #2 doesn't distinguish measurements from other interactions. The "preferred basis" is just the observer's choice which observable s/he likes to measure. That's the only meaning in which a basis has to be chosen to evaluate probabilities: To get the probabilities for the outcome of measurements you need the eigenbasis of the self-adjoint operator representing this observable. It's not more special then the observer's choice in classical physics too. If I measure the position of a particle I need another device than when I measure its momentum.
As was said in a previous comment, a measurement device plays two different roles in the minimal interpretation:
In role #1, there is no distinction between measurement devices and any other physical system. In role #2, there is a big distinction. Probabilities only appear in quantum mechanics if you have measurements, not for any other interactions.
I don't understand why, when the issue is #2, people keep bringing up that #1 doesn't distinguish measurements from other interactions.I see a reason to bring #1 into discussing #2:
By observer equivalence i expect that the non probabilistical hamiltonian of a complex system is "explainable" terms of the transformed views from the inside views which would contain probabilistic components due to internal measurements. So the deterministic evolution of the state vector should have a probabiliatic explanation that removes the classical baggage. Then we would have a complete duality between interaction and measurement.
But it is not yet known of course. But this to me suggests revision on qm.
/Fredrik
This must be the very first chapter, where exactly the contrary is stated to the claim you make. They say that classical behavior of macroscopic devices is understood as an approximative limit of QT. They also bring the very example of the cloud-chamber traces for an electron that I also stated in this thread. In other words, what I read in LL is completely consistent with what I stated. They are very careful and only use a very weak version of the collapse postulate, more careful than most other books following the Copenhagen doctrine. I guess that's because Landau is following more Bohr's than Heisenberg's opinion.Of course classical mechanics is a limit of quantum mechanics. One can see this in the saddle point approximation to the path integral.
However, what you are not understanding and which Landau and Lifshitz state clearly, is that quantum mechanics cannot be formulated without "classical concepts" also in its assumptions. It is not possible to derive classical physics from "purely quantum" assumptions.
One can use different language to state this assumption, but they are all essentially equivalent – measurement has a different status than the interactions described in the Hamiltonian.
As was said in a previous comment, a measurement device plays two different roles in the minimal interpretation:
In role #1, there is no distinction between measurement devices and any other physical system. In role #2, there is a big distinction. Probabilities only appear in quantum mechanics if you have measurements, not for any other interactions.
I don't understand why, when the issue is #2, people keep bringing up that #1 doesn't distinguish measurements from other interactions. That's true, but it's only half the story.
This must be the very first chapter, where exactly the contrary is stated to the claim you make. They say that classical behavior of macroscopic devices is understood as an approximative limit of QT. They also bring the very example of the cloud-chamber traces for an electron that I also stated in this thread. In other words, what I read in LL is completely consistent with what I stated. They are very careful and only use a very weak version of the collapse postulate, more careful than most other books following the Copenhagen doctrine. I guess that's because Landau is following more Bohr's than Heisenberg's opinion.
Where in Landau Lifshitz is the claim you are not willing to state explicitly in this thread? It's strange that you always make these claims about measurement devices being outside of the standard laws of QT but never explaining in which sense you mean it and then point to textbooks and not giving the clear statement nor where to find it in these books. I've never heard such a statement nor read it in any serious textbook about QT, and LL for sure is one very serious textbook.page 3 in the 1991 reprint of the 1958 English translation
Where in Landau Lifshitz is the claim you are not willing to state explicitly in this thread? It's strange that you always make these claims about measurement devices being outside of the standard laws of QT but never explaining in which sense you mean it and then point to textbooks and not giving the clear statement nor where to find it in these books. I've never heard such a statement nor read it in any serious textbook about QT, and LL for sure is one very serious textbook.
Again you just claim this, but that's not what standard QT claims. According to standard QT the functioning of measurement apparati are completely consistent with the laws of physics valid for all matter observed yet (usually one assumes also that that's how all matter in the universe behaves, but that's of course an extrapolation which never can be tested empircially). My point is that there is no measurement problem at all. What's called a measurement problem is usually a metaphysical quest for an ontological interpretation of the notion of quantum states, which is however not subject of science but only of philosophy.You are completely wrong – see Landau and Lifshitz.
This is the salient point of your misunderstanding. My belief, or your belief, are inconsequential here. We are not discussing what you call philosophy (some taste based word salad). We are asserting the coherence of some statement, only on the merit of what is written, nothing more.
QM that you claim to appreciate for its coherence and simplicity (which is fine) does make the distinction between interaction and measurement. Why does it have to do that distinction, while the universe is obviously only make of the same "stuff" ? That's the measurement problem.Again you just claim this, but that's not what standard QT claims. According to standard QT the functioning of measurement apparati are completely consistent with the laws of physics valid for all matter observed yet (usually one assumes also that that's how all matter in the universe behaves, but that's of course an extrapolation which never can be tested empircially). My point is that there is no measurement problem at all. What's called a measurement problem is usually a metaphysical quest for an ontological interpretation of the notion of quantum states, which is however not subject of science but only of philosophy.
Then please precisely explain to me what you mean when you say "measurements are special" (within quantum mechanics). I have no clue what that should mean if you admit that measurement devices are usual "stuff" and thus behaves according to the generally valid physical laws. Indeed, measuring a force with a balance invokes the very laws the concept of force is based on within the theory (necessarily Newtonian mechanics, because the force concept only makes sense within Newtonian mechanics). I'm not doubting that, but you do, if I understand the statement "measurements are special". I'm arguing against this claim of the Copenhagen-like interpretation all the time.I think i repeat myself, but again I want to note that there is a circularity or chicken/egg situtuation here. And this is a key observation this is why i emphasise it.
The circulatory is also not of the circular reasoning kind, its of the evolutionary kind.
I agree with vanhees that there exists and equivalence between interactions and observations. But its also clear that current QM formalism, does not manifest this equivalence, except for the special case where the class of observers are only classical. Because the statistics are "objective" only (or at least at best) in the classical realm. But even there is non-trivial if we include the classical observers obeying also GR. If we stick to SR and particle physics in a lab if we ignore the problem of unification of forces other than GR which is still lacking.
Any well defined measurement sort of DEPENDS on the theory. It involves (depending on how you frame this) preparation of measurement devices, signal processing of the lets say "raw data" coming off an actual detector. All these things are the "baggage" that are essentially put in by hand as constraints in QM. None of it is "explained". In here lies the reference to the classical measurement device, it implicitly includes all these things. Without this "background" one can not define any definite expectations (probabilities) due to undefined references in the conditional probability.
But of equal importance, the origin of the current state of laws as we know them, are de facto a result of hundreds of years of human scientific work. So all the stuff we "put in by hand" are not as ad hoc as it seems. But if we take Vanhees equivalence of measurement and interactions seriously (and i do as well) then this must hold even for complex systems, such as physicists. So the observers abduction of laws in its own environment, must be a physical process and most probably a survival trait of any system. In here lies also the key to understand how symmetries are emergent as a result of evolving interacting systems. Symmetries are most liekely NOT god given constraints.
As i see it can be no other way. But our understanding of this process, ie the PHYSICAL process by which one system infers and encodes predictive rules about another system, in which this inference process, is the key to understand the interactions as well, is undeveloped.
/Fredrik
Then please precisely explain to me what you mean when you say "measurements are special" (within quantum mechanics)I mean precisely what is said in the minimal interpretation of QM. See post #127 or #142, and for that matter nearly ever other post in this thread.
I have no clue what that should mean if you admit that measurement devices are usual "stuff" and thus behaves according to the generally valid physical laws.This is the salient point of your misunderstanding. My belief, or your belief, are inconsequential here. We are not discussing what you call philosophy (some taste based word salad). We are asserting the coherence of some statement, only on the merit of what is written, nothing more.
QM that you claim to appreciate for its coherence and simplicity (which is fine) does make the distinction between interaction and measurement. Why does it have to do that distinction, while the universe is obviously only make of the same "stuff" ? That's the measurement problem.
I'm arguing against this claim of the Copenhagen-like interpretation all the time.OK, but then you argumentation must be based on a version off QM that does not contains the Born rule. Yours does.
All of physics is about phenomenology. Theory aims at ever more comprehensive and ever more precise description of phenomena that are objectively observable in Nature.That is insufficient, the theory must also predict new/unknown phenomena, and this is very important. I suppose you'll agree that the future of physics is not to find zillion of equivalent abstract phenomenology (like string theory, …)
Also maybe you don't think of quantum "field" to be "ontological" field. So far so good. But then by saying that QM is complete, you've just give up on positivism, by asserting that you cannot find anything better (no even bettering QM itself) without any shred of evidence.
Also to assert that nothing can be gained by trying to connect a phenomenology to some ontology (like strings) is also an anti-positivist claim. There is no evidence for that.
Again: QT is causal but not deterministic.OK, given the time you have taken to explain, I'll use that word like you do. Although that I don't think it's the correct word to encompass the uncertainty relations, because I have always read that the Schrodinger equation is all there is, and even if every knowledge cannot be known "at once", all is continuous and unitary (and thus determined)
What do you mean by that? I don't use Born's rule inside the Schrödinger equation.Really, all I wanted you is to admit the following and …
For me Born's rule is an independent postulate, necessary to give an interpretation to the wave function (in this very special case of systems, where a wave function is a sufficient description of the (pure) quantum states of the system) usable in the lab.
{…}
So far the Born rule seems to be an independent postulate, necessary to give a minimal interpretation needed to apply the QT formalism to real-world observations….that this independent postulate is only for the lab, which need a special interpretation. There is no such thing in classical mechanics, which treat all the stuff in the universe equally.
Which "distinction"?The same you've made above. We are on the same page now.
I have no clue, what you want to tell by this statements. The same units are used in QT as in classical physics.No. That's not even the same field. And going to real by using a modulo is one thing, but the squaring implies that the unit of the Hilbert space is "square root" of probabilities…
Probabilities are of course numbers between 0 and 1. I've no clue, why you think probabilities might be complex numbers.I don't, and that's exactly what I write ==> "As far as I known, probabilities are not complex numbers… even (0,0)"
Yes, because there is no physical interpretation to the wave function alone.Where would you put the cut in these examples?
Consider a classical stochastic process, e.g. a random walk. The observables are not deterministic, but given by a probabilistic law. Yet, observables have definite values at each time, irrespective of whether you measure them or not. I think the key to understand QM is to explain what exactly is the difference between QM and classical stochastic processes.Classical stochastic processes (as described, e.g., by Langevin equations) only occur, because we are not able to describe the detailed equations of motion of the classical system (e.g., the Brownian particle in a fluid consisting of very many classical particles, with which this Brownian particle interacts) and instead describe the interaction of a subsystem (e.g., the Brownian particles) in terms of a friction force and a randomly fluctuating force in the sense of classical statistical physics. The system as a whole is still deterministic, i.e., here we use probabilistic arguments to mimic the interactions which we cannot fully describe due to the complexity of the situation.
The probabilities in QT given by Born's rule are different: If you take QT as complete, then Nature is intrinsically random and not only by ingnoring too complicated deterministic dynamics, i.e., a particle doesn't take determined positions and momenta no matter how well I try to prepare the particle to have determined positions and momenta. Even worse, the Heisenberg-Robertson uncertainty relation tells us that if we attempt to make the position pretty well determined we must buy this at the prize that necessarily the momentum gets only very badly determined (and vice versa). Of course, I don't claim that QT is necessarily complete. We can't know, whether one day someone finds a deterministic non-local theory from which QT can be derived as a stochastic effective theory as with the above discussed example of a Brownian particle, where a stochastic theory is used to describe the effect of many unresolved degrees of freedom.
On the other hand classicality also follows via quantum theory. In my opinion this was beautifully already clarified very early in Mott's famous paper on the question, how ##alpha## particles from a radioactive nucleus can leave classical straight tracks in a cloud chamber. Indeed, you can view this at many science exhibitions: An ##alpha## particle's track can indeed be easily followed by eye, looking as if a classical particle moves with (almost constant) velocity in a straight line. Indeed, this behavior is nice explained in Mott's paper: The issue is that (a) we do not look too closely, i.e., we are satisfied with the finite resolution of the track in the cloud chamber and the corresponding finite resolution for the velocity (or momentum) of the particle you can get by measuring the speed by following the head of the track and (b) that the straight-line trajectory is the vastly most probable trajectory of the particle. Only the initial momentum of the ##alpha## particle is of course completely random, i.e., the direction of the track is indeed random, as one can see by watching more and more nuclei emit their ##alpha## particles. After a while you see a quite isotropic distribution of tracks. So the single ##alpha## particle indeed seems to look classical when you take the track's head as the momentaneous position of the particle, but that's a very much coarse-grained macroscopic quantity, which seems to behave classically. In this way one sees that there is no contradiction in an apparently classical behavior of a particle when (constantly) "observed" through it's interaction with a macroscopic "measurement apparatus", which here is the vapour in the cloud chamber, and the observation of the track by a human being is completely irrelevant. You can as well film the whole thing and watch the movie much later, i.e., the track is indeed there, no matter whether a conscious being is watching it or not. So there's indeed no necessity for any solipsistic collapse arguments of some flavors of Copenhagen interpretations (like the Princeton interpretation).
So is the cut necessary to predict that electric charge will be conserved?Yes, because there is no physical interpretation to the wave function alone.